Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 206515, 9 pages http://dx.doi.org/10.1155/2013/206515 Research Article Fixed Point Theorems on Ordered Metric Spaces through a Rational Contraction Poom Kumam, 1 Fayyaz Rouzkard, 2 Mohammad Imdad, 2 and Dhananjay Gopal 3 1 Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology onburi (KMUTT), Bang Mod, Bangkok 10140, ailand 2 Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India 3 Department of Applied Mathematics & Humanities, S. V. National Institute of Technology, Surat 395007, India Correspondence should be addressed to Poom Kumam; [email protected] Received 10 May 2013; Accepted 9 July 2013 Academic Editor: Salvador Hernandez Copyright © 2013 Poom Kumam 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. Ran and Reurings (2004) established an interesting analogue of Banach Contraction Principle in a complete metric space equipped with a partial ordering and also utilized the same oneto discuss the existence of solutions to matrix equations. Motivated by this paper, we prove results on coincidence points for a pair of weakly increasing mappings satisfying a nonlinear contraction condition described by a rational expression on an ordered complete metric space. e uniqueness of common fixed point is also discussed. Some examples are furnished to demonstrate the validity of the hypotheses of our results. As an application, we derive an existence theorem for the solution of an integral equation. 1. Introduction with Preliminaries A variety of generalizations of the Classical Banach Contrac- tion Principle [1] are available in the existing literature of metric fixed point theory. e majority of these generaliza- tions are obtained by improving the underlying contraction condition (e.g., [2]). Presently, there is vigorous research activity to prove existence results on complete metric spaces equipped with a partial ordering. In fact, various existence and uniqueness theorems on fixed and common fixed point for monotone mappings are of paramount importance in the study of nonlinear equations which generate natural interest to establish usable fixed point theorems in partial metric spaces (e.g., [1–24]). Very recently, Harjani et al. [25] proved a fixed point theo- rem in partially ordered metric spaces satisfying a contractive condition of rational type due to Jaggi [26]. e aim of this paper is to prove some results of Harjani et al. [25] type for a pair of self-mappings. We accomplish this using the concept of weakly increasing property due to Nashine and Samet [14] (also see [4, 27, 28]). Some examples are also furnished to demonstrate the validity of the hypotheses of our results. As an application, we establish the existence of solution to an integral equation (also see [2, 23, 29, 30]). Before presenting our results, we recall some notations, definitions, and examples required in our subsequent discus- sions. Definition 1. Let X be a nonempty set. en (X, , ⪯) is called an ordered (partial) metric space if (i) (X, ⪯) is a partially ordered set and (ii) (X, ) is a metric space. Definition 2. Let (X, ⪯) be a partially ordered set. en (a) elements , ∈ X are called comparable with respect to “⪯” if either ⪯ or ⪯; (b) a mapping T : X → X is called nondecreasing with respect to “⪯” if ⪯ implies T⪯ T. Let X be a nonempty set and R : X → X be a given mapping. For every ∈ X, we denote by R −1 (), the subset of X defined by R −1 () := { ∈ X | R = } . (1)
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 206515 9 pageshttpdxdoiorg1011552013206515
Research ArticleFixed Point Theorems on Ordered Metric Spaces through aRational Contraction
Poom Kumam1 Fayyaz Rouzkard2 Mohammad Imdad2 and Dhananjay Gopal3
1 Department of Mathematics Faculty of Science King Mongkutrsquos University of Technology Thonburi (KMUTT) Bang ModBangkok 10140 Thailand
2Department of Mathematics Aligarh Muslim University Aligarh 202002 India3 Department of Applied Mathematics amp Humanities S V National Institute of Technology Surat 395007 India
