Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 783731, 8 pages http://dx.doi.org/10.1155/2013/783731 Research Article Almost Sequence Spaces Derived by the Domain of the Matrix Ali Karaisa and ÜmJt KarabJyJk Department of Mathematics-Computer Science, Faculty of Sciences, Necmettin Erbakan University, Meram Yerles ¸kesi, Meram, 42090 Konya, Turkey Correspondence should be addressed to Ali Karaisa; [email protected] Received 9 May 2013; Revised 26 August 2013; Accepted 26 September 2013 Academic Editor: Feyzi Bas ¸ar Copyright © 2013 A. Karaisa and ¨ U. Karabıyık. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By using , we introduce the sequence spaces , 0 , and of normed space and -space and prove that , 0 , and are linearly isomorphic to the sequence spaces , 0 , and , respectively. Further, we give some inclusion relations concerning the spaces , 0 , and the nonexistence of Schauder basis of the spaces and is shown. Finally, we determine the - and -duals of the spaces and . Furthermore, the characterization of certain matrix classes on new almost convergent sequence and series spaces has exhaustively been examined. 1. Preliminaries, Background and Notation By , we will denote the space of all real or complex valued sequences. Any vector subspace of is called sequence space. We will write ℓ ∞ , 0 , , and ℓ for the spaces of all bounded, null, convergent, and absolutely -summable sequences, re- spectively, which are -space with the usual sup-norm de- fined by ‖‖ ∞ = sup | | and ‖‖ ℓ = (∑ | | ) 1/ , for 1< <∞, where, here and in what follows, the summation with- out limits runs from 0 to ∞. Further, we will write , for the spaces of all sequences associated with bounded and con- vergent series, respectively, which are -spaces with their natural norm [1]. Let and be two sequence spaces and = ( ) an infinite matrix of real or complex numbers , where , ∈ N. en, we say that defines a matrix mapping from into and we denote it by writing that : → and if for every sequence = ( )∈ the sequence = () , the -transform of is in , where () =∑ , ( ∈ N). (1) e notation ( : ) denotes the class of all matrices such that :→. us, ∈ ( : ) if and only if the series on the right hand side of (1) converges for each ∈ N and every ∈ and we have = {() } ∈N ∈ for all ∈. e matrix domain of an infinite matrix in a sequence space is defined by = { = ( ) ∈ : ∈ } . (2) e approach constructing a new sequence space by means of the matrix domain of a particular triangle has re- cently been employed by several authors in many research papers. For example, they introduced the sequence spaces () 1 = in [2], (ℓ ) = and (ℓ ∞ ) = ∞ in [3], = (, V; ) in [4], ( 0 ) Λ = 0 and Λ = in [5], and (ℓ ) = and (ℓ ∞ ) = ∞ in [6]. Recently, matrix domains of the gen- eralized difference matrix (, ) and triple band matrix (, , ) in the sets of almost null and almost convergent se- quences have been investigated by Bas ¸ar and Kiris ¸c ¸i [7] and S¨ onmez [8], respectively. Later, Kayaduman and S ¸eng¨ on¨ ul introduced some almost convergent spaces which are the matrix domains of the Riesz matrix and Ces` aro matrix of order 1 in the sets of almost null and almost convergent sequences (see [9, 10]). We now focus on the sets of almost convergent sequences. A continuous linear functional on ℓ ∞ is called a Banach limit if (i) () ⩾ 0 for = ( ) and ⩾0 for every , (ii) ( () ) = ( ), where is shiſt operator which is defined on by () = +1, and (iii) () = 1, where = (1, 1, 1, . . .). A sequence = ( )∈ℓ ∞ is said to be almost convergent to
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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 783731 8 pageshttpdxdoiorg1011552013783731
Research ArticleAlmost Sequence Spaces Derived by the Domain of the Matrix 119860119903
Ali Karaisa and UumlmJt KarabJyJk
Department of Mathematics-Computer Science Faculty of Sciences Necmettin Erbakan University Meram Yerleskesi Meram42090 Konya Turkey
Correspondence should be addressed to Ali Karaisa alikaraisahotmailcom
Received 9 May 2013 Revised 26 August 2013 Accepted 26 September 2013
Academic Editor Feyzi Basar
Copyright copy 2013 A Karaisa and U Karabıyık This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
By using 119860119903 we introduce the sequence spaces 119886119903119891 1198861199031198910 and 119886119903
119891119904of normed space and 119861119870-space and prove that 119886119903
119891 119886119903
1198910 and 119886119903
119891119904are
linearly isomorphic to the sequence spaces 119891 1198910 and 119891119904 respectively Further we give some inclusion relations concerning the
spaces 119886119903119891 1198861199031198910 and the nonexistence of Schauder basis of the spaces 119891119904 and 119886119903
119891119904is shown Finally we determine the 120573- and 120574-duals
of the spaces 119886119903119891and 119886119903
119891119904 Furthermore the characterization of certain matrix classes on new almost convergent sequence and series
spaces has exhaustively been examined
1 Preliminaries Background and Notation
By 119908 we will denote the space of all real or complex valuedsequences Any vector subspace of119908 is called sequence spaceWe will write ℓ
infin 1198880 119888 and ℓ
119901for the spaces of all bounded
null convergent and absolutely 119901-summable sequences re-spectively which are 119861119870-space with the usual sup-norm de-fined by 119909
infin= sup
119896|119909119896| and 119909
ℓ119901= (sum119896|119909119896|119901
)1119901 for 1 lt
119901 lt infin where here and inwhat follows the summationwith-out limits runs from 0 toinfin Further we will write 119887119904 119888119904 forthe spaces of all sequences associated with bounded and con-vergent series respectively which are 119861119870-spaces with theirnatural norm [1]
Let 120583 and 120574 be two sequence spaces and 119860 = (119886119899119896) an
infinite matrix of real or complex numbers 119886119899119896 where 119899 119896 isin
N Then we say that 119860 defines a matrix mapping from 120583 into120574 and we denote it by writing that 119860 120583 rarr 120574 and if forevery sequence 119909 = (119909
119896) isin 120583 the sequence 119860119909 = (119860119909)
119899 the
119860-transform of 119909 is in 120574 where
(119860119909)119899= sum
119896
119886119899119896119909119896 (119899 isin N) (1)
The notation (120583 120574) denotes the class of all matrices 119860such that119860 120583 rarr 120574Thus119860 isin (120583 120574) if and only if the serieson the right hand side of (1) converges for each 119899 isin N andevery 119909 isin 120583 and we have 119860119909 = (119860119909)
119899119899isinN isin 120574 for all 119909 isin 120583
The matrix domain 120583119860of an infinite matrix 119860 in a sequence
space 120583 is defined by
120583119860= 119909 = (119909
119896) isin 120596 119860119909 isin 120583 (2)
The approach constructing a new sequence space bymeans of the matrix domain of a particular triangle has re-cently been employed by several authors in many researchpapers For example they introduced the sequence spaces(119888)1198621= 119888 in [2] (ℓ
119901)119860119903 = 119886
119903
119901and (ℓ
infin)119860119903 = 119886
119903
infinin [3] 120583
119866=
119885(119906 V 120583) in [4] (1198880)Λ= 119888120582
0and 119888Λ= 119888120582 in [5] and (ℓ
119901)119864119903 = 119890119903
119901
and (ℓinfin)119864119903 = 119890119903
infinin [6] Recently matrix domains of the gen-
eralized difference matrix 119861(119903 119904) and triple band matrix119861(119903 119904 119905) in the sets of almost null and almost convergent se-quences have been investigated by Basar and Kirisci [7] andSonmez [8] respectively Later Kayaduman and Sengonulintroduced some almost convergent spaces which are thematrix domains of the Riesz matrix and Cesaro matrix oforder 1 in the sets of almost null and almost convergentsequences (see [9 10])
We now focus on the sets of almost convergent sequencesA continuous linear functional 120601 on ℓ
infinis called a Banach
limit if (i) 120601(119909) ⩾ 0 for 119909 = (119909119896) and 119909
119896⩾ 0 for every 119896 (ii)
120601(119909120590(119896)) = 120601(119909
119896) where 120590 is shift operator which is defined
on120596 by 120590(119896) = 119896+1 and (iii) 120601(119890) = 1 where 119890 = (1 1 1 )A sequence 119909 = (119909
119896) isin ℓinfin
is said to be almost convergent to
2 Abstract and Applied Analysis
the generalized limit 120572 if all Banach limits of 119909 are 120572 [11] anddenoted by 119891minus lim119909 = 120572 In other words 119891minus lim119909
119896= 120572 uni-
formly in 119899 if and only if
lim119898rarrinfin
1
119898 + 1
119898
sum
119896=0
119909119896+119899
uniformly in 119899 (3)
The characterization given above was proved by Lorentz in[11] We denote the sets of all almost convergent sequences 119891and series 119891119904 by
119891 = 119909 = (119909119896) isin 120596 lim
119898rarrinfin
119905119898119899(119909) = 120572 uniformly in 119899
(4)
where
119905119898119899(119909) =
119898
sum
119896=0
1
119898 + 1
119909119896+119899 119905minus1119899
= 0
119891119904 =
119909 = (119909119896) isin 120596
exist119897 isin C ni lim119898rarrinfin
119898
sum
119896=0
119899+119896
sum
119895=0
119909119895
119898 + 1
= 119897
uniformly in 119899
(5)We know that the inclusions 119888 sub 119891 sub ℓ
infinstrictly hold Be-
cause of these inclusions norms sdot 119891and sdot
infinof the spaces
119891 and ℓinfin
are equivalent So the sets 119891 and 1198910are BK-spaces
with the norm 119909119891= sup
119898119899|119905119898119899(119909)|
The rest of this paper is organized as follows We giveforeknowledge on the main argument of this study and nota-tions in this section In Section 2 we introduce the almostconvergent sequence and series spaces 119886119903
119891119904and 119886119903119891which are
thematrix domains of the119860119903matrix in the almost convergentsequence and series spaces119891119904 and119891 respectively In additionwe give some inclusion relations concerning the spaces 119886119903
119891
119886119903
1198910
and the non-existence of Schauder basis of the spaces 119891119904and 119886119903119891119904is shown to give certain theorems related to behavior
of some sequences In Section 3 we determine the beta- andgamma-duals of the spaces 119886119903
119891and 119886119903
119891119904and characterize the
classes (120574 119886119903119891) (119886119903119891 120583) (120575 119886119903
119891119904) and (119886119903
119891 120579) where 120574 isin
119888(119901) 1198880(119901) ℓinfin(119901) 119888119904 119887119904 119891119904 119891 119888 ℓ
infin 120583 isin 119888119904 119888 ℓ
infin 120575 isin
119888119904 119891119904 119887119904 and 120579 isin 119891 119888 119891119904 ℓinfin where 119888(119901) 119888
0(119901) and ℓ
infin(119901)
denote the space of Maddox convergent null and boundedsequence spaces defined by Maddox [12]
Lemma 1 (see [13]) The set 119891119904 has no Schauder basis
2 The Sequence Spaces 119886119903119891
1198861199031198910
and 119886119903119891119904
Derivedby the Domain of the Matrix 119860119903
In the present section we introduce the sequence spaces 119886119903119891
119886119903
1198910
and 119886119903119891119904as the set of all sequences such that119860119903-transforms
of them are in the spaces 119891 1198910 and 119891119904 respectively Further
this section is devoted to examination of the basic topologicalproperties of the sets 119886119903
119891 1198861199031198910
and 119886119903119891119904 Recently Aydın and
Basar [14] studied the sequence spaces 119886119903119888and 1198861199030
119886119903
119888= 119909 = (119909
119896) isin 120596 lim
119899rarrinfin
1
119899 + 1
119899
sum
119896=0
(1 + 119903119896
) 119909119896exists
119886119903
0= 119909 = (119909
119896) isin 120596 lim
119899rarrinfin
1
119899 + 1
119899
sum
119896=0
(1 + 119903119896
) 119909119896= 0
(6)
where 119860119903 denotes the matrix 119860119903 = (119886119903119899119896) defined by
119886119903
119899119896=
1 + 119903119896
119899 + 1
(0 ⩽ 119896 ⩽ 119899)
0 (119896 gt 119899)
(7)
Now we introduce the sequence spaces 119886119903119891 119886119903119891 and 119886119903
119891119904as
the sets of all sequences such that their 119860119903-transforms are inthe spaces 119891 119891
0 and 119891119904 respectively that is
119886119903
119891= 119909 = (119909
119896) isin 120596
exist120572 isin C ni lim119898rarrinfin
119898
sum
119896=0
1
119898 + 1
119896
sum
119894=0
1
119896 + 1
(1 + 119903119894
) 119909119899+119894
= 120572 uniformly in 119899
119886119903
1198910
= 119909 = (119909119896) isin 120596
lim119898rarrinfin
119898
sum
119896=0
1
119898 + 1
119896
sum
119894=0
1 + 119903119894
119896 + 1
119909119899+119894= 0
uniformly in 119899
119886119903
119891119904= 119909 = (119909
119896) isin 120596
exist120573 isin C ni lim119898rarrinfin
119898
sum
119896=0
119899+119896
sum
119895=0
119895
sum
119894=0
1 + 119903119895
119894 + 1
119909119899+119895
= 120573 uniformly in 119899
(8)
We can redefine the spaces 119886119903119891119904 119886119903119891 and 119886119903
1198910
by the notation of(2)
119886119903
1198910
= (1198910)119860119903 119886
119903
119891= 119891119860119903 119886
119903
119891119904= (119891119904)
119860119903 (9)
It is known by Basar [15] that the method is regular for 0 lt119903 lt 1 We assume unless stated otherwise that 0 lt 119903 lt 1
Abstract and Applied Analysis 3
Define the sequence 119910 = (119910119896) which will be frequently
used as the 119860119903-transform of a sequence 119909 = (119909119896) that is
119910119896(119903) =
119896
sum
119894=0
1 + 119903119894
119896 + 1
119909119894(119896 isin N) (10)
Theorem 2 The spaces 119886119903119891and 119886119903119891119904have no Schauder basis
Proof Since it is known that the matrix domain 120583119860of a
normed sequence space120583has a basis if and only if120583has a basiswhenever 119860 = (119886
119899119896) is a triangle [16 Remark 24] and the
space 119891 has no Schauder basis by [7 Corollary 33] we havethat 119886119903119891has no Schauder basis Since the set 119891119904 has no basis in
Lemma 1 119886119903119891119904has no Schauder basis
Theorem 3 The following statements hold
(i) The sets 119886119903119891and 119886119903
1198910
are linear spaces with the coordi-natewise addition and scalar multiplication which are119861119870-spaces with the norm
119909119886119903
119891
= sup119898
1003816100381610038161003816100381610038161003816100381610038161003816
119898
sum
119896=0
1
119898 + 1
119896
sum
119894=0
1 + 119903119894
119896 + 1
119909119894+119899
1003816100381610038161003816100381610038161003816100381610038161003816
(11)
(ii) The set 119886119903119891119904
is a linear space with the coordinatewiseaddition and scalar multiplication which is a 119861119870-spacewith the norm
119909119886119903
119891119904
= sup119898
10038161003816100381610038161003816100381610038161003816100381610038161003816
119898
sum
119896=0
1
119898 + 1
119896+119899
sum
119895=0
119895
sum
119894=0
1 + 119903119894
119895 + 1
119909119894
10038161003816100381610038161003816100381610038161003816100381610038161003816
(12)
Proof Since the second part can be similarly proved we onlyfocus on the first part Since the sequence spaces119891 and119891
0en-
dowed with the norm sdot infin
are 119861119870-spaces (see [1 Example732(b)]) and thematrix119860119903 = (119886119903
119899119896) is normalTheorem 432
of Wilansky [17 p61] gives the fact that the spaces 119886119903119891and 1198861199031198910
are 119861119870-spaces with the norm in (11)
Now we may give the following theorem concerning theisomorphism between our spaces and the sets 119891 119891
0 and 119891119904
Theorem 4 The sequence spaces 119886119903119891 1198861199031198910
and 119886119903119891119904are linearly
isomorphic to the sequence spaces 119891 1198910 and 119891119904 respectively
that is 119886119903119891cong 119891 119886119903
1198910
cong 1198910 and 119886119903
119891119904cong 119891119904
Proof To prove the fact that 119886119903119891cong 119891 we should show the exis-
tence of a linear bijection between the spaces 119886119903119891and 119891 Con-
sider the transformation 119879 defined with the notation of (2)from 119886
119903
119891to119891 by 119909 997891rarr 119910 = 119879119909 = 119860
119903
119909The linearity of119879 is clearFurther it is clear that119909 = 120579whenever119879119909 = 120579 and hence119879 isinjective
Let 119910 = (119910119896) isin 119886119903
119891 and define the sequence 119909 = (119909
119896(119903)) by
119909119896=
1
1 + 119903119896
[(119896 + 1) 119910119896minus 119896119910119896minus1] for each 119896 isin N
(13)
whence119891119860119903 minus lim119909
= lim119898rarrinfin
119898
sum
119896=0
1
119898 + 1
times
119896
sum
119894=0
(1 + 119903119894
) 119909119894+119899
1 + 119896
uniformly in 119899
= lim119898rarrinfin
119898
sum
119896=0
1
119898 + 1
times
119896
sum
119894=0
(1 + 119903119894
) [1 (1 + 119903119894
) (119910119894+119899(119896 + 1) minus 119910
119894+119899minus1119896)]
1 + 119896
uniformly in 119899
= lim119898rarrinfin
1
119898 + 1
119898
sum
119896=0
119910119896+119899
uniformly in 119899
= 119891 minus lim119910(14)
which implies that119909 isin 119886119903119891 As a result119879 is surjectiveHence119879
is a linear bijection which implies that the spaces 119886119903119891and119891 are
linearly isomorphic as desired Similarly the isomorphisms119886119903
1198910
cong 1198910and 119886119903119891119904cong 119891119904 can be proved
Theorem 5 The inclusion 119891 sub 119886119903119891strictly holds
Proof Let 119909 = (119909119896) isin 119888 Since 119888 sub 119891 119909 isin 119891 Because 119860119903 is
regular for 0 lt 119903 lt 1119860119903119909 isin 119888Therefore since lim119860119903119909 = 119891minuslim119860119903119909 we see that 119909 isin 119886119903
119891 So we have that the inclusion119891 sub
119886119903
119891holds Further consider the sequence 119905 = (119905
119896(119903)) defined
by 119905119896(119903) = (2119896 + 1)(1 + 119903
119896
)(minus1)119896
forall119896 isin N Then since 119860119903t =(minus1)119899
isin 119891 119909 isin 119886119903119891 One can easily see that 119905 notin 119891Thus 119905 isin 119886119903
119891
119891 and this completes the proof
Theorem 6 The sequence spaces 119886119903119891and ℓinfin
overlap but nei-ther of them contains the other
Proof Let us consider the sequence 119906 = (119906119896(119903)) defined by
119906119896(119903) = 1(1+119903
119896
) for allNThen since119860119903119906 = 119890 isin 119891 119906 isin 119886119903119891 It
is clear that 119906 isin ℓinfin This means that the sequence spaces 119886119903
119891
and ℓinfinare not disjointNowwe show that the sequence space
119886119903
119891and ℓinfin
do not include each other Let us consider the se-quence 119905 = (119905
119896(119903)) defined as in proof of Theorem 5 above
and 119911 = (119911119896(119903)) = (0 0 1(1 + 119903
101
) 1(1 + 119903110
)
0 0 1(1 + 119903211
) 1(1 + 119903231
) 0 0 ) where theblocks of 0rsquos are increasing by factors of 100 and the blocks of1(1 + 119903
119896
)rsquos are increasing by factors of 10 Then since 119860119903119905 =(minus1)119899
isin 119891 119905 isin 119886119903119891 but 119905 notin ℓ
infinTherefore 119905 isin 119886119903
119891ℓinfin Also the
sequence 119911 notin 119886119903119891since119860119903119911 = (0 0 1 1 0 0 1
1 0 0 ) notin 119891 where the blocks of 0rsquos are increasing byfactors of 100 and the blocks of 1rsquos are increasing by factors of10 but 119911 is boundedThis means that 119911 isin ℓ
infin 119886119903
119891 Hence the
4 Abstract and Applied Analysis
sequence spaces 119886119903119891and ℓinfinoverlap but neither of them con-
tains the other This completes the proof
Theorem 7 Let the spaces 1198861199031198910
119886119903119888 and 119886119903
119891be given Then
(i) 1198861199031198910
sub 119886119903
119891strictly hold
(ii) 119886119903119888sub 119886119903
119891strictly hold
Proof (i) Let 119909 = (119909119896) isin 119886
119903
1198910
which means that 119860119903119909 isin 1198910
Since1198910sub 119891119860119903119909 isin 119891This implies that119909 isin 119886119903
119891Thuswe have
119886119903
1198910
sub 119886119903
119891
Now we show that this inclusion is strict Let us considerthe sequence 119906 = (119906
119896(119903)) defined as in proof ofTheorem 6 for
all 119896 isin N Consider the following
119891119860119903 minus lim 119906 = lim
119898rarrinfin
119898
sum
119896=0
1
119898 + 1
119896
sum
119894=0
1 + 119903119894
119896 + 1
119906119894+119899
= lim119898rarrinfin
119898
sum
119896=0
1
119898 + 1
= 119890 = (1 1 )
(15)
which means that 119906 isin 119886119903119891 119886119903
1198910
that is to say the inclusion isstrict
(ii) Let 119909 = (119909119896) isin 119886119903
119888whichmeans that119860119903119909 isin 119888 Since 119888 sub
119891 119860119903119909 isin 119891 This implies that 119909 isin 119886119903119891 Thus we have
119886119903
119888sub 119886119903
119891 Furthermore let us consider the sequence
119905 = 119905119896(119903) defined as in proof of Theorem 5 for all
119896 isin N Then since 119860119903119905 = (minus1)119899 isin 119891 119888 119905 isin 119886119903119891 119886119903
119888
This completes the proof
3 Certain Matrix Mappings on the Sets 119886119903119891
119886119903119891119904
and Some Duals
In this section we will characterize some matrix transforma-tions between the spaces of 119860119903 almost convergent sequenceand almost convergent series in addition to paranormed andclassical sequence spaces after giving 120573- and 120574-duals of thespaces 119886119903
119891119904and 119886119903119891We start with the definition of the beta- and
gamma-dualsIf119909 and119910 are sequences and119883 and119884 are subsets of120596 then
we write 119909 sdot 119910 = (119909119896119910119896)infin
119896=0 119909minus1 lowast 119884 = 119886 isin 120596 119886 sdot 119909 isin 119884 and
119872(119883119884) = ⋂
119909isin119883
119909minus1
lowast 119884 = 119886 119886 sdot 119909 isin 119884 forall119909 isin 119883 (16)
for themultiplier space of119883 and119884 One can easily observe fora sequence space 119885 with 119884 sub 119885 and 119885 sub 119883 that inclusions119872(119883119884) sub 119872(119883119885) and119872(119883119884) sub 119872(119885 119884) hold respec-tively The 120572- 120573- and 120574-duals of a sequence space which arerespectively denoted by119883120572 119883120573 and119883120574 are defined by
119883120572
= 119872(119883 ℓ1) 119883
120573
= 119872(119883 119888119904) 119883120574
= 119872(119883 119887119904)
(17)
It is obvious that 119883120572 sub 119883120573 sub 119883120574 Also it can easily be seenthat the inclusions 119883120572 sub 119884120572 119883120573 sub 119884120573 and 119883120574 sub 119884
120574 holdwhenever 119884 sub 119883
Lemma 8 (see [18]) 119860 = (119886119899119896) isin (119891 ℓ
infin) if and only if
sup119899
sum
119896
1003816100381610038161003816119886119899119896
1003816100381610038161003816lt infin (18)
Lemma 9 (see [18]) 119860 = (119886119899119896) isin (119891 119888) if and only if (18)
holds and there are 120572 120572119896isin C such that
lim119899rarrinfin
119886119899119896= 120572119896
forallk isin N (19)
lim119899rarrinfin
sum
119896
119886119899119896= 120572 (20)
lim119899rarrinfin
sum
119896
1003816100381610038161003816Δ (119886119899119896minus 120572119896)1003816100381610038161003816= 0 (21)
Theorem 10 Define the sets 1199051199031and 1199051199032by
119905119903
1= 119886 = (119886
119896) isin 120596 sum
119896
1003816100381610038161003816100381610038161003816
Δ (
119886119896
1 + 119903119896
) (119896 + 1)
1003816100381610038161003816100381610038161003816
lt infin
119905119903
2= 119886 = (119886
119896) isin 120596 sup
119896
10038161003816100381610038161003816100381610038161003816
119886119896(119896 + 1)
1 + 119903119896
10038161003816100381610038161003816100381610038161003816
lt infin
(22)
where Δ(119886119896(1 + 119903
119896
)) = 119886119896(1 + 119903
119896
) minus 119886119896+1(1 + 119903
119896+1
) for all 119896 isinN Then (119886119903
119891)120574
= 119905119903
1cap 119905119903
2
Proof Take any sequence 119886 = (119886119896) isin 120596 and consider the fol-
lowing equality119899
sum
119896=0
119886119896119909119896=
119899
sum
119896=0
119886119896[
119896
sum
119894=119896minus1
(minus1)119896minus119895
119894 + 1
1 + 119903119894119910119894]
=
119899minus1
sum
119896=0
Δ(
119886119896
1 + 119903119896
) (119896 + 1) 119910119896
+
119899 + 1
1 + 119903119899119886119899119910119899
= (119879119910)119899
(23)
where 119879 = 119905119903119899119896 is
119905119903
119899119896=
Δ(
119886119896
1 + 119903119896
) (119896 + 1) (0 ⩽ 119896 ⩽ 119899 minus 1)
119899 + 1
