Hindawi Publishing Corporation e Scientific World Journal Volume 2013, Article ID 948472, 7 pages http://dx.doi.org/10.1155/2013/948472 Research Article Some Endpoint Results for -Generalized Weak Contractive Multifunctions H. Alikhani, 1 D. Gopal, 2 M. A. Miandaragh, 1 Sh. Rezapour, 1 and N. Shahzad 3 1 Department of Mathematics, Azarbaijan University of Shahid Madani, Azarshahr, Tabriz, Iran 2 Department of Applied Mathematics and Humanities, S. V. National Institute of Technology, Surat, Gujarat 395007, India 3 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21859, Saudi Arabia Correspondence should be addressed to N. Shahzad; [email protected] Received 3 August 2013; Accepted 29 August 2013 Academic Editors: M. M. Cavalcanti and N. Herisanu Copyright © 2013 H. Alikhani et al. 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. We introduce -generalized weak contractive multifunctions and give some results about endpoints of the multifunctions. Also, we give some results about role of a point in the existence of endpoints. 1. Introduction Let (, ) be a metric space, () the collection of all nonempty bounded and closed subsets of , and the Hausdorff metric with respect to ; that is, (, ) = max{sup ∈ (, ), sup ∈ (, )} for all , ∈ (), where (, ) = inf ∈ (, ). Let : → 2 be a multifunction. An element ∈ is said to be a fixed point of whenever ∈. Also, an element ∈ is said to be an endpoint of whenever = {} [1]. We say that has the approximate endpoint property whenever inf ∈ sup ∈ (, ) = 0 [1]. Let : → be a mapping. We say that has the approximate endpoint property whenever inf ∈ (,) = 0 [1]. Also, the function : R → R is called upper semicontinuous whenever lim sup →∞ ( ) ≤ () for all sequences { } ≥1 with → [2]. In 2010, Amini-Harandi defined the concept of approximate endpoint property for multifunctions and proved the following result (see [1]). eorem 1. Let : [0, ∞) → [0, ∞) be an upper semicontin- uous function such that () < and lim inf →∞ ( − ()) > 0 for all >0, (, ) a complete metric space, and :→ () a multifunction satisfing (, ) ≤ ((, )) for all , ∈ . en has a unique endpoint if and only if has the approximate endpoint property. en Moradi and Khojasteh introduced the concept of generalized weak contractive multifunctions and improved eorem 1 by providing the following result [3]. eorem 2. Let : [0, ∞) → [0, ∞) be an upper semi- continuous function such that () < and lim inf →∞ ( − ()) > 0 for all >0, (, ) a complete metric space, and : → () a generalized weak contractive multifunction; that is, satisfies (, ) ≤ ((, )) for all , ∈ , where (, ) = max{(, ), (, ), (, ), ((, ) + (, ))/2}. en has a unique endpoint if and only if has the approximate endpoint property. In this paper, we introduce -generalized weak con- tractive multifunctions, and by adding some conditions to assumptions of the results, we give some results about endpoints of -generalized weak contractive multifunctions. In 2012, the technique of --contractive mappings was introduced by Samet et al. [4]. Later, some authors used it for some subjects in fixed point theory (see for example [5– 8]) or generalized it by using the method of --contractive multifunctions (see e.g., [9–12]). Let (, ) be a metric space and : 2 ×2 → [0, ∞) a mapping. A multifunction : → 2 is called -generalized weak contraction whenever there
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Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 948472 7 pageshttpdxdoiorg1011552013948472
Research ArticleSome Endpoint Results for 120573-Generalized Weak ContractiveMultifunctions
H Alikhani1 D Gopal2 M A Miandaragh1 Sh Rezapour1 and N Shahzad3
1 Department of Mathematics Azarbaijan University of Shahid Madani Azarshahr Tabriz Iran2Department of Applied Mathematics and Humanities S V National Institute of TechnologySurat Gujarat 395007 India
3 Department of Mathematics King Abdulaziz University PO Box 80203 Jeddah 21859 Saudi Arabia
Correspondence should be addressed to N Shahzad nshahzadkauedusa
Received 3 August 2013 Accepted 29 August 2013
Academic Editors M M Cavalcanti and N Herisanu
Copyright copy 2013 H Alikhani et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We introduce 120573-generalized weak contractive multifunctions and give some results about endpoints of the multifunctions Alsowe give some results about role of a point in the existence of endpoints
1 Introduction
Let (119883 119889) be a metric space 119862119861(119883) the collection of allnonempty bounded and closed subsets of 119883 and 119867 theHausdorff metric with respect to 119889 that is 119867(119860 119861) =maxsup
119909isin119860119889(119909 119861) sup
119910isin119861119889(119910 119860) for all 119860 119861 isin 119862119861(119883)
where 119889(119909 119861) = inf119910isin119861
119889(119909 119910) Let 119879 119883 rarr 2119883 bea multifunction An element 119909 isin 119883 is said to be a fixedpoint of 119879 whenever 119909 isin 119879119909 Also an element 119909 isin 119883 issaid to be an endpoint of 119879 whenever 119879119909 = 119909 [1] Wesay that 119879 has the approximate endpoint property wheneverinf119909isin119883
sup119910isin119879119909
119889(119909 119910) = 0 [1] Let 119891 119883 rarr 119883 bea mapping We say that 119891 has the approximate endpointproperty whenever inf
119909isin119883119889(119909 119891119909) = 0 [1] Also the function
119892 R rarr R is called upper semicontinuous wheneverlim sup
119899rarrinfin119892(120582119899) le 119892(120582) for all sequences 120582
119899119899ge1
with120582119899
rarr 120582 [2] In 2010 Amini-Harandi defined the conceptof approximate endpoint property for multifunctions andproved the following result (see [1])
Theorem1 Let120595 [0infin) rarr [0infin) be anupper semicontin-uous function such that120595(119905) lt 119905 and lim inf
119905rarrinfin(119905minus120595(119905)) gt 0
for all 119905 gt 0 (119883 119889) a complete metric space and 119879 119883 rarr119862119861(119883) a multifunction satisfing 119867(119879119909 119879119910) le 120595(119889(119909 119910)) forall 119909 119910 isin 119883 Then 119879 has a unique endpoint if and only if 119879 hasthe approximate endpoint property
Then Moradi and Khojasteh introduced the concept ofgeneralized weak contractive multifunctions and improvedTheorem 1 by providing the following result [3]
Theorem 2 Let 120595 [0infin) rarr [0infin) be an upper semi-continuous function such that 120595(119905) lt 119905 and lim inf
119905rarrinfin(119905 minus
120595(119905)) gt 0 for all 119905 gt 0 (119883 119889) a complete metric space and 119879 119883 rarr 119862119861(119883) a generalized weak contractive multifunctionthat is 119879 satisfies 119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883where119873(119909 119910) = max119889(119909 119910) 119889(119909 119879119909) 119889(119910 119879119910) (119889(119909 119879119910) +119889(119910 119879119909))2Then119879 has a unique endpoint if and only if119879 hasthe approximate endpoint property
In this paper we introduce 120573-generalized weak con-tractive multifunctions and by adding some conditionsto assumptions of the results we give some results aboutendpoints of 120573-generalized weak contractive multifunctionsIn 2012 the technique of 120572-120595-contractive mappings wasintroduced by Samet et al [4] Later some authors used itfor some subjects in fixed point theory (see for example [5ndash8]) or generalized it by using the method of 120573-120595-contractivemultifunctions (see eg [9ndash12])
Let (119883 119889) be a metric space and 120573 2119883 times 2119883 rarr
[0infin) a mapping A multifunction 119879 119883 rarr 2119883
is called 120573-generalized weak contraction whenever there
2 The Scientific World Journal
exists a nondecreasing upper semicontinuous function 120595 [0 +infin) rarr [0 +infin) such that 120595(119905) lt 119905 for all 119905 gt 0 and
120573 (119879119909 119879119910)119867 (119879119909 119879119910) le 120595 (119873 (119909 119910)) (1)
for all 119909 119910 isin 119883 We say that 119879 is 120573-admissible whenever120573(119860 119861) ge 1 implies that 120573(119879119909 119879119910) ge 1 for all 119909 isin 119860 and119910 isin 119861 where 119860 and 119861 are subsets of 119883 We say that 119879 hasthe property (119877)whenever for each convergent sequence 119909
119899
in 119883 with 119909119899
rarr 119909 and 120573(119879119909119899minus1
119879119909119899) ge 1 for all 119899 ge 1
we have 120573(119879119909119899 119879119909) ge 1 One can find idea of the property
(119877) for mappings in [13] We say that 119879 has the property (119870)whenever for each sequence 119909
119899 in119883with 120573(119879119909
119899minus1 119879119909119899) ge 1
for all 119899 ge 1 there exists a natural number 119896 such that120573(119879119909119898 119879119909119899) ge 1 for all119898 gt 119899 ge 119896 Finally we say that 119879 has
the property (119867) whenever for each 120576 gt 0 there exists 119911 isin 119883such that sup
119886isin119879119911119889(119911 119886) lt 120576 implies that for every 119909 isin 119883
there exists 119910 isin 119879119909 such that119867(119879119909 119879119910) = sup119887isin119879119910
119889(119910 119887) Amultifunction119879 119883 rarr 2119883 is called lower semicontinuous at1199090isin 119883 whenever for each sequence 119909
119899 in119883 with 119909
119899rarr 1199090
and every 119910 isin 1198791199090 there exists a sequence 119910
119899 in 119883 with
119910119899isin 119879119909119899for all 119899 ge 1 such that 119910
119899rarr 119910 [14]
2 Main Results
Now we are ready to state and prove our main results
Theorem 3 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a120573-admissible 120573-generalized weak contractive multifunctionwhich has the properties (119877) (119870) and (119867) Suppose that thereexist a subset 119860 of 119883 and 119909
0isin 119860 such that 120573(119860 119879119909
0) ge 1
Then 119879 has an endpoint if and only if 119879 has the approximateendpoint property
Proof It is clear that if 119879 has an endpoint then 119879 has theapproximate endpoint property Conversely suppose that 119879has the approximate endpoint property Choose 119860 sub 119883 and1199090isin 119860 such that 120573(119860 119879119909
0) ge 1 Since 119879 has the approximate
endpoint property for each 120576 gt 0 there exists 119911 isin 119883 such thatsup119886isin119879119911
119889(119911 119886) lt 120576 Now by using the condition (H) choose1199091isin 1198791199090such that 119867(119879119909
0 1198791199091) = sup
119886isin1198791199091
119889(1199091 119886) Also
choose 1199092isin 1198791199091such that 119867(119879119909
1 1198791199092) = sup
119886isin1198791199092
119889(1199092 119886)
and by continuing this process we find a sequence 119909119899 in 119883
such that 119909119899isin 119879119909119899minus1
and
119867(119879119909119899minus1
119879119909119899) = sup119886isin119879119909119899
119889 (119909119899 119886) (2)
for all 119899 ge 1 Since 120573(119860 1198791199090) ge 1 and 119879 is 120573-admissible
120573(1198791199090 1198791199091) ge 1 By using induction it is easy to see that
120573(119879119909119899minus1
119879119909119899) ge 1 for all 119899 ge 1 Thus we obtain
119889 (119909119899 119909119899+1
) le sup119886isin119879119909119899
119889 (119909119899 119886) = 119867 (119879119909
119899minus1 119879119909119899)
le 120573 (119879119909119899minus1
119879119909119899)119867 (119879119909
119899minus1 119879119909119899)
le 120595 (119873 (119909119899minus1
119909119899))
(3)
for all 119899 ge 1 If119873(119909119899minus1
119909119899) = 119889(119909
119899minus1 119909119899) then
119889 (119909119899 119909119899+1
) le 120595 (119889 (119909119899minus1
119909119899)) (4)
If119873(119909119899minus1
119909119899) = 119889(119909
119899minus1 119879119909119899minus1
) then
119889 (119909119899 119909119899+1
) le 120595 (119889 (119909119899minus1
119879119909119899minus1
)) le 120595 (119889 (119909119899minus1
119909119899)) (5)
If119873(119909119899minus1
119909119899) = 119889(119909
119899 119879119909119899) then
119889 (119909119899 119909119899+1
) le 120595 (119889 (119909119899 119879119909119899)) le 120595 (119889 (119909
119899 119909119899+1
)) (6)
and so 119889(119909119899 119909119899+1
) = 0 Thus 119889(119909119899 119909119899+1
) le 120595(119889(119909119899minus1
119909119899)) If
119873(119909119899minus1
119909119899) =
119889 (119909119899 119879119909119899minus1
) + 119889 (119909119899minus1
119879119909119899)
2
=119889 (119909119899minus1
119879119909119899)
2
119889 (119909119899minus1
119879119909119899)
2
le119889 (119909119899minus1
119909119899+1
)
2
le119889 (119909119899minus1
119909119899) + 119889 (119909
119899 119909119899+1
)
2
le max 119889 (119909119899minus1
119909119899) 119889 (119909
119899 119909119899+1
)
(7)
then 119889(119909119899 119909119899+1
) le 120595(119889(119909119899minus1
119909119899)) (other case implies that
119889(119909119899 119909119899+1
) = 0) Thus
119889 (119909119899 119909119899+1
) le 120595 (119889 (119909119899minus1
119909119899)) (8)
for all 119899 ge 1We claim that120595(0) = 0 If120595(0) gt 0 then1205952(0) ge120595(0) gt 0 because 120595 is nondecreasing On the other handsince 120595(119905) lt 119905 for all 119905 gt 0 we have 1205952(0) lt 120595(0) which is acontradiction Hence 120595(0) = 0 Let 119889
119899= 119889(119909
119899 119909119899+1
) for all 119899If there exists a natural number 119899
0such that 119889
1198990
= 0 then it iseasy to see that 119889
119899= 0 for all 119899 ge 119899
0 and so lim
119899rarrinfin119889119899= 0
Now suppose that 119889119899
= 0 for all 119899 In this case we have 119889119899le
120595(119889119899minus1
) lt 119889119899minus1
for all 119899 Hence 119889119899 is a decreasing sequence
and so there exists 119889 ge 0 such that lim119899rarrinfin
119889119899= 119889 If 119889 gt
0 then 119889119899gt 0 for all 119899 and so 119889
119899le 120595(119889
119899minus1) lt 119889
119899minus1for
all 119899 Since 120595 is upper and semicontinuous we obtain 119889 =lim119899rarrinfin
119889119899le lim
119899rarrinfin120595(119889119899minus1
) le 120595(lim119899rarrinfin
119889119899minus1
) = 120595(119889) lt119889 which is a contradiction Thus lim
119899rarrinfin119889119899= 0 Now we
prove that 119909119899 is a Cauchy sequence If 119909
119899 is not a Cauchy
sequence then there exist 120576 gt 0 and natural numbers 119898119896 119899119896
such that119898119896gt 119899119896ge 119896 and 119889(119909
119898119896
119909119899119896
) ge 120576 for all 119896 ge 1 Alsowe choose119898
119896as small as possible such that
119889 (119909119898119896minus1 119909119899119896
) lt 120576 (9)
The Scientific World Journal 3
Thus 120576 le 119889(119909119898119896
119909119899119896
) le 119889(119909119898119896
119909119898119896minus1) + 119889(119909
119898119896minus1 119909119899119896
) le119889119898119896minus1
+ 120576 for all 119896 Hence lim119896rarrinfin
119889(119909119898119896
119909119899119896
) = 120576 Since 119879has the property (119870) we obtain
119889 (119909119898119896
119909119899119896
) le 119889 (119909119898119896
119909119898119896+1) + 119889 (119909
119898119896+1 119909119899119896+1)
+ 119889 (119909119899119896+1 119909119899119896
)
le 119889119898119896
+ 119867(119879119909119898119896
119879119909119899119896
) + 119889119899119896
le 119889119898119896
+ 120573 (119879119909119898119896
119879119909119899119896
)
times 119867(119879119909119898119896
119879119909119899119896
) + 119889119899119896
le 119889119898119896
+ 120595 (119873(119909119898119896
119909119899119896
)) + 119889119899119896
(lowast)
for all 119896 Since lim119896rarrinfin
119889(119909119898119896
119909119899119896
) = 120576 lim119896rarrinfin
119873(119909119898119896
119909119899119896
) = 120576 In fact
119889 (119909119898119896
119909119899119896
)
le 119873(119909119898119896
119909119899119896
)
= max119889 (119909119898119896
119909119899119896
) 119889 (119909119898119896
119879119909119898119896
) 119889 (119909119899119896
119879119909119899119896
)
119889 (119909119898119896
119879119909119899119896
) + 119889 (119909119899119896
119879119909119898119896
)
2
le max119889 (119909119898119896
119909119899119896
) 119889 (119909119898119896
119909119898119896+1
) 119889 (119909119899119896
119909119899119896+1
)
119889 (119909119898119896
119909119899119896+1
) + 119889 (119909119899119896
119909119898119896+1
)
2
le max 119889 (119909119898119896
119909119899119896
) 119889 (119909119898119896
119909119898119896+1
) 119889 (119909119899119896
119909119899119896+1
)
(119889 (119909119898119896
119909119899119896
) + 119889 (119909119899119896
119909119899119896+1
)
+119889 (119909119899119896
119909119898119896
) + 119889 (119909119898119896
119909119898k+1
))
times (2)minus1
(10)
and so 120576 = lim119896rarr+infin
119889(119909119898119896
119909119899119896
) le lim119896rarrinfin
119873(119909119898119896
119909119899119896
) le 120576Since 120595 is upper semicontinuous by using (lowast) we obtain
120576 = lim119896rarrinfin
119889 (119909119898119896
119909119899119896
)
le lim119896rarrinfin
120595 (119873(119909119898119896
119909119899119896
)) le 120595 (120576) lt 120576(11)
which is a contradiction and so 119909119899 is a Cauchy sequence
Choose 119909lowast isin 119883 such that 119909119899rarr 119909lowast Now note that
119867(119909119899 119879119909119899) = max119889 (119909
119899 119879119909119899) sup119910isin119879119909
119899
119889 (119909119899 119910)
= 119867 (119879119909119899minus1
119879119909119899)
(12)
for all 119899 and so
119867(119909119899 119879119909119899) = 119867 (119879119909
119899minus1 119879119909119899)
le 120573 (119879119909119899minus1
119879119909119899)119867 (119879119909
119899minus1 119879119909119899)
le 120595 (119873 (119909119899minus1
119909119899))
le 120595 (119889 (119909119899minus1
119909119899)) le 119889 (119909
119899minus1 119909119899)
(13)
for all 119899 and so lim119899rarrinfin
119867(119909119899 119879119909119899) = 0 Since 119879 has the
property (119877) we obtain
119867(119909lowast 119879119909lowast) le 119889 (119909lowast 119909119899)
+ 119867 (119909119899 119879119909119899) + 119867 (119879119909
119899 119879119909lowast)
le 119889 (119909lowast 119909119899) + 119867 (119909
119899 119879119909119899)
+ 120573 (119879119909119899 119879119909lowast)119867 (119879119909
119899 119879119909lowast)
le 119889 (119909lowast 119909119899) + 119867 (119909
119899 119879119909119899)
+ 120595 (119873 (119909119899 119909lowast))
(14)
for all 119899 If119873(119909119899 119909lowast) = 119889(119909lowast 119879119909lowast) then we have
119867(119909lowast 119879119909lowast)
le 119889 (119909lowast 119909119899) + 119867 (119909
119899 119879119909119899) + 120595 (119867 (119909lowast 119879119909lowast))
(15)
for all 119899 This implies that 119867(119909lowast 119879119909lowast) le 120595(119867(119909lowast 119879119909lowast))and so
119867(119909lowast 119879119909lowast) = 0 (16)
If119873(119909119899 119909lowast) = 119889(119909
119899 119909lowast) or119873(119909
119899 119909lowast) le 119889(119909
119899 119909119899+1
) then it iseasy to see that119867(119909lowast 119879119909lowast) = 0 Thus 119909lowast is an endpoint of119879
Next example shows that a 120573-generalized weak con-tractive multifunction is not necessarily a generalized weakcontractive multifunction