Correspondence should be addressed to Poom Kumam poomkumkmuttacth
Received 10 May 2013 Accepted 9 July 2013
Academic Editor Salvador Hernandez
Copyright copy 2013 Poom Kumam 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
Ran and Reurings (2004) established an interesting analogue of Banach Contraction Principle in a complete metric space equippedwith a partial ordering and also utilized the same oneto discuss the existence of solutions to matrix equations Motivated by thispaper we prove results on coincidence points for a pair of weakly increasing mappings satisfying a nonlinear contraction conditiondescribed by a rational expression on an ordered complete metric space The uniqueness of common fixed point is also discussedSome examples are furnished to demonstrate the validity of the hypotheses of our results As an application we derive an existencetheorem for the solution of an integral equation
1 Introduction with Preliminaries
A variety of generalizations of the Classical Banach Contrac-tion Principle [1] are available in the existing literature ofmetric fixed point theory The majority of these generaliza-tions are obtained by improving the underlying contractioncondition (eg [2]) Presently there is vigorous researchactivity to prove existence results on complete metric spacesequipped with a partial ordering In fact various existenceand uniqueness theorems on fixed and common fixed pointfor monotone mappings are of paramount importance in thestudy of nonlinear equations which generate natural interestto establish usable fixed point theorems in partial metricspaces (eg [1ndash24])
Very recentlyHarjani et al [25] proved a fixed point theo-rem in partially orderedmetric spaces satisfying a contractivecondition of rational type due to Jaggi [26] The aim of thispaper is to prove some results of Harjani et al [25] type for apair of self-mappings We accomplish this using the conceptof weakly increasing property due to Nashine and Samet [14](also see [4 27 28]) Some examples are also furnished todemonstrate the validity of the hypotheses of our results As
an application we establish the existence of solution to anintegral equation (also see [2 23 29 30])
Before presenting our results we recall some notationsdefinitions and examples required in our subsequent discus-sions
Definition 1 LetX be a nonempty setThen (X 119889 ⪯) is calledan ordered (partial) metric space if
(i) (X ⪯) is a partially ordered set and (ii) (X 119889) is ametric space
Definition 2 Let (X ⪯) be a partially ordered set Then
(a) elements 119909 119910 isin X are called comparable with respectto ldquo⪯rdquo if either 119909 ⪯ 119910 or 119910 ⪯ 119909
(b) a mappingT X rarr X is called nondecreasing withrespect to ldquo⪯rdquo if 119909 ⪯ 119910 impliesT119909 ⪯ T119910
Let X be a nonempty set and R X rarr X be a givenmapping For every 119909 isin X we denote byRminus1(119909) the subsetofX defined by
Rminus1
(119909) = 119906 isin X | R119906 = 119909 (1)
2 Abstract and Applied Analysis
Definition 3 Let (X ⪯) be a partially ordered set and letTR X rarr X be two mappings such that TX sube RXWe say that T is weakly increasing with respect to R if andonly if for all 119909 isin X one has
T119909 ⪯ T119910 forall119910 isin Rminus1
(T119909) (2)
Remark 4 If R X rarr X is the identity mapping on Xthen T is weakly increasing with respect to R if and only ifT(119909) ⪯ TT(119909) for all 119909 isin X
As mentioned earlier the notion of weakly increasingmappings was introduced in [4] which is presently in use (eg[27 28]) In what follows we furnish a relatively new exampleto demonstrate the preceding definition
Example 5 Consider X = [0infin) endowed with the naturalorder le Define two mappingsTR X rarr X as
T119909 = 1199092 + 1 0 le 119909 lt 1
1 1 le 119909
R119909 = 1199093
+ 1 0 le 119909 lt 1
1 1 le 119909
(3)