1 + 119903119899119886119899
(119896 = 119899)
0 (119896 gt 119899)
(24)
for all 119896 119899 isin NThus we deduce from (23) that 119886119909 = (119886119896119909119896) isin
119887119904 whenever 119909 = (119909119896) isin 119886119903
119891if and only if 119879119910 isin ℓ
infinwhenever
119910 = (119910119896) isin 119891 where 119879 = 119905119903
119899119896 is defined in (24) Therefore
with the help of Lemma 8 (119886119903119891)120574
= 119905119903
1cap 119905119903
2
Theorem 11 The 120573-dual of the space 119886119903119891is the intersection of
the sets
119905119903
3= 119886 = (119886
119896) isin 120596 lim
119899rarrinfin
sum
119896
1003816100381610038161003816Δ (119905119903
119899119896minus 120572119896)1003816100381610038161003816= 0
119905119903
4= 119886 = (119886
119896) isin 120596 (
119896 + 1
1 + 119903119896
119886119896) isin 119888119904
(25)
where lim119899rarrinfin
119905119903
119899119896= 120572119896for all 119896 isin N Then (119886119903
119891)120573
= 119905119903
3cap 119905119903
4
Abstract and Applied Analysis 5
Proof Let us take any sequence 119886 isin 120596 By (23) 119886119909 = (119886119896119909119896) isin
119888119904whenever 119909 = (119909119896) isin 119886119903
119891if and only if 119879119910 isin 119888whenever 119910 =
(119910119896) isin 119891 It is obvious that the columns of that matrix 119879 in 119888
where119879 = 119905119903119899119896 defined in (24) we derive the consequence by
Lemma 9 that (119886119903119891)120573
= 119905119903
3cap 119905119903
4
Theorem 12 The 120574-dual of the space 119886119903119891119904is the intersection of
the sets
119888119903
1= 119886 = (119886
119896) isin 120596 sum
119896
1003816100381610038161003816100381610038161003816
Δ [Δ(
119886119896
1 + 119903119896
) (119896 + 1)
+
119886119896
1 + 119903119896(119896 + 1)]
1003816100381610038161003816100381610038161003816
lt infin
119888119903
2= 119886 = (119886
119896) isin 120596 (
119886119896(119896 + 1)
1 + 119903119896
) isin 1198880
(26)
In other words we have (119886119903119891119904)120574
= 119888119903
1cap 119888119903
2
Proof We obtain from (23) that 119886119909 = (119886119896119909119896) isin 119887119904 whenever
119909 = (119909119896) isin 119886119903
119891119904if and only if119879119910 isin ℓ
infinwhenever 119910 = (119910
119896) isin 119891119904
where 119879 = 119905119903
119899119896 is defined in (24) Therefore by Lemma 19
(viii) (119886119903119891119904)120574
= 119888119903
1cap 119888119903
2
Theorem 13 Define the set 1198881199033by
119888119903
3= 119886 = (119886
119896) isin 120596 lim
119899rarrinfin
sum
119896
10038161003816100381610038161003816Δ2
(119905119903
119899119896)
10038161003816100381610038161003816exists (27)
Then (119886119903119891119904)120573
= 119888119903
1cap 119888119903
2cap 119888119903
3
Proof This may be obtained in the same way as mentionedin the proof of Theorem 12 with Lemma 19(viii) instead ofLemma 19(vii) So we omit details
For the sake of brevity the following notations will beused
119886 (119899 119896119898) =
1
119898 + 1
119898
sum
119894=0
119886119899+119894119896
119886 (119899 119896) =
119899
sum
119894=0
119886119894119896
119886119899119896= Δ(
119886119899119896
1 + 119903119896
) (119896 + 1) = (
119886119899119896
1 + 119903119896
minus
119886119899119896+1
1 + 119903119896+1
) (119896 + 1)
Δ119886119899119896= 119886119899119896minus 119886119899119896+1
119886119899119896=
119899
sum
119895=0
(1 + 119903119895
) 119890119895119896
119899 + 1
(28)
for all 119896 119899 isin N Assume that the infinite matrices 119860 = (119886119899119896)
and119861 = (119887119899119896)map the sequences119909 = (119909
119896) and119910 = (119910
119896)which
are connectedwith relation (10) to the sequences 119906 = (119906119899) and
V = (V119899) respectively that is
119906119899= (119860119909)
119899= sum
119896
119886119899119896119909119896
forall119899 isin N (29)
V119899= (119861119910)
119899= sum
119896
119887119899119896119910119896
forall119899 isin N (30)
One can easily conclude here that the method 119860 is directlyapplied to the terms of the sequence 119909 = (119909
119896) while the
method 119861 is applied to the 119860119903-transform of the sequence 119909 =(119909119896) So the methods 119860 and 119861 are essentially differentNow suppose that the matrix product 119861119860119903 exists which is
amuch weaker assumption than the conditions on thematrix119861 belonging to anymatrix class in general It is not difficult tosee that the sequence in (30) reduces to the sequence in (29)under the application of formal summation by parts Thisleads us to the fact that 119861119860119903 exists and is equal to 119860 and(119861119860119903
)119909 = 119861(119860119903
119909) formally holds if one side exists This state-ment is equivalent to the following relation between the en-tries of the matrices 119860 = (119886
119899119896) and 119861 = (119887
119899119896) which are con-
nected with the relation
119886119899119896= 119887119899119896= Δ(
119886119899119896
1 + 119903119896
) (119896 + 1) or
119886119899119896= (1 + 119903
119896
)
infin
sum
119895=119896
119887119899119895
1 + 119895
forall119896 119899 isin N
(31)
Note that the methods 119860 and 119861 are not necessarily equi-valent since the order of summationmay not be reversedWenow give the following fundamental theorem connected withthematrixmappings oninto the almost convergent spaces 119886119903
119891
and 119886119903119891s
Theorem 14 Suppose that the entries of the infinite matrices119860 = (119886
119899119896) and 119861 = (119887
119899119896) are connected with relation (31) for all
119896 119899 isin N and let 120582 be any given sequence spaceThen119860 isin (119886119903119891
120582) if and only if
119861 isin (119891 120582)
119899 + 1
1 + 119903119896
119886119899119896
119896isinNisin 1198880
(32)
Proof Suppose that 119860 = (119886119899119896) and 119861 = (119887
119899119896) are connected
with the relation (31) and let 120582 be any given sequence spaceand keep in mind that the spaces 119886119903
119891and 119891 are norm iso-
morphicLet 119860 isin (119886119903
119891 120582) and take any sequence 119909 isin 119886119903
119891 and keep
in mind that 119910 = 119860119903119909 Then (119886119899119896)119896isinN isin (119886
119903
119891)120573 that is (32)
holds for all 119899 isin N and 119861119860119903 exists which implies that(119887119899119896)119896isinN isin ℓ1 = 119891
120573 for each 119899 isin N Thus 119861119910 exists for all 119910 isin119891 and thus we have119898 rarr infin in the equality
119898
sum
119896=0
119887119899119896119910119896=
119898
sum
119896=0
119898
sum
119895=119896
(1 + 119903119896
)
119887119899119895
1 + 119895
119909119896 (33)
for all119898 119899 isin N and we have (31) 119861119910 = 119860119909 which means that119861 isin (119891 120582) On the other hand assume that (32) holds and119861 isin (119891 120582) Then we have (119887
119899119896)119896isinN isin ℓ1 for all 119899 isin N which
6 Abstract and Applied Analysis
gives together with (120592119899119896)119896isinN isin (119886
119903
119891)120573 for each 119899 isin N that 119860119909
exists Then we obtain from the equality
119898
sum
119896=0
119886119899119896119909119896=
119898minus1
sum
119896=0
Δ(
119886119899119896
1 + 119903119896
) (119896 + 1) 119910119896+
119898 + 1
1 + 119903119898119886119899119898119910119898
=
119898
sum
119896=0
119887119899119896119910119896
(34)
for all 119898 119899 isin N as 119898 rarr infin that 119860119909 = 119861119910 and this showsthat 119860 isin (119886119903
119891 120582)
Theorem 15 Suppose that the entries of the infinite matrices119864 = (119890
119899119896) and 119865 = (119891
119899119896) are connected with the relation
119891119899119896= 119886119899119896
(35)
for all 119898 119899 isin N and 120582 is any given sequence space Then 119864 isin(120582 119886119903
119891) if and only if 119865 isin (120582 119891)
Proof Let 119909 = (119909119896) isin 120582 and consider the following equality
119899
sum
119895=0
119898
sum
119896=0
1 + 119903119895
119899 + 1
119890119895119896119909119896=
119898
sum
119896=0
119891119899119896119909119896 (36)
for all 119896119898 119899 isin N which yields as 119898 rarr infin that 119864119909 isin 119886119903119891
whenever 119909 isin 120582 if and only if 119865119909 isin 119891 whenever 119909 isin 120582 Thisstep completes the proof
Theorem 16 Let120582 be any given sequence space and thematri-ces119860 = (119886
119899119896) and119861 = (119887
119899119896) are connectedwith the relation (31)
Then 119860 isin (119886119903
119891119904 120582) if and only if 119861 isin (119891119904 120582) and (119886
119899119896)119896isinN isin
(119886119903
119891119904)120573 for all 119899 isin N
Proof The proof is based on the proof of Theorem 14
Theorem 17 Let 120582 be any given sequence space and the ele-ments of the infinite matrices 119864 = (119890
119899119896) and 119865 = (119891
119899119896) are con-
nectedwith relation (35)Then119864 = (119890119899119896) isin (120582 119886
119903
119891119904) if and only
if 119865 isin (120582 119891119904)
Proof The proof is based on the proof of Theorem 15
By Theorems 14 15 16 and 17 we have quite a few out-comes depending on the choice of the space 120582 to characterizecertain matrix mappings Hence by the help of these theo-rems the necessary and sufficient conditions for the classes(119886119903
119891 120582) (120582 119886119903
119891) (119886119903119891119904 120582) and (120582 119886119903
119891119904) may be derived by
replacing the entries of119860 and 119861 by those of 119861 = 119860(119860119903)minus1 and119865 = 119860
119903
119864 respectively where the necessary and sufficient con-ditions on the matrices 119864 and 119865 are read from the concerningresults in the existing literature
Lemma 18 Let 119860 = (119886119899119896) be an infinite matrix Then the fol-
lowing statements hold
(i) 119860 isin (1198880(119901) 119891) if and only if
exist119873 gt 1 ni sup119898isinN
sum
119896
|119886 (119899 119896 119898)|1198731119901119896
lt infin forall119899 isin N
exist120572119896isin C forall119896 isin N ni lim
119898rarrinfin
119886 (119899 119896119898) = 120572119896
uniformly in 119899(37)
(ii) 119860 isin (119888(119901) 119891) if and only if (37) and
exist120572 isin C ni lim119898rarrinfin
sum
119896
119886 (119899 119896119898) = 120572 uniformly in 119899 (38)
(iii) 119860 isin (ℓinfin(119901) 119891) if and only if (37) and
exist119873 gt 1 ni lim119898rarrinfin
sum
119896
1003816100381610038161003816119886 (119899 119896 119898) minus 120572
119896
10038161003816100381610038161198731119901119896
= 0
uniformly in 119899(39)
Lemma 19 Let 119860 = (119886119899119896) be an infinite matrix Then the fol-
lowing statements hold
(i) (Duran [19])119860 isin (ℓinfin 119891) if and only if (18) holds and
119891 minus lim119886119899119896= 120572119896exists for each fixed 119896 (40)
lim119898rarrinfin
sum
119896
1003816100381610038161003816119886 (119899 119896 119898) minus 120572
119896
1003816100381610038161003816= 0 uniformly in 119899 (41)
(ii) (King [20]) 119860 isin (119888 119891) if and only if (18) (40) holdand
119891 minus limsum119896
119886119899119896= 120572 (42)
(iii) (Basar and Colak [21])119860 isin (119888119904 119891) if and only if (40)holds and
sup119899isinN
sum
119896
1003816100381610038161003816Δ119886119899119896
1003816100381610038161003816lt infin (43)
(iv) (Basar and Colak [21])119860 isin (119887119904 119891) if and only if (40)(43) hold and
lim119896
119886119899119896= 0 exists for each fixed 119899 (44)
lim119902rarrinfin
sum
119896
1
119902 + 1
119902
sum
119894=0
1003816100381610038161003816Δ [119886 (119899 + 119894 119896) minus 120572
119896]1003816100381610038161003816= 0 uniformly in 119899
(45)
(v) (Duran [19]) 119860 isin (119891 119891) if and only if (18) (40) and(42) hold and
lim119898rarrinfin
sum
119896
1003816100381610038161003816Δ [119886 (119899 119896119898) minus 120572
119896]1003816100381610038161003816= 0 uniformly in 119899 (46)
(vi) (Basar [22])119860 isin (119891119904 119891) if and only if (40) (44) (46)and (45) hold
Abstract and Applied Analysis 7
(vii) (Ozturk [23])119860 isin (119891119904 119888) if and only if (19) (43) and(44) hold and
lim119899rarrinfin
sum
119896
10038161003816100381610038161003816Δ2
119886119899119896
10038161003816100381610038161003816= 120572 (47)
(viii) 119860 isin (119891119904 ℓinfin) if and only if (43) and (44) hold
(ix) (Basar and Solak [24]) 119860 isin (119887119904 119891119904) if and only if(44) (45) hold and
sup119899isinN
sum
119896
|Δ119886 (119899 119896)| lt infin
119891 minus lim 119886 (119899 119896) = 120572119896exists for each fixed 119896
(48)
(x) (Basar [22])119860 isin (119891119904 119891119904) if and only if (45) (48) holdand
lim119902rarrinfin
sum
119896
1
119902 + 1
119902
sum
119894=0
10038161003816100381610038161003816Δ2
[119886 (119899 + 119894 119896) minus 120572119896]
10038161003816100381610038161003816= 0 uniformly in 119899
(49)
(xi) (Basar and Colak [21])119860 isin (119888119904 119891119904) if and only if (48)holds
(xii) (Basar [25]) 119860 isin (119891 119888119904) if and only if
sup119899isinN
sum
119896
|119886 (119899 119896)| lt infin (50)
sum
119899
119886119899119896= 120572119896
exists for each fixed 119896 (51)
sum
119899
sum
119896
119886119899119896= 120572 (52)
lim119898rarrinfin
sum
119896
1003816100381610038161003816Δ [119886 (119899 119896) minus 120572
119896]1003816100381610038161003816= 0 (53)
Now we give our main results which are related to matrixmappings oninto the spaces of almost convergent series 119886119903
119891119904
and sequences 119886119903119891
Corollary 20 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (119886119903119891119904 119891) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin
N and (40) (44) holdwith 119886119899119896instead of 119886
119899119896 (46) holds
with 119886(119899 119896119898) instead of 119886(119899 119896119898) and (45) holdswith 119886(119899 119896) instead of 119886(119899 119896)
(ii) 119860 isin (119886119903119891119904 119888) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin N
and (19) (43) (44) and (47) hold with 119886119899119896
instead of119886119899119896
(iii) 119860 isin (119886119903119891119904 ℓinfin) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin
N and (43) and (44) hold with 119886119899119896instead of 119886
119899119896
(iv) 119860 isin (119886119903119891119904 119891119904) if and only if 119886
119899119896119896isinN isin (119886
119903
119891119904)120573 for all 119899 isin
N and (45) (48) and (49) hold with 119886(119899 119896) instead of119886(119899 119896)
(v) 119860 isin (119888119904 119886119903119891119904) if and only if (48) holds with 119886(119899 119896) in-
stead of 119886(119899 119896)
(vi) 119860 isin (119887119904 119886119903119891119904) if and only if (44) holds with 119886
119899119896instead
of 119886119899119896and (45) (48) holdwith 119886(119899 119896) instead of 119886(119899 119896)
(vii) 119860 isin (119891119904 119886119903
119891119904) if and only if (45) (48) and (49) hold
with 119886(119899 119896) instead of 119886(119899 119896)
Corollary 21 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (119888(119901) 119886119903
119891119904) if and only if (37)and (38) hold with
119886(119899 119896119898) instead of 119886(119899 119896119898)
(ii) 119860 isin (1198880(119901) 119886
119903
119891119904) if and only if (37) holds with 119886(119899
119896119898) instead of 119886(119899 119896119898)
(iii) 119860 isin (ℓinfin(119901) 119886
119903
119891119904) if and only if (37) and (39) hold with
119886(119899 119896119898) instead of 119886(119899 119896119898)
Corollary 22 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (119886119903
119891 ℓinfin) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all
119899 isin N and (18) holds with 119886119899119896instead of 119886
119899119896
(ii) 119860 isin (119886119903119891 119888) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin N
and (18) (19) (20) and (21) hold with 119886119899119896
instead of119886119899119896
(iii) 119860 isin (119886119903119891 119888119904) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin
N and (50)(53) hold with 119886(119899 119896) instead of 119886(119899 119896) and(51)(52) hold with 119886
119899119896instead of 119886
119899119896
Corollary 23 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (ℓinfin 119886119903
119891) if and only if (18) (40) hold with 119886
119899119896in-
stead of 119886119899119896
and (41) holds with 119886(119899 119896119898) instead of119886(119899 119896119898)
(ii) 119860 isin (119891 119886119903119891) if and only if (18) (40) and (46) holdwith
119886(119899 119896119898) instead of 119886(119899 119896119898) and (42) holds with 119886119899119896
instead of 119886119899119896
(iii) 119860 isin (119888 119886119903119891) if and only if (18) (40) and (42) hold with
119886119899119896instead of 119886
119899119896
(iv) 119860 isin (119887119904 119886119903
119891) if and only if (40) (43) and (44) hold
with 119886119899119896
instead of 119886119899119896
and (45) holds with 119886(119899 119896)instead of 119886(119899 119896)
(v) 119860 isin (119891119904 119886119903
119891) if and only if (40) (44) hold with 119886
119899119896
instead of 119886119899119896 (46) holds with 119886(119899 119896119898) instead of 119886(119899
119896119898) and (45) holds with 119886(119899 119896) instead of 119886(119899 119896)
(vi) 119860 isin (119888119904 119886119903119891) if and only if (40) and (43) hold with 119886
119899119896
instead of 119886119899119896
Remark 24 Characterization of the classes (119886119903119891 119891infin) (119891infin
119886119903
119891) (119886119903119891119904 119891infin) and (119891
infin 119886119903
119891119904) is redundant since the spaces of
almost bounded sequences 119891infin
and ℓinfin
are equal
8 Abstract and Applied Analysis
Acknowledgment
The authors thank the referees for their careful reading of theoriginal paper and for the valuable comments
References
[1] J Boos Classical and Modern Methods in Summability OxfordUniversity Press New York NY USA 2000
[2] M Sengonul and F Basar ldquoSomenewCesaro sequence spaces ofnon-absolute type which include the spaces 119888
0and 119888rdquo Soochow
Journal of Mathematics vol 31 no 1 pp 107ndash119 2005[3] C Aydın and F Basar ldquoSome new sequence spaces which in-
clude the spaces 119897infinand 119897119875rdquo Demonstratio Mathematica vol 38
no 3 pp 641ndash656 2005[4] E Malkowsky and E Savas ldquoMatrix transformations between
sequence spaces of generalized weightedmeansrdquoAppliedMath-ematics and Computation vol 147 no 2 pp 333ndash345 2004
[5] MMursaleen andAKNoman ldquoOn the spaces of120582-convergentand bounded sequencesrdquo Thai Journal of Mathematics vol 8no 2 pp 311ndash329 2010
[6] B Altay F Basar and M Mursaleen ldquoOn the Euler sequencespaces which include the spaces 119897
119901and 119897infinIrdquo Information Sci-
ences vol 176 no 10 pp 1450ndash1462 2006[7] F Basar and M Kirisci ldquoAlmost convergence and generalized
differencematrixrdquoComputers ampMathematics with Applicationsvol 61 no 3 pp 602ndash611 2011
[8] A Sonmez ldquoAlmost convergence and triple bandmatrixrdquoMath-ematical and Computer Modelling vol 57 no 9-10 pp 2393ndash2402 2013
[9] K Kayaduman andM S Sengonul ldquoThe space of Cesaro almostconvergent sequence and core theoremsrdquo Acta MathematicaScientia vol 6 pp 2265ndash2278 2012
[10] M Sengonul and K Kayaduman ldquoOn the Riesz almost conver-gent sequences spacerdquo Abstract and Applied Analysis vol 2012Article ID 691694 18 pages 2012
[11] G G Lorentz ldquoA contribution to the theory of divergent se-quencesrdquo Acta Mathematica vol 80 pp 167ndash190 1948
[12] I J Maddox ldquoParanormed sequence spaces generated by infi-nite matricesrdquo Proceedings of the Cambridge Philosophical Soci-ety vol 64 pp 335ndash340 1968
[13] A Karaisa and F Ozger ldquoAlmost difference sequence space de-rived by using a generalized weightedmeanrdquoActaMathematicaScientia In review
[14] C Aydın and F Basar ldquoOn the new sequence spaces which in-clude the spaces 119888
0and 119888rdquo Hokkaido Mathematical Journal vol
33 no 2 pp 383ndash398 2004[15] F Basar ldquoA note on the triangle limitation methodsrdquo Fırat Uni-
versitesi Muhendislik Bilimleri Dergisi vol 5 no 1 pp 113ndash1171993
[16] AM Al-Jarrah and EMalkowsky ldquoBK spaces bases and linearoperatorsrdquo in Proceedings of the 3rd International Conference onFunctional Analysis and ApproximationTheory vol 1 no 52 pp177ndash191 1998
[17] A Wilansky Summability through Functional Analysis vol 85North-Holland Amsterdam The Netherlands 1984
[18] J A Sıddıqı ldquoInfinite matrices summing every almost periodicsequencerdquo Pacific Journal of Mathematics vol 39 pp 235ndash2511971
[19] J P Duran ldquoInfinite matrices and almost-convergencerdquo Math-ematische Zeitschrift vol 128 pp 75ndash83 1972
[20] J P King ldquoAlmost summable sequencesrdquo Proceedings of theAmerican Mathematical Society vol 17 pp 1219ndash1225 1966
[21] F Basar and R Colak ldquoAlmost-conservativematrix transforma-tionsrdquo Turkish Journal of Mathematics vol 13 no 3 pp 91ndash1001989
[22] F Basar ldquo119891-conservative matrix sequencesrdquo Tamkang Journalof Mathematics vol 22 no 2 pp 205ndash212 1991
[23] E Ozturk ldquoOn strongly regular dual summability methodsrdquoCommunications de la Faculte des Sciences de lrsquoUniversite drsquoAn-kara vol 32 no 1 pp 1ndash5 1983
[24] F Basar and I Solak ldquoAlmost-coercivematrix transformationsrdquoRendiconti di Matematica e delle sue Applicazioni vol 11 no 2pp 249ndash256 1991
[25] F Basar ldquoStrongly-conservative sequence-to-series matrixtransformationsrdquo Erciyes Universitesi Fen Bilimleri Dergisi vol5 no 12 pp 888ndash893 1989
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Journal ofApplied Mathematics
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Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Advances in
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Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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Volume 2014
International Journal of
Combinatorics
OperationsResearch
Advances in
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Journal of Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Advances in
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Volume 2014
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Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
the generalized limit 120572 if all Banach limits of 119909 are 120572 [11] anddenoted by 119891minus lim119909 = 120572 In other words 119891minus lim119909
119896= 120572 uni-
formly in 119899 if and only if
lim119898rarrinfin
1
119898 + 1
119898
sum
119896=0
119909119896+119899
uniformly in 119899 (3)
The characterization given above was proved by Lorentz in[11] We denote the sets of all almost convergent sequences 119891and series 119891119904 by
119891 = 119909 = (119909119896) isin 120596 lim
119898rarrinfin
119905119898119899(119909) = 120572 uniformly in 119899
(4)
where
119905119898119899(119909) =
119898
sum
119896=0
1
119898 + 1
119909119896+119899 119905minus1119899
= 0
119891119904 =
119909 = (119909119896) isin 120596
exist119897 isin C ni lim119898rarrinfin
119898
sum
119896=0
119899+119896
sum
119895=0
119909119895
119898 + 1
= 119897
uniformly in 119899
(5)We know that the inclusions 119888 sub 119891 sub ℓ
infinstrictly hold Be-
cause of these inclusions norms sdot 119891and sdot
infinof the spaces
119891 and ℓinfin
are equivalent So the sets 119891 and 1198910are BK-spaces
with the norm 119909119891= sup
119898119899|119905119898119899(119909)|
The rest of this paper is organized as follows We giveforeknowledge on the main argument of this study and nota-tions in this section In Section 2 we introduce the almostconvergent sequence and series spaces 119886119903
119891119904and 119886119903119891which are
thematrix domains of the119860119903matrix in the almost convergentsequence and series spaces119891119904 and119891 respectively In additionwe give some inclusion relations concerning the spaces 119886119903
119891
119886119903
1198910
and the non-existence of Schauder basis of the spaces 119891119904and 119886119903119891119904is shown to give certain theorems related to behavior
of some sequences In Section 3 we determine the beta- andgamma-duals of the spaces 119886119903
119891and 119886119903
119891119904and characterize the
classes (120574 119886119903119891) (119886119903119891 120583) (120575 119886119903
119891119904) and (119886119903
119891 120579) where 120574 isin
119888(119901) 1198880(119901) ℓinfin(119901) 119888119904 119887119904 119891119904 119891 119888 ℓ