Example 4 Let 119883 = R Define 119879 119883 rarr 119862119861(119883) by119879119909 = [119909 119909 + 2] for all 119909 isin 119883 Suppose that 120595 [0 +infin) rarr[0 +infin) is an arbitrary upper semicontinuous function suchthat 120595(119905) lt 119905 for all 119905 gt 0 If 119909 = 0 and 119910 = 2 then119867(119879119909 119879119910) = 119867([0 2] [2 4]) = 2 and119873(119909 119910) = 2 Hence
119867(119879119909 119879119910) = 2 ≩ 120595 (2) = 120595 (119873 (119909 119910)) (17)
Thus 119879 is not a generalized weak contractive multifunctionNow suppose that 120595(119905) = 1199052 for all 119905 ge 0 and define 120573
2119883 times 2119883 rarr [0infin) by 120573(119860 119861) = 12 for all subsets 119860 and 119861of119883 Then we have
120573 (119879119909 119879119910)119867 (119879119909 119879119910) =1
2119889 (119909 119910)
= 120595 (119889 (119909 119910)) = 120595 (119873 (119909 119910))
(18)
for all 119909 119910 isin R Thus 119879 is a 120573-generalized weak contractivemultifunction
4 The Scientific World Journal
Next example shows that there are multifunctions whichsatisfy the conditions of Theorem 3 while they are notgeneralized weak contractive multifunctions
Example 5 Let 119883 = [0 92] and let 119889(119909 119910) = |119909 minus 119910| Define119879 119883 rarr 119862119861(119883) by
119879119909 =
119909
2 0 le 119909 le 1
4119909 minus3
2 1 lt 119909 le
3
2
03
2lt 119909 le
9
2
(19)
If 119909 = 1 and 119910 = 32 then
119867(119879119909 119879119910) = 119867(1
2
9
2) = 4 gt 3
= 119873 (119909 119910) gt 120595 (119873 (119909 119910))
(20)
where 120595 [0 +infin) rarr [0 +infin) is an arbitrary uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Thus 119879 is not a generalized weak contractive multifunctionNow we show that119879 satisfies all conditions ofTheorem 3 Forthis aim define120595(119905) = 1199052 and 120573(119860 119861) = 1whenever119860 and119861are subsets of [0 1] and 120573(119860 119861) = 0 otherwise First supposethat 119909 notin [0 1] or that 119910 notin [0 1] If 119909 119910 isin (32 92] then119879119909 119879119910 sub [0 1] and 120573(119879119909 119879119910) = 1 But 119867(119879119909 119879119910) = 0 andso 120573(119879119909 119879119910)119867(119879119909 119879119910) le 120595(119873(119909 119910)) If 119909 isin (1 32] or 119910 isin(1 32] then119879119909 sube [0 1] or119879119910 sube [0 1] and so 120573(119879119909 119879119910) = 0Hence 120573(119879119909 119879119910)119867(119879119909 119879119910) le 120595(119873(119909 119910)) Now supposethat 119909 119910 isin [0 1] In this case we have 120573(119879119909 119879119910) ge1 119867(119879119909 119879119910) = 119867(1199092 1199102) = (12)119889(119909 119910) and119873(119909 119910) = max119889(119909 119910) 1199092 1199102 (119889(119909 1199102) + 119889(119910 1199092))2Thus 119889(119909 119910) le 119873(119909 119910) and so
120573 (119879119909 119879119910)119867 (119879119909 119879119910) =1
2119889 (119909 119910)
le 120595 (119889 (119909 119910)) le 120595 (119873 (119909 119910))
(21)
Therefore 119879 is a 120573-generalized weak contractive multifunc-tionNowwe show that119879 is120573-admissible If120573(119860 119861) ge 1 then119860 119861 sub [0 1] and so 119879119909 = 1199092 isin [0 1] and 119879119910 = 1199102 isin[0 1] for all 119909 isin 119860 and 119910 isin 119861 Thus 120573(119879119909 119879119910) ge 1 for all119909 isin 119860 and 119910 isin 119861 Now suppose 119860 = [0 12] and 119909
0= 14
Then 1198791199090= 18 isin [0 1] and [0 12] sub [0 1] Hence
120573(119860 1198791199090) ge 1 Now we show that 119879 satisfies the condition
(119867) First note that for each 120576 gt 0 there exists 119911 isin 119883 suchthat sup
119886isin119879119911119889(119911 119886) lt 120576 Now we show that for each 119909 isin 119883
there exists 119910 isin 119879119909 such that 119867(119879119909 119879119910) = sup119887isin119879119910
119889(119910 119887) If0 le 119909 le 1 then 119879119909 = 1199092 119879(1199092) = 1199094 and
119867(119879119909 119879 (119909
2)) = 119867(
119909
2
119909
4)
=119909
4= sup119887isin119879(1199092)
119889 (119909
2 119887)
(22)
Since for 1 lt 119909 le 32 we have 52 lt 4119909 minus (32) le 92119879(4119909 minus (32)) = 0 Thus
119867(119879119909 119879 (4119909 minus3
2)) = 119867(4119909 minus
3
2 0)
= 4119909 minus3
2= sup119887isin119879(4119909minus32)
119889(4119909 minus3
2 119887)
(23)
If 32 lt 119909 le 92 then 119879119909 = 0 and 119879(0) = 0 Hence
119867(119879119909 119879 (0)) = 119867 (0 0)
= 0 = sup119887isin119879(0)
119889 (0 119887) (24)
It is easy to check that 119879 satisfies the conditions (119877) and (119870)Note that 0 is the endpoint of 119879
Now we add an assumption to obtain uniqueness ofendpoint In this way we introduce a new notion Let 119883 bea set and 120573 2119883 times 2119883 rarr [0infin) a map We say that the set119883has the property (119866
120573) whenever 120573(119860 119861) ge 1 for all subsets 119860
and 119861 of119883 with 119860 sube 119861 or 119861 sube 119860
Corollary 6 Let (119883 119889) be a complete metric space 120573
2119883 times 2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883)a 120573-admissible 120573-generalized weak contractive multifunctionwhich has the properties (119877) (119870) and (119867) Suppose that thereexist a subset 119860 of119883 and 119909
0isin 119860 such that 120573(119860 119879119909
0) ge 1 If 119879
has the approximate endpoint property and119883 has the property(119866120573) then 119879 has a unique endpoint
Proof By using Theorem 3 119879 has a endpoint If 119879 hastwo distinct endpoints 119909lowast and 119910lowast then 120573(119879119909lowast 119879119910lowast) =120573(119909lowast 119910lowast) ge 1 because119883 has the property (119866
120573) Hence
119889 (119909lowast 119910lowast) le 119867 (119879119909lowast 119879119910lowast)
le 120573 (119879119909lowast 119879119910lowast)119867 (119879119909lowast 119879119910lowast)
le 120595 (119873 (119909lowast 119910lowast)) lt 119873 (119909lowast 119910lowast)
= 119889 (119909lowast 119910lowast)
(25)
which is a contradictionThus 119879 has a unique endpoint
In Example 5 119879 has a unique endpoint while119883 does hasnot the property (119866
120573) Also119879 has the property (119877) while119879 is
not lower semicontinuous To see this consider the sequence119909119899 defined by
119909119899=
1 minus1
119899119899 = 2119896
1 +1
119899119899 = 2119896 minus 1
(26)
for 119896 ge 1 and put 119910 = 12 and 1199090= 1 Then 119909
119899rarr 1 and 119910 isin
1198791199090= 12 Let 119910
119899 be an arbitrary sequence in119883 such that
119910119899isin 119879119909119899for all 119899 ge 1 Then 119910
2119896minus1isin 1198791199092119896minus1
and 1199102119896
isin 1198791199092119896
for all 119896 But 1199102119896minus1
= 41199092119896minus1
minus (32) for sufficiently large 119896and 119910
2119896= 11990921198962 for all 119896 since 119910
2119896minus1rarr 52 119910
119899999424999426999456 12 This
implies that 119879 is not lower semicontinuous
The Scientific World Journal 5
Corollary 7 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a 120573-admissiblemultifunctionwhich has the properties (119877) (119870) and(119867) Suppose that 119883 has the property (119866
120573) and there exist a
subset 119860 of X 1199090isin 119860 and 119896 isin [0 1) such that 120573(119860 119879119909
0) ge 1
and 120573(119879119909 119879119910)119867(119879119909 119879119910) le 119896119873(119909 119910) for all 119909 119910 isin 119883 Then119879 has a unique endpoint if and only if 119879 has the approximateendpoint property
Proof It is sufficient that we define 120595(119905) = 119896119905 for all 119905 ge 0Then Theorem 3 and Corollary 6 guarantee the result
It has been proved that lower semicontinuity of themulti-function 119879 and the property (119877) are independent conditions[9]We can replace lower semicontinuity of themultifunctioninstead of the property (119877) to obtain the next result Its proofis similar to the proof of Theorem 3
Theorem 8 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a lowersemicontinuous 120573-admissible 120573-generalized weak contractivemultifunction which has the properties (119870) and (119867) Supposethat there exist a subset 119860 of 119883 and 119909
0isin 119860 such that
120573(119860 1198791199090) ge 1 Then 119879 has the approximate endpoint property
if and only if 119879 has an endpoint
Corollary 9 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a lowersemicontinuous 120573-admissible 120573-generalized weak contractivemultifunction which has the properties (119870) and (119867) Supposethat there exist a subset 119860 of 119883 and 119909
0isin 119860 such that
120573(119860 1198791199090) ge 1 If 119879 has the approximate endpoint property
and 119883 has the property (119866120573) then 119879 has a unique endpoint
Corollary 10 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a 120573-admissiblemultifunctionwhich has the properties (119877) (119870) and(119867) Suppose that 119883 has the property (119866
120573) and there exist a
subset 119860 of 119883 1199090isin 119860 and 119896 isin [0 1) such that 120573(119860 119879119909
0) ge 1
and 120573(119879119909 119879119910)119867(119879119909 119879119910) le 119896119873(119909 119910) for all 119909 119910 isin 119883 If 119879 hasthe approximate endpoint property then Fix(119879) = End(119879) =119909
Proof If we put 120595(119905) = 119896119905 then by using Theorem 210 in[9]119879 has a fixed point Since119879 has the approximate endpointproperty by using Corollary 7 119879 has a unique endpoint such119909 Let 119910 isin Fix(119879) If 119879119909 = 119879119910 then 119910 = 119909 If 119879119909 = 119879119910 then120573(119879119909 119879119910) ge 1 because119883 has the property (119866
120573) Also we have
119889 (119909 119910) le 119867 (119909 119879119910) = 119867 (119879119909 119879119910)
le 120573 (119879119909 119879119910)119867 (119879119909 119879119910)
le 119896119873 (119909 119910)
(27)
But 119873(119909 119910) = max119889(119909 119910) 119889(119909 119879119909) 119889(119910 119879119910) (119889(119909 119879119910) +119889(119910 119879119909))2 = 119889(119909 119910) Thus 119889(119909 119910) = 0 and so Fix(119879) =End(119879) = 119909
Next corollary shows us the role of a point in the existenceof endpoints
Corollary 11 Let (119883 119889) be a complete metric space 119909lowast isin 119883 afixed element and 119879 119883 rarr 119862119861(119883) a multifunction such that119879 has the property (119867) and 119909lowast isin 119879119909cap119879119910 for all subsets119860 and119861 of119883 with 119909lowast isin 119860 cap 119861 and all 119909 isin 119860 and 119910 isin 119861 Assume that119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119909lowast isin 119879119909 cap 119879119910where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Suppose that there exist a subset 119860
0of 119883 and 119909
0isin 1198600such
that 119909lowast isin 1198600cap1198791199090 Assume that for each convergent sequence
119909119899 in 119883 with 119909
119899rarr 119909 and 119909lowast isin 119879119909
119899minus1cap 119879119909119899 for all 119899 ge 1
one has 119909lowast isin 119879119909119899cap 119879119909 Also for each sequence 119909
119899 in 119883
with 119909lowast isin 119879119909119899minus1
cap 119879119909119899for all 119899 ge 1 there exists a natural
number 119896 such that 119909lowast isin 119879119909119898cap119879119909119899for all119898 gt 119899 ge 119896 Then 119879
has an endpoint if and only if 119879 has the approximate endpointproperty
Proof It is sufficient we define 120573 2119883 times 2119883 rarr [0infin) by120573(119860 119861) = 1 whenever 119909lowast isin 119860cap119861 and 120573(119860 119861) = 0 otherwiseand then we use Theorem 3
Corollary 12 Let (119883 119889) be a complete metric space xlowast isin 119883a fixed element and 119879 119883 rarr 119862119861(119883) a lower semicontinuousmultifunction such that 119879 has the property (119867) and 119909lowast isin 119879119909cap119879119910 for all subsets 119860 and 119861 of119883 with 119909lowast isin 119860 cap 119861 and all 119909 isin 119860and 119910 isin 119861 Assume that
119867(119879119909 119879119910) le 120595 (119873 (119909 119910)) (28)
for all 119909 119910 isin 119883 with 119909lowast isin 119879119909 cap 119879119910 where 120595 [0 +infin) rarr[0 +infin) is a nondecreasing upper semicontinuous functionsuch that 120595(119905) lt 119905 for all 119905 gt 0 Suppose that there exist asubset 119860
0of119883 and 119909
0isin 1198600such that 119909lowast isin 119860
0cap 1198791199090 Assume
that for each convergent sequence 119909119899 in119883 with 119909
119899rarr 119909 and
119909lowast isin 119879119909119899minus1
cap119879119909119899for all 119899 ge 1 we have 119909lowast isin 119879119909
119899cap119879119909 Then 119879
has an endpoint if and only if 119879 has the approximate endpointproperty
Proof It is sufficient to define 120573 2119883 times 2119883 rarr [0infin) by120573(119860 119861) = 1 whenever 119909lowast isin 119860cap119861 and 120573(119860 119861) = 0 otherwiseand then we use Theorem 8
Let (119883 119889 le) be an ordered metric space Define the order⪯ on arbitrary subsets 119860 and 119861 of 119883 by 119860 ⪯ 119861 if and only iffor each 119886 isin 119860 there exists 119887 isin 119861 such that 119886 le 119887 It is easy tocheck that (119862119861(119883) ⪯) is a partially ordered set
Theorem 13 Let (119883 119889 le) be a complete ordered metric spaceand 119879 a closed and bounded valued multifunction on 119883 suchthat 119879 has the property (119867) and 119879119909 ⪯ 119879119910 for all subsets 119860and 119861 of 119883 with 119860 ⪯ 119861 and all 119909 isin 119860 and 119910 isin 119861 Assumethat 119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119879119909 ⪯ 119879119910where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Suppose that there exist a subset 119860
0of 119883 and 119909
0isin 1198600such
that 1198600
⪯ 1198791199090 Assume that for each convergent sequence
119909119899 in 119883 with 119909
119899rarr 119909 and 119879119909
119899minus1⪯ 119879119909
119899 for all 119899 ge 1
one has 119879119909119899
⪯ 119879119909 Also for each sequence 119909119899 in 119883 with
119879119909119899minus1
⪯ 119879119909119899for all 119899 ge 1 there exists a natural number 119896 such
that 119879119909119898⪯ 119879119909119899for all 119898 gt 119899 ge 119896 Then 119879 has an endpoint if
and only if 119879 has the approximate endpoint property
6 The Scientific World Journal
Proof Define 120573(119860 119861) = 1 whenever 119860 ⪯ 119861 and 120573(119860 119861) = 0otherwise and then we use Theorem 3
Corollary 14 Let (119883 119889 le) be a complete ordered metric spaceand 119879 a closed and bounded valued multifunction on 119883 suchthat 119879 has the property (119867) and 119879119909 ⪯ 119879119910 for all subsets 119860and 119861 of 119883 with 119860 ⪯ 119861 all 119909 isin 119860 and 119910 isin 119861 Assume that119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119879119909 ⪯ 119879119910where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Suppose that there exist a subset 119860
0of 119883 and 119909
0isin 1198600such
that 1198600
⪯ 1198791199090 Assume that for each convergent sequence
119909119899 in 119883 with 119909
119899rarr 119909 and 119879119909
119899minus1⪯ 119879119909
119899 for all 119899 ge 1
one has 119879119909119899
⪯ 119879119909 Also for each sequence 119909119899 in 119883 with
119879119909119899minus1
⪯ 119879119909119899for all 119899 ge 1 there exists a natural number 119896 such
that 119879119909119898⪯ 119879119909119899for all 119898 gt 119899 ge 119896 If 119879 has the approximate
endpoint property and 119860 ⪯ 119861 for all subsets119860 and 119861 of119883 with119860 sube 119861 or 119861 sube 119860 then 119879 has a unique endpoint
Proof Define 120573(119860 119861) = 1 whenever 119860 ⪯ 119861 and 120573(119860 119861) = 0otherwise and then we use Corollary 6
Let (119883 119889) be a metric space and 119879 119883 rarr 2119883
a multifunction We say that 119879 is an 119867119878-multifunctionwhenever for each 119909 isin 119883 there exists 119910 isin 119879119909 such that119867(119879119909 119879119910) = sup
119887isin119879y119889(119910 119887) It is obvious that each 119867119878-multifunction is an multifunction which has the property(119867) Thus one can conclude similar results to above ones for119867119878-multifunctions Here we provide some ones Althoughby considering 119867119878-multifunction we restrict ourselves weobtain strange results with respect to above ones One canprove the following by reading exactly the proofs of similarabove results
Theorem 15 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a 120573-admissible 120573-generalized weak contractive 119867119878-multifunctionwhich has the properties (119877) and (119870) Suppose that there exist asubset119860 of119883 and 119909
0isin 119860 such that 120573(119860 119879119909
0) ge 1 Then 119879 has
an endpoint and so 119879 has the approximate endpoint property
Theorem 16 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) alower semicontinuous 120573-admissible and 120573-generalized weakcontractive 119867119878-multifunction which has the property (119870)Suppose that there exist a subset 119860 of 119883 and 119909
0isin 119860 such
that 120573(119860 1198791199090) ge 1 Then 119879 has an endpoint and so 119879 has
the approximate endpoint property
The next result is a consequence of Theorem 15
Corollary 17 Let (119883 119889) be a complete metric space 119909lowast isin 119883 afixed element and 119879 119883 rarr 119862119861(119883) an 119867119878-multifunctionsuch that 119909lowast isin 119879119909 cap 119879119910 for all subsets 119860 and 119861 of 119883 with119909lowast isin 119860 cap 119861 all 119909 isin 119860 and 119910 isin 119861 Assume that 119867(119879119909 119879119910) le120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119909lowast isin 119879119909 cap 119879119910 where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing upper semicontinuousfunction such that120595(119905) lt 119905 for all 119905 gt 0 Suppose that there exista subset119860
0of119883 and 119909
0isin 1198600such that 119909lowast isin 119860
0cap1198791199090 Assume
that for each convergent sequence 119909119899 in119883 with 119909
119899rarr 119909 and
119909lowast isin 119879119909119899minus1
cap119879119909119899for all 119899 ge 1 one has 119909lowast isin 119879119909
119899cap119879119909 Also for
each sequence 119909119899 in 119883 with 119909lowast isin 119879119909
119899minus1cap 119879119909119899for all 119899 ge 1
there exists a natural number 119896 such that 119909lowast isin 119879119909119898cap 119879119909119899
for all 119898 gt 119899 ge 119896 Then 119879 has an endpoint and so 119879 has theapproximate endpoint property
The next result is a consequence of Theorem 16