In order to show that the mappingT is weakly increasingwith respect to mappingR we distinguish three cases
Firstly we consider the case 119909 = 0 Let 119910 isin
Rminus1(T(0)) that isR(119910) = T(0) = 1 so that R(119910) = 1 andhenceforth 119910 = 0 or 119910 ge 1 By the definitions ofT we have
1 = T (0) le T(119910) = 1 (4)
Secondly we argue the case 119909 ge 1 Let 119910 isin Rminus1(T(119909)) thatisR(119910) = T(119909) = 1 which amounts to say (in view ofdefinition 119877) that 119910 = 0 or 119910 ge 1 implying thereby
1 = T (119909) le T(119910) = 1 (5)
Finally we need to consider the case 0 lt 119909 lt 1 Let 119910 isin
Rminus1(T(119909)) that isR(119910) = T(119909) By the definition ofT wehaveT(119909) = 1199092 + 1 so thatR(119910) = 1199092 + 1 Now in view ofdefinition ofR we haveR(119910) = 1199103+1 so that1199103+1 = 1199092+1yielding thereby 119910 = (119909)
23 Now we have
1199092
+ 1 = T(119909) le T (119910) = T((119909)23
) = (119909)43
+ 1 (6)
Thus we have shown thatT is weakly increasing with respecttoR
Definition 6 Let (X ⪯) be an ordered metric space We saythatX is regular if and only if the following hypothesis holds
if 119911119899
is a nondecreasing sequence inXwith respect to ⪯such that lim
119899rarrinfin
119911119899
= 119911 isin X then 119911119899
⪯ 119911 for all 119899 isin N
Definition 7 A pair (RT) of self-mappings of a met-ric space (X 119889) is said to be compatible if and only iflim119899rarr+infin
119889(TR119909119899
RT119909119899
) = 0whenever 119909119899
is a sequenceinX such that lim
119899rarr+infin
R119909119899
= lim119899rarr+infin
T119909119899
= 119911 isin X
Definition 8 (see [18]) A pair (RT) of self-mappings of ametric space (X 119889) is said to be reciprocally continuous ifand only if lim
119899rarrinfin
RT119909119899
= R119911 and lim119899rarrinfin
TR119909119899
= T119911
for every sequence 119909119899
inX satisfying
lim119899rarrinfin
R119909119899
= lim119899rarrinfin
T119909119899
= 119911 (7)
for some 119911 isin XNotice that a pair of continuous mappings is always
reciprocally continuous but not conversely as substantiatedby examples in [18]
Definition 9 (see [19]) A pair (RT) of self mappingsof a metric space (X 119889) is said to be weakly reciprocallycontinuous if and only if lim
119899rarrinfin
RT119909119899
= R119911 for everysequence 119909
119899
inX satisfying
lim119899rarrinfin
R119909119899
= lim119899rarrinfin
T119909119899
= 119911 (8)
for some 119911 isin XEvidently every pair of reciprocally continuousmappings
is always weakly reciprocally but not conversely as demon-strated in Pant et al [19]
2 Results
Themain result of this paper runs as follows
Theorem 10 Let (X ⪯) be a partially ordered set equippedwith a metric 119889 on X such that (X 119889) is a complete metricspace Let TR X rarr X be two mappings satisfying (forpairs (119909 119910) isin X timesX whereinR119909 andR119910 are comparable)
119889 (T119909T119910) le 120572119889 (R119909T119909) sdot 119889 (R119910T119910)
1 + 119889 (R119909R119910)
+ 120573119889 (R119909R119910)
(9)
where 120572 120573 are nonnegative real numbers with 120572 + 120573 lt 1Suppose that
(a) X is regular andT is weakly increasing withR
(b) the pair (RT) is commuting as well as weaklyreciprocally continuous
Then T and R have a coincidence point That is there exists119906 isin X such thatR119906 = T119906
Proof Let 1199090
be an arbitrary point in X Since TX sube RXone can inductively construct a sequence 119909
119899
in X definedby
R119909119899
= T119909119899minus1
forall119899 isin N (10)
As 1199091
isin Rminus1(T1199090
) and 1199092
isin Rminus1(T1199091
) using weakly in-creasing property ofT with respect toR we obtain
R1199091
= T1199090
⪯ T1199091
= R1199092
⪯ T1199092
= R1199093
(11)
Abstract and Applied Analysis 3
Continuing this process indefinitely we get
R1199091
⪯ R1199092
⪯ R1199093
⪯ sdot sdot sdot ⪯ R119909119899
⪯ R119909119899+1
⪯ sdot sdot sdot (12)