infin 120583 isin 119888119904 119888 ℓ
infin 120575 isin
119888119904 119891119904 119887119904 and 120579 isin 119891 119888 119891119904 ℓinfin where 119888(119901) 119888
0(119901) and ℓ
infin(119901)
denote the space of Maddox convergent null and boundedsequence spaces defined by Maddox [12]
Lemma 1 (see [13]) The set 119891119904 has no Schauder basis
2 The Sequence Spaces 119886119903119891
1198861199031198910
and 119886119903119891119904
Derivedby the Domain of the Matrix 119860119903
In the present section we introduce the sequence spaces 119886119903119891
119886119903
1198910
and 119886119903119891119904as the set of all sequences such that119860119903-transforms
of them are in the spaces 119891 1198910 and 119891119904 respectively Further
this section is devoted to examination of the basic topologicalproperties of the sets 119886119903
119891 1198861199031198910
and 119886119903119891119904 Recently Aydın and
Basar [14] studied the sequence spaces 119886119903119888and 1198861199030
119886119903
119888= 119909 = (119909
119896) isin 120596 lim
119899rarrinfin
1
119899 + 1
119899
sum
119896=0
(1 + 119903119896
) 119909119896exists
119886119903
0= 119909 = (119909
119896) isin 120596 lim
119899rarrinfin
1
119899 + 1
119899
sum
119896=0
(1 + 119903119896
) 119909119896= 0
(6)
where 119860119903 denotes the matrix 119860119903 = (119886119903119899119896) defined by
119886119903
119899119896=
1 + 119903119896
119899 + 1
(0 ⩽ 119896 ⩽ 119899)
0 (119896 gt 119899)
(7)
Now we introduce the sequence spaces 119886119903119891 119886119903119891 and 119886119903
119891119904as
the sets of all sequences such that their 119860119903-transforms are inthe spaces 119891 119891
0 and 119891119904 respectively that is
119886119903
119891= 119909 = (119909
119896) isin 120596
exist120572 isin C ni lim119898rarrinfin
119898
sum
119896=0
1
119898 + 1
119896
sum
119894=0
1
119896 + 1
(1 + 119903119894
) 119909119899+119894
= 120572 uniformly in 119899
119886119903
1198910
= 119909 = (119909119896) isin 120596
lim119898rarrinfin
119898
sum
119896=0
1
119898 + 1
119896
sum
119894=0
1 + 119903119894
119896 + 1
119909119899+119894= 0
uniformly in 119899
119886119903
119891119904= 119909 = (119909
119896) isin 120596
exist120573 isin C ni lim119898rarrinfin
119898
sum
119896=0
119899+119896
sum
119895=0
119895
sum
119894=0
1 + 119903119895
119894 + 1
119909119899+119895
= 120573 uniformly in 119899
(8)
We can redefine the spaces 119886119903119891119904 119886119903119891 and 119886119903
1198910
by the notation of(2)
119886119903
1198910
= (1198910)119860119903 119886
119903
119891= 119891119860119903 119886
119903
119891119904= (119891119904)
119860119903 (9)
It is known by Basar [15] that the method is regular for 0 lt119903 lt 1 We assume unless stated otherwise that 0 lt 119903 lt 1
Abstract and Applied Analysis 3
Define the sequence 119910 = (119910119896) which will be frequently
used as the 119860119903-transform of a sequence 119909 = (119909119896) that is
119910119896(119903) =
119896
sum
119894=0
1 + 119903119894
119896 + 1
119909119894(119896 isin N) (10)
Theorem 2 The spaces 119886119903119891and 119886119903119891119904have no Schauder basis
Proof Since it is known that the matrix domain 120583119860of a
normed sequence space120583has a basis if and only if120583has a basiswhenever 119860 = (119886
119899119896) is a triangle [16 Remark 24] and the
space 119891 has no Schauder basis by [7 Corollary 33] we havethat 119886119903119891has no Schauder basis Since the set 119891119904 has no basis in
Lemma 1 119886119903119891119904has no Schauder basis
Theorem 3 The following statements hold
(i) The sets 119886119903119891and 119886119903
1198910
are linear spaces with the coordi-natewise addition and scalar multiplication which are119861119870-spaces with the norm
119909119886119903
119891
= sup119898
1003816100381610038161003816100381610038161003816100381610038161003816
119898
sum
119896=0
1
119898 + 1
119896
sum
119894=0
1 + 119903119894
119896 + 1
119909119894+119899
1003816100381610038161003816100381610038161003816100381610038161003816
(11)
(ii) The set 119886119903119891119904
is a linear space with the coordinatewiseaddition and scalar multiplication which is a 119861119870-spacewith the norm
119909119886119903
119891119904
= sup119898
10038161003816100381610038161003816100381610038161003816100381610038161003816
119898
sum
119896=0
1
119898 + 1
119896+119899
sum
119895=0
119895
sum
119894=0
1 + 119903119894
119895 + 1
119909119894
10038161003816100381610038161003816100381610038161003816100381610038161003816
(12)
Proof Since the second part can be similarly proved we onlyfocus on the first part Since the sequence spaces119891 and119891
0en-
dowed with the norm sdot infin
are 119861119870-spaces (see [1 Example732(b)]) and thematrix119860119903 = (119886119903
119899119896) is normalTheorem 432
of Wilansky [17 p61] gives the fact that the spaces 119886119903119891and 1198861199031198910
are 119861119870-spaces with the norm in (11)
Now we may give the following theorem concerning theisomorphism between our spaces and the sets 119891 119891
0 and 119891119904
Theorem 4 The sequence spaces 119886119903119891 1198861199031198910
and 119886119903119891119904are linearly
isomorphic to the sequence spaces 119891 1198910 and 119891119904 respectively
that is 119886119903119891cong 119891 119886119903
1198910
cong 1198910 and 119886119903
119891119904cong 119891119904
Proof To prove the fact that 119886119903119891cong 119891 we should show the exis-
tence of a linear bijection between the spaces 119886119903119891and 119891 Con-
sider the transformation 119879 defined with the notation of (2)from 119886
119903
119891to119891 by 119909 997891rarr 119910 = 119879119909 = 119860
119903
119909The linearity of119879 is clearFurther it is clear that119909 = 120579whenever119879119909 = 120579 and hence119879 isinjective
Let 119910 = (119910119896) isin 119886119903
119891 and define the sequence 119909 = (119909
119896(119903)) by
119909119896=
1
1 + 119903119896
[(119896 + 1) 119910119896minus 119896119910119896minus1] for each 119896 isin N
(13)
whence119891119860119903 minus lim119909
= lim119898rarrinfin
119898
sum
119896=0
1
119898 + 1
times
119896
sum
119894=0
(1 + 119903119894
) 119909119894+119899
1 + 119896
uniformly in 119899
= lim119898rarrinfin
119898
sum
119896=0
1
119898 + 1
times
119896
sum
119894=0
(1 + 119903119894
) [1 (1 + 119903119894
) (119910119894+119899(119896 + 1) minus 119910
119894+119899minus1119896)]
1 + 119896
uniformly in 119899
= lim119898rarrinfin
1
119898 + 1
119898
sum
119896=0
119910119896+119899
uniformly in 119899
= 119891 minus lim119910(14)
which implies that119909 isin 119886119903119891 As a result119879 is surjectiveHence119879
is a linear bijection which implies that the spaces 119886119903119891and119891 are
linearly isomorphic as desired Similarly the isomorphisms119886119903
1198910
cong 1198910and 119886119903119891119904cong 119891119904 can be proved
Theorem 5 The inclusion 119891 sub 119886119903119891strictly holds
Proof Let 119909 = (119909119896) isin 119888 Since 119888 sub 119891 119909 isin 119891 Because 119860119903 is
regular for 0 lt 119903 lt 1119860119903119909 isin 119888Therefore since lim119860119903119909 = 119891minuslim119860119903119909 we see that 119909 isin 119886119903
119891 So we have that the inclusion119891 sub
119886119903
119891holds Further consider the sequence 119905 = (119905
119896(119903)) defined
by 119905119896(119903) = (2119896 + 1)(1 + 119903
119896
)(minus1)119896
forall119896 isin N Then since 119860119903t =(minus1)119899
isin 119891 119909 isin 119886119903119891 One can easily see that 119905 notin 119891Thus 119905 isin 119886119903
119891
119891 and this completes the proof
Theorem 6 The sequence spaces 119886119903119891and ℓinfin
overlap but nei-ther of them contains the other
Proof Let us consider the sequence 119906 = (119906119896(119903)) defined by
119906119896(119903) = 1(1+119903
119896
) for allNThen since119860119903119906 = 119890 isin 119891 119906 isin 119886119903119891 It
is clear that 119906 isin ℓinfin This means that the sequence spaces 119886119903
119891
and ℓinfinare not disjointNowwe show that the sequence space
119886119903
119891and ℓinfin
do not include each other Let us consider the se-quence 119905 = (119905
119896(119903)) defined as in proof of Theorem 5 above
and 119911 = (119911119896(119903)) = (0 0 1(1 + 119903
101
) 1(1 + 119903110
)
0 0 1(1 + 119903211
) 1(1 + 119903231
) 0 0 ) where theblocks of 0rsquos are increasing by factors of 100 and the blocks of1(1 + 119903
119896
)rsquos are increasing by factors of 10 Then since 119860119903119905 =(minus1)119899
isin 119891 119905 isin 119886119903119891 but 119905 notin ℓ
infinTherefore 119905 isin 119886119903
119891ℓinfin Also the
sequence 119911 notin 119886119903119891since119860119903119911 = (0 0 1 1 0 0 1
1 0 0 ) notin 119891 where the blocks of 0rsquos are increasing byfactors of 100 and the blocks of 1rsquos are increasing by factors of10 but 119911 is boundedThis means that 119911 isin ℓ
infin 119886119903
119891 Hence the
4 Abstract and Applied Analysis
sequence spaces 119886119903119891and ℓinfinoverlap but neither of them con-
tains the other This completes the proof
Theorem 7 Let the spaces 1198861199031198910
119886119903119888 and 119886119903
119891be given Then
(i) 1198861199031198910
sub 119886119903
119891strictly hold
(ii) 119886119903119888sub 119886119903
119891strictly hold
Proof (i) Let 119909 = (119909119896) isin 119886
119903
1198910
which means that 119860119903119909 isin 1198910
Since1198910sub 119891119860119903119909 isin 119891This implies that119909 isin 119886119903
119891Thuswe have
119886119903
1198910
sub 119886119903
119891
Now we show that this inclusion is strict Let us considerthe sequence 119906 = (119906
119896(119903)) defined as in proof ofTheorem 6 for
all 119896 isin N Consider the following
119891119860119903 minus lim 119906 = lim
119898rarrinfin
119898
sum
119896=0
1
119898 + 1
119896
sum
119894=0
1 + 119903119894
119896 + 1
119906119894+119899
= lim119898rarrinfin
119898
sum
119896=0
1
119898 + 1
= 119890 = (1 1 )
(15)
which means that 119906 isin 119886119903119891 119886119903
1198910
that is to say the inclusion isstrict
(ii) Let 119909 = (119909119896) isin 119886119903
119888whichmeans that119860119903119909 isin 119888 Since 119888 sub
119891 119860119903119909 isin 119891 This implies that 119909 isin 119886119903119891 Thus we have
119886119903
119888sub 119886119903
119891 Furthermore let us consider the sequence
119905 = 119905119896(119903) defined as in proof of Theorem 5 for all
119896 isin N Then since 119860119903119905 = (minus1)119899 isin 119891 119888 119905 isin 119886119903119891 119886119903
119888
This completes the proof
3 Certain Matrix Mappings on the Sets 119886119903119891
119886119903119891119904
and Some Duals
In this section we will characterize some matrix transforma-tions between the spaces of 119860119903 almost convergent sequenceand almost convergent series in addition to paranormed andclassical sequence spaces after giving 120573- and 120574-duals of thespaces 119886119903
119891119904and 119886119903119891We start with the definition of the beta- and
gamma-dualsIf119909 and119910 are sequences and119883 and119884 are subsets of120596 then
we write 119909 sdot 119910 = (119909119896119910119896)infin
119896=0 119909minus1 lowast 119884 = 119886 isin 120596 119886 sdot 119909 isin 119884 and
119872(119883119884) = ⋂
119909isin119883
119909minus1
lowast 119884 = 119886 119886 sdot 119909 isin 119884 forall119909 isin 119883 (16)
for themultiplier space of119883 and119884 One can easily observe fora sequence space 119885 with 119884 sub 119885 and 119885 sub 119883 that inclusions119872(119883119884) sub 119872(119883119885) and119872(119883119884) sub 119872(119885 119884) hold respec-tively The 120572- 120573- and 120574-duals of a sequence space which arerespectively denoted by119883120572 119883120573 and119883120574 are defined by
119883120572
= 119872(119883 ℓ1) 119883
120573
= 119872(119883 119888119904) 119883120574
= 119872(119883 119887119904)
(17)
It is obvious that 119883120572 sub 119883120573 sub 119883120574 Also it can easily be seenthat the inclusions 119883120572 sub 119884120572 119883120573 sub 119884120573 and 119883120574 sub 119884
120574 holdwhenever 119884 sub 119883
Lemma 8 (see [18]) 119860 = (119886119899119896) isin (119891 ℓ
infin) if and only if
sup119899
sum
119896
1003816100381610038161003816119886119899119896
1003816100381610038161003816lt infin (18)
Lemma 9 (see [18]) 119860 = (119886119899119896) isin (119891 119888) if and only if (18)
holds and there are 120572 120572119896isin C such that
lim119899rarrinfin
119886119899119896= 120572119896
forallk isin N (19)
lim119899rarrinfin
sum
119896
119886119899119896= 120572 (20)
lim119899rarrinfin
sum
119896
1003816100381610038161003816Δ (119886119899119896minus 120572119896)1003816100381610038161003816= 0 (21)
Theorem 10 Define the sets 1199051199031and 1199051199032by
119905119903
1= 119886 = (119886
119896) isin 120596 sum
119896
1003816100381610038161003816100381610038161003816
Δ (
119886119896
1 + 119903119896
) (119896 + 1)
1003816100381610038161003816100381610038161003816
lt infin
119905119903
2= 119886 = (119886
119896) isin 120596 sup
119896
10038161003816100381610038161003816100381610038161003816
119886119896(119896 + 1)
1 + 119903119896
10038161003816100381610038161003816100381610038161003816
lt infin
(22)
where Δ(119886119896(1 + 119903
119896
)) = 119886119896(1 + 119903
119896
) minus 119886119896+1(1 + 119903
119896+1
) for all 119896 isinN Then (119886119903
119891)120574
= 119905119903
1cap 119905119903
2
Proof Take any sequence 119886 = (119886119896) isin 120596 and consider the fol-
lowing equality119899
sum
119896=0
119886119896119909119896=
119899
sum
119896=0
119886119896[
119896
sum
119894=119896minus1
(minus1)119896minus119895
119894 + 1
1 + 119903119894119910119894]
=
119899minus1
sum
119896=0
Δ(
119886119896
1 + 119903119896
) (119896 + 1) 119910119896
+
119899 + 1
1 + 119903119899119886119899119910119899
= (119879119910)119899
(23)
where 119879 = 119905119903119899119896 is
119905119903
119899119896=
Δ(
119886119896
1 + 119903119896
) (119896 + 1) (0 ⩽ 119896 ⩽ 119899 minus 1)
119899 + 1
1 + 119903119899119886119899
(119896 = 119899)
0 (119896 gt 119899)
(24)
for all 119896 119899 isin NThus we deduce from (23) that 119886119909 = (119886119896119909119896) isin
119887119904 whenever 119909 = (119909119896) isin 119886119903
119891if and only if 119879119910 isin ℓ
infinwhenever
119910 = (119910119896) isin 119891 where 119879 = 119905119903
119899119896 is defined in (24) Therefore
with the help of Lemma 8 (119886119903119891)120574
= 119905119903
1cap 119905119903
2
Theorem 11 The 120573-dual of the space 119886119903119891is the intersection of
the sets
119905119903
3= 119886 = (119886
119896) isin 120596 lim
119899rarrinfin
sum
119896
1003816100381610038161003816Δ (119905119903
119899119896minus 120572119896)1003816100381610038161003816= 0
119905119903
4= 119886 = (119886
119896) isin 120596 (
119896 + 1
1 + 119903119896
119886119896) isin 119888119904
(25)
where lim119899rarrinfin
119905119903
119899119896= 120572119896for all 119896 isin N Then (119886119903
119891)120573
= 119905119903
3cap 119905119903
4
Abstract and Applied Analysis 5
Proof Let us take any sequence 119886 isin 120596 By (23) 119886119909 = (119886119896119909119896) isin
119888119904whenever 119909 = (119909119896) isin 119886119903
119891if and only if 119879119910 isin 119888whenever 119910 =
(119910119896) isin 119891 It is obvious that the columns of that matrix 119879 in 119888
where119879 = 119905119903119899119896 defined in (24) we derive the consequence by
Lemma 9 that (119886119903119891)120573
= 119905119903
3cap 119905119903
4
Theorem 12 The 120574-dual of the space 119886119903119891119904is the intersection of
the sets
119888119903
1= 119886 = (119886
119896) isin 120596 sum
119896
1003816100381610038161003816100381610038161003816
Δ [Δ(
119886119896
1 + 119903119896
) (119896 + 1)
+
119886119896
1 + 119903119896(119896 + 1)]
1003816100381610038161003816100381610038161003816
lt infin
119888119903
2= 119886 = (119886
119896) isin 120596 (
119886119896(119896 + 1)
1 + 119903119896
) isin 1198880
(26)
In other words we have (119886119903119891119904)120574
= 119888119903
1cap 119888119903
2
Proof We obtain from (23) that 119886119909 = (119886119896119909119896) isin 119887119904 whenever
119909 = (119909119896) isin 119886119903
119891119904if and only if119879119910 isin ℓ
infinwhenever 119910 = (119910
119896) isin 119891119904
where 119879 = 119905119903
119899119896 is defined in (24) Therefore by Lemma 19
(viii) (119886119903119891119904)120574
= 119888119903
1cap 119888119903
2
Theorem 13 Define the set 1198881199033by
119888119903
3= 119886 = (119886
119896) isin 120596 lim
119899rarrinfin
sum
119896
10038161003816100381610038161003816Δ2
(119905119903
119899119896)
10038161003816100381610038161003816exists (27)
Then (119886119903119891119904)120573
= 119888119903
1cap 119888119903
2cap 119888119903
3
Proof This may be obtained in the same way as mentionedin the proof of Theorem 12 with Lemma 19(viii) instead ofLemma 19(vii) So we omit details
For the sake of brevity the following notations will beused
119886 (119899 119896119898) =
1
119898 + 1
119898
sum
119894=0
119886119899+119894119896
119886 (119899 119896) =
119899
sum
119894=0
119886119894119896
119886119899119896= Δ(
119886119899119896
1 + 119903119896
) (119896 + 1) = (
119886119899119896
1 + 119903119896
minus
119886119899119896+1
1 + 119903119896+1
) (119896 + 1)
Δ119886119899119896= 119886119899119896minus 119886119899119896+1
119886119899119896=
119899
sum
119895=0
(1 + 119903119895
) 119890119895119896
119899 + 1
(28)
for all 119896 119899 isin N Assume that the infinite matrices 119860 = (119886119899119896)
and119861 = (119887119899119896)map the sequences119909 = (119909
119896) and119910 = (119910
119896)which
are connectedwith relation (10) to the sequences 119906 = (119906119899) and
V = (V119899) respectively that is
119906119899= (119860119909)
119899= sum
119896
119886119899119896119909119896
forall119899 isin N (29)
V119899= (119861119910)
119899= sum
119896
119887119899119896119910119896
forall119899 isin N (30)
One can easily conclude here that the method 119860 is directlyapplied to the terms of the sequence 119909 = (119909
119896) while the
method 119861 is applied to the 119860119903-transform of the sequence 119909 =(119909119896) So the methods 119860 and 119861 are essentially differentNow suppose that the matrix product 119861119860119903 exists which is
amuch weaker assumption than the conditions on thematrix119861 belonging to anymatrix class in general It is not difficult tosee that the sequence in (30) reduces to the sequence in (29)under the application of formal summation by parts Thisleads us to the fact that 119861119860119903 exists and is equal to 119860 and(119861119860119903
)119909 = 119861(119860119903
119909) formally holds if one side exists This state-ment is equivalent to the following relation between the en-tries of the matrices 119860 = (119886
119899119896) and 119861 = (119887
119899119896) which are con-
nected with the relation
119886119899119896= 119887119899119896= Δ(
119886119899119896
1 + 119903119896
) (119896 + 1) or
119886119899119896= (1 + 119903
119896
)
infin
sum
119895=119896
119887119899119895
1 + 119895
forall119896 119899 isin N
(31)
Note that the methods 119860 and 119861 are not necessarily equi-valent since the order of summationmay not be reversedWenow give the following fundamental theorem connected withthematrixmappings oninto the almost convergent spaces 119886119903
119891
and 119886119903119891s
Theorem 14 Suppose that the entries of the infinite matrices119860 = (119886
119899119896) and 119861 = (119887
119899119896) are connected with relation (31) for all
119896 119899 isin N and let 120582 be any given sequence spaceThen119860 isin (119886119903119891
120582) if and only if
119861 isin (119891 120582)
119899 + 1
1 + 119903119896
119886119899119896
119896isinNisin 1198880
(32)
Proof Suppose that 119860 = (119886119899119896) and 119861 = (119887
119899119896) are connected
with the relation (31) and let 120582 be any given sequence spaceand keep in mind that the spaces 119886119903
119891and 119891 are norm iso-
morphicLet 119860 isin (119886119903
119891 120582) and take any sequence 119909 isin 119886119903
119891 and keep
in mind that 119910 = 119860119903119909 Then (119886119899119896)119896isinN isin (119886
119903
119891)120573 that is (32)
holds for all 119899 isin N and 119861119860119903 exists which implies that(119887119899119896)119896isinN isin ℓ1 = 119891
120573 for each 119899 isin N Thus 119861119910 exists for all 119910 isin119891 and thus we have119898 rarr infin in the equality
119898
sum
119896=0
119887119899119896119910119896=
119898
sum
119896=0
119898
sum
119895=119896
(1 + 119903119896
)
119887119899119895
1 + 119895
119909119896 (33)
for all119898 119899 isin N and we have (31) 119861119910 = 119860119909 which means that119861 isin (119891 120582) On the other hand assume that (32) holds and119861 isin (119891 120582) Then we have (119887
119899119896)119896isinN isin ℓ1 for all 119899 isin N which
6 Abstract and Applied Analysis
gives together with (120592119899119896)119896isinN isin (119886
119903
119891)120573 for each 119899 isin N that 119860119909
exists Then we obtain from the equality
119898
sum
119896=0
119886119899119896119909119896=
119898minus1
sum
119896=0
Δ(
119886119899119896
1 + 119903119896
) (119896 + 1) 119910119896+
119898 + 1
1 + 119903119898119886119899119898119910119898
=
119898
sum
119896=0
119887119899119896119910119896
(34)
for all 119898 119899 isin N as 119898 rarr infin that 119860119909 = 119861119910 and this showsthat 119860 isin (119886119903
119891 120582)
Theorem 15 Suppose that the entries of the infinite matrices119864 = (119890
119899119896) and 119865 = (119891
119899119896) are connected with the relation
119891119899119896= 119886119899119896
(35)
for all 119898 119899 isin N and 120582 is any given sequence space Then 119864 isin(120582 119886119903
119891) if and only if 119865 isin (120582 119891)
Proof Let 119909 = (119909119896) isin 120582 and consider the following equality
119899
sum
119895=0
119898
sum
119896=0
1 + 119903119895
119899 + 1
119890119895119896119909119896=
119898
sum
119896=0
119891119899119896119909119896 (36)
for all 119896119898 119899 isin N which yields as 119898 rarr infin that 119864119909 isin 119886119903119891
whenever 119909 isin 120582 if and only if 119865119909 isin 119891 whenever 119909 isin 120582 Thisstep completes the proof
Theorem 16 Let120582 be any given sequence space and thematri-ces119860 = (119886
119899119896) and119861 = (119887
119899119896) are connectedwith the relation (31)
Then 119860 isin (119886119903
119891119904 120582) if and only if 119861 isin (119891119904 120582) and (119886
119899119896)119896isinN isin
(119886119903
119891119904)120573 for all 119899 isin N
Proof The proof is based on the proof of Theorem 14
Theorem 17 Let 120582 be any given sequence space and the ele-ments of the infinite matrices 119864 = (119890
119899119896) and 119865 = (119891
119899119896) are con-
nectedwith relation (35)Then119864 = (119890119899119896) isin (120582 119886
119903
119891119904) if and only
if 119865 isin (120582 119891119904)
Proof The proof is based on the proof of Theorem 15
By Theorems 14 15 16 and 17 we have quite a few out-comes depending on the choice of the space 120582 to characterizecertain matrix mappings Hence by the help of these theo-rems the necessary and sufficient conditions for the classes(119886119903
119891 120582) (120582 119886119903
119891) (119886119903119891119904 120582) and (120582 119886119903
119891119904) may be derived by
replacing the entries of119860 and 119861 by those of 119861 = 119860(119860119903)minus1 and119865 = 119860
119903
119864 respectively where the necessary and sufficient con-ditions on the matrices 119864 and 119865 are read from the concerningresults in the existing literature
Lemma 18 Let 119860 = (119886119899119896) be an infinite matrix Then the fol-
lowing statements hold
(i) 119860 isin (1198880(119901) 119891) if and only if
exist119873 gt 1 ni sup119898isinN
sum
119896
|119886 (119899 119896 119898)|1198731119901119896
lt infin forall119899 isin N
exist120572119896isin C forall119896 isin N ni lim
119898rarrinfin
119886 (119899 119896119898) = 120572119896
uniformly in 119899(37)
(ii) 119860 isin (119888(119901) 119891) if and only if (37) and
exist120572 isin C ni lim119898rarrinfin
sum
119896
119886 (119899 119896119898) = 120572 uniformly in 119899 (38)
(iii) 119860 isin (ℓinfin(119901) 119891) if and only if (37) and
exist119873 gt 1 ni lim119898rarrinfin
sum