Corollary 18 Let (119883 119889 le) be a complete ordered metric spaceand 119879 a closed and bounded valued lower semicontinuous119867119878-multifunction on 119883 such that 119879119909 ⪯ 119879119910 for all subsets 119860 and119861 of 119883 with 119860 ⪯ 119861 all 119909 isin 119860 and 119910 isin 119861 Assume that119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119879119909 ⪯ 119879119910where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Suppose that there exist a subset 119860
0of 119883 and 119909
0isin 1198600such
that 1198600⪯ 1198791199090 Assume that for each sequence 119909
119899 in 119883 with
119879119909119899minus1
⪯ 119879119909119899for all 119899 ge 1 there exists a natural number 119896 such
that 119879119909119898
⪯ 119879119909119899for all 119898 gt 119899 ge 119896 Then 119879 has an endpoint
and so 119879 has the approximate endpoint property
Acknowledgments
This work was completed while the second author (DrGopal) was visiting the Azarbaijan University of ShahidMadani Azarshahr Tabriz Iran during the summer of 2012He thanks Professor Sh Rezapour and the University fortheir hospitality and support The second author gratefullyacknowledges the support from the CSIR govternment ofIndia Grant no-25(0215)13EMR-II
References
[1] A Amini-Harandi ldquoEndpoints of set-valued contractions inmetric spacesrdquo Nonlinear Analysis Theory Methods and Appli-cations vol 72 no 1 pp 132ndash134 2010
[2] R P Agarwal D OrsquoRegan and D R Sahu Fixed Point Theoryfor Lipschitzian-TypeMappings with Applications Springer NewYork NY USA 2009
[3] S Moradi and F Khojasteh ldquoEndpoints of multi-valued gener-alized weak contraction mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 74 no 6 pp 2170ndash2174 2011
[4] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012
[5] D BaleanuHMohammadi and Sh Rezapour ldquoSome existenceresults on nonlinear fractional differential equationsrdquo Philo-sophical Transactions of the Royal Society vol 371 Article ID20120144 2013
[6] M Jleli and B Samet ldquoBest proximity points for 120572-120595-proximalcontractive type mappings and applicationsrdquo Bulletin des Sci-ences Mathematiques 2013
[7] M A Miandaragh M Postolache and Sh Rezapour ldquoSomeapproximate fixed point results for generalized-contractivemappings rdquo To appear in Scientific Bulletin-UniversityPolitehnica of Bucharest A
[8] Sh Rezapour and J H Asl ldquoA simple method for obtainingcoupled fixed points of 120572-120595-contractive type mappingsrdquo Inter-national Journal of Analysis Article ID 438029 7 pages 2013
The Scientific World Journal 7
[9] H Alikhani V Rakocevic Sh Rezapour and N ShahzadldquoFixed points of proximinal valued 120573-120595-contractive multifunc-tionsrdquo To appear in Journal of Nonlinear and Convex Analysis
[10] H Alikhani Sh Rezapour and N Shahzad ldquoFixed points of anew type contractive mappings and multifunctionsrdquo To appearin Filomat
[11] S M A Aleomraninejad Sh Rezapour and N Shahzad ldquoOnfixed points of 120572-120595-contractive multifunctionsrdquo Fixed PointTheory and Applications vol 2012 article 212 8 pages 2012
[12] B Mohammadi Sh Rezapour and N Shahzad ldquoSome resultson fixed points of 120572-120595-ciric generalized multifunctionsrdquo FixedPoint Theory and Applications vol 2013 article 24 2013
[13] S M A Aleomraninejad Sh Rezapour andN Shahzad ldquoSomefixed point results on ametric space with a graphrdquoTopology andIts Applications vol 159 no 3 pp 659ndash663 2012
[14] J M Borwein and A S Lewis Convex Analysis and NonlinearOptimization Theory and Examples Springer New York NYUSA 2000
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2 The Scientific World Journal
exists a nondecreasing upper semicontinuous function 120595 [0 +infin) rarr [0 +infin) such that 120595(119905) lt 119905 for all 119905 gt 0 and
120573 (119879119909 119879119910)119867 (119879119909 119879119910) le 120595 (119873 (119909 119910)) (1)
for all 119909 119910 isin 119883 We say that 119879 is 120573-admissible whenever120573(119860 119861) ge 1 implies that 120573(119879119909 119879119910) ge 1 for all 119909 isin 119860 and119910 isin 119861 where 119860 and 119861 are subsets of 119883 We say that 119879 hasthe property (119877)whenever for each convergent sequence 119909
119899
in 119883 with 119909119899
rarr 119909 and 120573(119879119909119899minus1
119879119909119899) ge 1 for all 119899 ge 1
we have 120573(119879119909119899 119879119909) ge 1 One can find idea of the property
(119877) for mappings in [13] We say that 119879 has the property (119870)whenever for each sequence 119909
119899 in119883with 120573(119879119909
119899minus1 119879119909119899) ge 1
for all 119899 ge 1 there exists a natural number 119896 such that120573(119879119909119898 119879119909119899) ge 1 for all119898 gt 119899 ge 119896 Finally we say that 119879 has
the property (119867) whenever for each 120576 gt 0 there exists 119911 isin 119883such that sup
119886isin119879119911119889(119911 119886) lt 120576 implies that for every 119909 isin 119883
there exists 119910 isin 119879119909 such that119867(119879119909 119879119910) = sup119887isin119879119910
119889(119910 119887) Amultifunction119879 119883 rarr 2119883 is called lower semicontinuous at1199090isin 119883 whenever for each sequence 119909
119899 in119883 with 119909
119899rarr 1199090
and every 119910 isin 1198791199090 there exists a sequence 119910
119899 in 119883 with
119910119899isin 119879119909119899for all 119899 ge 1 such that 119910
119899rarr 119910 [14]
2 Main Results
Now we are ready to state and prove our main results
Theorem 3 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a120573-admissible 120573-generalized weak contractive multifunctionwhich has the properties (119877) (119870) and (119867) Suppose that thereexist a subset 119860 of 119883 and 119909
0isin 119860 such that 120573(119860 119879119909
0) ge 1
Then 119879 has an endpoint if and only if 119879 has the approximateendpoint property
Proof It is clear that if 119879 has an endpoint then 119879 has theapproximate endpoint property Conversely suppose that 119879has the approximate endpoint property Choose 119860 sub 119883 and1199090isin 119860 such that 120573(119860 119879119909
0) ge 1 Since 119879 has the approximate
endpoint property for each 120576 gt 0 there exists 119911 isin 119883 such thatsup119886isin119879119911
119889(119911 119886) lt 120576 Now by using the condition (H) choose1199091isin 1198791199090such that 119867(119879119909
0 1198791199091) = sup
119886isin1198791199091
119889(1199091 119886) Also
choose 1199092isin 1198791199091such that 119867(119879119909
1 1198791199092) = sup
119886isin1198791199092
119889(1199092 119886)
and by continuing this process we find a sequence 119909119899 in 119883
such that 119909119899isin 119879119909119899minus1
and
119867(119879119909119899minus1
119879119909119899) = sup119886isin119879119909119899
119889 (119909119899 119886) (2)
for all 119899 ge 1 Since 120573(119860 1198791199090) ge 1 and 119879 is 120573-admissible
120573(1198791199090 1198791199091) ge 1 By using induction it is easy to see that
120573(119879119909119899minus1
119879119909119899) ge 1 for all 119899 ge 1 Thus we obtain
119889 (119909119899 119909119899+1
) le sup119886isin119879119909119899
119889 (119909119899 119886) = 119867 (119879119909
119899minus1 119879119909119899)
le 120573 (119879119909119899minus1
119879119909119899)119867 (119879119909
119899minus1 119879119909119899)
le 120595 (119873 (119909119899minus1
119909119899))
(3)
for all 119899 ge 1 If119873(119909119899minus1
119909119899) = 119889(119909
119899minus1 119909119899) then
119889 (119909119899 119909119899+1
) le 120595 (119889 (119909119899minus1
119909119899)) (4)
If119873(119909119899minus1
119909119899) = 119889(119909
119899minus1 119879119909119899minus1
) then
119889 (119909119899 119909119899+1
) le 120595 (119889 (119909119899minus1
119879119909119899minus1
)) le 120595 (119889 (119909119899minus1
119909119899)) (5)
If119873(119909119899minus1
119909119899) = 119889(119909
119899 119879119909119899) then
119889 (119909119899 119909119899+1
) le 120595 (119889 (119909119899 119879119909119899)) le 120595 (119889 (119909
119899 119909119899+1
)) (6)
and so 119889(119909119899 119909119899+1
) = 0 Thus 119889(119909119899 119909119899+1
) le 120595(119889(119909119899minus1
119909119899)) If
119873(119909119899minus1
119909119899) =
119889 (119909119899 119879119909119899minus1
) + 119889 (119909119899minus1
119879119909119899)
2
=119889 (119909119899minus1
119879119909119899)
2
119889 (119909119899minus1
119879119909119899)
2
le119889 (119909119899minus1
119909119899+1
)
2
le119889 (119909119899minus1
119909119899) + 119889 (119909
119899 119909119899+1
)
2
le max 119889 (119909119899minus1
119909119899) 119889 (119909
119899 119909119899+1
)
(7)
then 119889(119909119899 119909119899+1
) le 120595(119889(119909119899minus1
119909119899)) (other case implies that
119889(119909119899 119909119899+1
) = 0) Thus
119889 (119909119899 119909119899+1
) le 120595 (119889 (119909119899minus1
119909119899)) (8)
for all 119899 ge 1We claim that120595(0) = 0 If120595(0) gt 0 then1205952(0) ge120595(0) gt 0 because 120595 is nondecreasing On the other handsince 120595(119905) lt 119905 for all 119905 gt 0 we have 1205952(0) lt 120595(0) which is acontradiction Hence 120595(0) = 0 Let 119889
119899= 119889(119909
119899 119909119899+1
) for all 119899If there exists a natural number 119899
0such that 119889
1198990
= 0 then it iseasy to see that 119889
119899= 0 for all 119899 ge 119899
0 and so lim
119899rarrinfin119889119899= 0
Now suppose that 119889119899
= 0 for all 119899 In this case we have 119889119899le
120595(119889119899minus1
) lt 119889119899minus1
for all 119899 Hence 119889119899 is a decreasing sequence
and so there exists 119889 ge 0 such that lim119899rarrinfin
119889119899= 119889 If 119889 gt
0 then 119889119899gt 0 for all 119899 and so 119889
119899le 120595(119889
119899minus1) lt 119889
119899minus1for
all 119899 Since 120595 is upper and semicontinuous we obtain 119889 =lim119899rarrinfin
119889119899le lim
119899rarrinfin120595(119889119899minus1
) le 120595(lim119899rarrinfin
119889119899minus1
) = 120595(119889) lt119889 which is a contradiction Thus lim
119899rarrinfin119889119899= 0 Now we
prove that 119909119899 is a Cauchy sequence If 119909
119899 is not a Cauchy
sequence then there exist 120576 gt 0 and natural numbers 119898119896 119899119896
such that119898119896gt 119899119896ge 119896 and 119889(119909
119898119896
119909119899119896
) ge 120576 for all 119896 ge 1 Alsowe choose119898
119896as small as possible such that
119889 (119909119898119896minus1 119909119899119896
) lt 120576 (9)
The Scientific World Journal 3
Thus 120576 le 119889(119909119898119896
119909119899119896
) le 119889(119909119898119896
119909119898119896minus1) + 119889(119909
119898119896minus1 119909119899119896
) le119889119898119896minus1
+ 120576 for all 119896 Hence lim119896rarrinfin
119889(119909119898119896
119909119899119896
) = 120576 Since 119879has the property (119870) we obtain
119889 (119909119898119896
119909119899119896
) le 119889 (119909119898119896
119909119898119896+1) + 119889 (119909
119898119896+1 119909119899119896+1)
+ 119889 (119909119899119896+1 119909119899119896
)
le 119889119898119896
+ 119867(119879119909119898119896
119879119909119899119896
) + 119889119899119896
le 119889119898119896
+ 120573 (119879119909119898119896
119879119909119899119896
)
times 119867(119879119909119898119896
119879119909119899119896
) + 119889119899119896
le 119889119898119896
+ 120595 (119873(119909119898119896
119909119899119896
)) + 119889119899119896
(lowast)
for all 119896 Since lim119896rarrinfin
119889(119909119898119896
119909119899119896
) = 120576 lim119896rarrinfin
119873(119909119898119896
119909119899119896
) = 120576 In fact
119889 (119909119898119896
119909119899119896
)
le 119873(119909119898119896
119909119899119896
)
= max119889 (119909119898119896
119909119899119896
) 119889 (119909119898119896
119879119909119898119896
) 119889 (119909119899119896
119879119909119899119896
)
119889 (119909119898119896
119879119909119899119896
) + 119889 (119909119899119896
119879119909119898119896
)
2
le max119889 (119909119898119896
119909119899119896
) 119889 (119909119898119896
119909119898119896+1
) 119889 (119909119899119896
119909119899119896+1
)
119889 (119909119898119896
119909119899119896+1
) + 119889 (119909119899119896
119909119898119896+1
)
2
le max 119889 (119909119898119896
119909119899119896
) 119889 (119909119898119896
119909119898119896+1
) 119889 (119909119899119896
119909119899119896+1
)
(119889 (119909119898119896
119909119899119896
) + 119889 (119909119899119896
119909119899119896+1
)
+119889 (119909119899119896
119909119898119896
) + 119889 (119909119898119896
119909119898k+1
))
times (2)minus1
(10)
and so 120576 = lim119896rarr+infin
119889(119909119898119896
119909119899119896
) le lim119896rarrinfin
119873(119909119898119896
119909119899119896
) le 120576Since 120595 is upper semicontinuous by using (lowast) we obtain
120576 = lim119896rarrinfin
119889 (119909119898119896
119909119899119896
)
le lim119896rarrinfin
120595 (119873(119909119898119896
119909119899119896
)) le 120595 (120576) lt 120576(11)
which is a contradiction and so 119909119899 is a Cauchy sequence
Choose 119909lowast isin 119883 such that 119909119899rarr 119909lowast Now note that
119867(119909119899 119879119909119899) = max119889 (119909
119899 119879119909119899) sup119910isin119879119909
119899
119889 (119909119899 119910)
= 119867 (119879119909119899minus1
119879119909119899)
(12)
for all 119899 and so
119867(119909119899 119879119909119899) = 119867 (119879119909
119899minus1 119879119909119899)
le 120573 (119879119909119899minus1
119879119909119899)119867 (119879119909
119899minus1 119879119909119899)
le 120595 (119873 (119909119899minus1
119909119899))
le 120595 (119889 (119909119899minus1
119909119899)) le 119889 (119909
119899minus1 119909119899)
(13)
for all 119899 and so lim119899rarrinfin
119867(119909119899 119879119909119899) = 0 Since 119879 has the
property (119877) we obtain
119867(119909lowast 119879119909lowast) le 119889 (119909lowast 119909119899)
+ 119867 (119909119899 119879119909119899) + 119867 (119879119909
119899 119879119909lowast)
le 119889 (119909lowast 119909119899) + 119867 (119909
119899 119879119909119899)
+ 120573 (119879119909119899 119879119909lowast)119867 (119879119909
119899 119879119909lowast)
le 119889 (119909lowast 119909119899) + 119867 (119909
119899 119879119909119899)
+ 120595 (119873 (119909119899 119909lowast))
(14)
for all 119899 If119873(119909119899 119909lowast) = 119889(119909lowast 119879119909lowast) then we have
119867(119909lowast 119879119909lowast)
le 119889 (119909lowast 119909119899) + 119867 (119909
119899 119879119909119899) + 120595 (119867 (119909lowast 119879119909lowast))
(15)
for all 119899 This implies that 119867(119909lowast 119879119909lowast) le 120595(119867(119909lowast 119879119909lowast))and so
119867(119909lowast 119879119909lowast) = 0 (16)
If119873(119909119899 119909lowast) = 119889(119909
119899 119909lowast) or119873(119909
119899 119909lowast) le 119889(119909
119899 119909119899+1
) then it iseasy to see that119867(119909lowast 119879119909lowast) = 0 Thus 119909lowast is an endpoint of119879
Next example shows that a 120573-generalized weak con-tractive multifunction is not necessarily a generalized weakcontractive multifunction
Example 4 Let 119883 = R Define 119879 119883 rarr 119862119861(119883) by119879119909 = [119909 119909 + 2] for all 119909 isin 119883 Suppose that 120595 [0 +infin) rarr[0 +infin) is an arbitrary upper semicontinuous function suchthat 120595(119905) lt 119905 for all 119905 gt 0 If 119909 = 0 and 119910 = 2 then119867(119879119909 119879119910) = 119867([0 2] [2 4]) = 2 and119873(119909 119910) = 2 Hence
119867(119879119909 119879119910) = 2 ≩ 120595 (2) = 120595 (119873 (119909 119910)) (17)
Thus 119879 is not a generalized weak contractive multifunctionNow suppose that 120595(119905) = 1199052 for all 119905 ge 0 and define 120573
2119883 times 2119883 rarr [0infin) by 120573(119860 119861) = 12 for all subsets 119860 and 119861of119883 Then we have
120573 (119879119909 119879119910)119867 (119879119909 119879119910) =1
2119889 (119909 119910)
= 120595 (119889 (119909 119910)) = 120595 (119873 (119909 119910))
(18)
for all 119909 119910 isin R Thus 119879 is a 120573-generalized weak contractivemultifunction
4 The Scientific World Journal
Next example shows that there are multifunctions whichsatisfy the conditions of Theorem 3 while they are notgeneralized weak contractive multifunctions
Example 5 Let 119883 = [0 92] and let 119889(119909 119910) = |119909 minus 119910| Define119879 119883 rarr 119862119861(119883) by
119879119909 =
119909
2 0 le 119909 le 1
4119909 minus3
2 1 lt 119909 le
3
2
03
2lt 119909 le
9
2
(19)
If 119909 = 1 and 119910 = 32 then
119867(119879119909 119879119910) = 119867(1
2
9
2) = 4 gt 3
= 119873 (119909 119910) gt 120595 (119873 (119909 119910))
(20)
where 120595 [0 +infin) rarr [0 +infin) is an arbitrary uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Thus 119879 is not a generalized weak contractive multifunctionNow we show that119879 satisfies all conditions ofTheorem 3 Forthis aim define120595(119905) = 1199052 and 120573(119860 119861) = 1whenever119860 and119861are subsets of [0 1] and 120573(119860 119861) = 0 otherwise First supposethat 119909 notin [0 1] or that 119910 notin [0 1] If 119909 119910 isin (32 92] then119879119909 119879119910 sub [0 1] and 120573(119879119909 119879119910) = 1 But 119867(119879119909 119879119910) = 0 andso 120573(119879119909 119879119910)119867(119879119909 119879119910) le 120595(119873(119909 119910)) If 119909 isin (1 32] or 119910 isin(1 32] then119879119909 sube [0 1] or119879119910 sube [0 1] and so 120573(119879119909 119879119910) = 0Hence 120573(119879119909 119879119910)119867(119879119909 119879119910) le 120595(119873(119909 119910)) Now supposethat 119909 119910 isin [0 1] In this case we have 120573(119879119909 119879119910) ge1 119867(119879119909 119879119910) = 119867(1199092 1199102) = (12)119889(119909 119910) and119873(119909 119910) = max119889(119909 119910) 1199092 1199102 (119889(119909 1199102) + 119889(119910 1199092))2Thus 119889(119909 119910) le 119873(119909 119910) and so
120573 (119879119909 119879119910)119867 (119879119909 119879119910) =1
2119889 (119909 119910)
le 120595 (119889 (119909 119910)) le 120595 (119873 (119909 119910))
(21)
Therefore 119879 is a 120573-generalized weak contractive multifunc-tionNowwe show that119879 is120573-admissible If120573(119860 119861) ge 1 then119860 119861 sub [0 1] and so 119879119909 = 1199092 isin [0 1] and 119879119910 = 1199102 isin[0 1] for all 119909 isin 119860 and 119910 isin 119861 Thus 120573(119879119909 119879119910) ge 1 for all119909 isin 119860 and 119910 isin 119861 Now suppose 119860 = [0 12] and 119909
0= 14
Then 1198791199090= 18 isin [0 1] and [0 12] sub [0 1] Hence
120573(119860 1198791199090) ge 1 Now we show that 119879 satisfies the condition
(119867) First note that for each 120576 gt 0 there exists 119911 isin 119883 suchthat sup
119886isin119879119911119889(119911 119886) lt 120576 Now we show that for each 119909 isin 119883
there exists 119910 isin 119879119909 such that 119867(119879119909 119879119910) = sup119887isin119879119910
119889(119910 119887) If0 le 119909 le 1 then 119879119909 = 1199092 119879(1199092) = 1199094 and
119867(119879119909 119879 (119909
2)) = 119867(
119909
2
119909
4)
=119909
4= sup119887isin119879(1199092)
119889 (119909
2 119887)
(22)