Now we proceed to show that for 119899 isin N
119889 (R (119909119899+1
) R (119909119899+2
)) le (120573
1 minus 120572)
119899
119889 (R (1199091
) R (1199092
))
(13)
AsR(1199092
) ⪰ R(1199091
) using (9) we have
119889 (R (1199092
) R (1199093
))
= 119889 (T (1199091
) T (1199092
))
le 120572119889 (R (119909
1
) T (1199091
)) sdot 119889 (R (1199092
) T (1199092
))
1 + 119889 (R (1199091
) R (1199092
))
+ 120573119889 (R (1199091
) R (1199092
))
(owing to 119889 (R (1199091
) R (1199092
))
le 1 + 119889 (R (1199091
) R (1199092
)))
le 120572119889 (R (1199092
) R (1199093
)) + 120573119889 (R (1199091
) R (1199092
))
(14)
which implies that
(1 minus 120572) 119889 (R (1199092
) R (1199093
)) le 120573119889 (R (1199091
) R (1199092
)) (15)
so that
119889 (R (1199092
) R (1199093
)) le (120573
1 minus 120572)119889 (R (119909
1
) R (1199092
)) (16)
Let (13) holds for all 119899 gt 0 As R(119909119899+1
) ⪰ R(119909119899
) using(9) we have
119889 (R (119909119899+2
) R (119909119899+3
)) = 119889 (T (119909119899+1
) T (119909119899+2
))
le 120572119889 (R (119909
119899+1
) T (119909119899+1
)) 119889 (R (119909119899+2
) T (119909119899+2
))
1 + 119889 (R (119909119899+1
) R (119909119899+2
))
+ 120573119889 (R (119909119899+1
) R (119909119899+2
))
(owing to 119889 (R (119909119899+1
) R (119909119899+2
)))
le 1 + 119889 (R (119909119899+1
) R (119909119899+2
))
le 120572119889 (R (119909119899+2
) R (119909119899+3
))
+ 120573119889 (R (119909119899+1
) R (119909119899+2
))
(17)
so that
119889 (R (119909119899+2
) R (119909119899+3
))
le (120573
1 minus 120572)119889 (R (119909
119899+1
) R (119909119899+2
))
le (120573
1 minus 120572)
119899+1
119889 (R1199091
R1199092
)
(18)
Now with ℎ = 120573(1 minus 120572) for any119898 gt 119899 we have
119889 (R (119909119899
) R (119909119898
))
le 119889 (R (119909119899
) R (119909119899+1
))
+ 119889 (R (119909119899+1
) R (119909119899+2
))
+ sdot sdot sdot 119889 (R (119909119898minus1
) R (119909119898
))
le [ℎ119899
+ ℎ119899+1
+ sdot sdot sdot ℎ119898minus1
] 119889 (R1199091
R1199092
)
le [ℎ119899
1 minus ℎ] 119889 (R119909
1
R1199092
)
(19)
As 0 lt ℎ = 120573(1minus120572)lt 1 we have lim119899rarr+infin
119889(R(119909119899
)R(119909119898
)) =
0 that is R119909119899
is Cauchy sequences in the complete metricspace (X 119889) and hence there exists some 119909 isin X such that
lim119899rarr+infin
T (119909119899
) = lim119899rarr+infin
R (119909119899
) = 119909 (20)
Owing to commutativity ofT withR and by using (10) (foreach 119899 ge 1) we have
R (R119909119899+1
) = R (T119909119899
) = T (R119909119899
)
that is Rminus1 (T (R119909119899
)) = R119909119899+1
(21)
SinceT is weakly increasing withR we can write
R (R119909119899+1
) = T (R119909119899
) ⪯ T (R119909119899+1
) = R (R119909119899+2
) (22)
so that R(R119909119899
) is nondecreasing As the maps R andT are weakly reciprocally continuous lim
119899rarrinfin
R(R119909119899
) =
lim119899rarrinfin
R(T119909119899minus1
) = R119909 which together with regularity ofX gives rise toR(R119909
119899
) ⪯ R119909 that isR(R119909119899
) andR119909 arecomparable On using condition (9) we have
119889 (R119909T119909)
le 119889 (R119909R (R119909119899+1
)) + 119889 (R (R119909119899+1
) T119909)
= 119889 (R119909R (R119909119899+1
)) + 119889 (R (T119909119899
) T119909)
= 119889 (R119909R (R119909119899+1
)) + 119889 (T (R119909119899
) T119909)
= 119889 (R119909R (R119909119899+1
)) + 119889 (T119909T (R119909119899
))
le 119889 (R119909R (R119909119899+1
))
+ 120572119889 (R119909T119909)119889 (R (R119909
119899
) T (R119909119899
))
1 + 119889 (R119909R (R119909119899
))
+ 120573119889 (R119909R (R119909119899
))
(23)
On making 119899 rarr infin in the preceeding inequality one gets
119889 (R119909T119909) = 0 (24)
so thatT119909 = R119909 Thus we have shown thatT andR havea coincidence point This completes the proof
Setting 120572 = 0 in Theorem 10 we deduce the following
4 Abstract and Applied Analysis