119896
1003816100381610038161003816119886 (119899 119896 119898) minus 120572
119896
10038161003816100381610038161198731119901119896
= 0
uniformly in 119899(39)
Lemma 19 Let 119860 = (119886119899119896) be an infinite matrix Then the fol-
lowing statements hold
(i) (Duran [19])119860 isin (ℓinfin 119891) if and only if (18) holds and
119891 minus lim119886119899119896= 120572119896exists for each fixed 119896 (40)
lim119898rarrinfin
sum
119896
1003816100381610038161003816119886 (119899 119896 119898) minus 120572
119896
1003816100381610038161003816= 0 uniformly in 119899 (41)
(ii) (King [20]) 119860 isin (119888 119891) if and only if (18) (40) holdand
119891 minus limsum119896
119886119899119896= 120572 (42)
(iii) (Basar and Colak [21])119860 isin (119888119904 119891) if and only if (40)holds and
sup119899isinN
sum
119896
1003816100381610038161003816Δ119886119899119896
1003816100381610038161003816lt infin (43)
(iv) (Basar and Colak [21])119860 isin (119887119904 119891) if and only if (40)(43) hold and
lim119896
119886119899119896= 0 exists for each fixed 119899 (44)
lim119902rarrinfin
sum
119896
1
119902 + 1
119902
sum
119894=0
1003816100381610038161003816Δ [119886 (119899 + 119894 119896) minus 120572
119896]1003816100381610038161003816= 0 uniformly in 119899
(45)
(v) (Duran [19]) 119860 isin (119891 119891) if and only if (18) (40) and(42) hold and
lim119898rarrinfin
sum
119896
1003816100381610038161003816Δ [119886 (119899 119896119898) minus 120572
119896]1003816100381610038161003816= 0 uniformly in 119899 (46)
(vi) (Basar [22])119860 isin (119891119904 119891) if and only if (40) (44) (46)and (45) hold
Abstract and Applied Analysis 7
(vii) (Ozturk [23])119860 isin (119891119904 119888) if and only if (19) (43) and(44) hold and
lim119899rarrinfin
sum
119896
10038161003816100381610038161003816Δ2
119886119899119896
10038161003816100381610038161003816= 120572 (47)
(viii) 119860 isin (119891119904 ℓinfin) if and only if (43) and (44) hold
(ix) (Basar and Solak [24]) 119860 isin (119887119904 119891119904) if and only if(44) (45) hold and
sup119899isinN
sum
119896
|Δ119886 (119899 119896)| lt infin
119891 minus lim 119886 (119899 119896) = 120572119896exists for each fixed 119896
(48)
(x) (Basar [22])119860 isin (119891119904 119891119904) if and only if (45) (48) holdand
lim119902rarrinfin
sum
119896
1
119902 + 1
119902
sum
119894=0
10038161003816100381610038161003816Δ2
[119886 (119899 + 119894 119896) minus 120572119896]
10038161003816100381610038161003816= 0 uniformly in 119899
(49)
(xi) (Basar and Colak [21])119860 isin (119888119904 119891119904) if and only if (48)holds
(xii) (Basar [25]) 119860 isin (119891 119888119904) if and only if
sup119899isinN
sum
119896
|119886 (119899 119896)| lt infin (50)
sum
119899
119886119899119896= 120572119896
exists for each fixed 119896 (51)
sum
119899
sum
119896
119886119899119896= 120572 (52)
lim119898rarrinfin
sum
119896
1003816100381610038161003816Δ [119886 (119899 119896) minus 120572
119896]1003816100381610038161003816= 0 (53)
Now we give our main results which are related to matrixmappings oninto the spaces of almost convergent series 119886119903
119891119904
and sequences 119886119903119891
Corollary 20 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (119886119903119891119904 119891) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin
N and (40) (44) holdwith 119886119899119896instead of 119886
119899119896 (46) holds
with 119886(119899 119896119898) instead of 119886(119899 119896119898) and (45) holdswith 119886(119899 119896) instead of 119886(119899 119896)
(ii) 119860 isin (119886119903119891119904 119888) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin N
and (19) (43) (44) and (47) hold with 119886119899119896
instead of119886119899119896
(iii) 119860 isin (119886119903119891119904 ℓinfin) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin
N and (43) and (44) hold with 119886119899119896instead of 119886
119899119896
(iv) 119860 isin (119886119903119891119904 119891119904) if and only if 119886
119899119896119896isinN isin (119886
119903
119891119904)120573 for all 119899 isin
N and (45) (48) and (49) hold with 119886(119899 119896) instead of119886(119899 119896)
(v) 119860 isin (119888119904 119886119903119891119904) if and only if (48) holds with 119886(119899 119896) in-
stead of 119886(119899 119896)
(vi) 119860 isin (119887119904 119886119903119891119904) if and only if (44) holds with 119886
119899119896instead
of 119886119899119896and (45) (48) holdwith 119886(119899 119896) instead of 119886(119899 119896)
(vii) 119860 isin (119891119904 119886119903
119891119904) if and only if (45) (48) and (49) hold
with 119886(119899 119896) instead of 119886(119899 119896)
Corollary 21 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (119888(119901) 119886119903
119891119904) if and only if (37)and (38) hold with
119886(119899 119896119898) instead of 119886(119899 119896119898)
(ii) 119860 isin (1198880(119901) 119886
119903
119891119904) if and only if (37) holds with 119886(119899
119896119898) instead of 119886(119899 119896119898)
(iii) 119860 isin (ℓinfin(119901) 119886
119903
119891119904) if and only if (37) and (39) hold with
119886(119899 119896119898) instead of 119886(119899 119896119898)
Corollary 22 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (119886119903
119891 ℓinfin) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all
119899 isin N and (18) holds with 119886119899119896instead of 119886
119899119896
(ii) 119860 isin (119886119903119891 119888) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin N
and (18) (19) (20) and (21) hold with 119886119899119896
instead of119886119899119896
(iii) 119860 isin (119886119903119891 119888119904) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin
N and (50)(53) hold with 119886(119899 119896) instead of 119886(119899 119896) and(51)(52) hold with 119886
119899119896instead of 119886
119899119896
Corollary 23 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (ℓinfin 119886119903
119891) if and only if (18) (40) hold with 119886
119899119896in-
stead of 119886119899119896
and (41) holds with 119886(119899 119896119898) instead of119886(119899 119896119898)
(ii) 119860 isin (119891 119886119903119891) if and only if (18) (40) and (46) holdwith
119886(119899 119896119898) instead of 119886(119899 119896119898) and (42) holds with 119886119899119896
instead of 119886119899119896
(iii) 119860 isin (119888 119886119903119891) if and only if (18) (40) and (42) hold with
119886119899119896instead of 119886
119899119896
(iv) 119860 isin (119887119904 119886119903
119891) if and only if (40) (43) and (44) hold
with 119886119899119896
instead of 119886119899119896
and (45) holds with 119886(119899 119896)instead of 119886(119899 119896)
(v) 119860 isin (119891119904 119886119903
119891) if and only if (40) (44) hold with 119886
119899119896
instead of 119886119899119896 (46) holds with 119886(119899 119896119898) instead of 119886(119899
119896119898) and (45) holds with 119886(119899 119896) instead of 119886(119899 119896)
(vi) 119860 isin (119888119904 119886119903119891) if and only if (40) and (43) hold with 119886
119899119896
instead of 119886119899119896
Remark 24 Characterization of the classes (119886119903119891 119891infin) (119891infin
119886119903
119891) (119886119903119891119904 119891infin) and (119891
infin 119886119903
119891119904) is redundant since the spaces of
almost bounded sequences 119891infin
and ℓinfin
are equal
8 Abstract and Applied Analysis
Acknowledgment
The authors thank the referees for their careful reading of theoriginal paper and for the valuable comments
References
[1] J Boos Classical and Modern Methods in Summability OxfordUniversity Press New York NY USA 2000
[2] M Sengonul and F Basar ldquoSomenewCesaro sequence spaces ofnon-absolute type which include the spaces 119888
0and 119888rdquo Soochow
Journal of Mathematics vol 31 no 1 pp 107ndash119 2005[3] C Aydın and F Basar ldquoSome new sequence spaces which in-
clude the spaces 119897infinand 119897119875rdquo Demonstratio Mathematica vol 38
no 3 pp 641ndash656 2005[4] E Malkowsky and E Savas ldquoMatrix transformations between
sequence spaces of generalized weightedmeansrdquoAppliedMath-ematics and Computation vol 147 no 2 pp 333ndash345 2004
[5] MMursaleen andAKNoman ldquoOn the spaces of120582-convergentand bounded sequencesrdquo Thai Journal of Mathematics vol 8no 2 pp 311ndash329 2010
[6] B Altay F Basar and M Mursaleen ldquoOn the Euler sequencespaces which include the spaces 119897
119901and 119897infinIrdquo Information Sci-
ences vol 176 no 10 pp 1450ndash1462 2006[7] F Basar and M Kirisci ldquoAlmost convergence and generalized
differencematrixrdquoComputers ampMathematics with Applicationsvol 61 no 3 pp 602ndash611 2011
[8] A Sonmez ldquoAlmost convergence and triple bandmatrixrdquoMath-ematical and Computer Modelling vol 57 no 9-10 pp 2393ndash2402 2013
[9] K Kayaduman andM S Sengonul ldquoThe space of Cesaro almostconvergent sequence and core theoremsrdquo Acta MathematicaScientia vol 6 pp 2265ndash2278 2012
[10] M Sengonul and K Kayaduman ldquoOn the Riesz almost conver-gent sequences spacerdquo Abstract and Applied Analysis vol 2012Article ID 691694 18 pages 2012
[11] G G Lorentz ldquoA contribution to the theory of divergent se-quencesrdquo Acta Mathematica vol 80 pp 167ndash190 1948
[12] I J Maddox ldquoParanormed sequence spaces generated by infi-nite matricesrdquo Proceedings of the Cambridge Philosophical Soci-ety vol 64 pp 335ndash340 1968
[13] A Karaisa and F Ozger ldquoAlmost difference sequence space de-rived by using a generalized weightedmeanrdquoActaMathematicaScientia In review
[14] C Aydın and F Basar ldquoOn the new sequence spaces which in-clude the spaces 119888
0and 119888rdquo Hokkaido Mathematical Journal vol
33 no 2 pp 383ndash398 2004[15] F Basar ldquoA note on the triangle limitation methodsrdquo Fırat Uni-
versitesi Muhendislik Bilimleri Dergisi vol 5 no 1 pp 113ndash1171993
[16] AM Al-Jarrah and EMalkowsky ldquoBK spaces bases and linearoperatorsrdquo in Proceedings of the 3rd International Conference onFunctional Analysis and ApproximationTheory vol 1 no 52 pp177ndash191 1998
[17] A Wilansky Summability through Functional Analysis vol 85North-Holland Amsterdam The Netherlands 1984
[18] J A Sıddıqı ldquoInfinite matrices summing every almost periodicsequencerdquo Pacific Journal of Mathematics vol 39 pp 235ndash2511971
[19] J P Duran ldquoInfinite matrices and almost-convergencerdquo Math-ematische Zeitschrift vol 128 pp 75ndash83 1972
[20] J P King ldquoAlmost summable sequencesrdquo Proceedings of theAmerican Mathematical Society vol 17 pp 1219ndash1225 1966
[21] F Basar and R Colak ldquoAlmost-conservativematrix transforma-tionsrdquo Turkish Journal of Mathematics vol 13 no 3 pp 91ndash1001989
[22] F Basar ldquo119891-conservative matrix sequencesrdquo Tamkang Journalof Mathematics vol 22 no 2 pp 205ndash212 1991
[23] E Ozturk ldquoOn strongly regular dual summability methodsrdquoCommunications de la Faculte des Sciences de lrsquoUniversite drsquoAn-kara vol 32 no 1 pp 1ndash5 1983
[24] F Basar and I Solak ldquoAlmost-coercivematrix transformationsrdquoRendiconti di Matematica e delle sue Applicazioni vol 11 no 2pp 249ndash256 1991
[25] F Basar ldquoStrongly-conservative sequence-to-series matrixtransformationsrdquo Erciyes Universitesi Fen Bilimleri Dergisi vol5 no 12 pp 888ndash893 1989
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Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
Define the sequence 119910 = (119910119896) which will be frequently
used as the 119860119903-transform of a sequence 119909 = (119909119896) that is
119910119896(119903) =
119896
sum
119894=0
1 + 119903119894
119896 + 1
119909119894(119896 isin N) (10)
Theorem 2 The spaces 119886119903119891and 119886119903119891119904have no Schauder basis
Proof Since it is known that the matrix domain 120583119860of a
normed sequence space120583has a basis if and only if120583has a basiswhenever 119860 = (119886
119899119896) is a triangle [16 Remark 24] and the
space 119891 has no Schauder basis by [7 Corollary 33] we havethat 119886119903119891has no Schauder basis Since the set 119891119904 has no basis in
Lemma 1 119886119903119891119904has no Schauder basis
Theorem 3 The following statements hold
(i) The sets 119886119903119891and 119886119903
1198910
are linear spaces with the coordi-natewise addition and scalar multiplication which are119861119870-spaces with the norm
119909119886119903
119891
= sup119898
1003816100381610038161003816100381610038161003816100381610038161003816
119898
sum
119896=0
1
119898 + 1
119896
sum
119894=0
1 + 119903119894
119896 + 1
119909119894+119899
1003816100381610038161003816100381610038161003816100381610038161003816
(11)
(ii) The set 119886119903119891119904
is a linear space with the coordinatewiseaddition and scalar multiplication which is a 119861119870-spacewith the norm
119909119886119903
119891119904
= sup119898
10038161003816100381610038161003816100381610038161003816100381610038161003816
119898
sum
119896=0
1
119898 + 1
119896+119899
sum
119895=0
119895
sum
119894=0
1 + 119903119894
119895 + 1
119909119894
10038161003816100381610038161003816100381610038161003816100381610038161003816
(12)
Proof Since the second part can be similarly proved we onlyfocus on the first part Since the sequence spaces119891 and119891
0en-
dowed with the norm sdot infin
are 119861119870-spaces (see [1 Example732(b)]) and thematrix119860119903 = (119886119903
119899119896) is normalTheorem 432
of Wilansky [17 p61] gives the fact that the spaces 119886119903119891and 1198861199031198910
are 119861119870-spaces with the norm in (11)
Now we may give the following theorem concerning theisomorphism between our spaces and the sets 119891 119891
0 and 119891119904
Theorem 4 The sequence spaces 119886119903119891 1198861199031198910
and 119886119903119891119904are linearly
isomorphic to the sequence spaces 119891 1198910 and 119891119904 respectively
that is 119886119903119891cong 119891 119886119903
1198910
cong 1198910 and 119886119903
119891119904cong 119891119904
Proof To prove the fact that 119886119903119891cong 119891 we should show the exis-
tence of a linear bijection between the spaces 119886119903119891and 119891 Con-
sider the transformation 119879 defined with the notation of (2)from 119886
119903
119891to119891 by 119909 997891rarr 119910 = 119879119909 = 119860
119903
119909The linearity of119879 is clearFurther it is clear that119909 = 120579whenever119879119909 = 120579 and hence119879 isinjective
Let 119910 = (119910119896) isin 119886119903
119891 and define the sequence 119909 = (119909
119896(119903)) by
119909119896=
1
1 + 119903119896
[(119896 + 1) 119910119896minus 119896119910119896minus1] for each 119896 isin N
(13)
whence119891119860119903 minus lim119909
= lim119898rarrinfin
119898
sum
119896=0
1
119898 + 1
times
119896
sum
119894=0
(1 + 119903119894
) 119909119894+119899
1 + 119896
uniformly in 119899
= lim119898rarrinfin
119898
sum
119896=0
1
119898 + 1
times
119896
sum
119894=0
(1 + 119903119894
) [1 (1 + 119903119894
) (119910119894+119899(119896 + 1) minus 119910
119894+119899minus1119896)]
1 + 119896
uniformly in 119899
= lim119898rarrinfin
1
119898 + 1
119898
sum
119896=0
119910119896+119899
uniformly in 119899
= 119891 minus lim119910(14)
which implies that119909 isin 119886119903119891 As a result119879 is surjectiveHence119879
is a linear bijection which implies that the spaces 119886119903119891and119891 are
linearly isomorphic as desired Similarly the isomorphisms119886119903
1198910
cong 1198910and 119886119903119891119904cong 119891119904 can be proved
Theorem 5 The inclusion 119891 sub 119886119903119891strictly holds
Proof Let 119909 = (119909119896) isin 119888 Since 119888 sub 119891 119909 isin 119891 Because 119860119903 is
regular for 0 lt 119903 lt 1119860119903119909 isin 119888Therefore since lim119860119903119909 = 119891minuslim119860119903119909 we see that 119909 isin 119886119903
119891 So we have that the inclusion119891 sub
119886119903
119891holds Further consider the sequence 119905 = (119905
119896(119903)) defined
by 119905119896(119903) = (2119896 + 1)(1 + 119903
119896
)(minus1)119896
forall119896 isin N Then since 119860119903t =(minus1)119899
isin 119891 119909 isin 119886119903119891 One can easily see that 119905 notin 119891Thus 119905 isin 119886119903
119891
119891 and this completes the proof
Theorem 6 The sequence spaces 119886119903119891and ℓinfin
overlap but nei-ther of them contains the other
Proof Let us consider the sequence 119906 = (119906119896(119903)) defined by
119906119896(119903) = 1(1+119903
119896
) for allNThen since119860119903119906 = 119890 isin 119891 119906 isin 119886119903119891 It
is clear that 119906 isin ℓinfin This means that the sequence spaces 119886119903
119891
and ℓinfinare not disjointNowwe show that the sequence space
119886119903
119891and ℓinfin
do not include each other Let us consider the se-quence 119905 = (119905
119896(119903)) defined as in proof of Theorem 5 above
and 119911 = (119911119896(119903)) = (0 0 1(1 + 119903
101
) 1(1 + 119903110
)
0 0 1(1 + 119903211
) 1(1 + 119903231
) 0 0 ) where theblocks of 0rsquos are increasing by factors of 100 and the blocks of1(1 + 119903
119896
)rsquos are increasing by factors of 10 Then since 119860119903119905 =(minus1)119899
isin 119891 119905 isin 119886119903119891 but 119905 notin ℓ
infinTherefore 119905 isin 119886119903
119891ℓinfin Also the
sequence 119911 notin 119886119903119891since119860119903119911 = (0 0 1 1 0 0 1
1 0 0 ) notin 119891 where the blocks of 0rsquos are increasing byfactors of 100 and the blocks of 1rsquos are increasing by factors of10 but 119911 is boundedThis means that 119911 isin ℓ
infin 119886119903
119891 Hence the
4 Abstract and Applied Analysis
sequence spaces 119886119903119891and ℓinfinoverlap but neither of them con-
tains the other This completes the proof
Theorem 7 Let the spaces 1198861199031198910
119886119903119888 and 119886119903
119891be given Then
(i) 1198861199031198910
sub 119886119903
119891strictly hold
(ii) 119886119903119888sub 119886119903
119891strictly hold
Proof (i) Let 119909 = (119909119896) isin 119886
119903
1198910
which means that 119860119903119909 isin 1198910
Since1198910sub 119891119860119903119909 isin 119891This implies that119909 isin 119886119903
119891Thuswe have
119886119903
1198910
sub 119886119903
119891
Now we show that this inclusion is strict Let us considerthe sequence 119906 = (119906
119896(119903)) defined as in proof ofTheorem 6 for
all 119896 isin N Consider the following
119891119860119903 minus lim 119906 = lim
119898rarrinfin
119898
sum
119896=0
1
119898 + 1
119896
sum
119894=0
1 + 119903119894
119896 + 1
119906119894+119899
= lim119898rarrinfin
119898
sum
119896=0
1
119898 + 1
= 119890 = (1 1 )
(15)
which means that 119906 isin 119886119903119891 119886119903
1198910
that is to say the inclusion isstrict
(ii) Let 119909 = (119909119896) isin 119886119903
119888whichmeans that119860119903119909 isin 119888 Since 119888 sub
119891 119860119903119909 isin 119891 This implies that 119909 isin 119886119903119891 Thus we have
119886119903
119888sub 119886119903
119891 Furthermore let us consider the sequence
119905 = 119905119896(119903) defined as in proof of Theorem 5 for all
119896 isin N Then since 119860119903119905 = (minus1)119899 isin 119891 119888 119905 isin 119886119903119891 119886119903
119888
This completes the proof
3 Certain Matrix Mappings on the Sets 119886119903119891
119886119903119891119904
and Some Duals
In this section we will characterize some matrix transforma-tions between the spaces of 119860119903 almost convergent sequenceand almost convergent series in addition to paranormed andclassical sequence spaces after giving 120573- and 120574-duals of thespaces 119886119903
119891119904and 119886119903119891We start with the definition of the beta- and
gamma-dualsIf119909 and119910 are sequences and119883 and119884 are subsets of120596 then
we write 119909 sdot 119910 = (119909119896119910119896)infin
119896=0 119909minus1 lowast 119884 = 119886 isin 120596 119886 sdot 119909 isin 119884 and
119872(119883119884) = ⋂
119909isin119883
119909minus1
lowast 119884 = 119886 119886 sdot 119909 isin 119884 forall119909 isin 119883 (16)
for themultiplier space of119883 and119884 One can easily observe fora sequence space 119885 with 119884 sub 119885 and 119885 sub 119883 that inclusions119872(119883119884) sub 119872(119883119885) and119872(119883119884) sub 119872(119885 119884) hold respec-tively The 120572- 120573- and 120574-duals of a sequence space which arerespectively denoted by119883120572 119883120573 and119883120574 are defined by
119883120572
= 119872(119883 ℓ1) 119883
120573
= 119872(119883 119888119904) 119883120574
= 119872(119883 119887119904)
(17)
It is obvious that 119883120572 sub 119883120573 sub 119883120574 Also it can easily be seenthat the inclusions 119883120572 sub 119884120572 119883120573 sub 119884120573 and 119883120574 sub 119884
120574 holdwhenever 119884 sub 119883
Lemma 8 (see [18]) 119860 = (119886119899119896) isin (119891 ℓ
infin) if and only if
sup119899
sum
119896
1003816100381610038161003816119886119899119896
1003816100381610038161003816lt infin (18)
Lemma 9 (see [18]) 119860 = (119886119899119896) isin (119891 119888) if and only if (18)
holds and there are 120572 120572119896isin C such that
lim119899rarrinfin
119886119899119896= 120572119896
forallk isin N (19)
lim119899rarrinfin
sum
119896
119886119899119896= 120572 (20)
lim119899rarrinfin
sum
119896
1003816100381610038161003816Δ (119886119899119896minus 120572119896)1003816100381610038161003816= 0 (21)
Theorem 10 Define the sets 1199051199031and 1199051199032by
119905119903
1= 119886 = (119886
119896) isin 120596 sum
119896
1003816100381610038161003816100381610038161003816
Δ (
119886119896
1 + 119903119896
) (119896 + 1)
1003816100381610038161003816100381610038161003816
lt infin
119905119903
2= 119886 = (119886
119896) isin 120596 sup
119896
10038161003816100381610038161003816100381610038161003816
119886119896(119896 + 1)
1 + 119903119896
10038161003816100381610038161003816100381610038161003816
lt infin
(22)
where Δ(119886119896(1 + 119903
119896
)) = 119886119896(1 + 119903
119896
) minus 119886119896+1(1 + 119903
119896+1
) for all 119896 isinN Then (119886119903
119891)120574
= 119905119903
1cap 119905119903
2
Proof Take any sequence 119886 = (119886119896) isin 120596 and consider the fol-
lowing equality119899
sum
119896=0
119886119896119909119896=
119899
sum
119896=0
119886119896[
119896
sum
119894=119896minus1
(minus1)119896minus119895
119894 + 1
1 + 119903119894119910119894]
=
119899minus1
sum
119896=0
Δ(
119886119896
1 + 119903119896
) (119896 + 1) 119910119896
+
119899 + 1
1 + 119903119899119886119899119910119899
= (119879119910)119899
(23)
where 119879 = 119905119903119899119896 is
119905119903
119899119896=
Δ(
119886119896
1 + 119903119896
) (119896 + 1) (0 ⩽ 119896 ⩽ 119899 minus 1)
119899 + 1
1 + 119903119899119886119899
(119896 = 119899)
0 (119896 gt 119899)
(24)
for all 119896 119899 isin NThus we deduce from (23) that 119886119909 = (119886119896119909119896) isin
119887119904 whenever 119909 = (119909119896) isin 119886119903
119891if and only if 119879119910 isin ℓ
infinwhenever
119910 = (119910119896) isin 119891 where 119879 = 119905119903
119899119896 is defined in (24) Therefore
with the help of Lemma 8 (119886119903119891)120574
= 119905119903
1cap 119905119903
2
Theorem 11 The 120573-dual of the space 119886119903119891is the intersection of
the sets
119905119903
3= 119886 = (119886
119896) isin 120596 lim
119899rarrinfin
sum
119896
1003816100381610038161003816Δ (119905119903
119899119896minus 120572119896)1003816100381610038161003816= 0
119905119903
4= 119886 = (119886
119896) isin 120596 (
119896 + 1
1 + 119903119896
119886119896) isin 119888119904
(25)
where lim119899rarrinfin
119905119903
119899119896= 120572119896for all 119896 isin N Then (119886119903
119891)120573
= 119905119903
3cap 119905119903
4
Abstract and Applied Analysis 5
Proof Let us take any sequence 119886 isin 120596 By (23) 119886119909 = (119886119896119909119896) isin
119888119904whenever 119909 = (119909119896) isin 119886119903
119891if and only if 119879119910 isin 119888whenever 119910 =
(119910119896) isin 119891 It is obvious that the columns of that matrix 119879 in 119888
where119879 = 119905119903119899119896 defined in (24) we derive the consequence by
Lemma 9 that (119886119903119891)120573
= 119905119903
3cap 119905119903
4
Theorem 12 The 120574-dual of the space 119886119903119891119904is the intersection of
the sets
119888119903
1= 119886 = (119886
119896) isin 120596 sum
119896
1003816100381610038161003816100381610038161003816
Δ [Δ(
119886119896
1 + 119903119896
) (119896 + 1)
+
119886119896
1 + 119903119896(119896 + 1)]
1003816100381610038161003816100381610038161003816
lt infin
119888119903
2= 119886 = (119886
119896) isin 120596 (