Since for 1 lt 119909 le 32 we have 52 lt 4119909 minus (32) le 92119879(4119909 minus (32)) = 0 Thus
119867(119879119909 119879 (4119909 minus3
2)) = 119867(4119909 minus
3
2 0)
= 4119909 minus3
2= sup119887isin119879(4119909minus32)
119889(4119909 minus3
2 119887)
(23)
If 32 lt 119909 le 92 then 119879119909 = 0 and 119879(0) = 0 Hence
119867(119879119909 119879 (0)) = 119867 (0 0)
= 0 = sup119887isin119879(0)
119889 (0 119887) (24)
It is easy to check that 119879 satisfies the conditions (119877) and (119870)Note that 0 is the endpoint of 119879
Now we add an assumption to obtain uniqueness ofendpoint In this way we introduce a new notion Let 119883 bea set and 120573 2119883 times 2119883 rarr [0infin) a map We say that the set119883has the property (119866
120573) whenever 120573(119860 119861) ge 1 for all subsets 119860
and 119861 of119883 with 119860 sube 119861 or 119861 sube 119860
Corollary 6 Let (119883 119889) be a complete metric space 120573
2119883 times 2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883)a 120573-admissible 120573-generalized weak contractive multifunctionwhich has the properties (119877) (119870) and (119867) Suppose that thereexist a subset 119860 of119883 and 119909
0isin 119860 such that 120573(119860 119879119909
0) ge 1 If 119879
has the approximate endpoint property and119883 has the property(119866120573) then 119879 has a unique endpoint
Proof By using Theorem 3 119879 has a endpoint If 119879 hastwo distinct endpoints 119909lowast and 119910lowast then 120573(119879119909lowast 119879119910lowast) =120573(119909lowast 119910lowast) ge 1 because119883 has the property (119866
120573) Hence
119889 (119909lowast 119910lowast) le 119867 (119879119909lowast 119879119910lowast)
le 120573 (119879119909lowast 119879119910lowast)119867 (119879119909lowast 119879119910lowast)
le 120595 (119873 (119909lowast 119910lowast)) lt 119873 (119909lowast 119910lowast)
= 119889 (119909lowast 119910lowast)
(25)
which is a contradictionThus 119879 has a unique endpoint
In Example 5 119879 has a unique endpoint while119883 does hasnot the property (119866
120573) Also119879 has the property (119877) while119879 is
not lower semicontinuous To see this consider the sequence119909119899 defined by
119909119899=
1 minus1
119899119899 = 2119896
1 +1
119899119899 = 2119896 minus 1
(26)
for 119896 ge 1 and put 119910 = 12 and 1199090= 1 Then 119909
119899rarr 1 and 119910 isin
1198791199090= 12 Let 119910
119899 be an arbitrary sequence in119883 such that
119910119899isin 119879119909119899for all 119899 ge 1 Then 119910
2119896minus1isin 1198791199092119896minus1
and 1199102119896
isin 1198791199092119896
for all 119896 But 1199102119896minus1
= 41199092119896minus1
minus (32) for sufficiently large 119896and 119910
2119896= 11990921198962 for all 119896 since 119910
2119896minus1rarr 52 119910
119899999424999426999456 12 This
implies that 119879 is not lower semicontinuous
The Scientific World Journal 5
Corollary 7 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a 120573-admissiblemultifunctionwhich has the properties (119877) (119870) and(119867) Suppose that 119883 has the property (119866
120573) and there exist a
subset 119860 of X 1199090isin 119860 and 119896 isin [0 1) such that 120573(119860 119879119909
0) ge 1
and 120573(119879119909 119879119910)119867(119879119909 119879119910) le 119896119873(119909 119910) for all 119909 119910 isin 119883 Then119879 has a unique endpoint if and only if 119879 has the approximateendpoint property
Proof It is sufficient that we define 120595(119905) = 119896119905 for all 119905 ge 0Then Theorem 3 and Corollary 6 guarantee the result
It has been proved that lower semicontinuity of themulti-function 119879 and the property (119877) are independent conditions[9]We can replace lower semicontinuity of themultifunctioninstead of the property (119877) to obtain the next result Its proofis similar to the proof of Theorem 3
Theorem 8 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a lowersemicontinuous 120573-admissible 120573-generalized weak contractivemultifunction which has the properties (119870) and (119867) Supposethat there exist a subset 119860 of 119883 and 119909
0isin 119860 such that
120573(119860 1198791199090) ge 1 Then 119879 has the approximate endpoint property
if and only if 119879 has an endpoint
Corollary 9 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a lowersemicontinuous 120573-admissible 120573-generalized weak contractivemultifunction which has the properties (119870) and (119867) Supposethat there exist a subset 119860 of 119883 and 119909
0isin 119860 such that
120573(119860 1198791199090) ge 1 If 119879 has the approximate endpoint property
and 119883 has the property (119866120573) then 119879 has a unique endpoint
Corollary 10 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a 120573-admissiblemultifunctionwhich has the properties (119877) (119870) and(119867) Suppose that 119883 has the property (119866
120573) and there exist a
subset 119860 of 119883 1199090isin 119860 and 119896 isin [0 1) such that 120573(119860 119879119909
0) ge 1
and 120573(119879119909 119879119910)119867(119879119909 119879119910) le 119896119873(119909 119910) for all 119909 119910 isin 119883 If 119879 hasthe approximate endpoint property then Fix(119879) = End(119879) =119909
Proof If we put 120595(119905) = 119896119905 then by using Theorem 210 in[9]119879 has a fixed point Since119879 has the approximate endpointproperty by using Corollary 7 119879 has a unique endpoint such119909 Let 119910 isin Fix(119879) If 119879119909 = 119879119910 then 119910 = 119909 If 119879119909 = 119879119910 then120573(119879119909 119879119910) ge 1 because119883 has the property (119866
120573) Also we have
119889 (119909 119910) le 119867 (119909 119879119910) = 119867 (119879119909 119879119910)
le 120573 (119879119909 119879119910)119867 (119879119909 119879119910)
le 119896119873 (119909 119910)
(27)
But 119873(119909 119910) = max119889(119909 119910) 119889(119909 119879119909) 119889(119910 119879119910) (119889(119909 119879119910) +119889(119910 119879119909))2 = 119889(119909 119910) Thus 119889(119909 119910) = 0 and so Fix(119879) =End(119879) = 119909
Next corollary shows us the role of a point in the existenceof endpoints
Corollary 11 Let (119883 119889) be a complete metric space 119909lowast isin 119883 afixed element and 119879 119883 rarr 119862119861(119883) a multifunction such that119879 has the property (119867) and 119909lowast isin 119879119909cap119879119910 for all subsets119860 and119861 of119883 with 119909lowast isin 119860 cap 119861 and all 119909 isin 119860 and 119910 isin 119861 Assume that119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119909lowast isin 119879119909 cap 119879119910where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Suppose that there exist a subset 119860
0of 119883 and 119909
0isin 1198600such
that 119909lowast isin 1198600cap1198791199090 Assume that for each convergent sequence
119909119899 in 119883 with 119909
119899rarr 119909 and 119909lowast isin 119879119909
119899minus1cap 119879119909119899 for all 119899 ge 1
one has 119909lowast isin 119879119909119899cap 119879119909 Also for each sequence 119909
119899 in 119883
with 119909lowast isin 119879119909119899minus1
cap 119879119909119899for all 119899 ge 1 there exists a natural
number 119896 such that 119909lowast isin 119879119909119898cap119879119909119899for all119898 gt 119899 ge 119896 Then 119879
has an endpoint if and only if 119879 has the approximate endpointproperty
Proof It is sufficient we define 120573 2119883 times 2119883 rarr [0infin) by120573(119860 119861) = 1 whenever 119909lowast isin 119860cap119861 and 120573(119860 119861) = 0 otherwiseand then we use Theorem 3
Corollary 12 Let (119883 119889) be a complete metric space xlowast isin 119883a fixed element and 119879 119883 rarr 119862119861(119883) a lower semicontinuousmultifunction such that 119879 has the property (119867) and 119909lowast isin 119879119909cap119879119910 for all subsets 119860 and 119861 of119883 with 119909lowast isin 119860 cap 119861 and all 119909 isin 119860and 119910 isin 119861 Assume that
119867(119879119909 119879119910) le 120595 (119873 (119909 119910)) (28)
for all 119909 119910 isin 119883 with 119909lowast isin 119879119909 cap 119879119910 where 120595 [0 +infin) rarr[0 +infin) is a nondecreasing upper semicontinuous functionsuch that 120595(119905) lt 119905 for all 119905 gt 0 Suppose that there exist asubset 119860
0of119883 and 119909
0isin 1198600such that 119909lowast isin 119860
0cap 1198791199090 Assume
that for each convergent sequence 119909119899 in119883 with 119909
119899rarr 119909 and
119909lowast isin 119879119909119899minus1
cap119879119909119899for all 119899 ge 1 we have 119909lowast isin 119879119909
119899cap119879119909 Then 119879
has an endpoint if and only if 119879 has the approximate endpointproperty
Proof It is sufficient to define 120573 2119883 times 2119883 rarr [0infin) by120573(119860 119861) = 1 whenever 119909lowast isin 119860cap119861 and 120573(119860 119861) = 0 otherwiseand then we use Theorem 8
Let (119883 119889 le) be an ordered metric space Define the order⪯ on arbitrary subsets 119860 and 119861 of 119883 by 119860 ⪯ 119861 if and only iffor each 119886 isin 119860 there exists 119887 isin 119861 such that 119886 le 119887 It is easy tocheck that (119862119861(119883) ⪯) is a partially ordered set
Theorem 13 Let (119883 119889 le) be a complete ordered metric spaceand 119879 a closed and bounded valued multifunction on 119883 suchthat 119879 has the property (119867) and 119879119909 ⪯ 119879119910 for all subsets 119860and 119861 of 119883 with 119860 ⪯ 119861 and all 119909 isin 119860 and 119910 isin 119861 Assumethat 119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119879119909 ⪯ 119879119910where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Suppose that there exist a subset 119860
0of 119883 and 119909
0isin 1198600such
that 1198600
⪯ 1198791199090 Assume that for each convergent sequence
119909119899 in 119883 with 119909
119899rarr 119909 and 119879119909
119899minus1⪯ 119879119909
119899 for all 119899 ge 1
one has 119879119909119899
⪯ 119879119909 Also for each sequence 119909119899 in 119883 with
119879119909119899minus1
⪯ 119879119909119899for all 119899 ge 1 there exists a natural number 119896 such
that 119879119909119898⪯ 119879119909119899for all 119898 gt 119899 ge 119896 Then 119879 has an endpoint if
and only if 119879 has the approximate endpoint property
6 The Scientific World Journal
Proof Define 120573(119860 119861) = 1 whenever 119860 ⪯ 119861 and 120573(119860 119861) = 0otherwise and then we use Theorem 3
Corollary 14 Let (119883 119889 le) be a complete ordered metric spaceand 119879 a closed and bounded valued multifunction on 119883 suchthat 119879 has the property (119867) and 119879119909 ⪯ 119879119910 for all subsets 119860and 119861 of 119883 with 119860 ⪯ 119861 all 119909 isin 119860 and 119910 isin 119861 Assume that119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119879119909 ⪯ 119879119910where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Suppose that there exist a subset 119860
0of 119883 and 119909
0isin 1198600such
that 1198600
⪯ 1198791199090 Assume that for each convergent sequence
119909119899 in 119883 with 119909
119899rarr 119909 and 119879119909
119899minus1⪯ 119879119909
119899 for all 119899 ge 1
one has 119879119909119899
⪯ 119879119909 Also for each sequence 119909119899 in 119883 with
119879119909119899minus1
⪯ 119879119909119899for all 119899 ge 1 there exists a natural number 119896 such
that 119879119909119898⪯ 119879119909119899for all 119898 gt 119899 ge 119896 If 119879 has the approximate
endpoint property and 119860 ⪯ 119861 for all subsets119860 and 119861 of119883 with119860 sube 119861 or 119861 sube 119860 then 119879 has a unique endpoint
Proof Define 120573(119860 119861) = 1 whenever 119860 ⪯ 119861 and 120573(119860 119861) = 0otherwise and then we use Corollary 6
Let (119883 119889) be a metric space and 119879 119883 rarr 2119883
a multifunction We say that 119879 is an 119867119878-multifunctionwhenever for each 119909 isin 119883 there exists 119910 isin 119879119909 such that119867(119879119909 119879119910) = sup
119887isin119879y119889(119910 119887) It is obvious that each 119867119878-multifunction is an multifunction which has the property(119867) Thus one can conclude similar results to above ones for119867119878-multifunctions Here we provide some ones Althoughby considering 119867119878-multifunction we restrict ourselves weobtain strange results with respect to above ones One canprove the following by reading exactly the proofs of similarabove results
Theorem 15 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a 120573-admissible 120573-generalized weak contractive 119867119878-multifunctionwhich has the properties (119877) and (119870) Suppose that there exist asubset119860 of119883 and 119909
0isin 119860 such that 120573(119860 119879119909
0) ge 1 Then 119879 has
an endpoint and so 119879 has the approximate endpoint property
Theorem 16 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) alower semicontinuous 120573-admissible and 120573-generalized weakcontractive 119867119878-multifunction which has the property (119870)Suppose that there exist a subset 119860 of 119883 and 119909
0isin 119860 such
that 120573(119860 1198791199090) ge 1 Then 119879 has an endpoint and so 119879 has
the approximate endpoint property
The next result is a consequence of Theorem 15
Corollary 17 Let (119883 119889) be a complete metric space 119909lowast isin 119883 afixed element and 119879 119883 rarr 119862119861(119883) an 119867119878-multifunctionsuch that 119909lowast isin 119879119909 cap 119879119910 for all subsets 119860 and 119861 of 119883 with119909lowast isin 119860 cap 119861 all 119909 isin 119860 and 119910 isin 119861 Assume that 119867(119879119909 119879119910) le120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119909lowast isin 119879119909 cap 119879119910 where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing upper semicontinuousfunction such that120595(119905) lt 119905 for all 119905 gt 0 Suppose that there exista subset119860
0of119883 and 119909
0isin 1198600such that 119909lowast isin 119860
0cap1198791199090 Assume
that for each convergent sequence 119909119899 in119883 with 119909
119899rarr 119909 and
119909lowast isin 119879119909119899minus1
cap119879119909119899for all 119899 ge 1 one has 119909lowast isin 119879119909
119899cap119879119909 Also for
each sequence 119909119899 in 119883 with 119909lowast isin 119879119909
119899minus1cap 119879119909119899for all 119899 ge 1
there exists a natural number 119896 such that 119909lowast isin 119879119909119898cap 119879119909119899
for all 119898 gt 119899 ge 119896 Then 119879 has an endpoint and so 119879 has theapproximate endpoint property
The next result is a consequence of Theorem 16
Corollary 18 Let (119883 119889 le) be a complete ordered metric spaceand 119879 a closed and bounded valued lower semicontinuous119867119878-multifunction on 119883 such that 119879119909 ⪯ 119879119910 for all subsets 119860 and119861 of 119883 with 119860 ⪯ 119861 all 119909 isin 119860 and 119910 isin 119861 Assume that119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119879119909 ⪯ 119879119910where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Suppose that there exist a subset 119860
0of 119883 and 119909
0isin 1198600such
that 1198600⪯ 1198791199090 Assume that for each sequence 119909
119899 in 119883 with
119879119909119899minus1
⪯ 119879119909119899for all 119899 ge 1 there exists a natural number 119896 such
that 119879119909119898
⪯ 119879119909119899for all 119898 gt 119899 ge 119896 Then 119879 has an endpoint
and so 119879 has the approximate endpoint property
Acknowledgments
This work was completed while the second author (DrGopal) was visiting the Azarbaijan University of ShahidMadani Azarshahr Tabriz Iran during the summer of 2012He thanks Professor Sh Rezapour and the University fortheir hospitality and support The second author gratefullyacknowledges the support from the CSIR govternment ofIndia Grant no-25(0215)13EMR-II
References
[1] A Amini-Harandi ldquoEndpoints of set-valued contractions inmetric spacesrdquo Nonlinear Analysis Theory Methods and Appli-cations vol 72 no 1 pp 132ndash134 2010
[2] R P Agarwal D OrsquoRegan and D R Sahu Fixed Point Theoryfor Lipschitzian-TypeMappings with Applications Springer NewYork NY USA 2009
[3] S Moradi and F Khojasteh ldquoEndpoints of multi-valued gener-alized weak contraction mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 74 no 6 pp 2170ndash2174 2011
[4] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012
[5] D BaleanuHMohammadi and Sh Rezapour ldquoSome existenceresults on nonlinear fractional differential equationsrdquo Philo-sophical Transactions of the Royal Society vol 371 Article ID20120144 2013
[6] M Jleli and B Samet ldquoBest proximity points for 120572-120595-proximalcontractive type mappings and applicationsrdquo Bulletin des Sci-ences Mathematiques 2013
[7] M A Miandaragh M Postolache and Sh Rezapour ldquoSomeapproximate fixed point results for generalized-contractivemappings rdquo To appear in Scientific Bulletin-UniversityPolitehnica of Bucharest A
[8] Sh Rezapour and J H Asl ldquoA simple method for obtainingcoupled fixed points of 120572-120595-contractive type mappingsrdquo Inter-national Journal of Analysis Article ID 438029 7 pages 2013
The Scientific World Journal 7
[9] H Alikhani V Rakocevic Sh Rezapour and N ShahzadldquoFixed points of proximinal valued 120573-120595-contractive multifunc-tionsrdquo To appear in Journal of Nonlinear and Convex Analysis
[10] H Alikhani Sh Rezapour and N Shahzad ldquoFixed points of anew type contractive mappings and multifunctionsrdquo To appearin Filomat
[11] S M A Aleomraninejad Sh Rezapour and N Shahzad ldquoOnfixed points of 120572-120595-contractive multifunctionsrdquo Fixed PointTheory and Applications vol 2012 article 212 8 pages 2012
[12] B Mohammadi Sh Rezapour and N Shahzad ldquoSome resultson fixed points of 120572-120595-ciric generalized multifunctionsrdquo FixedPoint Theory and Applications vol 2013 article 24 2013
[13] S M A Aleomraninejad Sh Rezapour andN Shahzad ldquoSomefixed point results on ametric space with a graphrdquoTopology andIts Applications vol 159 no 3 pp 659ndash663 2012
[14] J M Borwein and A S Lewis Convex Analysis and NonlinearOptimization Theory and Examples Springer New York NYUSA 2000
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Differential EquationsInternational Journal of
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The Scientific World Journal 3
Thus 120576 le 119889(119909119898119896
119909119899119896
) le 119889(119909119898119896
119909119898119896minus1) + 119889(119909
119898119896minus1 119909119899119896
) le119889119898119896minus1
+ 120576 for all 119896 Hence lim119896rarrinfin
119889(119909119898119896
119909119899119896
) = 120576 Since 119879has the property (119870) we obtain
119889 (119909119898119896
119909119899119896
) le 119889 (119909119898119896
119909119898119896+1) + 119889 (119909
119898119896+1 119909119899119896+1)
+ 119889 (119909119899119896+1 119909119899119896
)
le 119889119898119896
+ 119867(119879119909119898119896
119879119909119899119896
) + 119889119899119896
le 119889119898119896