Corollary 11 Let (X ⪯) be a partially ordered set on whichthere is a metric 119889 on X such that (X 119889) is a complete metricspace Let TR X rarr X be given mappings satisfying (forpairs (119909 119910) isin X timesX whereinR119909 andR119910 are comparable)
119889 (T119909T119910) le 120573119889 (R119909R119910) (25)
where 120573 is a nonnegative real number with 120573 lt 1 Suppose that
(a) X is regular andT is weakly increasing withR(b) the pair (RT) is commuting as well as weakly
reciprocally continuous
Then T and R have a coincidence point That is there exists119906 isin X such thatR119906 = T119906
Theorem 12 Let (X ⪯) be a partially ordered set equippedwith a metric 119889 on X such that (X 119889) is a complete metricspace Let TR X rarr X be two mappings satisfying (forpairs (119909 119910) isin X timesX wherein R119909 and R119910 are comparable)
119889 (T119909T119910) le 120572119889 (R119909T119909) sdot 119889 (R119910T119910)
1 + 119889 (R119909R119910)
+ 120573119889 (R119909R119910)
(26)
where 120572 120573 are non-negative real numbers with 120572 + 120573 lt 1Suppose that
(a) T is weakly increasing withR(b) the pair TR is compatible and reciprocally contin-
uous
Then T and R have a coincidence point That is there exists119906 isin X such thatR119906 = T119906
Proof Proceeding on the lines of the proof of Theorem 10one can furnish a sequence 119909
119899
such that
lim119899rarr+infin
T (119909119899
) = lim119899rarr+infin
R (119909119899
) = 119909 (27)
Now it remains to show that 119909 is a coincidence point of Tand R To accomplish this as the pair TR is compatibleas well as reciprocally continuous
lim119899rarrinfin
119889 (R (T (119909119899
)) T (R (119909119899
))) = 0 (28)
R (119909) = lim119899rarrinfin
R (T (119909119899
)) T (119909) = lim119899rarrinfin
T (R (119909119899
))
(29)
whenever
lim119899rarr+infin
T (119909119899
) = lim119899rarr+infin
R (119909119899
) = 119909 (30)
By using (29) in (28) we have 119889(T119909R119909) = 0 so thatT119909 =
R119909
By appealing Theorems 10 and 12 one can also have thefollowing natural theorem
Theorem 13 Let (X ⪯) be a partially ordered set on whichthere is a metric 119889 onX such that (X 119889) is a complete metricspace Let T X rarr X be nondecreasing mapping satisfying(for pairs (119909 119910) isin XtimesXwhereinR119909 andR119910 are comparable)
119889 (T119909T119910) le 120572119889 (119909T119909) sdot 119889 (119910T119910)
1 + 119889 (119909 119910)+ 120573119889 (119909 119910) (31)
where 120572 120573 are non-negative real numbers such that 120572 + 120573 lt 1Suppose that
(I) T119909 ⪯ T(T119909) for all 119909 isin X(II) eitherT is continuous orX is regular
ThenT has a fixed point
Proof The proof of this theorem can be outlined on the linesof the proof of Theorem 10 realizing R to be the identitymapping onX
Now we introduce the following property which will beutilized in our next theorem
Property (A) IfR(119909119899
) is a nondecreasing sequence in119883 suchthat lim
119899rarr+infin
R(119909119899
) = 119909 thenR(119909119899
) is comparable toR(119909)for all 119899 isin 119873
Theorem 14 Let (X ⪯) be a partially ordered set on whichthere exists a metric 119889 on X such that (X 119889) is a completemetric space Let TR X rarr X be given mappingssatisfying (for pairs (119909 119910) isin X timesX whereinR119909 andR119910 arecomparable)
119889 (T119909T119910) le 120572119889 (R119909T119909) sdot 119889 (R119910T119910)
1 + 119889 (R119909R119910)
+ 120573119889 (R119909R119910)
(32)
where 120572 120573 are non-negative real numbers with 120572 + 120573 lt 1Suppose that
(a) X is regular andT is weakly increasing withR(b) the pair (RT) is commuting as well as weakly
reciprocally continuous(c) R satisfies Property (A)
ThenR andT have a common fixed point
Proof Proceeding on the lines of the proof of Theorem 10one can inductively construct nondecreasing sequenceR119909119899