119886119896(119896 + 1)
1 + 119903119896
) isin 1198880
(26)
In other words we have (119886119903119891119904)120574
= 119888119903
1cap 119888119903
2
Proof We obtain from (23) that 119886119909 = (119886119896119909119896) isin 119887119904 whenever
119909 = (119909119896) isin 119886119903
119891119904if and only if119879119910 isin ℓ
infinwhenever 119910 = (119910
119896) isin 119891119904
where 119879 = 119905119903
119899119896 is defined in (24) Therefore by Lemma 19
(viii) (119886119903119891119904)120574
= 119888119903
1cap 119888119903
2
Theorem 13 Define the set 1198881199033by
119888119903
3= 119886 = (119886
119896) isin 120596 lim
119899rarrinfin
sum
119896
10038161003816100381610038161003816Δ2
(119905119903
119899119896)
10038161003816100381610038161003816exists (27)
Then (119886119903119891119904)120573
= 119888119903
1cap 119888119903
2cap 119888119903
3
Proof This may be obtained in the same way as mentionedin the proof of Theorem 12 with Lemma 19(viii) instead ofLemma 19(vii) So we omit details
For the sake of brevity the following notations will beused
119886 (119899 119896119898) =
1
119898 + 1
119898
sum
119894=0
119886119899+119894119896
119886 (119899 119896) =
119899
sum
119894=0
119886119894119896
119886119899119896= Δ(
119886119899119896
1 + 119903119896
) (119896 + 1) = (
119886119899119896
1 + 119903119896
minus
119886119899119896+1
1 + 119903119896+1
) (119896 + 1)
Δ119886119899119896= 119886119899119896minus 119886119899119896+1
119886119899119896=
119899
sum
119895=0
(1 + 119903119895
) 119890119895119896
119899 + 1
(28)
for all 119896 119899 isin N Assume that the infinite matrices 119860 = (119886119899119896)
and119861 = (119887119899119896)map the sequences119909 = (119909
119896) and119910 = (119910
119896)which
are connectedwith relation (10) to the sequences 119906 = (119906119899) and
V = (V119899) respectively that is
119906119899= (119860119909)
119899= sum
119896
119886119899119896119909119896
forall119899 isin N (29)
V119899= (119861119910)
119899= sum
119896
119887119899119896119910119896
forall119899 isin N (30)
One can easily conclude here that the method 119860 is directlyapplied to the terms of the sequence 119909 = (119909
119896) while the
method 119861 is applied to the 119860119903-transform of the sequence 119909 =(119909119896) So the methods 119860 and 119861 are essentially differentNow suppose that the matrix product 119861119860119903 exists which is
amuch weaker assumption than the conditions on thematrix119861 belonging to anymatrix class in general It is not difficult tosee that the sequence in (30) reduces to the sequence in (29)under the application of formal summation by parts Thisleads us to the fact that 119861119860119903 exists and is equal to 119860 and(119861119860119903
)119909 = 119861(119860119903
119909) formally holds if one side exists This state-ment is equivalent to the following relation between the en-tries of the matrices 119860 = (119886
119899119896) and 119861 = (119887
119899119896) which are con-
nected with the relation
119886119899119896= 119887119899119896= Δ(
119886119899119896
1 + 119903119896
) (119896 + 1) or
119886119899119896= (1 + 119903
119896
)
infin
sum
119895=119896
119887119899119895
1 + 119895
forall119896 119899 isin N
(31)
Note that the methods 119860 and 119861 are not necessarily equi-valent since the order of summationmay not be reversedWenow give the following fundamental theorem connected withthematrixmappings oninto the almost convergent spaces 119886119903
119891
and 119886119903119891s
Theorem 14 Suppose that the entries of the infinite matrices119860 = (119886
119899119896) and 119861 = (119887
119899119896) are connected with relation (31) for all
119896 119899 isin N and let 120582 be any given sequence spaceThen119860 isin (119886119903119891
120582) if and only if
119861 isin (119891 120582)
119899 + 1
1 + 119903119896
119886119899119896
119896isinNisin 1198880
(32)
Proof Suppose that 119860 = (119886119899119896) and 119861 = (119887
119899119896) are connected
with the relation (31) and let 120582 be any given sequence spaceand keep in mind that the spaces 119886119903
119891and 119891 are norm iso-
morphicLet 119860 isin (119886119903
119891 120582) and take any sequence 119909 isin 119886119903
119891 and keep
in mind that 119910 = 119860119903119909 Then (119886119899119896)119896isinN isin (119886
119903
119891)120573 that is (32)
holds for all 119899 isin N and 119861119860119903 exists which implies that(119887119899119896)119896isinN isin ℓ1 = 119891
120573 for each 119899 isin N Thus 119861119910 exists for all 119910 isin119891 and thus we have119898 rarr infin in the equality
119898
sum
119896=0
119887119899119896119910119896=
119898
sum
119896=0
119898
sum
119895=119896
(1 + 119903119896
)
119887119899119895
1 + 119895
119909119896 (33)
for all119898 119899 isin N and we have (31) 119861119910 = 119860119909 which means that119861 isin (119891 120582) On the other hand assume that (32) holds and119861 isin (119891 120582) Then we have (119887
119899119896)119896isinN isin ℓ1 for all 119899 isin N which
6 Abstract and Applied Analysis
gives together with (120592119899119896)119896isinN isin (119886
119903
119891)120573 for each 119899 isin N that 119860119909
exists Then we obtain from the equality
119898
sum
119896=0
119886119899119896119909119896=
119898minus1
sum
119896=0
Δ(
119886119899119896
1 + 119903119896
) (119896 + 1) 119910119896+
119898 + 1
1 + 119903119898119886119899119898119910119898
=
119898
sum
119896=0
119887119899119896119910119896
(34)
for all 119898 119899 isin N as 119898 rarr infin that 119860119909 = 119861119910 and this showsthat 119860 isin (119886119903
119891 120582)
Theorem 15 Suppose that the entries of the infinite matrices119864 = (119890
119899119896) and 119865 = (119891
119899119896) are connected with the relation
119891119899119896= 119886119899119896
(35)
for all 119898 119899 isin N and 120582 is any given sequence space Then 119864 isin(120582 119886119903
119891) if and only if 119865 isin (120582 119891)
Proof Let 119909 = (119909119896) isin 120582 and consider the following equality
119899
sum
119895=0
119898
sum
119896=0
1 + 119903119895
119899 + 1
119890119895119896119909119896=
119898
sum
119896=0
119891119899119896119909119896 (36)
for all 119896119898 119899 isin N which yields as 119898 rarr infin that 119864119909 isin 119886119903119891
whenever 119909 isin 120582 if and only if 119865119909 isin 119891 whenever 119909 isin 120582 Thisstep completes the proof
Theorem 16 Let120582 be any given sequence space and thematri-ces119860 = (119886
119899119896) and119861 = (119887
119899119896) are connectedwith the relation (31)
Then 119860 isin (119886119903
119891119904 120582) if and only if 119861 isin (119891119904 120582) and (119886
119899119896)119896isinN isin
(119886119903
119891119904)120573 for all 119899 isin N
Proof The proof is based on the proof of Theorem 14
Theorem 17 Let 120582 be any given sequence space and the ele-ments of the infinite matrices 119864 = (119890
119899119896) and 119865 = (119891
119899119896) are con-
nectedwith relation (35)Then119864 = (119890119899119896) isin (120582 119886
119903
119891119904) if and only
if 119865 isin (120582 119891119904)
Proof The proof is based on the proof of Theorem 15
By Theorems 14 15 16 and 17 we have quite a few out-comes depending on the choice of the space 120582 to characterizecertain matrix mappings Hence by the help of these theo-rems the necessary and sufficient conditions for the classes(119886119903
119891 120582) (120582 119886119903
119891) (119886119903119891119904 120582) and (120582 119886119903
119891119904) may be derived by
replacing the entries of119860 and 119861 by those of 119861 = 119860(119860119903)minus1 and119865 = 119860
119903
119864 respectively where the necessary and sufficient con-ditions on the matrices 119864 and 119865 are read from the concerningresults in the existing literature
Lemma 18 Let 119860 = (119886119899119896) be an infinite matrix Then the fol-
lowing statements hold
(i) 119860 isin (1198880(119901) 119891) if and only if
exist119873 gt 1 ni sup119898isinN
sum
119896
|119886 (119899 119896 119898)|1198731119901119896
lt infin forall119899 isin N
exist120572119896isin C forall119896 isin N ni lim
119898rarrinfin
119886 (119899 119896119898) = 120572119896
uniformly in 119899(37)
(ii) 119860 isin (119888(119901) 119891) if and only if (37) and
exist120572 isin C ni lim119898rarrinfin
sum
119896
119886 (119899 119896119898) = 120572 uniformly in 119899 (38)
(iii) 119860 isin (ℓinfin(119901) 119891) if and only if (37) and
exist119873 gt 1 ni lim119898rarrinfin
sum
119896
1003816100381610038161003816119886 (119899 119896 119898) minus 120572
119896
10038161003816100381610038161198731119901119896
= 0
uniformly in 119899(39)
Lemma 19 Let 119860 = (119886119899119896) be an infinite matrix Then the fol-
lowing statements hold
(i) (Duran [19])119860 isin (ℓinfin 119891) if and only if (18) holds and
119891 minus lim119886119899119896= 120572119896exists for each fixed 119896 (40)
lim119898rarrinfin
sum
119896
1003816100381610038161003816119886 (119899 119896 119898) minus 120572
119896
1003816100381610038161003816= 0 uniformly in 119899 (41)
(ii) (King [20]) 119860 isin (119888 119891) if and only if (18) (40) holdand
119891 minus limsum119896
119886119899119896= 120572 (42)
(iii) (Basar and Colak [21])119860 isin (119888119904 119891) if and only if (40)holds and
sup119899isinN
sum
119896
1003816100381610038161003816Δ119886119899119896
1003816100381610038161003816lt infin (43)
(iv) (Basar and Colak [21])119860 isin (119887119904 119891) if and only if (40)(43) hold and
lim119896
119886119899119896= 0 exists for each fixed 119899 (44)
lim119902rarrinfin
sum
119896
1
119902 + 1
119902
sum
119894=0
1003816100381610038161003816Δ [119886 (119899 + 119894 119896) minus 120572
119896]1003816100381610038161003816= 0 uniformly in 119899
(45)
(v) (Duran [19]) 119860 isin (119891 119891) if and only if (18) (40) and(42) hold and
lim119898rarrinfin
sum
119896
1003816100381610038161003816Δ [119886 (119899 119896119898) minus 120572
119896]1003816100381610038161003816= 0 uniformly in 119899 (46)
(vi) (Basar [22])119860 isin (119891119904 119891) if and only if (40) (44) (46)and (45) hold
Abstract and Applied Analysis 7
(vii) (Ozturk [23])119860 isin (119891119904 119888) if and only if (19) (43) and(44) hold and
lim119899rarrinfin
sum
119896
10038161003816100381610038161003816Δ2
119886119899119896
10038161003816100381610038161003816= 120572 (47)
(viii) 119860 isin (119891119904 ℓinfin) if and only if (43) and (44) hold
(ix) (Basar and Solak [24]) 119860 isin (119887119904 119891119904) if and only if(44) (45) hold and
sup119899isinN
sum
119896
|Δ119886 (119899 119896)| lt infin
119891 minus lim 119886 (119899 119896) = 120572119896exists for each fixed 119896
(48)
(x) (Basar [22])119860 isin (119891119904 119891119904) if and only if (45) (48) holdand
lim119902rarrinfin
sum
119896
1
119902 + 1
119902
sum
119894=0
10038161003816100381610038161003816Δ2
[119886 (119899 + 119894 119896) minus 120572119896]
10038161003816100381610038161003816= 0 uniformly in 119899
(49)
(xi) (Basar and Colak [21])119860 isin (119888119904 119891119904) if and only if (48)holds
(xii) (Basar [25]) 119860 isin (119891 119888119904) if and only if
sup119899isinN
sum
119896
|119886 (119899 119896)| lt infin (50)
sum
119899
119886119899119896= 120572119896
exists for each fixed 119896 (51)
sum
119899
sum
119896
119886119899119896= 120572 (52)
lim119898rarrinfin
sum
119896
1003816100381610038161003816Δ [119886 (119899 119896) minus 120572
119896]1003816100381610038161003816= 0 (53)
Now we give our main results which are related to matrixmappings oninto the spaces of almost convergent series 119886119903
119891119904
and sequences 119886119903119891
Corollary 20 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (119886119903119891119904 119891) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin
N and (40) (44) holdwith 119886119899119896instead of 119886
119899119896 (46) holds
with 119886(119899 119896119898) instead of 119886(119899 119896119898) and (45) holdswith 119886(119899 119896) instead of 119886(119899 119896)
(ii) 119860 isin (119886119903119891119904 119888) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin N
and (19) (43) (44) and (47) hold with 119886119899119896
instead of119886119899119896
(iii) 119860 isin (119886119903119891119904 ℓinfin) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin
N and (43) and (44) hold with 119886119899119896instead of 119886
119899119896
(iv) 119860 isin (119886119903119891119904 119891119904) if and only if 119886
119899119896119896isinN isin (119886
119903
119891119904)120573 for all 119899 isin
N and (45) (48) and (49) hold with 119886(119899 119896) instead of119886(119899 119896)
(v) 119860 isin (119888119904 119886119903119891119904) if and only if (48) holds with 119886(119899 119896) in-
stead of 119886(119899 119896)
(vi) 119860 isin (119887119904 119886119903119891119904) if and only if (44) holds with 119886
119899119896instead
of 119886119899119896and (45) (48) holdwith 119886(119899 119896) instead of 119886(119899 119896)
(vii) 119860 isin (119891119904 119886119903
119891119904) if and only if (45) (48) and (49) hold
with 119886(119899 119896) instead of 119886(119899 119896)
Corollary 21 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (119888(119901) 119886119903
119891119904) if and only if (37)and (38) hold with
119886(119899 119896119898) instead of 119886(119899 119896119898)
(ii) 119860 isin (1198880(119901) 119886
119903
119891119904) if and only if (37) holds with 119886(119899
119896119898) instead of 119886(119899 119896119898)
(iii) 119860 isin (ℓinfin(119901) 119886
119903
119891119904) if and only if (37) and (39) hold with
119886(119899 119896119898) instead of 119886(119899 119896119898)
Corollary 22 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (119886119903
119891 ℓinfin) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all
119899 isin N and (18) holds with 119886119899119896instead of 119886
119899119896
(ii) 119860 isin (119886119903119891 119888) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin N
and (18) (19) (20) and (21) hold with 119886119899119896
instead of119886119899119896
(iii) 119860 isin (119886119903119891 119888119904) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin
N and (50)(53) hold with 119886(119899 119896) instead of 119886(119899 119896) and(51)(52) hold with 119886
119899119896instead of 119886
119899119896
Corollary 23 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (ℓinfin 119886119903
119891) if and only if (18) (40) hold with 119886
119899119896in-
stead of 119886119899119896
and (41) holds with 119886(119899 119896119898) instead of119886(119899 119896119898)
(ii) 119860 isin (119891 119886119903119891) if and only if (18) (40) and (46) holdwith
119886(119899 119896119898) instead of 119886(119899 119896119898) and (42) holds with 119886119899119896
instead of 119886119899119896
(iii) 119860 isin (119888 119886119903119891) if and only if (18) (40) and (42) hold with
119886119899119896instead of 119886
119899119896
(iv) 119860 isin (119887119904 119886119903
119891) if and only if (40) (43) and (44) hold
with 119886119899119896
instead of 119886119899119896
and (45) holds with 119886(119899 119896)instead of 119886(119899 119896)
(v) 119860 isin (119891119904 119886119903
119891) if and only if (40) (44) hold with 119886
119899119896
instead of 119886119899119896 (46) holds with 119886(119899 119896119898) instead of 119886(119899
119896119898) and (45) holds with 119886(119899 119896) instead of 119886(119899 119896)
(vi) 119860 isin (119888119904 119886119903119891) if and only if (40) and (43) hold with 119886
119899119896
instead of 119886119899119896
Remark 24 Characterization of the classes (119886119903119891 119891infin) (119891infin
119886119903
119891) (119886119903119891119904 119891infin) and (119891
infin 119886119903
119891119904) is redundant since the spaces of
almost bounded sequences 119891infin
and ℓinfin
are equal
8 Abstract and Applied Analysis
Acknowledgment
The authors thank the referees for their careful reading of theoriginal paper and for the valuable comments
References
[1] J Boos Classical and Modern Methods in Summability OxfordUniversity Press New York NY USA 2000
[2] M Sengonul and F Basar ldquoSomenewCesaro sequence spaces ofnon-absolute type which include the spaces 119888
0and 119888rdquo Soochow
Journal of Mathematics vol 31 no 1 pp 107ndash119 2005[3] C Aydın and F Basar ldquoSome new sequence spaces which in-
clude the spaces 119897infinand 119897119875rdquo Demonstratio Mathematica vol 38
no 3 pp 641ndash656 2005[4] E Malkowsky and E Savas ldquoMatrix transformations between
sequence spaces of generalized weightedmeansrdquoAppliedMath-ematics and Computation vol 147 no 2 pp 333ndash345 2004
[5] MMursaleen andAKNoman ldquoOn the spaces of120582-convergentand bounded sequencesrdquo Thai Journal of Mathematics vol 8no 2 pp 311ndash329 2010
[6] B Altay F Basar and M Mursaleen ldquoOn the Euler sequencespaces which include the spaces 119897
119901and 119897infinIrdquo Information Sci-
ences vol 176 no 10 pp 1450ndash1462 2006[7] F Basar and M Kirisci ldquoAlmost convergence and generalized
differencematrixrdquoComputers ampMathematics with Applicationsvol 61 no 3 pp 602ndash611 2011
[8] A Sonmez ldquoAlmost convergence and triple bandmatrixrdquoMath-ematical and Computer Modelling vol 57 no 9-10 pp 2393ndash2402 2013
[9] K Kayaduman andM S Sengonul ldquoThe space of Cesaro almostconvergent sequence and core theoremsrdquo Acta MathematicaScientia vol 6 pp 2265ndash2278 2012
[10] M Sengonul and K Kayaduman ldquoOn the Riesz almost conver-gent sequences spacerdquo Abstract and Applied Analysis vol 2012Article ID 691694 18 pages 2012
[11] G G Lorentz ldquoA contribution to the theory of divergent se-quencesrdquo Acta Mathematica vol 80 pp 167ndash190 1948
[12] I J Maddox ldquoParanormed sequence spaces generated by infi-nite matricesrdquo Proceedings of the Cambridge Philosophical Soci-ety vol 64 pp 335ndash340 1968
[13] A Karaisa and F Ozger ldquoAlmost difference sequence space de-rived by using a generalized weightedmeanrdquoActaMathematicaScientia In review
[14] C Aydın and F Basar ldquoOn the new sequence spaces which in-clude the spaces 119888
0and 119888rdquo Hokkaido Mathematical Journal vol
33 no 2 pp 383ndash398 2004[15] F Basar ldquoA note on the triangle limitation methodsrdquo Fırat Uni-
versitesi Muhendislik Bilimleri Dergisi vol 5 no 1 pp 113ndash1171993
[16] AM Al-Jarrah and EMalkowsky ldquoBK spaces bases and linearoperatorsrdquo in Proceedings of the 3rd International Conference onFunctional Analysis and ApproximationTheory vol 1 no 52 pp177ndash191 1998
[17] A Wilansky Summability through Functional Analysis vol 85North-Holland Amsterdam The Netherlands 1984
[18] J A Sıddıqı ldquoInfinite matrices summing every almost periodicsequencerdquo Pacific Journal of Mathematics vol 39 pp 235ndash2511971
[19] J P Duran ldquoInfinite matrices and almost-convergencerdquo Math-ematische Zeitschrift vol 128 pp 75ndash83 1972
[20] J P King ldquoAlmost summable sequencesrdquo Proceedings of theAmerican Mathematical Society vol 17 pp 1219ndash1225 1966
[21] F Basar and R Colak ldquoAlmost-conservativematrix transforma-tionsrdquo Turkish Journal of Mathematics vol 13 no 3 pp 91ndash1001989
[22] F Basar ldquo119891-conservative matrix sequencesrdquo Tamkang Journalof Mathematics vol 22 no 2 pp 205ndash212 1991
[23] E Ozturk ldquoOn strongly regular dual summability methodsrdquoCommunications de la Faculte des Sciences de lrsquoUniversite drsquoAn-kara vol 32 no 1 pp 1ndash5 1983
[24] F Basar and I Solak ldquoAlmost-coercivematrix transformationsrdquoRendiconti di Matematica e delle sue Applicazioni vol 11 no 2pp 249ndash256 1991
[25] F Basar ldquoStrongly-conservative sequence-to-series matrixtransformationsrdquo Erciyes Universitesi Fen Bilimleri Dergisi vol5 no 12 pp 888ndash893 1989
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Journal ofApplied Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Advances in
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Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
Combinatorics
OperationsResearch
Advances in
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Journal of Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
DecisionSciences
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Discrete MathematicsJournal of
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Volume 2014
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Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
sequence spaces 119886119903119891and ℓinfinoverlap but neither of them con-
tains the other This completes the proof
Theorem 7 Let the spaces 1198861199031198910
119886119903119888 and 119886119903
119891be given Then
(i) 1198861199031198910
sub 119886119903
119891strictly hold
(ii) 119886119903119888sub 119886119903
119891strictly hold
Proof (i) Let 119909 = (119909119896) isin 119886
119903
1198910
which means that 119860119903119909 isin 1198910
Since1198910sub 119891119860119903119909 isin 119891This implies that119909 isin 119886119903
119891Thuswe have
119886119903
1198910
sub 119886119903
119891
Now we show that this inclusion is strict Let us considerthe sequence 119906 = (119906
119896(119903)) defined as in proof ofTheorem 6 for
all 119896 isin N Consider the following
119891119860119903 minus lim 119906 = lim
119898rarrinfin
119898
sum
119896=0
1
119898 + 1
119896
sum
119894=0
1 + 119903119894
119896 + 1
119906119894+119899
= lim119898rarrinfin
119898
sum
119896=0
1
119898 + 1
= 119890 = (1 1 )
(15)
which means that 119906 isin 119886119903119891 119886119903
1198910
that is to say the inclusion isstrict
(ii) Let 119909 = (119909119896) isin 119886119903
119888whichmeans that119860119903119909 isin 119888 Since 119888 sub
119891 119860119903119909 isin 119891 This implies that 119909 isin 119886119903119891 Thus we have
119886119903
119888sub 119886119903
119891 Furthermore let us consider the sequence
119905 = 119905119896(119903) defined as in proof of Theorem 5 for all
119896 isin N Then since 119860119903119905 = (minus1)119899 isin 119891 119888 119905 isin 119886119903119891 119886119903
119888
This completes the proof
3 Certain Matrix Mappings on the Sets 119886119903119891
119886119903119891119904
and Some Duals
In this section we will characterize some matrix transforma-tions between the spaces of 119860119903 almost convergent sequenceand almost convergent series in addition to paranormed andclassical sequence spaces after giving 120573- and 120574-duals of thespaces 119886119903
119891119904and 119886119903119891We start with the definition of the beta- and
gamma-dualsIf119909 and119910 are sequences and119883 and119884 are subsets of120596 then
we write 119909 sdot 119910 = (119909119896119910119896)infin
119896=0 119909minus1 lowast 119884 = 119886 isin 120596 119886 sdot 119909 isin 119884 and
119872(119883119884) = ⋂
119909isin119883
119909minus1
lowast 119884 = 119886 119886 sdot 119909 isin 119884 forall119909 isin 119883 (16)
for themultiplier space of119883 and119884 One can easily observe fora sequence space 119885 with 119884 sub 119885 and 119885 sub 119883 that inclusions119872(119883119884) sub 119872(119883119885) and119872(119883119884) sub 119872(119885 119884) hold respec-tively The 120572- 120573- and 120574-duals of a sequence space which arerespectively denoted by119883120572 119883120573 and119883120574 are defined by
119883120572
= 119872(119883 ℓ1) 119883
120573
= 119872(119883 119888119904) 119883120574
= 119872(119883 119887119904)
(17)
It is obvious that 119883120572 sub 119883120573 sub 119883120574 Also it can easily be seenthat the inclusions 119883120572 sub 119884120572 119883120573 sub 119884120573 and 119883120574 sub 119884
120574 holdwhenever 119884 sub 119883
Lemma 8 (see [18]) 119860 = (119886119899119896) isin (119891 ℓ
infin) if and only if
sup119899
sum
119896
1003816100381610038161003816119886119899119896
1003816100381610038161003816lt infin (18)
Lemma 9 (see [18]) 119860 = (119886119899119896) isin (119891 119888) if and only if (18)
holds and there are 120572 120572119896isin C such that
lim119899rarrinfin
119886119899119896= 120572119896
forallk isin N (19)
lim119899rarrinfin
sum
119896
119886119899119896= 120572 (20)
lim119899rarrinfin
sum
119896
1003816100381610038161003816Δ (119886119899119896minus 120572119896)1003816100381610038161003816= 0 (21)
Theorem 10 Define the sets 1199051199031and 1199051199032by
119905119903
1= 119886 = (119886
119896) isin 120596 sum
119896
1003816100381610038161003816100381610038161003816
Δ (
119886119896
1 + 119903119896
) (119896 + 1)
1003816100381610038161003816100381610038161003816
lt infin
119905119903
2= 119886 = (119886
119896) isin 120596 sup
119896