+ 120573 (119879119909119898119896
119879119909119899119896
)
times 119867(119879119909119898119896
119879119909119899119896
) + 119889119899119896
le 119889119898119896
+ 120595 (119873(119909119898119896
119909119899119896
)) + 119889119899119896
(lowast)
for all 119896 Since lim119896rarrinfin
119889(119909119898119896
119909119899119896
) = 120576 lim119896rarrinfin
119873(119909119898119896
119909119899119896
) = 120576 In fact
119889 (119909119898119896
119909119899119896
)
le 119873(119909119898119896
119909119899119896
)
= max119889 (119909119898119896
119909119899119896
) 119889 (119909119898119896
119879119909119898119896
) 119889 (119909119899119896
119879119909119899119896
)
119889 (119909119898119896
119879119909119899119896
) + 119889 (119909119899119896
119879119909119898119896
)
2
le max119889 (119909119898119896
119909119899119896
) 119889 (119909119898119896
119909119898119896+1
) 119889 (119909119899119896
119909119899119896+1
)
119889 (119909119898119896
119909119899119896+1
) + 119889 (119909119899119896
119909119898119896+1
)
2
le max 119889 (119909119898119896
119909119899119896
) 119889 (119909119898119896
119909119898119896+1
) 119889 (119909119899119896
119909119899119896+1
)
(119889 (119909119898119896
119909119899119896
) + 119889 (119909119899119896
119909119899119896+1
)
+119889 (119909119899119896
119909119898119896
) + 119889 (119909119898119896
119909119898k+1
))
times (2)minus1
(10)
and so 120576 = lim119896rarr+infin
119889(119909119898119896
119909119899119896
) le lim119896rarrinfin
119873(119909119898119896
119909119899119896
) le 120576Since 120595 is upper semicontinuous by using (lowast) we obtain
120576 = lim119896rarrinfin
119889 (119909119898119896
119909119899119896
)
le lim119896rarrinfin
120595 (119873(119909119898119896
119909119899119896
)) le 120595 (120576) lt 120576(11)
which is a contradiction and so 119909119899 is a Cauchy sequence
Choose 119909lowast isin 119883 such that 119909119899rarr 119909lowast Now note that
119867(119909119899 119879119909119899) = max119889 (119909
119899 119879119909119899) sup119910isin119879119909
119899
119889 (119909119899 119910)
= 119867 (119879119909119899minus1
119879119909119899)
(12)
for all 119899 and so
119867(119909119899 119879119909119899) = 119867 (119879119909
119899minus1 119879119909119899)
le 120573 (119879119909119899minus1
119879119909119899)119867 (119879119909
119899minus1 119879119909119899)
le 120595 (119873 (119909119899minus1
119909119899))
le 120595 (119889 (119909119899minus1
119909119899)) le 119889 (119909
119899minus1 119909119899)
(13)
for all 119899 and so lim119899rarrinfin
119867(119909119899 119879119909119899) = 0 Since 119879 has the
property (119877) we obtain
119867(119909lowast 119879119909lowast) le 119889 (119909lowast 119909119899)
+ 119867 (119909119899 119879119909119899) + 119867 (119879119909
119899 119879119909lowast)
le 119889 (119909lowast 119909119899) + 119867 (119909
119899 119879119909119899)
+ 120573 (119879119909119899 119879119909lowast)119867 (119879119909
119899 119879119909lowast)
le 119889 (119909lowast 119909119899) + 119867 (119909
119899 119879119909119899)
+ 120595 (119873 (119909119899 119909lowast))
(14)
for all 119899 If119873(119909119899 119909lowast) = 119889(119909lowast 119879119909lowast) then we have
119867(119909lowast 119879119909lowast)
le 119889 (119909lowast 119909119899) + 119867 (119909
119899 119879119909119899) + 120595 (119867 (119909lowast 119879119909lowast))
(15)
for all 119899 This implies that 119867(119909lowast 119879119909lowast) le 120595(119867(119909lowast 119879119909lowast))and so
119867(119909lowast 119879119909lowast) = 0 (16)
If119873(119909119899 119909lowast) = 119889(119909
119899 119909lowast) or119873(119909
119899 119909lowast) le 119889(119909
119899 119909119899+1
) then it iseasy to see that119867(119909lowast 119879119909lowast) = 0 Thus 119909lowast is an endpoint of119879
Next example shows that a 120573-generalized weak con-tractive multifunction is not necessarily a generalized weakcontractive multifunction
Example 4 Let 119883 = R Define 119879 119883 rarr 119862119861(119883) by119879119909 = [119909 119909 + 2] for all 119909 isin 119883 Suppose that 120595 [0 +infin) rarr[0 +infin) is an arbitrary upper semicontinuous function suchthat 120595(119905) lt 119905 for all 119905 gt 0 If 119909 = 0 and 119910 = 2 then119867(119879119909 119879119910) = 119867([0 2] [2 4]) = 2 and119873(119909 119910) = 2 Hence
119867(119879119909 119879119910) = 2 ≩ 120595 (2) = 120595 (119873 (119909 119910)) (17)
Thus 119879 is not a generalized weak contractive multifunctionNow suppose that 120595(119905) = 1199052 for all 119905 ge 0 and define 120573
2119883 times 2119883 rarr [0infin) by 120573(119860 119861) = 12 for all subsets 119860 and 119861of119883 Then we have
120573 (119879119909 119879119910)119867 (119879119909 119879119910) =1
2119889 (119909 119910)
= 120595 (119889 (119909 119910)) = 120595 (119873 (119909 119910))
(18)
for all 119909 119910 isin R Thus 119879 is a 120573-generalized weak contractivemultifunction
4 The Scientific World Journal
Next example shows that there are multifunctions whichsatisfy the conditions of Theorem 3 while they are notgeneralized weak contractive multifunctions
Example 5 Let 119883 = [0 92] and let 119889(119909 119910) = |119909 minus 119910| Define119879 119883 rarr 119862119861(119883) by
119879119909 =
119909
2 0 le 119909 le 1
4119909 minus3
2 1 lt 119909 le
3
2
03
2lt 119909 le
9
2
(19)
If 119909 = 1 and 119910 = 32 then
119867(119879119909 119879119910) = 119867(1
2
9
2) = 4 gt 3
= 119873 (119909 119910) gt 120595 (119873 (119909 119910))
(20)
where 120595 [0 +infin) rarr [0 +infin) is an arbitrary uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Thus 119879 is not a generalized weak contractive multifunctionNow we show that119879 satisfies all conditions ofTheorem 3 Forthis aim define120595(119905) = 1199052 and 120573(119860 119861) = 1whenever119860 and119861are subsets of [0 1] and 120573(119860 119861) = 0 otherwise First supposethat 119909 notin [0 1] or that 119910 notin [0 1] If 119909 119910 isin (32 92] then119879119909 119879119910 sub [0 1] and 120573(119879119909 119879119910) = 1 But 119867(119879119909 119879119910) = 0 andso 120573(119879119909 119879119910)119867(119879119909 119879119910) le 120595(119873(119909 119910)) If 119909 isin (1 32] or 119910 isin(1 32] then119879119909 sube [0 1] or119879119910 sube [0 1] and so 120573(119879119909 119879119910) = 0Hence 120573(119879119909 119879119910)119867(119879119909 119879119910) le 120595(119873(119909 119910)) Now supposethat 119909 119910 isin [0 1] In this case we have 120573(119879119909 119879119910) ge1 119867(119879119909 119879119910) = 119867(1199092 1199102) = (12)119889(119909 119910) and119873(119909 119910) = max119889(119909 119910) 1199092 1199102 (119889(119909 1199102) + 119889(119910 1199092))2Thus 119889(119909 119910) le 119873(119909 119910) and so
120573 (119879119909 119879119910)119867 (119879119909 119879119910) =1
2119889 (119909 119910)
le 120595 (119889 (119909 119910)) le 120595 (119873 (119909 119910))
(21)
Therefore 119879 is a 120573-generalized weak contractive multifunc-tionNowwe show that119879 is120573-admissible If120573(119860 119861) ge 1 then119860 119861 sub [0 1] and so 119879119909 = 1199092 isin [0 1] and 119879119910 = 1199102 isin[0 1] for all 119909 isin 119860 and 119910 isin 119861 Thus 120573(119879119909 119879119910) ge 1 for all119909 isin 119860 and 119910 isin 119861 Now suppose 119860 = [0 12] and 119909
0= 14
Then 1198791199090= 18 isin [0 1] and [0 12] sub [0 1] Hence
120573(119860 1198791199090) ge 1 Now we show that 119879 satisfies the condition
(119867) First note that for each 120576 gt 0 there exists 119911 isin 119883 suchthat sup
119886isin119879119911119889(119911 119886) lt 120576 Now we show that for each 119909 isin 119883
there exists 119910 isin 119879119909 such that 119867(119879119909 119879119910) = sup119887isin119879119910
119889(119910 119887) If0 le 119909 le 1 then 119879119909 = 1199092 119879(1199092) = 1199094 and
119867(119879119909 119879 (119909
2)) = 119867(
119909
2
119909
4)
=119909
4= sup119887isin119879(1199092)
119889 (119909
2 119887)
(22)
Since for 1 lt 119909 le 32 we have 52 lt 4119909 minus (32) le 92119879(4119909 minus (32)) = 0 Thus
119867(119879119909 119879 (4119909 minus3
2)) = 119867(4119909 minus
3
2 0)
= 4119909 minus3
2= sup119887isin119879(4119909minus32)
119889(4119909 minus3
2 119887)
(23)
If 32 lt 119909 le 92 then 119879119909 = 0 and 119879(0) = 0 Hence
119867(119879119909 119879 (0)) = 119867 (0 0)
= 0 = sup119887isin119879(0)
119889 (0 119887) (24)
It is easy to check that 119879 satisfies the conditions (119877) and (119870)Note that 0 is the endpoint of 119879
Now we add an assumption to obtain uniqueness ofendpoint In this way we introduce a new notion Let 119883 bea set and 120573 2119883 times 2119883 rarr [0infin) a map We say that the set119883has the property (119866
120573) whenever 120573(119860 119861) ge 1 for all subsets 119860
and 119861 of119883 with 119860 sube 119861 or 119861 sube 119860
Corollary 6 Let (119883 119889) be a complete metric space 120573
2119883 times 2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883)a 120573-admissible 120573-generalized weak contractive multifunctionwhich has the properties (119877) (119870) and (119867) Suppose that thereexist a subset 119860 of119883 and 119909
0isin 119860 such that 120573(119860 119879119909
0) ge 1 If 119879
has the approximate endpoint property and119883 has the property(119866120573) then 119879 has a unique endpoint
Proof By using Theorem 3 119879 has a endpoint If 119879 hastwo distinct endpoints 119909lowast and 119910lowast then 120573(119879119909lowast 119879119910lowast) =120573(119909lowast 119910lowast) ge 1 because119883 has the property (119866
120573) Hence
119889 (119909lowast 119910lowast) le 119867 (119879119909lowast 119879119910lowast)
le 120573 (119879119909lowast 119879119910lowast)119867 (119879119909lowast 119879119910lowast)
le 120595 (119873 (119909lowast 119910lowast)) lt 119873 (119909lowast 119910lowast)
= 119889 (119909lowast 119910lowast)
(25)
which is a contradictionThus 119879 has a unique endpoint
In Example 5 119879 has a unique endpoint while119883 does hasnot the property (119866
120573) Also119879 has the property (119877) while119879 is
not lower semicontinuous To see this consider the sequence119909119899 defined by
119909119899=
1 minus1
119899119899 = 2119896
1 +1
119899119899 = 2119896 minus 1
(26)
for 119896 ge 1 and put 119910 = 12 and 1199090= 1 Then 119909
119899rarr 1 and 119910 isin
1198791199090= 12 Let 119910
119899 be an arbitrary sequence in119883 such that
119910119899isin 119879119909119899for all 119899 ge 1 Then 119910
2119896minus1isin 1198791199092119896minus1
and 1199102119896
isin 1198791199092119896
for all 119896 But 1199102119896minus1
= 41199092119896minus1
minus (32) for sufficiently large 119896and 119910
2119896= 11990921198962 for all 119896 since 119910
2119896minus1rarr 52 119910
119899999424999426999456 12 This
implies that 119879 is not lower semicontinuous
The Scientific World Journal 5
Corollary 7 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a 120573-admissiblemultifunctionwhich has the properties (119877) (119870) and(119867) Suppose that 119883 has the property (119866
120573) and there exist a
subset 119860 of X 1199090isin 119860 and 119896 isin [0 1) such that 120573(119860 119879119909
0) ge 1
and 120573(119879119909 119879119910)119867(119879119909 119879119910) le 119896119873(119909 119910) for all 119909 119910 isin 119883 Then119879 has a unique endpoint if and only if 119879 has the approximateendpoint property
Proof It is sufficient that we define 120595(119905) = 119896119905 for all 119905 ge 0Then Theorem 3 and Corollary 6 guarantee the result
It has been proved that lower semicontinuity of themulti-function 119879 and the property (119877) are independent conditions[9]We can replace lower semicontinuity of themultifunctioninstead of the property (119877) to obtain the next result Its proofis similar to the proof of Theorem 3
Theorem 8 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a lowersemicontinuous 120573-admissible 120573-generalized weak contractivemultifunction which has the properties (119870) and (119867) Supposethat there exist a subset 119860 of 119883 and 119909
0isin 119860 such that
120573(119860 1198791199090) ge 1 Then 119879 has the approximate endpoint property
if and only if 119879 has an endpoint
Corollary 9 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a lowersemicontinuous 120573-admissible 120573-generalized weak contractivemultifunction which has the properties (119870) and (119867) Supposethat there exist a subset 119860 of 119883 and 119909
0isin 119860 such that
120573(119860 1198791199090) ge 1 If 119879 has the approximate endpoint property
and 119883 has the property (119866120573) then 119879 has a unique endpoint
Corollary 10 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a 120573-admissiblemultifunctionwhich has the properties (119877) (119870) and(119867) Suppose that 119883 has the property (119866
120573) and there exist a
subset 119860 of 119883 1199090isin 119860 and 119896 isin [0 1) such that 120573(119860 119879119909
0) ge 1
and 120573(119879119909 119879119910)119867(119879119909 119879119910) le 119896119873(119909 119910) for all 119909 119910 isin 119883 If 119879 hasthe approximate endpoint property then Fix(119879) = End(119879) =119909
Proof If we put 120595(119905) = 119896119905 then by using Theorem 210 in[9]119879 has a fixed point Since119879 has the approximate endpointproperty by using Corollary 7 119879 has a unique endpoint such119909 Let 119910 isin Fix(119879) If 119879119909 = 119879119910 then 119910 = 119909 If 119879119909 = 119879119910 then120573(119879119909 119879119910) ge 1 because119883 has the property (119866
120573) Also we have
119889 (119909 119910) le 119867 (119909 119879119910) = 119867 (119879119909 119879119910)
le 120573 (119879119909 119879119910)119867 (119879119909 119879119910)
le 119896119873 (119909 119910)
(27)
But 119873(119909 119910) = max119889(119909 119910) 119889(119909 119879119909) 119889(119910 119879119910) (119889(119909 119879119910) +119889(119910 119879119909))2 = 119889(119909 119910) Thus 119889(119909 119910) = 0 and so Fix(119879) =End(119879) = 119909
Next corollary shows us the role of a point in the existenceof endpoints
Corollary 11 Let (119883 119889) be a complete metric space 119909lowast isin 119883 afixed element and 119879 119883 rarr 119862119861(119883) a multifunction such that119879 has the property (119867) and 119909lowast isin 119879119909cap119879119910 for all subsets119860 and119861 of119883 with 119909lowast isin 119860 cap 119861 and all 119909 isin 119860 and 119910 isin 119861 Assume that119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119909lowast isin 119879119909 cap 119879119910where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Suppose that there exist a subset 119860
0of 119883 and 119909
0isin 1198600such
that 119909lowast isin 1198600cap1198791199090 Assume that for each convergent sequence
119909119899 in 119883 with 119909
119899rarr 119909 and 119909lowast isin 119879119909
119899minus1cap 119879119909119899 for all 119899 ge 1
one has 119909lowast isin 119879119909119899cap 119879119909 Also for each sequence 119909
119899 in 119883
with 119909lowast isin 119879119909119899minus1
cap 119879119909119899for all 119899 ge 1 there exists a natural
number 119896 such that 119909lowast isin 119879119909119898cap119879119909119899for all119898 gt 119899 ge 119896 Then 119879
has an endpoint if and only if 119879 has the approximate endpointproperty
Proof It is sufficient we define 120573 2119883 times 2119883 rarr [0infin) by120573(119860 119861) = 1 whenever 119909lowast isin 119860cap119861 and 120573(119860 119861) = 0 otherwiseand then we use Theorem 3
Corollary 12 Let (119883 119889) be a complete metric space xlowast isin 119883a fixed element and 119879 119883 rarr 119862119861(119883) a lower semicontinuousmultifunction such that 119879 has the property (119867) and 119909lowast isin 119879119909cap119879119910 for all subsets 119860 and 119861 of119883 with 119909lowast isin 119860 cap 119861 and all 119909 isin 119860and 119910 isin 119861 Assume that
119867(119879119909 119879119910) le 120595 (119873 (119909 119910)) (28)
for all 119909 119910 isin 119883 with 119909lowast isin 119879119909 cap 119879119910 where 120595 [0 +infin) rarr[0 +infin) is a nondecreasing upper semicontinuous functionsuch that 120595(119905) lt 119905 for all 119905 gt 0 Suppose that there exist asubset 119860
0of119883 and 119909
0isin 1198600such that 119909lowast isin 119860
0cap 1198791199090 Assume
that for each convergent sequence 119909119899 in119883 with 119909
119899rarr 119909 and
119909lowast isin 119879119909119899minus1
cap119879119909119899for all 119899 ge 1 we have 119909lowast isin 119879119909
119899cap119879119909 Then 119879
has an endpoint if and only if 119879 has the approximate endpointproperty
Proof It is sufficient to define 120573 2119883 times 2119883 rarr [0infin) by120573(119860 119861) = 1 whenever 119909lowast isin 119860cap119861 and 120573(119860 119861) = 0 otherwiseand then we use Theorem 8
Let (119883 119889 le) be an ordered metric space Define the order⪯ on arbitrary subsets 119860 and 119861 of 119883 by 119860 ⪯ 119861 if and only iffor each 119886 isin 119860 there exists 119887 isin 119861 such that 119886 le 119887 It is easy tocheck that (119862119861(119883) ⪯) is a partially ordered set
Theorem 13 Let (119883 119889 le) be a complete ordered metric spaceand 119879 a closed and bounded valued multifunction on 119883 suchthat 119879 has the property (119867) and 119879119909 ⪯ 119879119910 for all subsets 119860and 119861 of 119883 with 119860 ⪯ 119861 and all 119909 isin 119860 and 119910 isin 119861 Assumethat 119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119879119909 ⪯ 119879119910where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Suppose that there exist a subset 119860
0of 119883 and 119909
0isin 1198600such
that 1198600
⪯ 1198791199090 Assume that for each convergent sequence
119909119899 in 119883 with 119909
119899rarr 119909 and 119879119909
119899minus1⪯ 119879119909
119899 for all 119899 ge 1
one has 119879119909119899
⪯ 119879119909 Also for each sequence 119909119899 in 119883 with
119879119909119899minus1
⪯ 119879119909119899for all 119899 ge 1 there exists a natural number 119896 such
that 119879119909119898⪯ 119879119909119899for all 119898 gt 119899 ge 119896 Then 119879 has an endpoint if
and only if 119879 has the approximate endpoint property
6 The Scientific World Journal
Proof Define 120573(119860 119861) = 1 whenever 119860 ⪯ 119861 and 120573(119860 119861) = 0otherwise and then we use Theorem 3
Corollary 14 Let (119883 119889 le) be a complete ordered metric spaceand 119879 a closed and bounded valued multifunction on 119883 suchthat 119879 has the property (119867) and 119879119909 ⪯ 119879119910 for all subsets 119860and 119861 of 119883 with 119860 ⪯ 119861 all 119909 isin 119860 and 119910 isin 119861 Assume that119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119879119909 ⪯ 119879119910where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Suppose that there exist a subset 119860
0of 119883 and 119909
0isin 1198600such