such that lim119899rarrinfin
T119909119899
= lim119899rarrinfin
R119909119899+1
= 119909 andR(119909) = T(119909) SinceR(119909
119899
) andR(119909) are comparable (for all119899 isin 119873) by using condition (32) we have
119889 (R119909R119909119899+1
) = 119889 (T119909T119909119899
)
le 120572119889 (R119909T119909)119889 (R119909
119899
T119909119899
)
1 + 119889 (R119909R119909119899
)
+ 120573119889 (R119909R119909119899
)
(33)
Abstract and Applied Analysis 5
Taking the limit as 119899 rarr infin one gets
119909 = R119909 = T119909 (34)
This completes the proof
Theorem 15 Let (X ⪯) be a partially ordered set on whichthere exists a metric 119889 on X such that (X 119889) is a completemetric space Let TR X rarr X be given mappingssatisfying (for pairs (119909 119910) isin X timesX whereinR119909 andR119910 arecomparable)
119889 (T119909T119910) le 120572119889 (R119909T119909) sdot 119889 (R119910T119910)
1 + 119889 (R119909R119910)
+ 120573119889 (R119909R119910)
(35)
where 120572 120573 are non-negative real numbers with 120572 + 120573 lt 1Suppose that
(a) T is weakly increasing withR(b) the pair TR is compatible and reciprocally contin-
uous(c) R satisfies Property (A)
ThenR andT have a common fixed point
Proof Proof is obvious in view of Theorems 12 and 14
3 Uniqueness Results
In what follows we investigate the conditions under whichTheorem 10 ensures the uniqueness of common fixed point
Theorem 16 If in addition to the hypotheses of Theorem 10every (119909 119909lowast) isin X timesX there exists a 119906 isin X such that T119906 isupper bound of T119909 119886119899119889 T119909lowast then T and R have a uniquecommon fixed point
Proof In view of Theorem 10 the set of coincidence pointsof the maps T and R is non-empty If 119909 and 119909lowast are twocoincidence points of the maps T and R (ie R119909 =
T119909 R119909lowast = T119909lowast) then we proceed to show that
R119909 = R119909lowast
(36)
In view of the additional hypothesis of this theorem thereexists 119906 isin X such thatT119906 is upper bound ofT119909 and T119909lowastPut 1199060
= 119906 and choose 1199061
isin X such thatR1199061
= T1199060
Nowproceeding on the lines of the proof of Theorem 10 one caninductively define sequence R119906
119899
such that
R119906119899
= T119906119899minus1
forall119899 (37)
wherein
R1199061
⪯ R1199062
⪯ R1199093
⪯ sdot sdot sdot ⪯ R119906119899
⪯ R119906119899+1
⪯ sdot sdot sdot (38)
Further setting 1199090
= 119909 and 119909lowast0
= 119909lowast one can also definethe sequences R119909
119899
and R119909lowast119899
such that
R119909119899
= T119909119899minus1
R119909lowast
119899
= T119909lowast
119899minus1
(39)
For every 119899 ge 1 we have
R119909119899
= T119909 R119909lowast
119899
= T119909lowast
forall119899 ge 1 (40)
SinceT119906 = R1199061
is upper bound ofT119909 = R1199091
andT119909lowast =
R119909lowast1
then
R1199091
= R119909 ⪯ R1199061
R119909lowast
1
= R119909lowast
⪯ R1199061
(41)
It is easy to show that R119909 ⪯ R119906119899
and R119909lowast ⪯ R119906119899
for all119899 ge 1 thenR119909 andR119909lowast are comparable withR119906
119899
On using(9) we have
119889 (R119909R119906119899+1
) = 119889 (T119909T119906119899
)
le 120572119889 (R119909T119909)119889 (R119906
119899
T119906119899
)
1 + 119889 (R119909R119906119899
)
+ 120573119889 (R119909R119906119899
)
(42)
or
119889 (R119909R119906119899+1
) le 120573119889 (R119909R119906119899
) (43)
Owing to (43) we can write
119889 (R119909R119906119899+1
) le 120573119889 (R119909R119906119899
)
le 1205732
119889 (R119909R119906119899minus1
) le sdot sdot sdot
le 120573119899+1
119889 (R119909R1199060
)
(44)
Taking the limit as 119899 rarr infin in (44) we get
lim119899rarrinfin
119889 (R119909R119906119899
) = 0 (45)
as 0 lt 120573 lt 1Similarly one can also show that
lim119899rarrinfin
119889 (R119909lowast
R119906119899
) = 0 (46)
On using (45) and (46) we can have
119889 (R119909R119909lowast