10038161003816100381610038161003816100381610038161003816
119886119896(119896 + 1)
1 + 119903119896
10038161003816100381610038161003816100381610038161003816
lt infin
(22)
where Δ(119886119896(1 + 119903
119896
)) = 119886119896(1 + 119903
119896
) minus 119886119896+1(1 + 119903
119896+1
) for all 119896 isinN Then (119886119903
119891)120574
= 119905119903
1cap 119905119903
2
Proof Take any sequence 119886 = (119886119896) isin 120596 and consider the fol-
lowing equality119899
sum
119896=0
119886119896119909119896=
119899
sum
119896=0
119886119896[
119896
sum
119894=119896minus1
(minus1)119896minus119895
119894 + 1
1 + 119903119894119910119894]
=
119899minus1
sum
119896=0
Δ(
119886119896
1 + 119903119896
) (119896 + 1) 119910119896
+
119899 + 1
1 + 119903119899119886119899119910119899
= (119879119910)119899
(23)
where 119879 = 119905119903119899119896 is
119905119903
119899119896=
Δ(
119886119896
1 + 119903119896
) (119896 + 1) (0 ⩽ 119896 ⩽ 119899 minus 1)
119899 + 1
1 + 119903119899119886119899
(119896 = 119899)
0 (119896 gt 119899)
(24)
for all 119896 119899 isin NThus we deduce from (23) that 119886119909 = (119886119896119909119896) isin
119887119904 whenever 119909 = (119909119896) isin 119886119903
119891if and only if 119879119910 isin ℓ
infinwhenever
119910 = (119910119896) isin 119891 where 119879 = 119905119903
119899119896 is defined in (24) Therefore
with the help of Lemma 8 (119886119903119891)120574
= 119905119903
1cap 119905119903
2
Theorem 11 The 120573-dual of the space 119886119903119891is the intersection of
the sets
119905119903
3= 119886 = (119886
119896) isin 120596 lim
119899rarrinfin
sum
119896
1003816100381610038161003816Δ (119905119903
119899119896minus 120572119896)1003816100381610038161003816= 0
119905119903
4= 119886 = (119886
119896) isin 120596 (
119896 + 1
1 + 119903119896
119886119896) isin 119888119904
(25)
where lim119899rarrinfin
119905119903
119899119896= 120572119896for all 119896 isin N Then (119886119903
119891)120573
= 119905119903
3cap 119905119903
4
Abstract and Applied Analysis 5
Proof Let us take any sequence 119886 isin 120596 By (23) 119886119909 = (119886119896119909119896) isin
119888119904whenever 119909 = (119909119896) isin 119886119903
119891if and only if 119879119910 isin 119888whenever 119910 =
(119910119896) isin 119891 It is obvious that the columns of that matrix 119879 in 119888
where119879 = 119905119903119899119896 defined in (24) we derive the consequence by
Lemma 9 that (119886119903119891)120573
= 119905119903
3cap 119905119903
4
Theorem 12 The 120574-dual of the space 119886119903119891119904is the intersection of
the sets
119888119903
1= 119886 = (119886
119896) isin 120596 sum
119896
1003816100381610038161003816100381610038161003816
Δ [Δ(
119886119896
1 + 119903119896
) (119896 + 1)
+
119886119896
1 + 119903119896(119896 + 1)]
1003816100381610038161003816100381610038161003816
lt infin
119888119903
2= 119886 = (119886
119896) isin 120596 (
119886119896(119896 + 1)
1 + 119903119896
) isin 1198880
(26)
In other words we have (119886119903119891119904)120574
= 119888119903
1cap 119888119903
2
Proof We obtain from (23) that 119886119909 = (119886119896119909119896) isin 119887119904 whenever
119909 = (119909119896) isin 119886119903
119891119904if and only if119879119910 isin ℓ
infinwhenever 119910 = (119910
119896) isin 119891119904
where 119879 = 119905119903
119899119896 is defined in (24) Therefore by Lemma 19
(viii) (119886119903119891119904)120574
= 119888119903
1cap 119888119903
2
Theorem 13 Define the set 1198881199033by
119888119903
3= 119886 = (119886
119896) isin 120596 lim
119899rarrinfin
sum
119896
10038161003816100381610038161003816Δ2
(119905119903
119899119896)
10038161003816100381610038161003816exists (27)
Then (119886119903119891119904)120573
= 119888119903
1cap 119888119903
2cap 119888119903
3
Proof This may be obtained in the same way as mentionedin the proof of Theorem 12 with Lemma 19(viii) instead ofLemma 19(vii) So we omit details
For the sake of brevity the following notations will beused
119886 (119899 119896119898) =
1
119898 + 1
119898
sum
119894=0
119886119899+119894119896
119886 (119899 119896) =
119899
sum
119894=0
119886119894119896
119886119899119896= Δ(
119886119899119896
1 + 119903119896
) (119896 + 1) = (
119886119899119896
1 + 119903119896
minus
119886119899119896+1
1 + 119903119896+1
) (119896 + 1)
Δ119886119899119896= 119886119899119896minus 119886119899119896+1
119886119899119896=
119899
sum
119895=0
(1 + 119903119895
) 119890119895119896
119899 + 1
(28)
for all 119896 119899 isin N Assume that the infinite matrices 119860 = (119886119899119896)
and119861 = (119887119899119896)map the sequences119909 = (119909
119896) and119910 = (119910
119896)which
are connectedwith relation (10) to the sequences 119906 = (119906119899) and
V = (V119899) respectively that is
119906119899= (119860119909)
119899= sum
119896
119886119899119896119909119896
forall119899 isin N (29)
V119899= (119861119910)
119899= sum
119896
119887119899119896119910119896
forall119899 isin N (30)
One can easily conclude here that the method 119860 is directlyapplied to the terms of the sequence 119909 = (119909
119896) while the
method 119861 is applied to the 119860119903-transform of the sequence 119909 =(119909119896) So the methods 119860 and 119861 are essentially differentNow suppose that the matrix product 119861119860119903 exists which is
amuch weaker assumption than the conditions on thematrix119861 belonging to anymatrix class in general It is not difficult tosee that the sequence in (30) reduces to the sequence in (29)under the application of formal summation by parts Thisleads us to the fact that 119861119860119903 exists and is equal to 119860 and(119861119860119903
)119909 = 119861(119860119903
119909) formally holds if one side exists This state-ment is equivalent to the following relation between the en-tries of the matrices 119860 = (119886
119899119896) and 119861 = (119887
119899119896) which are con-
nected with the relation
119886119899119896= 119887119899119896= Δ(
119886119899119896
1 + 119903119896
) (119896 + 1) or
119886119899119896= (1 + 119903
119896
)
infin
sum
119895=119896
119887119899119895
1 + 119895
forall119896 119899 isin N
(31)
Note that the methods 119860 and 119861 are not necessarily equi-valent since the order of summationmay not be reversedWenow give the following fundamental theorem connected withthematrixmappings oninto the almost convergent spaces 119886119903
119891
and 119886119903119891s
Theorem 14 Suppose that the entries of the infinite matrices119860 = (119886
119899119896) and 119861 = (119887
119899119896) are connected with relation (31) for all
119896 119899 isin N and let 120582 be any given sequence spaceThen119860 isin (119886119903119891
120582) if and only if
119861 isin (119891 120582)
119899 + 1
1 + 119903119896
119886119899119896
119896isinNisin 1198880
(32)
Proof Suppose that 119860 = (119886119899119896) and 119861 = (119887
119899119896) are connected
with the relation (31) and let 120582 be any given sequence spaceand keep in mind that the spaces 119886119903
119891and 119891 are norm iso-
morphicLet 119860 isin (119886119903
119891 120582) and take any sequence 119909 isin 119886119903
119891 and keep
in mind that 119910 = 119860119903119909 Then (119886119899119896)119896isinN isin (119886
119903
119891)120573 that is (32)
holds for all 119899 isin N and 119861119860119903 exists which implies that(119887119899119896)119896isinN isin ℓ1 = 119891
120573 for each 119899 isin N Thus 119861119910 exists for all 119910 isin119891 and thus we have119898 rarr infin in the equality
119898
sum
119896=0
119887119899119896119910119896=
119898
sum
119896=0
119898
sum
119895=119896
(1 + 119903119896
)
119887119899119895
1 + 119895
119909119896 (33)
for all119898 119899 isin N and we have (31) 119861119910 = 119860119909 which means that119861 isin (119891 120582) On the other hand assume that (32) holds and119861 isin (119891 120582) Then we have (119887
119899119896)119896isinN isin ℓ1 for all 119899 isin N which
6 Abstract and Applied Analysis
gives together with (120592119899119896)119896isinN isin (119886
119903
119891)120573 for each 119899 isin N that 119860119909
exists Then we obtain from the equality
119898
sum
119896=0
119886119899119896119909119896=
119898minus1
sum
119896=0
Δ(
119886119899119896
1 + 119903119896
) (119896 + 1) 119910119896+
119898 + 1
1 + 119903119898119886119899119898119910119898
=
119898
sum
119896=0
119887119899119896119910119896
(34)
for all 119898 119899 isin N as 119898 rarr infin that 119860119909 = 119861119910 and this showsthat 119860 isin (119886119903
119891 120582)
Theorem 15 Suppose that the entries of the infinite matrices119864 = (119890
119899119896) and 119865 = (119891
119899119896) are connected with the relation
119891119899119896= 119886119899119896
(35)
for all 119898 119899 isin N and 120582 is any given sequence space Then 119864 isin(120582 119886119903
119891) if and only if 119865 isin (120582 119891)
Proof Let 119909 = (119909119896) isin 120582 and consider the following equality
119899
sum
119895=0
119898
sum
119896=0
1 + 119903119895
119899 + 1
119890119895119896119909119896=
119898
sum
119896=0
119891119899119896119909119896 (36)
for all 119896119898 119899 isin N which yields as 119898 rarr infin that 119864119909 isin 119886119903119891
whenever 119909 isin 120582 if and only if 119865119909 isin 119891 whenever 119909 isin 120582 Thisstep completes the proof
Theorem 16 Let120582 be any given sequence space and thematri-ces119860 = (119886
119899119896) and119861 = (119887
119899119896) are connectedwith the relation (31)
Then 119860 isin (119886119903
119891119904 120582) if and only if 119861 isin (119891119904 120582) and (119886
119899119896)119896isinN isin
(119886119903
119891119904)120573 for all 119899 isin N
Proof The proof is based on the proof of Theorem 14
Theorem 17 Let 120582 be any given sequence space and the ele-ments of the infinite matrices 119864 = (119890
119899119896) and 119865 = (119891
119899119896) are con-
nectedwith relation (35)Then119864 = (119890119899119896) isin (120582 119886
119903
119891119904) if and only
if 119865 isin (120582 119891119904)
Proof The proof is based on the proof of Theorem 15
By Theorems 14 15 16 and 17 we have quite a few out-comes depending on the choice of the space 120582 to characterizecertain matrix mappings Hence by the help of these theo-rems the necessary and sufficient conditions for the classes(119886119903
119891 120582) (120582 119886119903
119891) (119886119903119891119904 120582) and (120582 119886119903
119891119904) may be derived by
replacing the entries of119860 and 119861 by those of 119861 = 119860(119860119903)minus1 and119865 = 119860
119903
119864 respectively where the necessary and sufficient con-ditions on the matrices 119864 and 119865 are read from the concerningresults in the existing literature
Lemma 18 Let 119860 = (119886119899119896) be an infinite matrix Then the fol-
lowing statements hold
(i) 119860 isin (1198880(119901) 119891) if and only if
exist119873 gt 1 ni sup119898isinN
sum
119896
|119886 (119899 119896 119898)|1198731119901119896
lt infin forall119899 isin N
exist120572119896isin C forall119896 isin N ni lim
119898rarrinfin
119886 (119899 119896119898) = 120572119896
uniformly in 119899(37)
(ii) 119860 isin (119888(119901) 119891) if and only if (37) and
exist120572 isin C ni lim119898rarrinfin
sum
119896
119886 (119899 119896119898) = 120572 uniformly in 119899 (38)
(iii) 119860 isin (ℓinfin(119901) 119891) if and only if (37) and
exist119873 gt 1 ni lim119898rarrinfin
sum
119896
1003816100381610038161003816119886 (119899 119896 119898) minus 120572
119896
10038161003816100381610038161198731119901119896
= 0
uniformly in 119899(39)
Lemma 19 Let 119860 = (119886119899119896) be an infinite matrix Then the fol-
lowing statements hold
(i) (Duran [19])119860 isin (ℓinfin 119891) if and only if (18) holds and
119891 minus lim119886119899119896= 120572119896exists for each fixed 119896 (40)
lim119898rarrinfin
sum
119896
1003816100381610038161003816119886 (119899 119896 119898) minus 120572
119896
1003816100381610038161003816= 0 uniformly in 119899 (41)
(ii) (King [20]) 119860 isin (119888 119891) if and only if (18) (40) holdand
119891 minus limsum119896
119886119899119896= 120572 (42)
(iii) (Basar and Colak [21])119860 isin (119888119904 119891) if and only if (40)holds and
sup119899isinN
sum
119896
1003816100381610038161003816Δ119886119899119896
1003816100381610038161003816lt infin (43)
(iv) (Basar and Colak [21])119860 isin (119887119904 119891) if and only if (40)(43) hold and
lim119896
119886119899119896= 0 exists for each fixed 119899 (44)
lim119902rarrinfin
sum
119896
1
119902 + 1
119902
sum
119894=0
1003816100381610038161003816Δ [119886 (119899 + 119894 119896) minus 120572
119896]1003816100381610038161003816= 0 uniformly in 119899
(45)
(v) (Duran [19]) 119860 isin (119891 119891) if and only if (18) (40) and(42) hold and
lim119898rarrinfin
sum
119896
1003816100381610038161003816Δ [119886 (119899 119896119898) minus 120572
119896]1003816100381610038161003816= 0 uniformly in 119899 (46)
(vi) (Basar [22])119860 isin (119891119904 119891) if and only if (40) (44) (46)and (45) hold
Abstract and Applied Analysis 7
(vii) (Ozturk [23])119860 isin (119891119904 119888) if and only if (19) (43) and(44) hold and
lim119899rarrinfin
sum
119896
10038161003816100381610038161003816Δ2
119886119899119896
10038161003816100381610038161003816= 120572 (47)
(viii) 119860 isin (119891119904 ℓinfin) if and only if (43) and (44) hold
(ix) (Basar and Solak [24]) 119860 isin (119887119904 119891119904) if and only if(44) (45) hold and
sup119899isinN
sum
119896
|Δ119886 (119899 119896)| lt infin
119891 minus lim 119886 (119899 119896) = 120572119896exists for each fixed 119896
(48)
(x) (Basar [22])119860 isin (119891119904 119891119904) if and only if (45) (48) holdand
lim119902rarrinfin
sum
119896
1
119902 + 1
119902
sum
119894=0
10038161003816100381610038161003816Δ2
[119886 (119899 + 119894 119896) minus 120572119896]
10038161003816100381610038161003816= 0 uniformly in 119899
(49)
(xi) (Basar and Colak [21])119860 isin (119888119904 119891119904) if and only if (48)holds
(xii) (Basar [25]) 119860 isin (119891 119888119904) if and only if
sup119899isinN
sum
119896
|119886 (119899 119896)| lt infin (50)
sum
119899
119886119899119896= 120572119896
exists for each fixed 119896 (51)
sum
119899
sum
119896
119886119899119896= 120572 (52)
lim119898rarrinfin
sum
119896
1003816100381610038161003816Δ [119886 (119899 119896) minus 120572
119896]1003816100381610038161003816= 0 (53)
Now we give our main results which are related to matrixmappings oninto the spaces of almost convergent series 119886119903
119891119904
and sequences 119886119903119891
Corollary 20 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (119886119903119891119904 119891) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin
N and (40) (44) holdwith 119886119899119896instead of 119886
119899119896 (46) holds
with 119886(119899 119896119898) instead of 119886(119899 119896119898) and (45) holdswith 119886(119899 119896) instead of 119886(119899 119896)
(ii) 119860 isin (119886119903119891119904 119888) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin N
and (19) (43) (44) and (47) hold with 119886119899119896
instead of119886119899119896
(iii) 119860 isin (119886119903119891119904 ℓinfin) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin
N and (43) and (44) hold with 119886119899119896instead of 119886
119899119896
(iv) 119860 isin (119886119903119891119904 119891119904) if and only if 119886
119899119896119896isinN isin (119886
119903
119891119904)120573 for all 119899 isin
N and (45) (48) and (49) hold with 119886(119899 119896) instead of119886(119899 119896)
(v) 119860 isin (119888119904 119886119903119891119904) if and only if (48) holds with 119886(119899 119896) in-
stead of 119886(119899 119896)
(vi) 119860 isin (119887119904 119886119903119891119904) if and only if (44) holds with 119886
119899119896instead
of 119886119899119896and (45) (48) holdwith 119886(119899 119896) instead of 119886(119899 119896)
(vii) 119860 isin (119891119904 119886119903
119891119904) if and only if (45) (48) and (49) hold
with 119886(119899 119896) instead of 119886(119899 119896)
Corollary 21 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (119888(119901) 119886119903
119891119904) if and only if (37)and (38) hold with
119886(119899 119896119898) instead of 119886(119899 119896119898)
(ii) 119860 isin (1198880(119901) 119886
119903
119891119904) if and only if (37) holds with 119886(119899
119896119898) instead of 119886(119899 119896119898)
(iii) 119860 isin (ℓinfin(119901) 119886
119903
119891119904) if and only if (37) and (39) hold with
119886(119899 119896119898) instead of 119886(119899 119896119898)
Corollary 22 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (119886119903
119891 ℓinfin) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all
119899 isin N and (18) holds with 119886119899119896instead of 119886
119899119896
(ii) 119860 isin (119886119903119891 119888) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin N
and (18) (19) (20) and (21) hold with 119886119899119896
instead of119886119899119896
(iii) 119860 isin (119886119903119891 119888119904) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin
N and (50)(53) hold with 119886(119899 119896) instead of 119886(119899 119896) and(51)(52) hold with 119886
119899119896instead of 119886
119899119896
Corollary 23 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (ℓinfin 119886119903
119891) if and only if (18) (40) hold with 119886
119899119896in-
stead of 119886119899119896
and (41) holds with 119886(119899 119896119898) instead of119886(119899 119896119898)
(ii) 119860 isin (119891 119886119903119891) if and only if (18) (40) and (46) holdwith
119886(119899 119896119898) instead of 119886(119899 119896119898) and (42) holds with 119886119899119896
instead of 119886119899119896
(iii) 119860 isin (119888 119886119903119891) if and only if (18) (40) and (42) hold with
119886119899119896instead of 119886
119899119896
(iv) 119860 isin (119887119904 119886119903
119891) if and only if (40) (43) and (44) hold
with 119886119899119896
instead of 119886119899119896
and (45) holds with 119886(119899 119896)instead of 119886(119899 119896)
(v) 119860 isin (119891119904 119886119903
119891) if and only if (40) (44) hold with 119886
119899119896
instead of 119886119899119896 (46) holds with 119886(119899 119896119898) instead of 119886(119899
119896119898) and (45) holds with 119886(119899 119896) instead of 119886(119899 119896)
(vi) 119860 isin (119888119904 119886119903119891) if and only if (40) and (43) hold with 119886
119899119896
instead of 119886119899119896
Remark 24 Characterization of the classes (119886119903119891 119891infin) (119891infin
119886119903
119891) (119886119903119891119904 119891infin) and (119891
infin 119886119903
119891119904) is redundant since the spaces of
almost bounded sequences 119891infin
and ℓinfin
are equal
8 Abstract and Applied Analysis
Acknowledgment
The authors thank the referees for their careful reading of theoriginal paper and for the valuable comments
References
[1] J Boos Classical and Modern Methods in Summability OxfordUniversity Press New York NY USA 2000
[2] M Sengonul and F Basar ldquoSomenewCesaro sequence spaces ofnon-absolute type which include the spaces 119888
0and 119888rdquo Soochow
Journal of Mathematics vol 31 no 1 pp 107ndash119 2005[3] C Aydın and F Basar ldquoSome new sequence spaces which in-
clude the spaces 119897infinand 119897119875rdquo Demonstratio Mathematica vol 38
no 3 pp 641ndash656 2005[4] E Malkowsky and E Savas ldquoMatrix transformations between
sequence spaces of generalized weightedmeansrdquoAppliedMath-ematics and Computation vol 147 no 2 pp 333ndash345 2004
[5] MMursaleen andAKNoman ldquoOn the spaces of120582-convergentand bounded sequencesrdquo Thai Journal of Mathematics vol 8no 2 pp 311ndash329 2010
[6] B Altay F Basar and M Mursaleen ldquoOn the Euler sequencespaces which include the spaces 119897
119901and 119897infinIrdquo Information Sci-
ences vol 176 no 10 pp 1450ndash1462 2006[7] F Basar and M Kirisci ldquoAlmost convergence and generalized
differencematrixrdquoComputers ampMathematics with Applicationsvol 61 no 3 pp 602ndash611 2011
[8] A Sonmez ldquoAlmost convergence and triple bandmatrixrdquoMath-ematical and Computer Modelling vol 57 no 9-10 pp 2393ndash2402 2013
[9] K Kayaduman andM S Sengonul ldquoThe space of Cesaro almostconvergent sequence and core theoremsrdquo Acta MathematicaScientia vol 6 pp 2265ndash2278 2012
[10] M Sengonul and K Kayaduman ldquoOn the Riesz almost conver-gent sequences spacerdquo Abstract and Applied Analysis vol 2012Article ID 691694 18 pages 2012
[11] G G Lorentz ldquoA contribution to the theory of divergent se-quencesrdquo Acta Mathematica vol 80 pp 167ndash190 1948
[12] I J Maddox ldquoParanormed sequence spaces generated by infi-nite matricesrdquo Proceedings of the Cambridge Philosophical Soci-ety vol 64 pp 335ndash340 1968
[13] A Karaisa and F Ozger ldquoAlmost difference sequence space de-rived by using a generalized weightedmeanrdquoActaMathematicaScientia In review
[14] C Aydın and F Basar ldquoOn the new sequence spaces which in-clude the spaces 119888
0and 119888rdquo Hokkaido Mathematical Journal vol
33 no 2 pp 383ndash398 2004[15] F Basar ldquoA note on the triangle limitation methodsrdquo Fırat Uni-
versitesi Muhendislik Bilimleri Dergisi vol 5 no 1 pp 113ndash1171993
[16] AM Al-Jarrah and EMalkowsky ldquoBK spaces bases and linearoperatorsrdquo in Proceedings of the 3rd International Conference onFunctional Analysis and ApproximationTheory vol 1 no 52 pp177ndash191 1998
[17] A Wilansky Summability through Functional Analysis vol 85North-Holland Amsterdam The Netherlands 1984
[18] J A Sıddıqı ldquoInfinite matrices summing every almost periodicsequencerdquo Pacific Journal of Mathematics vol 39 pp 235ndash2511971
[19] J P Duran ldquoInfinite matrices and almost-convergencerdquo Math-ematische Zeitschrift vol 128 pp 75ndash83 1972
[20] J P King ldquoAlmost summable sequencesrdquo Proceedings of theAmerican Mathematical Society vol 17 pp 1219ndash1225 1966
[21] F Basar and R Colak ldquoAlmost-conservativematrix transforma-tionsrdquo Turkish Journal of Mathematics vol 13 no 3 pp 91ndash1001989
[22] F Basar ldquo119891-conservative matrix sequencesrdquo Tamkang Journalof Mathematics vol 22 no 2 pp 205ndash212 1991
[23] E Ozturk ldquoOn strongly regular dual summability methodsrdquoCommunications de la Faculte des Sciences de lrsquoUniversite drsquoAn-kara vol 32 no 1 pp 1ndash5 1983
[24] F Basar and I Solak ldquoAlmost-coercivematrix transformationsrdquoRendiconti di Matematica e delle sue Applicazioni vol 11 no 2pp 249ndash256 1991
[25] F Basar ldquoStrongly-conservative sequence-to-series matrixtransformationsrdquo Erciyes Universitesi Fen Bilimleri Dergisi vol5 no 12 pp 888ndash893 1989
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
Proof Let us take any sequence 119886 isin 120596 By (23) 119886119909 = (119886119896119909119896) isin
119888119904whenever 119909 = (119909119896) isin 119886119903
119891if and only if 119879119910 isin 119888whenever 119910 =
(119910119896) isin 119891 It is obvious that the columns of that matrix 119879 in 119888
where119879 = 119905119903119899119896 defined in (24) we derive the consequence by
Lemma 9 that (119886119903119891)120573
= 119905119903
3cap 119905119903
4
Theorem 12 The 120574-dual of the space 119886119903119891119904is the intersection of
the sets
119888119903
1= 119886 = (119886
119896) isin 120596 sum
119896
1003816100381610038161003816100381610038161003816
Δ [Δ(
119886119896
1 + 119903119896
) (119896 + 1)
+
119886119896
1 + 119903119896(119896 + 1)]
1003816100381610038161003816100381610038161003816
lt infin
119888119903
2= 119886 = (119886
119896) isin 120596 (
119886119896(119896 + 1)
1 + 119903119896
) isin 1198880
(26)
In other words we have (119886119903119891119904)120574
= 119888119903
1cap 119888119903
2
Proof We obtain from (23) that 119886119909 = (119886119896119909119896) isin 119887119904 whenever
119909 = (119909119896) isin 119886119903
119891119904if and only if119879119910 isin ℓ
infinwhenever 119910 = (119910
119896) isin 119891119904
where 119879 = 119905119903
119899119896 is defined in (24) Therefore by Lemma 19
(viii) (119886119903119891119904)120574
= 119888119903
1cap 119888119903
2
Theorem 13 Define the set 1198881199033by
119888119903
3= 119886 = (119886
119896) isin 120596 lim
119899rarrinfin
sum
119896
10038161003816100381610038161003816Δ2
(119905119903
119899119896)
10038161003816100381610038161003816exists (27)
Then (119886119903119891119904)120573
= 119888119903
1cap 119888119903
2cap 119888119903
3
Proof This may be obtained in the same way as mentionedin the proof of Theorem 12 with Lemma 19(viii) instead ofLemma 19(vii) So we omit details