that 1198600
⪯ 1198791199090 Assume that for each convergent sequence
119909119899 in 119883 with 119909
119899rarr 119909 and 119879119909
119899minus1⪯ 119879119909
119899 for all 119899 ge 1
one has 119879119909119899
⪯ 119879119909 Also for each sequence 119909119899 in 119883 with
119879119909119899minus1
⪯ 119879119909119899for all 119899 ge 1 there exists a natural number 119896 such
that 119879119909119898⪯ 119879119909119899for all 119898 gt 119899 ge 119896 If 119879 has the approximate
endpoint property and 119860 ⪯ 119861 for all subsets119860 and 119861 of119883 with119860 sube 119861 or 119861 sube 119860 then 119879 has a unique endpoint
Proof Define 120573(119860 119861) = 1 whenever 119860 ⪯ 119861 and 120573(119860 119861) = 0otherwise and then we use Corollary 6
Let (119883 119889) be a metric space and 119879 119883 rarr 2119883
a multifunction We say that 119879 is an 119867119878-multifunctionwhenever for each 119909 isin 119883 there exists 119910 isin 119879119909 such that119867(119879119909 119879119910) = sup
119887isin119879y119889(119910 119887) It is obvious that each 119867119878-multifunction is an multifunction which has the property(119867) Thus one can conclude similar results to above ones for119867119878-multifunctions Here we provide some ones Althoughby considering 119867119878-multifunction we restrict ourselves weobtain strange results with respect to above ones One canprove the following by reading exactly the proofs of similarabove results
Theorem 15 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a 120573-admissible 120573-generalized weak contractive 119867119878-multifunctionwhich has the properties (119877) and (119870) Suppose that there exist asubset119860 of119883 and 119909
0isin 119860 such that 120573(119860 119879119909
0) ge 1 Then 119879 has
an endpoint and so 119879 has the approximate endpoint property
Theorem 16 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) alower semicontinuous 120573-admissible and 120573-generalized weakcontractive 119867119878-multifunction which has the property (119870)Suppose that there exist a subset 119860 of 119883 and 119909
0isin 119860 such
that 120573(119860 1198791199090) ge 1 Then 119879 has an endpoint and so 119879 has
the approximate endpoint property
The next result is a consequence of Theorem 15
Corollary 17 Let (119883 119889) be a complete metric space 119909lowast isin 119883 afixed element and 119879 119883 rarr 119862119861(119883) an 119867119878-multifunctionsuch that 119909lowast isin 119879119909 cap 119879119910 for all subsets 119860 and 119861 of 119883 with119909lowast isin 119860 cap 119861 all 119909 isin 119860 and 119910 isin 119861 Assume that 119867(119879119909 119879119910) le120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119909lowast isin 119879119909 cap 119879119910 where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing upper semicontinuousfunction such that120595(119905) lt 119905 for all 119905 gt 0 Suppose that there exista subset119860
0of119883 and 119909
0isin 1198600such that 119909lowast isin 119860
0cap1198791199090 Assume
that for each convergent sequence 119909119899 in119883 with 119909
119899rarr 119909 and
119909lowast isin 119879119909119899minus1
cap119879119909119899for all 119899 ge 1 one has 119909lowast isin 119879119909
119899cap119879119909 Also for
each sequence 119909119899 in 119883 with 119909lowast isin 119879119909
119899minus1cap 119879119909119899for all 119899 ge 1
there exists a natural number 119896 such that 119909lowast isin 119879119909119898cap 119879119909119899
for all 119898 gt 119899 ge 119896 Then 119879 has an endpoint and so 119879 has theapproximate endpoint property
The next result is a consequence of Theorem 16
Corollary 18 Let (119883 119889 le) be a complete ordered metric spaceand 119879 a closed and bounded valued lower semicontinuous119867119878-multifunction on 119883 such that 119879119909 ⪯ 119879119910 for all subsets 119860 and119861 of 119883 with 119860 ⪯ 119861 all 119909 isin 119860 and 119910 isin 119861 Assume that119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119879119909 ⪯ 119879119910where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Suppose that there exist a subset 119860
0of 119883 and 119909
0isin 1198600such
that 1198600⪯ 1198791199090 Assume that for each sequence 119909
119899 in 119883 with
119879119909119899minus1
⪯ 119879119909119899for all 119899 ge 1 there exists a natural number 119896 such
that 119879119909119898
⪯ 119879119909119899for all 119898 gt 119899 ge 119896 Then 119879 has an endpoint
and so 119879 has the approximate endpoint property
Acknowledgments
This work was completed while the second author (DrGopal) was visiting the Azarbaijan University of ShahidMadani Azarshahr Tabriz Iran during the summer of 2012He thanks Professor Sh Rezapour and the University fortheir hospitality and support The second author gratefullyacknowledges the support from the CSIR govternment ofIndia Grant no-25(0215)13EMR-II
References
[1] A Amini-Harandi ldquoEndpoints of set-valued contractions inmetric spacesrdquo Nonlinear Analysis Theory Methods and Appli-cations vol 72 no 1 pp 132ndash134 2010
[2] R P Agarwal D OrsquoRegan and D R Sahu Fixed Point Theoryfor Lipschitzian-TypeMappings with Applications Springer NewYork NY USA 2009
[3] S Moradi and F Khojasteh ldquoEndpoints of multi-valued gener-alized weak contraction mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 74 no 6 pp 2170ndash2174 2011
[4] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012
[5] D BaleanuHMohammadi and Sh Rezapour ldquoSome existenceresults on nonlinear fractional differential equationsrdquo Philo-sophical Transactions of the Royal Society vol 371 Article ID20120144 2013
[6] M Jleli and B Samet ldquoBest proximity points for 120572-120595-proximalcontractive type mappings and applicationsrdquo Bulletin des Sci-ences Mathematiques 2013
[7] M A Miandaragh M Postolache and Sh Rezapour ldquoSomeapproximate fixed point results for generalized-contractivemappings rdquo To appear in Scientific Bulletin-UniversityPolitehnica of Bucharest A
[8] Sh Rezapour and J H Asl ldquoA simple method for obtainingcoupled fixed points of 120572-120595-contractive type mappingsrdquo Inter-national Journal of Analysis Article ID 438029 7 pages 2013
The Scientific World Journal 7
[9] H Alikhani V Rakocevic Sh Rezapour and N ShahzadldquoFixed points of proximinal valued 120573-120595-contractive multifunc-tionsrdquo To appear in Journal of Nonlinear and Convex Analysis
[10] H Alikhani Sh Rezapour and N Shahzad ldquoFixed points of anew type contractive mappings and multifunctionsrdquo To appearin Filomat
[11] S M A Aleomraninejad Sh Rezapour and N Shahzad ldquoOnfixed points of 120572-120595-contractive multifunctionsrdquo Fixed PointTheory and Applications vol 2012 article 212 8 pages 2012
[12] B Mohammadi Sh Rezapour and N Shahzad ldquoSome resultson fixed points of 120572-120595-ciric generalized multifunctionsrdquo FixedPoint Theory and Applications vol 2013 article 24 2013
[13] S M A Aleomraninejad Sh Rezapour andN Shahzad ldquoSomefixed point results on ametric space with a graphrdquoTopology andIts Applications vol 159 no 3 pp 659ndash663 2012
[14] J M Borwein and A S Lewis Convex Analysis and NonlinearOptimization Theory and Examples Springer New York NYUSA 2000
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Differential EquationsInternational Journal of
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4 The Scientific World Journal
Next example shows that there are multifunctions whichsatisfy the conditions of Theorem 3 while they are notgeneralized weak contractive multifunctions
Example 5 Let 119883 = [0 92] and let 119889(119909 119910) = |119909 minus 119910| Define119879 119883 rarr 119862119861(119883) by
119879119909 =
119909
2 0 le 119909 le 1
4119909 minus3
2 1 lt 119909 le
3
2
03
2lt 119909 le
9
2
(19)
If 119909 = 1 and 119910 = 32 then
119867(119879119909 119879119910) = 119867(1
2
9
2) = 4 gt 3
= 119873 (119909 119910) gt 120595 (119873 (119909 119910))
(20)
where 120595 [0 +infin) rarr [0 +infin) is an arbitrary uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Thus 119879 is not a generalized weak contractive multifunctionNow we show that119879 satisfies all conditions ofTheorem 3 Forthis aim define120595(119905) = 1199052 and 120573(119860 119861) = 1whenever119860 and119861are subsets of [0 1] and 120573(119860 119861) = 0 otherwise First supposethat 119909 notin [0 1] or that 119910 notin [0 1] If 119909 119910 isin (32 92] then119879119909 119879119910 sub [0 1] and 120573(119879119909 119879119910) = 1 But 119867(119879119909 119879119910) = 0 andso 120573(119879119909 119879119910)119867(119879119909 119879119910) le 120595(119873(119909 119910)) If 119909 isin (1 32] or 119910 isin(1 32] then119879119909 sube [0 1] or119879119910 sube [0 1] and so 120573(119879119909 119879119910) = 0Hence 120573(119879119909 119879119910)119867(119879119909 119879119910) le 120595(119873(119909 119910)) Now supposethat 119909 119910 isin [0 1] In this case we have 120573(119879119909 119879119910) ge1 119867(119879119909 119879119910) = 119867(1199092 1199102) = (12)119889(119909 119910) and119873(119909 119910) = max119889(119909 119910) 1199092 1199102 (119889(119909 1199102) + 119889(119910 1199092))2Thus 119889(119909 119910) le 119873(119909 119910) and so
120573 (119879119909 119879119910)119867 (119879119909 119879119910) =1
2119889 (119909 119910)
le 120595 (119889 (119909 119910)) le 120595 (119873 (119909 119910))
(21)
Therefore 119879 is a 120573-generalized weak contractive multifunc-tionNowwe show that119879 is120573-admissible If120573(119860 119861) ge 1 then119860 119861 sub [0 1] and so 119879119909 = 1199092 isin [0 1] and 119879119910 = 1199102 isin[0 1] for all 119909 isin 119860 and 119910 isin 119861 Thus 120573(119879119909 119879119910) ge 1 for all119909 isin 119860 and 119910 isin 119861 Now suppose 119860 = [0 12] and 119909
0= 14
Then 1198791199090= 18 isin [0 1] and [0 12] sub [0 1] Hence
120573(119860 1198791199090) ge 1 Now we show that 119879 satisfies the condition
(119867) First note that for each 120576 gt 0 there exists 119911 isin 119883 suchthat sup
119886isin119879119911119889(119911 119886) lt 120576 Now we show that for each 119909 isin 119883
there exists 119910 isin 119879119909 such that 119867(119879119909 119879119910) = sup119887isin119879119910
119889(119910 119887) If0 le 119909 le 1 then 119879119909 = 1199092 119879(1199092) = 1199094 and
119867(119879119909 119879 (119909
2)) = 119867(
119909
2
119909
4)
=119909
4= sup119887isin119879(1199092)
119889 (119909
2 119887)
(22)
Since for 1 lt 119909 le 32 we have 52 lt 4119909 minus (32) le 92119879(4119909 minus (32)) = 0 Thus
119867(119879119909 119879 (4119909 minus3
2)) = 119867(4119909 minus
3
2 0)
= 4119909 minus3
2= sup119887isin119879(4119909minus32)
119889(4119909 minus3
2 119887)
(23)
If 32 lt 119909 le 92 then 119879119909 = 0 and 119879(0) = 0 Hence
119867(119879119909 119879 (0)) = 119867 (0 0)
= 0 = sup119887isin119879(0)
119889 (0 119887) (24)
It is easy to check that 119879 satisfies the conditions (119877) and (119870)Note that 0 is the endpoint of 119879
Now we add an assumption to obtain uniqueness ofendpoint In this way we introduce a new notion Let 119883 bea set and 120573 2119883 times 2119883 rarr [0infin) a map We say that the set119883has the property (119866
120573) whenever 120573(119860 119861) ge 1 for all subsets 119860
and 119861 of119883 with 119860 sube 119861 or 119861 sube 119860
Corollary 6 Let (119883 119889) be a complete metric space 120573
2119883 times 2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883)a 120573-admissible 120573-generalized weak contractive multifunctionwhich has the properties (119877) (119870) and (119867) Suppose that thereexist a subset 119860 of119883 and 119909
0isin 119860 such that 120573(119860 119879119909
0) ge 1 If 119879
has the approximate endpoint property and119883 has the property(119866120573) then 119879 has a unique endpoint
Proof By using Theorem 3 119879 has a endpoint If 119879 hastwo distinct endpoints 119909lowast and 119910lowast then 120573(119879119909lowast 119879119910lowast) =120573(119909lowast 119910lowast) ge 1 because119883 has the property (119866
120573) Hence
119889 (119909lowast 119910lowast) le 119867 (119879119909lowast 119879119910lowast)
le 120573 (119879119909lowast 119879119910lowast)119867 (119879119909lowast 119879119910lowast)
le 120595 (119873 (119909lowast 119910lowast)) lt 119873 (119909lowast 119910lowast)
= 119889 (119909lowast 119910lowast)
(25)
which is a contradictionThus 119879 has a unique endpoint
In Example 5 119879 has a unique endpoint while119883 does hasnot the property (119866
120573) Also119879 has the property (119877) while119879 is
not lower semicontinuous To see this consider the sequence119909119899 defined by
119909119899=
1 minus1
119899119899 = 2119896
1 +1
119899119899 = 2119896 minus 1
(26)
for 119896 ge 1 and put 119910 = 12 and 1199090= 1 Then 119909
119899rarr 1 and 119910 isin
1198791199090= 12 Let 119910
119899 be an arbitrary sequence in119883 such that
119910119899isin 119879119909119899for all 119899 ge 1 Then 119910
2119896minus1isin 1198791199092119896minus1
and 1199102119896
isin 1198791199092119896
for all 119896 But 1199102119896minus1
= 41199092119896minus1
minus (32) for sufficiently large 119896and 119910
2119896= 11990921198962 for all 119896 since 119910
2119896minus1rarr 52 119910
119899999424999426999456 12 This
implies that 119879 is not lower semicontinuous
The Scientific World Journal 5
Corollary 7 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a 120573-admissiblemultifunctionwhich has the properties (119877) (119870) and(119867) Suppose that 119883 has the property (119866
120573) and there exist a
subset 119860 of X 1199090isin 119860 and 119896 isin [0 1) such that 120573(119860 119879119909
0) ge 1
and 120573(119879119909 119879119910)119867(119879119909 119879119910) le 119896119873(119909 119910) for all 119909 119910 isin 119883 Then119879 has a unique endpoint if and only if 119879 has the approximateendpoint property
Proof It is sufficient that we define 120595(119905) = 119896119905 for all 119905 ge 0Then Theorem 3 and Corollary 6 guarantee the result
It has been proved that lower semicontinuity of themulti-function 119879 and the property (119877) are independent conditions[9]We can replace lower semicontinuity of themultifunctioninstead of the property (119877) to obtain the next result Its proofis similar to the proof of Theorem 3
Theorem 8 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a lowersemicontinuous 120573-admissible 120573-generalized weak contractivemultifunction which has the properties (119870) and (119867) Supposethat there exist a subset 119860 of 119883 and 119909
0isin 119860 such that
120573(119860 1198791199090) ge 1 Then 119879 has the approximate endpoint property
if and only if 119879 has an endpoint
Corollary 9 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a lowersemicontinuous 120573-admissible 120573-generalized weak contractivemultifunction which has the properties (119870) and (119867) Supposethat there exist a subset 119860 of 119883 and 119909
0isin 119860 such that
120573(119860 1198791199090) ge 1 If 119879 has the approximate endpoint property
and 119883 has the property (119866120573) then 119879 has a unique endpoint
Corollary 10 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a 120573-admissiblemultifunctionwhich has the properties (119877) (119870) and(119867) Suppose that 119883 has the property (119866
120573) and there exist a
subset 119860 of 119883 1199090isin 119860 and 119896 isin [0 1) such that 120573(119860 119879119909
0) ge 1
and 120573(119879119909 119879119910)119867(119879119909 119879119910) le 119896119873(119909 119910) for all 119909 119910 isin 119883 If 119879 hasthe approximate endpoint property then Fix(119879) = End(119879) =119909
Proof If we put 120595(119905) = 119896119905 then by using Theorem 210 in[9]119879 has a fixed point Since119879 has the approximate endpointproperty by using Corollary 7 119879 has a unique endpoint such119909 Let 119910 isin Fix(119879) If 119879119909 = 119879119910 then 119910 = 119909 If 119879119909 = 119879119910 then120573(119879119909 119879119910) ge 1 because119883 has the property (119866
120573) Also we have
119889 (119909 119910) le 119867 (119909 119879119910) = 119867 (119879119909 119879119910)
le 120573 (119879119909 119879119910)119867 (119879119909 119879119910)
le 119896119873 (119909 119910)
(27)
But 119873(119909 119910) = max119889(119909 119910) 119889(119909 119879119909) 119889(119910 119879119910) (119889(119909 119879119910) +119889(119910 119879119909))2 = 119889(119909 119910) Thus 119889(119909 119910) = 0 and so Fix(119879) =End(119879) = 119909
Next corollary shows us the role of a point in the existenceof endpoints
Corollary 11 Let (119883 119889) be a complete metric space 119909lowast isin 119883 afixed element and 119879 119883 rarr 119862119861(119883) a multifunction such that119879 has the property (119867) and 119909lowast isin 119879119909cap119879119910 for all subsets119860 and119861 of119883 with 119909lowast isin 119860 cap 119861 and all 119909 isin 119860 and 119910 isin 119861 Assume that119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119909lowast isin 119879119909 cap 119879119910where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Suppose that there exist a subset 119860
0of 119883 and 119909
0isin 1198600such
that 119909lowast isin 1198600cap1198791199090 Assume that for each convergent sequence
119909119899 in 119883 with 119909
119899rarr 119909 and 119909lowast isin 119879119909
119899minus1cap 119879119909119899 for all 119899 ge 1
one has 119909lowast isin 119879119909119899cap 119879119909 Also for each sequence 119909
119899 in 119883
with 119909lowast isin 119879119909119899minus1
cap 119879119909119899for all 119899 ge 1 there exists a natural
number 119896 such that 119909lowast isin 119879119909119898cap119879119909119899for all119898 gt 119899 ge 119896 Then 119879
has an endpoint if and only if 119879 has the approximate endpointproperty
Proof It is sufficient we define 120573 2119883 times 2119883 rarr [0infin) by120573(119860 119861) = 1 whenever 119909lowast isin 119860cap119861 and 120573(119860 119861) = 0 otherwiseand then we use Theorem 3
Corollary 12 Let (119883 119889) be a complete metric space xlowast isin 119883a fixed element and 119879 119883 rarr 119862119861(119883) a lower semicontinuousmultifunction such that 119879 has the property (119867) and 119909lowast isin 119879119909cap119879119910 for all subsets 119860 and 119861 of119883 with 119909lowast isin 119860 cap 119861 and all 119909 isin 119860and 119910 isin 119861 Assume that
119867(119879119909 119879119910) le 120595 (119873 (119909 119910)) (28)
for all 119909 119910 isin 119883 with 119909lowast isin 119879119909 cap 119879119910 where 120595 [0 +infin) rarr[0 +infin) is a nondecreasing upper semicontinuous functionsuch that 120595(119905) lt 119905 for all 119905 gt 0 Suppose that there exist asubset 119860
0of119883 and 119909
0isin 1198600such that 119909lowast isin 119860
0cap 1198791199090 Assume
that for each convergent sequence 119909119899 in119883 with 119909
119899rarr 119909 and
119909lowast isin 119879119909119899minus1
cap119879119909119899for all 119899 ge 1 we have 119909lowast isin 119879119909