) le 119889 (R119909R119906119899+1
) + 119889 (R119906119899+1
R119909lowast
) (47)
so that lim119899rarrinfin
119889(R119909R119909lowast)0 that isR119909 = R119909lowast Thus wehave proved (36)
Since R119909 = T119909 owing to commutativity of T and Rone can write
R (R119909) = R (T119909) = T (R119909) (48)
which on insertingR119909 = 119911 gives rise to
R119911 = T119911 (49)
Thus 119911 is another coincidence point of the pair Now due toR119909 = R119909lowast (for every coincidence point119909 and119909lowast) and owingto the fact that 119911 is coincidence point of the pair RT itfollows thatR119911 = R119909 that is
R119911 = 119911 (50)
6 Abstract and Applied Analysis
On making use of (49) and (50) we can have
119911 = R119911 = T119911 (51)
which shows that 119911 is a common fixed point ofT andRTo prove the uniqueness let 119901 be another common fixed
point of the pair RT Since119901 and 119911 are coincidence pointsthenR119901 = R119911
Owing to that119901 and 119911 are commonfixed point of the pairsRT one can have
119901 = R119901 = R119911 = 119911 (52)
This completes the proof
The following simple example demonstrates Theorem 16
Example 17 ConsiderX = (1 0) (0 1) sub 1198772 equipped withnatural order (119909 119910) le (119911 119905) hArr 119909 le 119911 and 119910 le 119905 Thus (X le)is a partially ordered set wherein the two elements are notcomparable to each other Also (X le) is a complete metricspace under Euclideanmetric DefinedmappingsT and R
X rarr X as
T (119909 119910) = (1 0) (119909 119910) = (1 0)
(1 0) (119909 119910) = (0 1) (53)
R (119909 119910) = (1 0) (119909 119910) = (1 0)
(0 1) (119909 119910) = (0 1) (54)
As R(119909 119910) and R(119911 119905) in X are merely comparable tothemselves inequality (9) is vacuously satisfied for every 120572 120573
Notice that T(X) = (1 0) sub R(X) = X Also for(119909 119910) = (1 0)
R (119911 119905) = T (119909 119910) = T (1 0) = (1 0) 997904rArr (119911 119905) = (1 0)
(55)
so that
T (1 0) = T (119909 119910) le T (119911 119905) = T (1 0) (56)
Otherwise for (119911 119905) = (0 1)
R (119911 119905) = T (119909 119910) = T (0 1) = (1 0) 997904rArr (119911 119905) = (1 0)
(57)
so that
(1 0) = T (0 1) = T (119909 119910) le T (119911 119905) = T (1 0) = (1 0)
(58)
ThusT is weakly increasing with respect toRIf (119909119899
119910119899
) sub X is a nondecreasing sequence convergingto (119909 119910) isin X then necessarily (119909
119899
119910119899
) must be constantsequence that is (119909
119899
119910119899
) = (119909 119910) for all 119899 isin 119873 so that limit(119909 119910) is an upper bond for all the terms in the sequencewhichshows thatX is regular
By using definitions of the mapsT andR we have
(119909 119910) = R (119909 119910) = lim119899rarrinfin
T (119909119899
119910119899
) = lim119899rarrinfin
RT (119909119899
119910119899
)
(59)
wherein lim119899rarrinfin
R(119909119899
119910119899
) = lim119899rarrinfin
T(119909119899
119910119899
) = (119909 119910) sothat
R (119909 119910) = lim119899rarrinfin
RT (119909119899
119910119899
) (60)
which shows that the pair RT is weakly reciprocallycontinuous Also the pair RT is clearly commuting
Now we show that for every 119909 and 119909lowast in X there existsa 119906 isin X such that T119906 is upper bound of T119909 and T119909lowast If119909 = 119909lowast then choice of 119906 is obvious Otherwise if 119909 = (1 0)
and 119909lowast = (0 1) we can choose 119906 = (0 1) such that T119906 =
T(0 1) = (1 0) is upper bound of T119909 = (1 0) and T119909lowast =
(1 0)Thus we have shown that all the conditions ofTheorem 16
are satisfied and (119909 119910) = (1 0) is the unique common fixedpoint ofT andR
Theorem 18 If in addition to the hypotheses of Theorem 12for pairs (119909 119909lowast) isin XtimesX there exists a 119906 isin X such thatT119906 isupper bound of T119909 119886119899119889 T119909lowast then T and R have a uniquecommon fixed point