For the sake of brevity the following notations will beused
119886 (119899 119896119898) =
1
119898 + 1
119898
sum
119894=0
119886119899+119894119896
119886 (119899 119896) =
119899
sum
119894=0
119886119894119896
119886119899119896= Δ(
119886119899119896
1 + 119903119896
) (119896 + 1) = (
119886119899119896
1 + 119903119896
minus
119886119899119896+1
1 + 119903119896+1
) (119896 + 1)
Δ119886119899119896= 119886119899119896minus 119886119899119896+1
119886119899119896=
119899
sum
119895=0
(1 + 119903119895
) 119890119895119896
119899 + 1
(28)
for all 119896 119899 isin N Assume that the infinite matrices 119860 = (119886119899119896)
and119861 = (119887119899119896)map the sequences119909 = (119909
119896) and119910 = (119910
119896)which
are connectedwith relation (10) to the sequences 119906 = (119906119899) and
V = (V119899) respectively that is
119906119899= (119860119909)
119899= sum
119896
119886119899119896119909119896
forall119899 isin N (29)
V119899= (119861119910)
119899= sum
119896
119887119899119896119910119896
forall119899 isin N (30)
One can easily conclude here that the method 119860 is directlyapplied to the terms of the sequence 119909 = (119909
119896) while the
method 119861 is applied to the 119860119903-transform of the sequence 119909 =(119909119896) So the methods 119860 and 119861 are essentially differentNow suppose that the matrix product 119861119860119903 exists which is
amuch weaker assumption than the conditions on thematrix119861 belonging to anymatrix class in general It is not difficult tosee that the sequence in (30) reduces to the sequence in (29)under the application of formal summation by parts Thisleads us to the fact that 119861119860119903 exists and is equal to 119860 and(119861119860119903
)119909 = 119861(119860119903
119909) formally holds if one side exists This state-ment is equivalent to the following relation between the en-tries of the matrices 119860 = (119886
119899119896) and 119861 = (119887
119899119896) which are con-
nected with the relation
119886119899119896= 119887119899119896= Δ(
119886119899119896
1 + 119903119896
) (119896 + 1) or
119886119899119896= (1 + 119903
119896
)
infin
sum
119895=119896
119887119899119895
1 + 119895
forall119896 119899 isin N
(31)
Note that the methods 119860 and 119861 are not necessarily equi-valent since the order of summationmay not be reversedWenow give the following fundamental theorem connected withthematrixmappings oninto the almost convergent spaces 119886119903
119891
and 119886119903119891s
Theorem 14 Suppose that the entries of the infinite matrices119860 = (119886
119899119896) and 119861 = (119887
119899119896) are connected with relation (31) for all
119896 119899 isin N and let 120582 be any given sequence spaceThen119860 isin (119886119903119891
120582) if and only if
119861 isin (119891 120582)
119899 + 1
1 + 119903119896
119886119899119896
119896isinNisin 1198880
(32)
Proof Suppose that 119860 = (119886119899119896) and 119861 = (119887
119899119896) are connected
with the relation (31) and let 120582 be any given sequence spaceand keep in mind that the spaces 119886119903
119891and 119891 are norm iso-
morphicLet 119860 isin (119886119903
119891 120582) and take any sequence 119909 isin 119886119903
119891 and keep
in mind that 119910 = 119860119903119909 Then (119886119899119896)119896isinN isin (119886
119903
119891)120573 that is (32)
holds for all 119899 isin N and 119861119860119903 exists which implies that(119887119899119896)119896isinN isin ℓ1 = 119891
120573 for each 119899 isin N Thus 119861119910 exists for all 119910 isin119891 and thus we have119898 rarr infin in the equality
119898
sum
119896=0
119887119899119896119910119896=
119898
sum
119896=0
119898
sum
119895=119896
(1 + 119903119896
)
119887119899119895
1 + 119895
119909119896 (33)
for all119898 119899 isin N and we have (31) 119861119910 = 119860119909 which means that119861 isin (119891 120582) On the other hand assume that (32) holds and119861 isin (119891 120582) Then we have (119887
119899119896)119896isinN isin ℓ1 for all 119899 isin N which
6 Abstract and Applied Analysis
gives together with (120592119899119896)119896isinN isin (119886
119903
119891)120573 for each 119899 isin N that 119860119909
exists Then we obtain from the equality
119898
sum
119896=0
119886119899119896119909119896=
119898minus1
sum
119896=0
Δ(
119886119899119896
1 + 119903119896
) (119896 + 1) 119910119896+
119898 + 1
1 + 119903119898119886119899119898119910119898
=
119898
sum
119896=0
119887119899119896119910119896
(34)
for all 119898 119899 isin N as 119898 rarr infin that 119860119909 = 119861119910 and this showsthat 119860 isin (119886119903
119891 120582)
Theorem 15 Suppose that the entries of the infinite matrices119864 = (119890
119899119896) and 119865 = (119891
119899119896) are connected with the relation
119891119899119896= 119886119899119896
(35)
for all 119898 119899 isin N and 120582 is any given sequence space Then 119864 isin(120582 119886119903
119891) if and only if 119865 isin (120582 119891)
Proof Let 119909 = (119909119896) isin 120582 and consider the following equality
119899
sum
119895=0
119898
sum
119896=0
1 + 119903119895
119899 + 1
119890119895119896119909119896=
119898
sum
119896=0
119891119899119896119909119896 (36)
for all 119896119898 119899 isin N which yields as 119898 rarr infin that 119864119909 isin 119886119903119891
whenever 119909 isin 120582 if and only if 119865119909 isin 119891 whenever 119909 isin 120582 Thisstep completes the proof
Theorem 16 Let120582 be any given sequence space and thematri-ces119860 = (119886
119899119896) and119861 = (119887
119899119896) are connectedwith the relation (31)
Then 119860 isin (119886119903
119891119904 120582) if and only if 119861 isin (119891119904 120582) and (119886
119899119896)119896isinN isin
(119886119903
119891119904)120573 for all 119899 isin N
Proof The proof is based on the proof of Theorem 14
Theorem 17 Let 120582 be any given sequence space and the ele-ments of the infinite matrices 119864 = (119890
119899119896) and 119865 = (119891
119899119896) are con-
nectedwith relation (35)Then119864 = (119890119899119896) isin (120582 119886
119903
119891119904) if and only
if 119865 isin (120582 119891119904)
Proof The proof is based on the proof of Theorem 15
By Theorems 14 15 16 and 17 we have quite a few out-comes depending on the choice of the space 120582 to characterizecertain matrix mappings Hence by the help of these theo-rems the necessary and sufficient conditions for the classes(119886119903
119891 120582) (120582 119886119903
119891) (119886119903119891119904 120582) and (120582 119886119903
119891119904) may be derived by
replacing the entries of119860 and 119861 by those of 119861 = 119860(119860119903)minus1 and119865 = 119860
119903
119864 respectively where the necessary and sufficient con-ditions on the matrices 119864 and 119865 are read from the concerningresults in the existing literature
Lemma 18 Let 119860 = (119886119899119896) be an infinite matrix Then the fol-
lowing statements hold
(i) 119860 isin (1198880(119901) 119891) if and only if
exist119873 gt 1 ni sup119898isinN
sum
119896
|119886 (119899 119896 119898)|1198731119901119896
lt infin forall119899 isin N
exist120572119896isin C forall119896 isin N ni lim
119898rarrinfin
119886 (119899 119896119898) = 120572119896
uniformly in 119899(37)
(ii) 119860 isin (119888(119901) 119891) if and only if (37) and
exist120572 isin C ni lim119898rarrinfin
sum
119896
119886 (119899 119896119898) = 120572 uniformly in 119899 (38)
(iii) 119860 isin (ℓinfin(119901) 119891) if and only if (37) and
exist119873 gt 1 ni lim119898rarrinfin
sum
119896
1003816100381610038161003816119886 (119899 119896 119898) minus 120572
119896
10038161003816100381610038161198731119901119896
= 0
uniformly in 119899(39)
Lemma 19 Let 119860 = (119886119899119896) be an infinite matrix Then the fol-
lowing statements hold
(i) (Duran [19])119860 isin (ℓinfin 119891) if and only if (18) holds and
119891 minus lim119886119899119896= 120572119896exists for each fixed 119896 (40)
lim119898rarrinfin
sum
119896
1003816100381610038161003816119886 (119899 119896 119898) minus 120572
119896
1003816100381610038161003816= 0 uniformly in 119899 (41)
(ii) (King [20]) 119860 isin (119888 119891) if and only if (18) (40) holdand
119891 minus limsum119896
119886119899119896= 120572 (42)
(iii) (Basar and Colak [21])119860 isin (119888119904 119891) if and only if (40)holds and
sup119899isinN
sum
119896
1003816100381610038161003816Δ119886119899119896
1003816100381610038161003816lt infin (43)
(iv) (Basar and Colak [21])119860 isin (119887119904 119891) if and only if (40)(43) hold and
lim119896
119886119899119896= 0 exists for each fixed 119899 (44)
lim119902rarrinfin
sum
119896
1
119902 + 1
119902
sum
119894=0
1003816100381610038161003816Δ [119886 (119899 + 119894 119896) minus 120572
119896]1003816100381610038161003816= 0 uniformly in 119899
(45)
(v) (Duran [19]) 119860 isin (119891 119891) if and only if (18) (40) and(42) hold and
lim119898rarrinfin
sum
119896
1003816100381610038161003816Δ [119886 (119899 119896119898) minus 120572
119896]1003816100381610038161003816= 0 uniformly in 119899 (46)
(vi) (Basar [22])119860 isin (119891119904 119891) if and only if (40) (44) (46)and (45) hold
Abstract and Applied Analysis 7
(vii) (Ozturk [23])119860 isin (119891119904 119888) if and only if (19) (43) and(44) hold and
lim119899rarrinfin
sum
119896
10038161003816100381610038161003816Δ2
119886119899119896
10038161003816100381610038161003816= 120572 (47)
(viii) 119860 isin (119891119904 ℓinfin) if and only if (43) and (44) hold
(ix) (Basar and Solak [24]) 119860 isin (119887119904 119891119904) if and only if(44) (45) hold and
sup119899isinN
sum
119896
|Δ119886 (119899 119896)| lt infin
119891 minus lim 119886 (119899 119896) = 120572119896exists for each fixed 119896
(48)
(x) (Basar [22])119860 isin (119891119904 119891119904) if and only if (45) (48) holdand
lim119902rarrinfin
sum
119896
1
119902 + 1
119902
sum
119894=0
10038161003816100381610038161003816Δ2
[119886 (119899 + 119894 119896) minus 120572119896]
10038161003816100381610038161003816= 0 uniformly in 119899
(49)
(xi) (Basar and Colak [21])119860 isin (119888119904 119891119904) if and only if (48)holds
(xii) (Basar [25]) 119860 isin (119891 119888119904) if and only if
sup119899isinN
sum
119896
|119886 (119899 119896)| lt infin (50)
sum
119899
119886119899119896= 120572119896
exists for each fixed 119896 (51)
sum
119899
sum
119896
119886119899119896= 120572 (52)
lim119898rarrinfin
sum
119896
1003816100381610038161003816Δ [119886 (119899 119896) minus 120572
119896]1003816100381610038161003816= 0 (53)
Now we give our main results which are related to matrixmappings oninto the spaces of almost convergent series 119886119903
119891119904
and sequences 119886119903119891
Corollary 20 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (119886119903119891119904 119891) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin
N and (40) (44) holdwith 119886119899119896instead of 119886
119899119896 (46) holds
with 119886(119899 119896119898) instead of 119886(119899 119896119898) and (45) holdswith 119886(119899 119896) instead of 119886(119899 119896)
(ii) 119860 isin (119886119903119891119904 119888) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin N
and (19) (43) (44) and (47) hold with 119886119899119896
instead of119886119899119896
(iii) 119860 isin (119886119903119891119904 ℓinfin) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin
N and (43) and (44) hold with 119886119899119896instead of 119886
119899119896
(iv) 119860 isin (119886119903119891119904 119891119904) if and only if 119886
119899119896119896isinN isin (119886
119903
119891119904)120573 for all 119899 isin
N and (45) (48) and (49) hold with 119886(119899 119896) instead of119886(119899 119896)
(v) 119860 isin (119888119904 119886119903119891119904) if and only if (48) holds with 119886(119899 119896) in-
stead of 119886(119899 119896)
(vi) 119860 isin (119887119904 119886119903119891119904) if and only if (44) holds with 119886
119899119896instead
of 119886119899119896and (45) (48) holdwith 119886(119899 119896) instead of 119886(119899 119896)
(vii) 119860 isin (119891119904 119886119903
119891119904) if and only if (45) (48) and (49) hold
with 119886(119899 119896) instead of 119886(119899 119896)
Corollary 21 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (119888(119901) 119886119903
119891119904) if and only if (37)and (38) hold with
119886(119899 119896119898) instead of 119886(119899 119896119898)
(ii) 119860 isin (1198880(119901) 119886
119903
119891119904) if and only if (37) holds with 119886(119899
119896119898) instead of 119886(119899 119896119898)
(iii) 119860 isin (ℓinfin(119901) 119886
119903
119891119904) if and only if (37) and (39) hold with
119886(119899 119896119898) instead of 119886(119899 119896119898)
Corollary 22 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (119886119903
119891 ℓinfin) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all
119899 isin N and (18) holds with 119886119899119896instead of 119886
119899119896
(ii) 119860 isin (119886119903119891 119888) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin N
and (18) (19) (20) and (21) hold with 119886119899119896
instead of119886119899119896
(iii) 119860 isin (119886119903119891 119888119904) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin
N and (50)(53) hold with 119886(119899 119896) instead of 119886(119899 119896) and(51)(52) hold with 119886
119899119896instead of 119886
119899119896
Corollary 23 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (ℓinfin 119886119903
119891) if and only if (18) (40) hold with 119886
119899119896in-
stead of 119886119899119896
and (41) holds with 119886(119899 119896119898) instead of119886(119899 119896119898)
(ii) 119860 isin (119891 119886119903119891) if and only if (18) (40) and (46) holdwith
119886(119899 119896119898) instead of 119886(119899 119896119898) and (42) holds with 119886119899119896
instead of 119886119899119896
(iii) 119860 isin (119888 119886119903119891) if and only if (18) (40) and (42) hold with
119886119899119896instead of 119886
119899119896
(iv) 119860 isin (119887119904 119886119903
119891) if and only if (40) (43) and (44) hold
with 119886119899119896
instead of 119886119899119896
and (45) holds with 119886(119899 119896)instead of 119886(119899 119896)
(v) 119860 isin (119891119904 119886119903
119891) if and only if (40) (44) hold with 119886
119899119896
instead of 119886119899119896 (46) holds with 119886(119899 119896119898) instead of 119886(119899
119896119898) and (45) holds with 119886(119899 119896) instead of 119886(119899 119896)
(vi) 119860 isin (119888119904 119886119903119891) if and only if (40) and (43) hold with 119886
119899119896
instead of 119886119899119896
Remark 24 Characterization of the classes (119886119903119891 119891infin) (119891infin
119886119903
119891) (119886119903119891119904 119891infin) and (119891
infin 119886119903
119891119904) is redundant since the spaces of
almost bounded sequences 119891infin
and ℓinfin
are equal
8 Abstract and Applied Analysis
Acknowledgment
The authors thank the referees for their careful reading of theoriginal paper and for the valuable comments
References
[1] J Boos Classical and Modern Methods in Summability OxfordUniversity Press New York NY USA 2000
[2] M Sengonul and F Basar ldquoSomenewCesaro sequence spaces ofnon-absolute type which include the spaces 119888
0and 119888rdquo Soochow
Journal of Mathematics vol 31 no 1 pp 107ndash119 2005[3] C Aydın and F Basar ldquoSome new sequence spaces which in-
clude the spaces 119897infinand 119897119875rdquo Demonstratio Mathematica vol 38
no 3 pp 641ndash656 2005[4] E Malkowsky and E Savas ldquoMatrix transformations between
sequence spaces of generalized weightedmeansrdquoAppliedMath-ematics and Computation vol 147 no 2 pp 333ndash345 2004
[5] MMursaleen andAKNoman ldquoOn the spaces of120582-convergentand bounded sequencesrdquo Thai Journal of Mathematics vol 8no 2 pp 311ndash329 2010
[6] B Altay F Basar and M Mursaleen ldquoOn the Euler sequencespaces which include the spaces 119897
119901and 119897infinIrdquo Information Sci-
ences vol 176 no 10 pp 1450ndash1462 2006[7] F Basar and M Kirisci ldquoAlmost convergence and generalized
differencematrixrdquoComputers ampMathematics with Applicationsvol 61 no 3 pp 602ndash611 2011
[8] A Sonmez ldquoAlmost convergence and triple bandmatrixrdquoMath-ematical and Computer Modelling vol 57 no 9-10 pp 2393ndash2402 2013
[9] K Kayaduman andM S Sengonul ldquoThe space of Cesaro almostconvergent sequence and core theoremsrdquo Acta MathematicaScientia vol 6 pp 2265ndash2278 2012
[10] M Sengonul and K Kayaduman ldquoOn the Riesz almost conver-gent sequences spacerdquo Abstract and Applied Analysis vol 2012Article ID 691694 18 pages 2012
[11] G G Lorentz ldquoA contribution to the theory of divergent se-quencesrdquo Acta Mathematica vol 80 pp 167ndash190 1948
[12] I J Maddox ldquoParanormed sequence spaces generated by infi-nite matricesrdquo Proceedings of the Cambridge Philosophical Soci-ety vol 64 pp 335ndash340 1968
[13] A Karaisa and F Ozger ldquoAlmost difference sequence space de-rived by using a generalized weightedmeanrdquoActaMathematicaScientia In review
[14] C Aydın and F Basar ldquoOn the new sequence spaces which in-clude the spaces 119888
0and 119888rdquo Hokkaido Mathematical Journal vol
33 no 2 pp 383ndash398 2004[15] F Basar ldquoA note on the triangle limitation methodsrdquo Fırat Uni-
versitesi Muhendislik Bilimleri Dergisi vol 5 no 1 pp 113ndash1171993
[16] AM Al-Jarrah and EMalkowsky ldquoBK spaces bases and linearoperatorsrdquo in Proceedings of the 3rd International Conference onFunctional Analysis and ApproximationTheory vol 1 no 52 pp177ndash191 1998
[17] A Wilansky Summability through Functional Analysis vol 85North-Holland Amsterdam The Netherlands 1984
[18] J A Sıddıqı ldquoInfinite matrices summing every almost periodicsequencerdquo Pacific Journal of Mathematics vol 39 pp 235ndash2511971
[19] J P Duran ldquoInfinite matrices and almost-convergencerdquo Math-ematische Zeitschrift vol 128 pp 75ndash83 1972
[20] J P King ldquoAlmost summable sequencesrdquo Proceedings of theAmerican Mathematical Society vol 17 pp 1219ndash1225 1966
[21] F Basar and R Colak ldquoAlmost-conservativematrix transforma-tionsrdquo Turkish Journal of Mathematics vol 13 no 3 pp 91ndash1001989
[22] F Basar ldquo119891-conservative matrix sequencesrdquo Tamkang Journalof Mathematics vol 22 no 2 pp 205ndash212 1991
[23] E Ozturk ldquoOn strongly regular dual summability methodsrdquoCommunications de la Faculte des Sciences de lrsquoUniversite drsquoAn-kara vol 32 no 1 pp 1ndash5 1983
[24] F Basar and I Solak ldquoAlmost-coercivematrix transformationsrdquoRendiconti di Matematica e delle sue Applicazioni vol 11 no 2pp 249ndash256 1991
[25] F Basar ldquoStrongly-conservative sequence-to-series matrixtransformationsrdquo Erciyes Universitesi Fen Bilimleri Dergisi vol5 no 12 pp 888ndash893 1989
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Journal ofApplied Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ProbabilityandStatistics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Advances in
Mathematical Physics
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
Combinatorics
OperationsResearch
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
DecisionSciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
gives together with (120592119899119896)119896isinN isin (119886
119903
119891)120573 for each 119899 isin N that 119860119909
exists Then we obtain from the equality
119898
sum
119896=0
119886119899119896119909119896=
119898minus1
sum
119896=0
Δ(
119886119899119896
1 + 119903119896
) (119896 + 1) 119910119896+
119898 + 1
1 + 119903119898119886119899119898119910119898
=
119898
sum
119896=0
119887119899119896119910119896
(34)
for all 119898 119899 isin N as 119898 rarr infin that 119860119909 = 119861119910 and this showsthat 119860 isin (119886119903
119891 120582)
Theorem 15 Suppose that the entries of the infinite matrices119864 = (119890
119899119896) and 119865 = (119891
119899119896) are connected with the relation
119891119899119896= 119886119899119896
(35)
for all 119898 119899 isin N and 120582 is any given sequence space Then 119864 isin(120582 119886119903
119891) if and only if 119865 isin (120582 119891)
Proof Let 119909 = (119909119896) isin 120582 and consider the following equality
119899
sum
119895=0
119898
sum
119896=0
1 + 119903119895
119899 + 1
119890119895119896119909119896=
119898
sum
119896=0
119891119899119896119909119896 (36)
for all 119896119898 119899 isin N which yields as 119898 rarr infin that 119864119909 isin 119886119903119891
whenever 119909 isin 120582 if and only if 119865119909 isin 119891 whenever 119909 isin 120582 Thisstep completes the proof
Theorem 16 Let120582 be any given sequence space and thematri-ces119860 = (119886
119899119896) and119861 = (119887
119899119896) are connectedwith the relation (31)
Then 119860 isin (119886119903
119891119904 120582) if and only if 119861 isin (119891119904 120582) and (119886
119899119896)119896isinN isin
(119886119903
119891119904)120573 for all 119899 isin N
Proof The proof is based on the proof of Theorem 14
Theorem 17 Let 120582 be any given sequence space and the ele-ments of the infinite matrices 119864 = (119890
119899119896) and 119865 = (119891
119899119896) are con-
nectedwith relation (35)Then119864 = (119890119899119896) isin (120582 119886
119903
119891119904) if and only
if 119865 isin (120582 119891119904)
Proof The proof is based on the proof of Theorem 15
By Theorems 14 15 16 and 17 we have quite a few out-comes depending on the choice of the space 120582 to characterizecertain matrix mappings Hence by the help of these theo-rems the necessary and sufficient conditions for the classes(119886119903
119891 120582) (120582 119886119903
119891) (119886119903119891119904 120582) and (120582 119886119903
119891119904) may be derived by
replacing the entries of119860 and 119861 by those of 119861 = 119860(119860119903)minus1 and119865 = 119860
119903
119864 respectively where the necessary and sufficient con-ditions on the matrices 119864 and 119865 are read from the concerningresults in the existing literature
Lemma 18 Let 119860 = (119886119899119896) be an infinite matrix Then the fol-
lowing statements hold
(i) 119860 isin (1198880(119901) 119891) if and only if
exist119873 gt 1 ni sup119898isinN
sum
119896
|119886 (119899 119896 119898)|1198731119901119896
lt infin forall119899 isin N
exist120572119896isin C forall119896 isin N ni lim
119898rarrinfin
119886 (119899 119896119898) = 120572119896
uniformly in 119899(37)
(ii) 119860 isin (119888(119901) 119891) if and only if (37) and
exist120572 isin C ni lim119898rarrinfin
sum
119896
119886 (119899 119896119898) = 120572 uniformly in 119899 (38)
(iii) 119860 isin (ℓinfin(119901) 119891) if and only if (37) and
exist119873 gt 1 ni lim119898rarrinfin
sum
119896
1003816100381610038161003816119886 (119899 119896 119898) minus 120572
119896
10038161003816100381610038161198731119901119896
= 0
uniformly in 119899(39)
Lemma 19 Let 119860 = (119886119899119896) be an infinite matrix Then the fol-
lowing statements hold
(i) (Duran [19])119860 isin (ℓinfin 119891) if and only if (18) holds and
119891 minus lim119886119899119896= 120572119896exists for each fixed 119896 (40)
lim119898rarrinfin
sum
119896
1003816100381610038161003816119886 (119899 119896 119898) minus 120572
119896
1003816100381610038161003816= 0 uniformly in 119899 (41)
(ii) (King [20]) 119860 isin (119888 119891) if and only if (18) (40) holdand
119891 minus limsum119896
119886119899119896= 120572 (42)
(iii) (Basar and Colak [21])119860 isin (119888119904 119891) if and only if (40)holds and
sup119899isinN
sum
119896
1003816100381610038161003816Δ119886119899119896
1003816100381610038161003816lt infin (43)
(iv) (Basar and Colak [21])119860 isin (119887119904 119891) if and only if (40)(43) hold and
lim119896
119886119899119896= 0 exists for each fixed 119899 (44)
lim119902rarrinfin
sum
119896
1
119902 + 1
119902
sum
119894=0
1003816100381610038161003816Δ [119886 (119899 + 119894 119896) minus 120572
119896]1003816100381610038161003816= 0 uniformly in 119899
(45)
(v) (Duran [19]) 119860 isin (119891 119891) if and only if (18) (40) and(42) hold and
lim119898rarrinfin
sum
119896
1003816100381610038161003816Δ [119886 (119899 119896119898) minus 120572
119896]1003816100381610038161003816= 0 uniformly in 119899 (46)
(vi) (Basar [22])119860 isin (119891119904 119891) if and only if (40) (44) (46)and (45) hold
Abstract and Applied Analysis 7
(vii) (Ozturk [23])119860 isin (119891119904 119888) if and only if (19) (43) and(44) hold and
lim119899rarrinfin
sum
119896
10038161003816100381610038161003816Δ2
119886119899119896
10038161003816100381610038161003816= 120572 (47)
(viii) 119860 isin (119891119904 ℓinfin) if and only if (43) and (44) hold