119899cap119879119909 Then 119879
has an endpoint if and only if 119879 has the approximate endpointproperty
Proof It is sufficient to define 120573 2119883 times 2119883 rarr [0infin) by120573(119860 119861) = 1 whenever 119909lowast isin 119860cap119861 and 120573(119860 119861) = 0 otherwiseand then we use Theorem 8
Let (119883 119889 le) be an ordered metric space Define the order⪯ on arbitrary subsets 119860 and 119861 of 119883 by 119860 ⪯ 119861 if and only iffor each 119886 isin 119860 there exists 119887 isin 119861 such that 119886 le 119887 It is easy tocheck that (119862119861(119883) ⪯) is a partially ordered set
Theorem 13 Let (119883 119889 le) be a complete ordered metric spaceand 119879 a closed and bounded valued multifunction on 119883 suchthat 119879 has the property (119867) and 119879119909 ⪯ 119879119910 for all subsets 119860and 119861 of 119883 with 119860 ⪯ 119861 and all 119909 isin 119860 and 119910 isin 119861 Assumethat 119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119879119909 ⪯ 119879119910where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Suppose that there exist a subset 119860
0of 119883 and 119909
0isin 1198600such
that 1198600
⪯ 1198791199090 Assume that for each convergent sequence
119909119899 in 119883 with 119909
119899rarr 119909 and 119879119909
119899minus1⪯ 119879119909
119899 for all 119899 ge 1
one has 119879119909119899
⪯ 119879119909 Also for each sequence 119909119899 in 119883 with
119879119909119899minus1
⪯ 119879119909119899for all 119899 ge 1 there exists a natural number 119896 such
that 119879119909119898⪯ 119879119909119899for all 119898 gt 119899 ge 119896 Then 119879 has an endpoint if
and only if 119879 has the approximate endpoint property
6 The Scientific World Journal
Proof Define 120573(119860 119861) = 1 whenever 119860 ⪯ 119861 and 120573(119860 119861) = 0otherwise and then we use Theorem 3
Corollary 14 Let (119883 119889 le) be a complete ordered metric spaceand 119879 a closed and bounded valued multifunction on 119883 suchthat 119879 has the property (119867) and 119879119909 ⪯ 119879119910 for all subsets 119860and 119861 of 119883 with 119860 ⪯ 119861 all 119909 isin 119860 and 119910 isin 119861 Assume that119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119879119909 ⪯ 119879119910where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Suppose that there exist a subset 119860
0of 119883 and 119909
0isin 1198600such
that 1198600
⪯ 1198791199090 Assume that for each convergent sequence
119909119899 in 119883 with 119909
119899rarr 119909 and 119879119909
119899minus1⪯ 119879119909
119899 for all 119899 ge 1
one has 119879119909119899
⪯ 119879119909 Also for each sequence 119909119899 in 119883 with
119879119909119899minus1
⪯ 119879119909119899for all 119899 ge 1 there exists a natural number 119896 such
that 119879119909119898⪯ 119879119909119899for all 119898 gt 119899 ge 119896 If 119879 has the approximate
endpoint property and 119860 ⪯ 119861 for all subsets119860 and 119861 of119883 with119860 sube 119861 or 119861 sube 119860 then 119879 has a unique endpoint
Proof Define 120573(119860 119861) = 1 whenever 119860 ⪯ 119861 and 120573(119860 119861) = 0otherwise and then we use Corollary 6
Let (119883 119889) be a metric space and 119879 119883 rarr 2119883
a multifunction We say that 119879 is an 119867119878-multifunctionwhenever for each 119909 isin 119883 there exists 119910 isin 119879119909 such that119867(119879119909 119879119910) = sup
119887isin119879y119889(119910 119887) It is obvious that each 119867119878-multifunction is an multifunction which has the property(119867) Thus one can conclude similar results to above ones for119867119878-multifunctions Here we provide some ones Althoughby considering 119867119878-multifunction we restrict ourselves weobtain strange results with respect to above ones One canprove the following by reading exactly the proofs of similarabove results
Theorem 15 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a 120573-admissible 120573-generalized weak contractive 119867119878-multifunctionwhich has the properties (119877) and (119870) Suppose that there exist asubset119860 of119883 and 119909
0isin 119860 such that 120573(119860 119879119909
0) ge 1 Then 119879 has
an endpoint and so 119879 has the approximate endpoint property
Theorem 16 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) alower semicontinuous 120573-admissible and 120573-generalized weakcontractive 119867119878-multifunction which has the property (119870)Suppose that there exist a subset 119860 of 119883 and 119909
0isin 119860 such
that 120573(119860 1198791199090) ge 1 Then 119879 has an endpoint and so 119879 has
the approximate endpoint property
The next result is a consequence of Theorem 15
Corollary 17 Let (119883 119889) be a complete metric space 119909lowast isin 119883 afixed element and 119879 119883 rarr 119862119861(119883) an 119867119878-multifunctionsuch that 119909lowast isin 119879119909 cap 119879119910 for all subsets 119860 and 119861 of 119883 with119909lowast isin 119860 cap 119861 all 119909 isin 119860 and 119910 isin 119861 Assume that 119867(119879119909 119879119910) le120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119909lowast isin 119879119909 cap 119879119910 where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing upper semicontinuousfunction such that120595(119905) lt 119905 for all 119905 gt 0 Suppose that there exista subset119860
0of119883 and 119909
0isin 1198600such that 119909lowast isin 119860
0cap1198791199090 Assume
that for each convergent sequence 119909119899 in119883 with 119909
119899rarr 119909 and
119909lowast isin 119879119909119899minus1
cap119879119909119899for all 119899 ge 1 one has 119909lowast isin 119879119909
119899cap119879119909 Also for
each sequence 119909119899 in 119883 with 119909lowast isin 119879119909
119899minus1cap 119879119909119899for all 119899 ge 1
there exists a natural number 119896 such that 119909lowast isin 119879119909119898cap 119879119909119899
for all 119898 gt 119899 ge 119896 Then 119879 has an endpoint and so 119879 has theapproximate endpoint property
The next result is a consequence of Theorem 16
Corollary 18 Let (119883 119889 le) be a complete ordered metric spaceand 119879 a closed and bounded valued lower semicontinuous119867119878-multifunction on 119883 such that 119879119909 ⪯ 119879119910 for all subsets 119860 and119861 of 119883 with 119860 ⪯ 119861 all 119909 isin 119860 and 119910 isin 119861 Assume that119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119879119909 ⪯ 119879119910where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Suppose that there exist a subset 119860
0of 119883 and 119909
0isin 1198600such
that 1198600⪯ 1198791199090 Assume that for each sequence 119909
119899 in 119883 with
119879119909119899minus1
⪯ 119879119909119899for all 119899 ge 1 there exists a natural number 119896 such
that 119879119909119898
⪯ 119879119909119899for all 119898 gt 119899 ge 119896 Then 119879 has an endpoint
and so 119879 has the approximate endpoint property
Acknowledgments
This work was completed while the second author (DrGopal) was visiting the Azarbaijan University of ShahidMadani Azarshahr Tabriz Iran during the summer of 2012He thanks Professor Sh Rezapour and the University fortheir hospitality and support The second author gratefullyacknowledges the support from the CSIR govternment ofIndia Grant no-25(0215)13EMR-II
References
[1] A Amini-Harandi ldquoEndpoints of set-valued contractions inmetric spacesrdquo Nonlinear Analysis Theory Methods and Appli-cations vol 72 no 1 pp 132ndash134 2010
[2] R P Agarwal D OrsquoRegan and D R Sahu Fixed Point Theoryfor Lipschitzian-TypeMappings with Applications Springer NewYork NY USA 2009
[3] S Moradi and F Khojasteh ldquoEndpoints of multi-valued gener-alized weak contraction mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 74 no 6 pp 2170ndash2174 2011
[4] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012
[5] D BaleanuHMohammadi and Sh Rezapour ldquoSome existenceresults on nonlinear fractional differential equationsrdquo Philo-sophical Transactions of the Royal Society vol 371 Article ID20120144 2013
[6] M Jleli and B Samet ldquoBest proximity points for 120572-120595-proximalcontractive type mappings and applicationsrdquo Bulletin des Sci-ences Mathematiques 2013
[7] M A Miandaragh M Postolache and Sh Rezapour ldquoSomeapproximate fixed point results for generalized-contractivemappings rdquo To appear in Scientific Bulletin-UniversityPolitehnica of Bucharest A
[8] Sh Rezapour and J H Asl ldquoA simple method for obtainingcoupled fixed points of 120572-120595-contractive type mappingsrdquo Inter-national Journal of Analysis Article ID 438029 7 pages 2013
The Scientific World Journal 7
[9] H Alikhani V Rakocevic Sh Rezapour and N ShahzadldquoFixed points of proximinal valued 120573-120595-contractive multifunc-tionsrdquo To appear in Journal of Nonlinear and Convex Analysis
[10] H Alikhani Sh Rezapour and N Shahzad ldquoFixed points of anew type contractive mappings and multifunctionsrdquo To appearin Filomat
[11] S M A Aleomraninejad Sh Rezapour and N Shahzad ldquoOnfixed points of 120572-120595-contractive multifunctionsrdquo Fixed PointTheory and Applications vol 2012 article 212 8 pages 2012
[12] B Mohammadi Sh Rezapour and N Shahzad ldquoSome resultson fixed points of 120572-120595-ciric generalized multifunctionsrdquo FixedPoint Theory and Applications vol 2013 article 24 2013
[13] S M A Aleomraninejad Sh Rezapour andN Shahzad ldquoSomefixed point results on ametric space with a graphrdquoTopology andIts Applications vol 159 no 3 pp 659ndash663 2012
[14] J M Borwein and A S Lewis Convex Analysis and NonlinearOptimization Theory and Examples Springer New York NYUSA 2000
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The Scientific World Journal 5
Corollary 7 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a 120573-admissiblemultifunctionwhich has the properties (119877) (119870) and(119867) Suppose that 119883 has the property (119866
120573) and there exist a
subset 119860 of X 1199090isin 119860 and 119896 isin [0 1) such that 120573(119860 119879119909
0) ge 1
and 120573(119879119909 119879119910)119867(119879119909 119879119910) le 119896119873(119909 119910) for all 119909 119910 isin 119883 Then119879 has a unique endpoint if and only if 119879 has the approximateendpoint property
Proof It is sufficient that we define 120595(119905) = 119896119905 for all 119905 ge 0Then Theorem 3 and Corollary 6 guarantee the result
It has been proved that lower semicontinuity of themulti-function 119879 and the property (119877) are independent conditions[9]We can replace lower semicontinuity of themultifunctioninstead of the property (119877) to obtain the next result Its proofis similar to the proof of Theorem 3
Theorem 8 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a lowersemicontinuous 120573-admissible 120573-generalized weak contractivemultifunction which has the properties (119870) and (119867) Supposethat there exist a subset 119860 of 119883 and 119909
0isin 119860 such that
120573(119860 1198791199090) ge 1 Then 119879 has the approximate endpoint property
if and only if 119879 has an endpoint
Corollary 9 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a lowersemicontinuous 120573-admissible 120573-generalized weak contractivemultifunction which has the properties (119870) and (119867) Supposethat there exist a subset 119860 of 119883 and 119909
0isin 119860 such that
120573(119860 1198791199090) ge 1 If 119879 has the approximate endpoint property
and 119883 has the property (119866120573) then 119879 has a unique endpoint
Corollary 10 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a 120573-admissiblemultifunctionwhich has the properties (119877) (119870) and(119867) Suppose that 119883 has the property (119866
120573) and there exist a
subset 119860 of 119883 1199090isin 119860 and 119896 isin [0 1) such that 120573(119860 119879119909
0) ge 1
and 120573(119879119909 119879119910)119867(119879119909 119879119910) le 119896119873(119909 119910) for all 119909 119910 isin 119883 If 119879 hasthe approximate endpoint property then Fix(119879) = End(119879) =119909
Proof If we put 120595(119905) = 119896119905 then by using Theorem 210 in[9]119879 has a fixed point Since119879 has the approximate endpointproperty by using Corollary 7 119879 has a unique endpoint such119909 Let 119910 isin Fix(119879) If 119879119909 = 119879119910 then 119910 = 119909 If 119879119909 = 119879119910 then120573(119879119909 119879119910) ge 1 because119883 has the property (119866
120573) Also we have
119889 (119909 119910) le 119867 (119909 119879119910) = 119867 (119879119909 119879119910)
le 120573 (119879119909 119879119910)119867 (119879119909 119879119910)
le 119896119873 (119909 119910)
(27)
But 119873(119909 119910) = max119889(119909 119910) 119889(119909 119879119909) 119889(119910 119879119910) (119889(119909 119879119910) +119889(119910 119879119909))2 = 119889(119909 119910) Thus 119889(119909 119910) = 0 and so Fix(119879) =End(119879) = 119909
Next corollary shows us the role of a point in the existenceof endpoints
Corollary 11 Let (119883 119889) be a complete metric space 119909lowast isin 119883 afixed element and 119879 119883 rarr 119862119861(119883) a multifunction such that119879 has the property (119867) and 119909lowast isin 119879119909cap119879119910 for all subsets119860 and119861 of119883 with 119909lowast isin 119860 cap 119861 and all 119909 isin 119860 and 119910 isin 119861 Assume that119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119909lowast isin 119879119909 cap 119879119910where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Suppose that there exist a subset 119860
0of 119883 and 119909
0isin 1198600such
that 119909lowast isin 1198600cap1198791199090 Assume that for each convergent sequence
119909119899 in 119883 with 119909
119899rarr 119909 and 119909lowast isin 119879119909
119899minus1cap 119879119909119899 for all 119899 ge 1
one has 119909lowast isin 119879119909119899cap 119879119909 Also for each sequence 119909
119899 in 119883
with 119909lowast isin 119879119909119899minus1
cap 119879119909119899for all 119899 ge 1 there exists a natural
number 119896 such that 119909lowast isin 119879119909119898cap119879119909119899for all119898 gt 119899 ge 119896 Then 119879
has an endpoint if and only if 119879 has the approximate endpointproperty
Proof It is sufficient we define 120573 2119883 times 2119883 rarr [0infin) by120573(119860 119861) = 1 whenever 119909lowast isin 119860cap119861 and 120573(119860 119861) = 0 otherwiseand then we use Theorem 3
Corollary 12 Let (119883 119889) be a complete metric space xlowast isin 119883a fixed element and 119879 119883 rarr 119862119861(119883) a lower semicontinuousmultifunction such that 119879 has the property (119867) and 119909lowast isin 119879119909cap119879119910 for all subsets 119860 and 119861 of119883 with 119909lowast isin 119860 cap 119861 and all 119909 isin 119860and 119910 isin 119861 Assume that
119867(119879119909 119879119910) le 120595 (119873 (119909 119910)) (28)
for all 119909 119910 isin 119883 with 119909lowast isin 119879119909 cap 119879119910 where 120595 [0 +infin) rarr[0 +infin) is a nondecreasing upper semicontinuous functionsuch that 120595(119905) lt 119905 for all 119905 gt 0 Suppose that there exist asubset 119860
0of119883 and 119909
0isin 1198600such that 119909lowast isin 119860
0cap 1198791199090 Assume
that for each convergent sequence 119909119899 in119883 with 119909
119899rarr 119909 and
119909lowast isin 119879119909119899minus1
cap119879119909119899for all 119899 ge 1 we have 119909lowast isin 119879119909
119899cap119879119909 Then 119879
has an endpoint if and only if 119879 has the approximate endpointproperty
Proof It is sufficient to define 120573 2119883 times 2119883 rarr [0infin) by120573(119860 119861) = 1 whenever 119909lowast isin 119860cap119861 and 120573(119860 119861) = 0 otherwiseand then we use Theorem 8
Let (119883 119889 le) be an ordered metric space Define the order⪯ on arbitrary subsets 119860 and 119861 of 119883 by 119860 ⪯ 119861 if and only iffor each 119886 isin 119860 there exists 119887 isin 119861 such that 119886 le 119887 It is easy tocheck that (119862119861(119883) ⪯) is a partially ordered set
Theorem 13 Let (119883 119889 le) be a complete ordered metric spaceand 119879 a closed and bounded valued multifunction on 119883 suchthat 119879 has the property (119867) and 119879119909 ⪯ 119879119910 for all subsets 119860and 119861 of 119883 with 119860 ⪯ 119861 and all 119909 isin 119860 and 119910 isin 119861 Assumethat 119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119879119909 ⪯ 119879119910where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Suppose that there exist a subset 119860
0of 119883 and 119909
0isin 1198600such
that 1198600
⪯ 1198791199090 Assume that for each convergent sequence
119909119899 in 119883 with 119909
119899rarr 119909 and 119879119909
119899minus1⪯ 119879119909
119899 for all 119899 ge 1
one has 119879119909119899
⪯ 119879119909 Also for each sequence 119909119899 in 119883 with
119879119909119899minus1
⪯ 119879119909119899for all 119899 ge 1 there exists a natural number 119896 such
that 119879119909119898⪯ 119879119909119899for all 119898 gt 119899 ge 119896 Then 119879 has an endpoint if
and only if 119879 has the approximate endpoint property
6 The Scientific World Journal
Proof Define 120573(119860 119861) = 1 whenever 119860 ⪯ 119861 and 120573(119860 119861) = 0otherwise and then we use Theorem 3
Corollary 14 Let (119883 119889 le) be a complete ordered metric spaceand 119879 a closed and bounded valued multifunction on 119883 suchthat 119879 has the property (119867) and 119879119909 ⪯ 119879119910 for all subsets 119860and 119861 of 119883 with 119860 ⪯ 119861 all 119909 isin 119860 and 119910 isin 119861 Assume that119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119879119909 ⪯ 119879119910where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Suppose that there exist a subset 119860
0of 119883 and 119909
0isin 1198600such
that 1198600
⪯ 1198791199090 Assume that for each convergent sequence
119909119899 in 119883 with 119909
119899rarr 119909 and 119879119909
119899minus1⪯ 119879119909
119899 for all 119899 ge 1
one has 119879119909119899
⪯ 119879119909 Also for each sequence 119909119899 in 119883 with
119879119909119899minus1
⪯ 119879119909119899for all 119899 ge 1 there exists a natural number 119896 such
that 119879119909119898⪯ 119879119909119899for all 119898 gt 119899 ge 119896 If 119879 has the approximate
endpoint property and 119860 ⪯ 119861 for all subsets119860 and 119861 of119883 with119860 sube 119861 or 119861 sube 119860 then 119879 has a unique endpoint
Proof Define 120573(119860 119861) = 1 whenever 119860 ⪯ 119861 and 120573(119860 119861) = 0otherwise and then we use Corollary 6
Let (119883 119889) be a metric space and 119879 119883 rarr 2119883
a multifunction We say that 119879 is an 119867119878-multifunctionwhenever for each 119909 isin 119883 there exists 119910 isin 119879119909 such that119867(119879119909 119879119910) = sup
119887isin119879y119889(119910 119887) It is obvious that each 119867119878-multifunction is an multifunction which has the property(119867) Thus one can conclude similar results to above ones for119867119878-multifunctions Here we provide some ones Althoughby considering 119867119878-multifunction we restrict ourselves weobtain strange results with respect to above ones One canprove the following by reading exactly the proofs of similarabove results
Theorem 15 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a 120573-admissible 120573-generalized weak contractive 119867119878-multifunctionwhich has the properties (119877) and (119870) Suppose that there exist asubset119860 of119883 and 119909
0isin 119860 such that 120573(119860 119879119909
0) ge 1 Then 119879 has
an endpoint and so 119879 has the approximate endpoint property
Theorem 16 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) alower semicontinuous 120573-admissible and 120573-generalized weakcontractive 119867119878-multifunction which has the property (119870)Suppose that there exist a subset 119860 of 119883 and 119909
0isin 119860 such
that 120573(119860 1198791199090) ge 1 Then 119879 has an endpoint and so 119879 has
the approximate endpoint property
The next result is a consequence of Theorem 15
Corollary 17 Let (119883 119889) be a complete metric space 119909lowast isin 119883 afixed element and 119879 119883 rarr 119862119861(119883) an 119867119878-multifunctionsuch that 119909lowast isin 119879119909 cap 119879119910 for all subsets 119860 and 119861 of 119883 with119909lowast isin 119860 cap 119861 all 119909 isin 119860 and 119910 isin 119861 Assume that 119867(119879119909 119879119910) le120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119909lowast isin 119879119909 cap 119879119910 where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing upper semicontinuousfunction such that120595(119905) lt 119905 for all 119905 gt 0 Suppose that there exista subset119860
0of119883 and 119909
0isin 1198600such that 119909lowast isin 119860
0cap1198791199090 Assume
that for each convergent sequence 119909119899 in119883 with 119909
119899rarr 119909 and
119909lowast isin 119879119909119899minus1
cap119879119909119899for all 119899 ge 1 one has 119909lowast isin 119879119909
119899cap119879119909 Also for
each sequence 119909119899 in 119883 with 119909lowast isin 119879119909
119899minus1cap 119879119909119899for all 119899 ge 1
there exists a natural number 119896 such that 119909lowast isin 119879119909119898cap 119879119909119899
for all 119898 gt 119899 ge 119896 Then 119879 has an endpoint and so 119879 has theapproximate endpoint property
The next result is a consequence of Theorem 16
Corollary 18 Let (119883 119889 le) be a complete ordered metric spaceand 119879 a closed and bounded valued lower semicontinuous119867119878-multifunction on 119883 such that 119879119909 ⪯ 119879119910 for all subsets 119860 and119861 of 119883 with 119860 ⪯ 119861 all 119909 isin 119860 and 119910 isin 119861 Assume that119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119879119909 ⪯ 119879119910where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Suppose that there exist a subset 119860
0of 119883 and 119909
0isin 1198600such
that 1198600⪯ 1198791199090 Assume that for each sequence 119909
119899 in 119883 with
119879119909119899minus1
⪯ 119879119909119899for all 119899 ge 1 there exists a natural number 119896 such
that 119879119909119898
⪯ 119879119909119899for all 119898 gt 119899 ge 119896 Then 119879 has an endpoint
and so 119879 has the approximate endpoint property
Acknowledgments
This work was completed while the second author (DrGopal) was visiting the Azarbaijan University of ShahidMadani Azarshahr Tabriz Iran during the summer of 2012He thanks Professor Sh Rezapour and the University fortheir hospitality and support The second author gratefullyacknowledges the support from the CSIR govternment ofIndia Grant no-25(0215)13EMR-II
References
[1] A Amini-Harandi ldquoEndpoints of set-valued contractions inmetric spacesrdquo Nonlinear Analysis Theory Methods and Appli-cations vol 72 no 1 pp 132ndash134 2010
[2] R P Agarwal D OrsquoRegan and D R Sahu Fixed Point Theoryfor Lipschitzian-TypeMappings with Applications Springer NewYork NY USA 2009
[3] S Moradi and F Khojasteh ldquoEndpoints of multi-valued gener-alized weak contraction mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 74 no 6 pp 2170ndash2174 2011
[4] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012
[5] D BaleanuHMohammadi and Sh Rezapour ldquoSome existenceresults on nonlinear fractional differential equationsrdquo Philo-sophical Transactions of the Royal Society vol 371 Article ID20120144 2013
[6] M Jleli and B Samet ldquoBest proximity points for 120572-120595-proximalcontractive type mappings and applicationsrdquo Bulletin des Sci-ences Mathematiques 2013
[7] M A Miandaragh M Postolache and Sh Rezapour ldquoSomeapproximate fixed point results for generalized-contractivemappings rdquo To appear in Scientific Bulletin-UniversityPolitehnica of Bucharest A
[8] Sh Rezapour and J H Asl ldquoA simple method for obtainingcoupled fixed points of 120572-120595-contractive type mappingsrdquo Inter-national Journal of Analysis Article ID 438029 7 pages 2013
The Scientific World Journal 7
[9] H Alikhani V Rakocevic Sh Rezapour and N ShahzadldquoFixed points of proximinal valued 120573-120595-contractive multifunc-tionsrdquo To appear in Journal of Nonlinear and Convex Analysis
[10] H Alikhani Sh Rezapour and N Shahzad ldquoFixed points of anew type contractive mappings and multifunctionsrdquo To appearin Filomat
[11] S M A Aleomraninejad Sh Rezapour and N Shahzad ldquoOnfixed points of 120572-120595-contractive multifunctionsrdquo Fixed PointTheory and Applications vol 2012 article 212 8 pages 2012
[12] B Mohammadi Sh Rezapour and N Shahzad ldquoSome resultson fixed points of 120572-120595-ciric generalized multifunctionsrdquo FixedPoint Theory and Applications vol 2013 article 24 2013
[13] S M A Aleomraninejad Sh Rezapour andN Shahzad ldquoSomefixed point results on ametric space with a graphrdquoTopology andIts Applications vol 159 no 3 pp 659ndash663 2012
[14] J M Borwein and A S Lewis Convex Analysis and NonlinearOptimization Theory and Examples Springer New York NYUSA 2000
Submit your manuscripts athttpwwwhindawicom
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Volume 2013
Advances in
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ProbabilityandStatistics
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Advances in
DecisionSciences
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Stochastic AnalysisInternational Journal of
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The Scientific World Journal
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Discrete Mathematics
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Differential EquationsInternational Journal of
Volume 2013
6 The Scientific World Journal
Proof Define 120573(119860 119861) = 1 whenever 119860 ⪯ 119861 and 120573(119860 119861) = 0otherwise and then we use Theorem 3
Corollary 14 Let (119883 119889 le) be a complete ordered metric spaceand 119879 a closed and bounded valued multifunction on 119883 suchthat 119879 has the property (119867) and 119879119909 ⪯ 119879119910 for all subsets 119860and 119861 of 119883 with 119860 ⪯ 119861 all 119909 isin 119860 and 119910 isin 119861 Assume that119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119879119909 ⪯ 119879119910where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Suppose that there exist a subset 119860
0of 119883 and 119909
0isin 1198600such
that 1198600
⪯ 1198791199090 Assume that for each convergent sequence
119909119899 in 119883 with 119909
119899rarr 119909 and 119879119909
119899minus1⪯ 119879119909
119899 for all 119899 ge 1
one has 119879119909119899
⪯ 119879119909 Also for each sequence 119909119899 in 119883 with
119879119909119899minus1
⪯ 119879119909119899for all 119899 ge 1 there exists a natural number 119896 such
that 119879119909119898⪯ 119879119909119899for all 119898 gt 119899 ge 119896 If 119879 has the approximate
endpoint property and 119860 ⪯ 119861 for all subsets119860 and 119861 of119883 with119860 sube 119861 or 119861 sube 119860 then 119879 has a unique endpoint
Proof Define 120573(119860 119861) = 1 whenever 119860 ⪯ 119861 and 120573(119860 119861) = 0otherwise and then we use Corollary 6
Let (119883 119889) be a metric space and 119879 119883 rarr 2119883
a multifunction We say that 119879 is an 119867119878-multifunctionwhenever for each 119909 isin 119883 there exists 119910 isin 119879119909 such that119867(119879119909 119879119910) = sup
119887isin119879y119889(119910 119887) It is obvious that each 119867119878-multifunction is an multifunction which has the property(119867) Thus one can conclude similar results to above ones for119867119878-multifunctions Here we provide some ones Althoughby considering 119867119878-multifunction we restrict ourselves weobtain strange results with respect to above ones One canprove the following by reading exactly the proofs of similarabove results
Theorem 15 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) a 120573-admissible 120573-generalized weak contractive 119867119878-multifunctionwhich has the properties (119877) and (119870) Suppose that there exist asubset119860 of119883 and 119909
0isin 119860 such that 120573(119860 119879119909
0) ge 1 Then 119879 has
an endpoint and so 119879 has the approximate endpoint property
Theorem 16 Let (119883 119889) be a complete metric space 120573 2119883 times
2119883 rarr [0infin) a mapping and 119879 119883 rarr 119862119861(119883) alower semicontinuous 120573-admissible and 120573-generalized weakcontractive 119867119878-multifunction which has the property (119870)Suppose that there exist a subset 119860 of 119883 and 119909
0isin 119860 such
that 120573(119860 1198791199090) ge 1 Then 119879 has an endpoint and so 119879 has
the approximate endpoint property
The next result is a consequence of Theorem 15
Corollary 17 Let (119883 119889) be a complete metric space 119909lowast isin 119883 afixed element and 119879 119883 rarr 119862119861(119883) an 119867119878-multifunctionsuch that 119909lowast isin 119879119909 cap 119879119910 for all subsets 119860 and 119861 of 119883 with119909lowast isin 119860 cap 119861 all 119909 isin 119860 and 119910 isin 119861 Assume that 119867(119879119909 119879119910) le120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119909lowast isin 119879119909 cap 119879119910 where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing upper semicontinuousfunction such that120595(119905) lt 119905 for all 119905 gt 0 Suppose that there exista subset119860
0of119883 and 119909
0isin 1198600such that 119909lowast isin 119860
0cap1198791199090 Assume
that for each convergent sequence 119909119899 in119883 with 119909
119899rarr 119909 and
119909lowast isin 119879119909119899minus1
cap119879119909119899for all 119899 ge 1 one has 119909lowast isin 119879119909
119899cap119879119909 Also for
each sequence 119909119899 in 119883 with 119909lowast isin 119879119909
119899minus1cap 119879119909119899for all 119899 ge 1
there exists a natural number 119896 such that 119909lowast isin 119879119909119898cap 119879119909119899
for all 119898 gt 119899 ge 119896 Then 119879 has an endpoint and so 119879 has theapproximate endpoint property
The next result is a consequence of Theorem 16
Corollary 18 Let (119883 119889 le) be a complete ordered metric spaceand 119879 a closed and bounded valued lower semicontinuous119867119878-multifunction on 119883 such that 119879119909 ⪯ 119879119910 for all subsets 119860 and119861 of 119883 with 119860 ⪯ 119861 all 119909 isin 119860 and 119910 isin 119861 Assume that119867(119879119909 119879119910) le 120595(119873(119909 119910)) for all 119909 119910 isin 119883 with 119879119909 ⪯ 119879119910where 120595 [0 +infin) rarr [0 +infin) is a nondecreasing uppersemicontinuous function such that 120595(119905) lt 119905 for all 119905 gt 0Suppose that there exist a subset 119860
0of 119883 and 119909
0isin 1198600such
that 1198600⪯ 1198791199090 Assume that for each sequence 119909
119899 in 119883 with
119879119909119899minus1
⪯ 119879119909119899for all 119899 ge 1 there exists a natural number 119896 such
that 119879119909119898
⪯ 119879119909119899for all 119898 gt 119899 ge 119896 Then 119879 has an endpoint
and so 119879 has the approximate endpoint property
Acknowledgments
This work was completed while the second author (DrGopal) was visiting the Azarbaijan University of ShahidMadani Azarshahr Tabriz Iran during the summer of 2012He thanks Professor Sh Rezapour and the University fortheir hospitality and support The second author gratefullyacknowledges the support from the CSIR govternment ofIndia Grant no-25(0215)13EMR-II
References
[1] A Amini-Harandi ldquoEndpoints of set-valued contractions inmetric spacesrdquo Nonlinear Analysis Theory Methods and Appli-cations vol 72 no 1 pp 132ndash134 2010
[2] R P Agarwal D OrsquoRegan and D R Sahu Fixed Point Theoryfor Lipschitzian-TypeMappings with Applications Springer NewYork NY USA 2009
[3] S Moradi and F Khojasteh ldquoEndpoints of multi-valued gener-alized weak contraction mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 74 no 6 pp 2170ndash2174 2011
[4] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012
[5] D BaleanuHMohammadi and Sh Rezapour ldquoSome existenceresults on nonlinear fractional differential equationsrdquo Philo-sophical Transactions of the Royal Society vol 371 Article ID20120144 2013
[6] M Jleli and B Samet ldquoBest proximity points for 120572-120595-proximalcontractive type mappings and applicationsrdquo Bulletin des Sci-ences Mathematiques 2013
[7] M A Miandaragh M Postolache and Sh Rezapour ldquoSomeapproximate fixed point results for generalized-contractivemappings rdquo To appear in Scientific Bulletin-UniversityPolitehnica of Bucharest A
[8] Sh Rezapour and J H Asl ldquoA simple method for obtainingcoupled fixed points of 120572-120595-contractive type mappingsrdquo Inter-national Journal of Analysis Article ID 438029 7 pages 2013
The Scientific World Journal 7
[9] H Alikhani V Rakocevic Sh Rezapour and N ShahzadldquoFixed points of proximinal valued 120573-120595-contractive multifunc-tionsrdquo To appear in Journal of Nonlinear and Convex Analysis
[10] H Alikhani Sh Rezapour and N Shahzad ldquoFixed points of anew type contractive mappings and multifunctionsrdquo To appearin Filomat
[11] S M A Aleomraninejad Sh Rezapour and N Shahzad ldquoOnfixed points of 120572-120595-contractive multifunctionsrdquo Fixed PointTheory and Applications vol 2012 article 212 8 pages 2012
[12] B Mohammadi Sh Rezapour and N Shahzad ldquoSome resultson fixed points of 120572-120595-ciric generalized multifunctionsrdquo FixedPoint Theory and Applications vol 2013 article 24 2013
[13] S M A Aleomraninejad Sh Rezapour andN Shahzad ldquoSomefixed point results on ametric space with a graphrdquoTopology andIts Applications vol 159 no 3 pp 659ndash663 2012
[14] J M Borwein and A S Lewis Convex Analysis and NonlinearOptimization Theory and Examples Springer New York NYUSA 2000
Submit your manuscripts athttpwwwhindawicom
OperationsResearch
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Mathematical Problems in Engineering
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Applied Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2013
International Journal of
Combinatorics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Journal of Function Spaces and Applications
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Geometry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2013
Advances in
Mathematical Physics
ISRN Algebra
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ProbabilityandStatistics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Mathematical Analysis
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Journal ofApplied Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Advances in
DecisionSciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Stochastic AnalysisInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013
The Scientific World Journal
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Discrete Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2013
The Scientific World Journal 7
[9] H Alikhani V Rakocevic Sh Rezapour and N ShahzadldquoFixed points of proximinal valued 120573-120595-contractive multifunc-tionsrdquo To appear in Journal of Nonlinear and Convex Analysis
[10] H Alikhani Sh Rezapour and N Shahzad ldquoFixed points of anew type contractive mappings and multifunctionsrdquo To appearin Filomat
[11] S M A Aleomraninejad Sh Rezapour and N Shahzad ldquoOnfixed points of 120572-120595-contractive multifunctionsrdquo Fixed PointTheory and Applications vol 2012 article 212 8 pages 2012
[12] B Mohammadi Sh Rezapour and N Shahzad ldquoSome resultson fixed points of 120572-120595-ciric generalized multifunctionsrdquo FixedPoint Theory and Applications vol 2013 article 24 2013
[13] S M A Aleomraninejad Sh Rezapour andN Shahzad ldquoSomefixed point results on ametric space with a graphrdquoTopology andIts Applications vol 159 no 3 pp 659ndash663 2012
[14] J M Borwein and A S Lewis Convex Analysis and NonlinearOptimization Theory and Examples Springer New York NYUSA 2000
Submit your manuscripts athttpwwwhindawicom
OperationsResearch
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Mathematical Problems in Engineering
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Applied Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2013
International Journal of
Combinatorics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Journal of Function Spaces and Applications
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Geometry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2013
Advances in
Mathematical Physics
ISRN Algebra
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ProbabilityandStatistics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Mathematical Analysis
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Journal ofApplied Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Advances in
DecisionSciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Stochastic AnalysisInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013
The Scientific World Journal
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Discrete Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2013
Submit your manuscripts athttpwwwhindawicom
OperationsResearch
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Mathematical Problems in Engineering
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Applied Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2013
International Journal of
Combinatorics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Journal of Function Spaces and Applications
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Geometry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2013
Advances in
Mathematical Physics
ISRN Algebra
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ProbabilityandStatistics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Mathematical Analysis
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Journal ofApplied Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Advances in
DecisionSciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Stochastic AnalysisInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013
The Scientific World Journal
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Discrete Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2013
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