(ix) (Basar and Solak [24]) 119860 isin (119887119904 119891119904) if and only if(44) (45) hold and
sup119899isinN
sum
119896
|Δ119886 (119899 119896)| lt infin
119891 minus lim 119886 (119899 119896) = 120572119896exists for each fixed 119896
(48)
(x) (Basar [22])119860 isin (119891119904 119891119904) if and only if (45) (48) holdand
lim119902rarrinfin
sum
119896
1
119902 + 1
119902
sum
119894=0
10038161003816100381610038161003816Δ2
[119886 (119899 + 119894 119896) minus 120572119896]
10038161003816100381610038161003816= 0 uniformly in 119899
(49)
(xi) (Basar and Colak [21])119860 isin (119888119904 119891119904) if and only if (48)holds
(xii) (Basar [25]) 119860 isin (119891 119888119904) if and only if
sup119899isinN
sum
119896
|119886 (119899 119896)| lt infin (50)
sum
119899
119886119899119896= 120572119896
exists for each fixed 119896 (51)
sum
119899
sum
119896
119886119899119896= 120572 (52)
lim119898rarrinfin
sum
119896
1003816100381610038161003816Δ [119886 (119899 119896) minus 120572
119896]1003816100381610038161003816= 0 (53)
Now we give our main results which are related to matrixmappings oninto the spaces of almost convergent series 119886119903
119891119904
and sequences 119886119903119891
Corollary 20 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (119886119903119891119904 119891) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin
N and (40) (44) holdwith 119886119899119896instead of 119886
119899119896 (46) holds
with 119886(119899 119896119898) instead of 119886(119899 119896119898) and (45) holdswith 119886(119899 119896) instead of 119886(119899 119896)
(ii) 119860 isin (119886119903119891119904 119888) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin N
and (19) (43) (44) and (47) hold with 119886119899119896
instead of119886119899119896
(iii) 119860 isin (119886119903119891119904 ℓinfin) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin
N and (43) and (44) hold with 119886119899119896instead of 119886
119899119896
(iv) 119860 isin (119886119903119891119904 119891119904) if and only if 119886
119899119896119896isinN isin (119886
119903
119891119904)120573 for all 119899 isin
N and (45) (48) and (49) hold with 119886(119899 119896) instead of119886(119899 119896)
(v) 119860 isin (119888119904 119886119903119891119904) if and only if (48) holds with 119886(119899 119896) in-
stead of 119886(119899 119896)
(vi) 119860 isin (119887119904 119886119903119891119904) if and only if (44) holds with 119886
119899119896instead
of 119886119899119896and (45) (48) holdwith 119886(119899 119896) instead of 119886(119899 119896)
(vii) 119860 isin (119891119904 119886119903
119891119904) if and only if (45) (48) and (49) hold
with 119886(119899 119896) instead of 119886(119899 119896)
Corollary 21 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (119888(119901) 119886119903
119891119904) if and only if (37)and (38) hold with
119886(119899 119896119898) instead of 119886(119899 119896119898)
(ii) 119860 isin (1198880(119901) 119886
119903
119891119904) if and only if (37) holds with 119886(119899
119896119898) instead of 119886(119899 119896119898)
(iii) 119860 isin (ℓinfin(119901) 119886
119903
119891119904) if and only if (37) and (39) hold with
119886(119899 119896119898) instead of 119886(119899 119896119898)
Corollary 22 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (119886119903
119891 ℓinfin) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all
119899 isin N and (18) holds with 119886119899119896instead of 119886
119899119896
(ii) 119860 isin (119886119903119891 119888) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin N
and (18) (19) (20) and (21) hold with 119886119899119896
instead of119886119899119896
(iii) 119860 isin (119886119903119891 119888119904) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin
N and (50)(53) hold with 119886(119899 119896) instead of 119886(119899 119896) and(51)(52) hold with 119886
119899119896instead of 119886
119899119896
Corollary 23 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (ℓinfin 119886119903
119891) if and only if (18) (40) hold with 119886
119899119896in-
stead of 119886119899119896
and (41) holds with 119886(119899 119896119898) instead of119886(119899 119896119898)
(ii) 119860 isin (119891 119886119903119891) if and only if (18) (40) and (46) holdwith
119886(119899 119896119898) instead of 119886(119899 119896119898) and (42) holds with 119886119899119896
instead of 119886119899119896
(iii) 119860 isin (119888 119886119903119891) if and only if (18) (40) and (42) hold with
119886119899119896instead of 119886
119899119896
(iv) 119860 isin (119887119904 119886119903
119891) if and only if (40) (43) and (44) hold
with 119886119899119896
instead of 119886119899119896
and (45) holds with 119886(119899 119896)instead of 119886(119899 119896)
(v) 119860 isin (119891119904 119886119903
119891) if and only if (40) (44) hold with 119886
119899119896
instead of 119886119899119896 (46) holds with 119886(119899 119896119898) instead of 119886(119899
119896119898) and (45) holds with 119886(119899 119896) instead of 119886(119899 119896)
(vi) 119860 isin (119888119904 119886119903119891) if and only if (40) and (43) hold with 119886
119899119896
instead of 119886119899119896
Remark 24 Characterization of the classes (119886119903119891 119891infin) (119891infin
119886119903
119891) (119886119903119891119904 119891infin) and (119891
infin 119886119903
119891119904) is redundant since the spaces of
almost bounded sequences 119891infin
and ℓinfin
are equal
8 Abstract and Applied Analysis
Acknowledgment
The authors thank the referees for their careful reading of theoriginal paper and for the valuable comments
References
[1] J Boos Classical and Modern Methods in Summability OxfordUniversity Press New York NY USA 2000
[2] M Sengonul and F Basar ldquoSomenewCesaro sequence spaces ofnon-absolute type which include the spaces 119888
0and 119888rdquo Soochow
Journal of Mathematics vol 31 no 1 pp 107ndash119 2005[3] C Aydın and F Basar ldquoSome new sequence spaces which in-
clude the spaces 119897infinand 119897119875rdquo Demonstratio Mathematica vol 38
no 3 pp 641ndash656 2005[4] E Malkowsky and E Savas ldquoMatrix transformations between
sequence spaces of generalized weightedmeansrdquoAppliedMath-ematics and Computation vol 147 no 2 pp 333ndash345 2004
[5] MMursaleen andAKNoman ldquoOn the spaces of120582-convergentand bounded sequencesrdquo Thai Journal of Mathematics vol 8no 2 pp 311ndash329 2010
[6] B Altay F Basar and M Mursaleen ldquoOn the Euler sequencespaces which include the spaces 119897
119901and 119897infinIrdquo Information Sci-
ences vol 176 no 10 pp 1450ndash1462 2006[7] F Basar and M Kirisci ldquoAlmost convergence and generalized
differencematrixrdquoComputers ampMathematics with Applicationsvol 61 no 3 pp 602ndash611 2011
[8] A Sonmez ldquoAlmost convergence and triple bandmatrixrdquoMath-ematical and Computer Modelling vol 57 no 9-10 pp 2393ndash2402 2013
[9] K Kayaduman andM S Sengonul ldquoThe space of Cesaro almostconvergent sequence and core theoremsrdquo Acta MathematicaScientia vol 6 pp 2265ndash2278 2012
[10] M Sengonul and K Kayaduman ldquoOn the Riesz almost conver-gent sequences spacerdquo Abstract and Applied Analysis vol 2012Article ID 691694 18 pages 2012
[11] G G Lorentz ldquoA contribution to the theory of divergent se-quencesrdquo Acta Mathematica vol 80 pp 167ndash190 1948
[12] I J Maddox ldquoParanormed sequence spaces generated by infi-nite matricesrdquo Proceedings of the Cambridge Philosophical Soci-ety vol 64 pp 335ndash340 1968
[13] A Karaisa and F Ozger ldquoAlmost difference sequence space de-rived by using a generalized weightedmeanrdquoActaMathematicaScientia In review
[14] C Aydın and F Basar ldquoOn the new sequence spaces which in-clude the spaces 119888
0and 119888rdquo Hokkaido Mathematical Journal vol
33 no 2 pp 383ndash398 2004[15] F Basar ldquoA note on the triangle limitation methodsrdquo Fırat Uni-
versitesi Muhendislik Bilimleri Dergisi vol 5 no 1 pp 113ndash1171993
[16] AM Al-Jarrah and EMalkowsky ldquoBK spaces bases and linearoperatorsrdquo in Proceedings of the 3rd International Conference onFunctional Analysis and ApproximationTheory vol 1 no 52 pp177ndash191 1998
[17] A Wilansky Summability through Functional Analysis vol 85North-Holland Amsterdam The Netherlands 1984
[18] J A Sıddıqı ldquoInfinite matrices summing every almost periodicsequencerdquo Pacific Journal of Mathematics vol 39 pp 235ndash2511971
[19] J P Duran ldquoInfinite matrices and almost-convergencerdquo Math-ematische Zeitschrift vol 128 pp 75ndash83 1972
[20] J P King ldquoAlmost summable sequencesrdquo Proceedings of theAmerican Mathematical Society vol 17 pp 1219ndash1225 1966
[21] F Basar and R Colak ldquoAlmost-conservativematrix transforma-tionsrdquo Turkish Journal of Mathematics vol 13 no 3 pp 91ndash1001989
[22] F Basar ldquo119891-conservative matrix sequencesrdquo Tamkang Journalof Mathematics vol 22 no 2 pp 205ndash212 1991
[23] E Ozturk ldquoOn strongly regular dual summability methodsrdquoCommunications de la Faculte des Sciences de lrsquoUniversite drsquoAn-kara vol 32 no 1 pp 1ndash5 1983
[24] F Basar and I Solak ldquoAlmost-coercivematrix transformationsrdquoRendiconti di Matematica e delle sue Applicazioni vol 11 no 2pp 249ndash256 1991
[25] F Basar ldquoStrongly-conservative sequence-to-series matrixtransformationsrdquo Erciyes Universitesi Fen Bilimleri Dergisi vol5 no 12 pp 888ndash893 1989
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Journal ofApplied Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ProbabilityandStatistics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Advances in
Mathematical Physics
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
Combinatorics
OperationsResearch
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
DecisionSciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
(vii) (Ozturk [23])119860 isin (119891119904 119888) if and only if (19) (43) and(44) hold and
lim119899rarrinfin
sum
119896
10038161003816100381610038161003816Δ2
119886119899119896
10038161003816100381610038161003816= 120572 (47)
(viii) 119860 isin (119891119904 ℓinfin) if and only if (43) and (44) hold
(ix) (Basar and Solak [24]) 119860 isin (119887119904 119891119904) if and only if(44) (45) hold and
sup119899isinN
sum
119896
|Δ119886 (119899 119896)| lt infin
119891 minus lim 119886 (119899 119896) = 120572119896exists for each fixed 119896
(48)
(x) (Basar [22])119860 isin (119891119904 119891119904) if and only if (45) (48) holdand
lim119902rarrinfin
sum
119896
1
119902 + 1
119902
sum
119894=0
10038161003816100381610038161003816Δ2
[119886 (119899 + 119894 119896) minus 120572119896]
10038161003816100381610038161003816= 0 uniformly in 119899
(49)
(xi) (Basar and Colak [21])119860 isin (119888119904 119891119904) if and only if (48)holds
(xii) (Basar [25]) 119860 isin (119891 119888119904) if and only if
sup119899isinN
sum
119896
|119886 (119899 119896)| lt infin (50)
sum
119899
119886119899119896= 120572119896
exists for each fixed 119896 (51)
sum
119899
sum
119896
119886119899119896= 120572 (52)
lim119898rarrinfin
sum
119896
1003816100381610038161003816Δ [119886 (119899 119896) minus 120572
119896]1003816100381610038161003816= 0 (53)
Now we give our main results which are related to matrixmappings oninto the spaces of almost convergent series 119886119903
119891119904
and sequences 119886119903119891
Corollary 20 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (119886119903119891119904 119891) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin
N and (40) (44) holdwith 119886119899119896instead of 119886
119899119896 (46) holds
with 119886(119899 119896119898) instead of 119886(119899 119896119898) and (45) holdswith 119886(119899 119896) instead of 119886(119899 119896)
(ii) 119860 isin (119886119903119891119904 119888) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin N
and (19) (43) (44) and (47) hold with 119886119899119896
instead of119886119899119896
(iii) 119860 isin (119886119903119891119904 ℓinfin) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin
N and (43) and (44) hold with 119886119899119896instead of 119886
119899119896
(iv) 119860 isin (119886119903119891119904 119891119904) if and only if 119886
119899119896119896isinN isin (119886
119903
119891119904)120573 for all 119899 isin
N and (45) (48) and (49) hold with 119886(119899 119896) instead of119886(119899 119896)
(v) 119860 isin (119888119904 119886119903119891119904) if and only if (48) holds with 119886(119899 119896) in-
stead of 119886(119899 119896)
(vi) 119860 isin (119887119904 119886119903119891119904) if and only if (44) holds with 119886
119899119896instead
of 119886119899119896and (45) (48) holdwith 119886(119899 119896) instead of 119886(119899 119896)
(vii) 119860 isin (119891119904 119886119903
119891119904) if and only if (45) (48) and (49) hold
with 119886(119899 119896) instead of 119886(119899 119896)
Corollary 21 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (119888(119901) 119886119903
119891119904) if and only if (37)and (38) hold with
119886(119899 119896119898) instead of 119886(119899 119896119898)
(ii) 119860 isin (1198880(119901) 119886
119903
119891119904) if and only if (37) holds with 119886(119899
119896119898) instead of 119886(119899 119896119898)
(iii) 119860 isin (ℓinfin(119901) 119886
119903
119891119904) if and only if (37) and (39) hold with
119886(119899 119896119898) instead of 119886(119899 119896119898)
Corollary 22 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (119886119903
119891 ℓinfin) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all
119899 isin N and (18) holds with 119886119899119896instead of 119886
119899119896
(ii) 119860 isin (119886119903119891 119888) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin N
and (18) (19) (20) and (21) hold with 119886119899119896
instead of119886119899119896
(iii) 119860 isin (119886119903119891 119888119904) if and only if 119886
119899119896119896isinN isin (119886
119903
119891)120573 for all 119899 isin
N and (50)(53) hold with 119886(119899 119896) instead of 119886(119899 119896) and(51)(52) hold with 119886
119899119896instead of 119886
119899119896
Corollary 23 Let 119860 = (119886119899119896) be an infinite matrix Then the
following statements hold
(i) 119860 isin (ℓinfin 119886119903
119891) if and only if (18) (40) hold with 119886
119899119896in-
stead of 119886119899119896
and (41) holds with 119886(119899 119896119898) instead of119886(119899 119896119898)
(ii) 119860 isin (119891 119886119903119891) if and only if (18) (40) and (46) holdwith
119886(119899 119896119898) instead of 119886(119899 119896119898) and (42) holds with 119886119899119896
instead of 119886119899119896
(iii) 119860 isin (119888 119886119903119891) if and only if (18) (40) and (42) hold with
119886119899119896instead of 119886
119899119896
(iv) 119860 isin (119887119904 119886119903
119891) if and only if (40) (43) and (44) hold
with 119886119899119896
instead of 119886119899119896
and (45) holds with 119886(119899 119896)instead of 119886(119899 119896)
(v) 119860 isin (119891119904 119886119903
119891) if and only if (40) (44) hold with 119886
119899119896
instead of 119886119899119896 (46) holds with 119886(119899 119896119898) instead of 119886(119899
119896119898) and (45) holds with 119886(119899 119896) instead of 119886(119899 119896)
(vi) 119860 isin (119888119904 119886119903119891) if and only if (40) and (43) hold with 119886
119899119896
instead of 119886119899119896
Remark 24 Characterization of the classes (119886119903119891 119891infin) (119891infin
119886119903
119891) (119886119903119891119904 119891infin) and (119891
infin 119886119903
119891119904) is redundant since the spaces of
almost bounded sequences 119891infin
and ℓinfin
are equal
8 Abstract and Applied Analysis
Acknowledgment
The authors thank the referees for their careful reading of theoriginal paper and for the valuable comments
References
[1] J Boos Classical and Modern Methods in Summability OxfordUniversity Press New York NY USA 2000
[2] M Sengonul and F Basar ldquoSomenewCesaro sequence spaces ofnon-absolute type which include the spaces 119888
0and 119888rdquo Soochow
Journal of Mathematics vol 31 no 1 pp 107ndash119 2005[3] C Aydın and F Basar ldquoSome new sequence spaces which in-
clude the spaces 119897infinand 119897119875rdquo Demonstratio Mathematica vol 38
no 3 pp 641ndash656 2005[4] E Malkowsky and E Savas ldquoMatrix transformations between
sequence spaces of generalized weightedmeansrdquoAppliedMath-ematics and Computation vol 147 no 2 pp 333ndash345 2004
[5] MMursaleen andAKNoman ldquoOn the spaces of120582-convergentand bounded sequencesrdquo Thai Journal of Mathematics vol 8no 2 pp 311ndash329 2010
[6] B Altay F Basar and M Mursaleen ldquoOn the Euler sequencespaces which include the spaces 119897
119901and 119897infinIrdquo Information Sci-
ences vol 176 no 10 pp 1450ndash1462 2006[7] F Basar and M Kirisci ldquoAlmost convergence and generalized
differencematrixrdquoComputers ampMathematics with Applicationsvol 61 no 3 pp 602ndash611 2011
[8] A Sonmez ldquoAlmost convergence and triple bandmatrixrdquoMath-ematical and Computer Modelling vol 57 no 9-10 pp 2393ndash2402 2013
[9] K Kayaduman andM S Sengonul ldquoThe space of Cesaro almostconvergent sequence and core theoremsrdquo Acta MathematicaScientia vol 6 pp 2265ndash2278 2012
[10] M Sengonul and K Kayaduman ldquoOn the Riesz almost conver-gent sequences spacerdquo Abstract and Applied Analysis vol 2012Article ID 691694 18 pages 2012
[11] G G Lorentz ldquoA contribution to the theory of divergent se-quencesrdquo Acta Mathematica vol 80 pp 167ndash190 1948
[12] I J Maddox ldquoParanormed sequence spaces generated by infi-nite matricesrdquo Proceedings of the Cambridge Philosophical Soci-ety vol 64 pp 335ndash340 1968
[13] A Karaisa and F Ozger ldquoAlmost difference sequence space de-rived by using a generalized weightedmeanrdquoActaMathematicaScientia In review
[14] C Aydın and F Basar ldquoOn the new sequence spaces which in-clude the spaces 119888
0and 119888rdquo Hokkaido Mathematical Journal vol
33 no 2 pp 383ndash398 2004[15] F Basar ldquoA note on the triangle limitation methodsrdquo Fırat Uni-
versitesi Muhendislik Bilimleri Dergisi vol 5 no 1 pp 113ndash1171993
[16] AM Al-Jarrah and EMalkowsky ldquoBK spaces bases and linearoperatorsrdquo in Proceedings of the 3rd International Conference onFunctional Analysis and ApproximationTheory vol 1 no 52 pp177ndash191 1998
[17] A Wilansky Summability through Functional Analysis vol 85North-Holland Amsterdam The Netherlands 1984
[18] J A Sıddıqı ldquoInfinite matrices summing every almost periodicsequencerdquo Pacific Journal of Mathematics vol 39 pp 235ndash2511971
[19] J P Duran ldquoInfinite matrices and almost-convergencerdquo Math-ematische Zeitschrift vol 128 pp 75ndash83 1972
[20] J P King ldquoAlmost summable sequencesrdquo Proceedings of theAmerican Mathematical Society vol 17 pp 1219ndash1225 1966
[21] F Basar and R Colak ldquoAlmost-conservativematrix transforma-tionsrdquo Turkish Journal of Mathematics vol 13 no 3 pp 91ndash1001989
[22] F Basar ldquo119891-conservative matrix sequencesrdquo Tamkang Journalof Mathematics vol 22 no 2 pp 205ndash212 1991
[23] E Ozturk ldquoOn strongly regular dual summability methodsrdquoCommunications de la Faculte des Sciences de lrsquoUniversite drsquoAn-kara vol 32 no 1 pp 1ndash5 1983
[24] F Basar and I Solak ldquoAlmost-coercivematrix transformationsrdquoRendiconti di Matematica e delle sue Applicazioni vol 11 no 2pp 249ndash256 1991
[25] F Basar ldquoStrongly-conservative sequence-to-series matrixtransformationsrdquo Erciyes Universitesi Fen Bilimleri Dergisi vol5 no 12 pp 888ndash893 1989
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Journal ofApplied Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ProbabilityandStatistics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Advances in
Mathematical Physics
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
Combinatorics
OperationsResearch
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
DecisionSciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Abstract and Applied Analysis
Acknowledgment
The authors thank the referees for their careful reading of theoriginal paper and for the valuable comments
References
[1] J Boos Classical and Modern Methods in Summability OxfordUniversity Press New York NY USA 2000
[2] M Sengonul and F Basar ldquoSomenewCesaro sequence spaces ofnon-absolute type which include the spaces 119888
0and 119888rdquo Soochow
Journal of Mathematics vol 31 no 1 pp 107ndash119 2005[3] C Aydın and F Basar ldquoSome new sequence spaces which in-
clude the spaces 119897infinand 119897119875rdquo Demonstratio Mathematica vol 38
no 3 pp 641ndash656 2005[4] E Malkowsky and E Savas ldquoMatrix transformations between
sequence spaces of generalized weightedmeansrdquoAppliedMath-ematics and Computation vol 147 no 2 pp 333ndash345 2004
[5] MMursaleen andAKNoman ldquoOn the spaces of120582-convergentand bounded sequencesrdquo Thai Journal of Mathematics vol 8no 2 pp 311ndash329 2010
[6] B Altay F Basar and M Mursaleen ldquoOn the Euler sequencespaces which include the spaces 119897
119901and 119897infinIrdquo Information Sci-
ences vol 176 no 10 pp 1450ndash1462 2006[7] F Basar and M Kirisci ldquoAlmost convergence and generalized
differencematrixrdquoComputers ampMathematics with Applicationsvol 61 no 3 pp 602ndash611 2011
[8] A Sonmez ldquoAlmost convergence and triple bandmatrixrdquoMath-ematical and Computer Modelling vol 57 no 9-10 pp 2393ndash2402 2013
[9] K Kayaduman andM S Sengonul ldquoThe space of Cesaro almostconvergent sequence and core theoremsrdquo Acta MathematicaScientia vol 6 pp 2265ndash2278 2012
[10] M Sengonul and K Kayaduman ldquoOn the Riesz almost conver-gent sequences spacerdquo Abstract and Applied Analysis vol 2012Article ID 691694 18 pages 2012
[11] G G Lorentz ldquoA contribution to the theory of divergent se-quencesrdquo Acta Mathematica vol 80 pp 167ndash190 1948
[12] I J Maddox ldquoParanormed sequence spaces generated by infi-nite matricesrdquo Proceedings of the Cambridge Philosophical Soci-ety vol 64 pp 335ndash340 1968
[13] A Karaisa and F Ozger ldquoAlmost difference sequence space de-rived by using a generalized weightedmeanrdquoActaMathematicaScientia In review
[14] C Aydın and F Basar ldquoOn the new sequence spaces which in-clude the spaces 119888
0and 119888rdquo Hokkaido Mathematical Journal vol
33 no 2 pp 383ndash398 2004[15] F Basar ldquoA note on the triangle limitation methodsrdquo Fırat Uni-
versitesi Muhendislik Bilimleri Dergisi vol 5 no 1 pp 113ndash1171993
[16] AM Al-Jarrah and EMalkowsky ldquoBK spaces bases and linearoperatorsrdquo in Proceedings of the 3rd International Conference onFunctional Analysis and ApproximationTheory vol 1 no 52 pp177ndash191 1998
[17] A Wilansky Summability through Functional Analysis vol 85North-Holland Amsterdam The Netherlands 1984
[18] J A Sıddıqı ldquoInfinite matrices summing every almost periodicsequencerdquo Pacific Journal of Mathematics vol 39 pp 235ndash2511971
[19] J P Duran ldquoInfinite matrices and almost-convergencerdquo Math-ematische Zeitschrift vol 128 pp 75ndash83 1972
[20] J P King ldquoAlmost summable sequencesrdquo Proceedings of theAmerican Mathematical Society vol 17 pp 1219ndash1225 1966
[21] F Basar and R Colak ldquoAlmost-conservativematrix transforma-tionsrdquo Turkish Journal of Mathematics vol 13 no 3 pp 91ndash1001989
[22] F Basar ldquo119891-conservative matrix sequencesrdquo Tamkang Journalof Mathematics vol 22 no 2 pp 205ndash212 1991
[23] E Ozturk ldquoOn strongly regular dual summability methodsrdquoCommunications de la Faculte des Sciences de lrsquoUniversite drsquoAn-kara vol 32 no 1 pp 1ndash5 1983
[24] F Basar and I Solak ldquoAlmost-coercivematrix transformationsrdquoRendiconti di Matematica e delle sue Applicazioni vol 11 no 2pp 249ndash256 1991
[25] F Basar ldquoStrongly-conservative sequence-to-series matrixtransformationsrdquo Erciyes Universitesi Fen Bilimleri Dergisi vol5 no 12 pp 888ndash893 1989
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Journal ofApplied Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ProbabilityandStatistics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Advances in
Mathematical Physics
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
Combinatorics
OperationsResearch
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
DecisionSciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Journal ofApplied Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ProbabilityandStatistics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Advances in
Mathematical Physics
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
Combinatorics
OperationsResearch
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
DecisionSciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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