Proof Proof is obvious in view of Theorems 12 and 16
Corollary 19 In addition to the hypotheses of Theorem 16 (orTheorem 18) suppose that for every (119909 119909lowast) isin XtimesX there existsa 119906 isin X such thatT119906 is upper bound ofT119909 119886119899119889 T119909lowast ThenT has a unique fixed point that is there exists a unique 119909 isin Xsuch that 119909 = T119909
Proof In Theorem 16 ifR = 119868 (the identity mapping on 119883)we have the result
The following example demonstrates Theorem 18
Example 20 Consider X = [0infin) equipped with the usualmetric and natural order le Define two mappings TR
X rarr X by
T119909 = 1199093 + 2 0 le 119909 lt 1
3 1 le 119909
R119909 = 21199093 + 1 0 le 119909 lt 1
3 1 le 119909
(61)
Then evidently (X le) is a partially ordered set and the mapsT andR satisfy inequality (9) with 120573 = 12 and 120572 = 13 for119909 119910 isin [0infin)
To show that T is weakly increasing with respect to Rnotice thatT(X) = [2 3] sub [1 3] = R(X) Firstly we arguethe case 119909 = 0 Let 119910 isin Rminus1(1198790) that isR(119910) = T(0) = 2
so that 21199103 + 1 = 2 or 119910 = 1213 Using definitions ofT andR we have
2 = T (0) le T (119910) = T(1
213) =
1
2+ 2 (62)
Secondly if 119909 ge 1 then we haveR(119910) = T(119909) = 3 By usingdefinition ofR we have 119910 ge 1 so that
3 = T (119909) le T (119910) = 3 (63)
Abstract and Applied Analysis 7
Finally we consider the case 0 lt 119909 lt 1 Let 119910 isin
Rminus1(T(119909)) that isR(119910) = T(119909) In view of definition ofT we have T(119909) = 1199093 + 2 so that R(119910) = 1199093 + 2 Byusing the definition of R we have R(119910) = 21199103 + 1 so that21199103 + 1 = 1199093 + 2 or 119910 = ((119909
3
+ 1)2)13 Thus we have
1199093
+ 2 = T (119909) le T (119910) = T((1199093 + 1
2)
13
) =1199093 + 1
2+ 2
(64)
ThereforeT is weakly increasing with respect toRSince T and R are continuous therefore this pair of
maps is reciprocally continuousNow we show that maps T and R are compatible
If lim119899rarrinfin
119909119899
= 119896 and 119896 ge 1 then lim119899rarrinfin
T119909119899
=
lim119899rarrinfin
R119909119899
= 3 and henceforth
lim119899rarrinfin
TR119909119899
= lim119899rarrinfin
RT119909119899
= 3 (65)
implying thereby lim119899rarrinfin
119889(RT119909119899
TR119909119899
) = 0 so thatT and R are compatible
Thus we have shown that all the conditions ofTheorem 18are satisfied and 119909 = 3 is the unique fixed point ofT and R
4 An Application
In this section we present an application of Theorem 13 andused the idea of Ciric et al [29] to define a partial order andprove an existence theorem for the solution of an integralequation
Theorem 21 Consider the integral equation
119909 (119905) = int119879
0
119870 (119905 119904 119909 (119904)) 119889119904 + 119892 (119905) 119905 isin [0 119879] (66)
with 119879 gt 0 wherein
(i) the functions 119870 [0 119879] times [0 119879] times R119899 rarr R119899 and 119892
R119899 rarr R119899 are continuous(ii) for each 119905 119904 isin [0 119879]
119870 (119905 119904 119909 (119904)) ≪ 119870(119905 119904 int119879
0
119870 (119904 120591 119909 (120591)) 119889120591 + 119892 (119904)) (67)
where≪ denotes a partial order relation on R119899(iii) there exists a continuous function 119901 [0 119879]times[0 119879] rarr
R+
such that
|119870 (119905 119904 119906) minus 119870 (119905 119904 V)| le 119901 (119905 119904) |119906 minus V| (68)
for each 119905 119904 isin [0 119879] and is also comparable 119906 V isin R119899
(iv) sup119905isin[0119879]
int119879
0
119901(119905 119904)119889119904 le 120573 lt 1
Then the integral equation (66) has a unique solution 119909lowast in119862([0 119879]R119899)
Proof ConsiderX = 119862([0 119879]R119899)with the usual supremumnorm that is
119909 = max119905isin[0119879]
|119909 (119905)| (69)
for 119909 119910 isin 119862([0 119879]R119899) Define onX a partial order as follows(119909 119910 isin 119862([0 119879]R119899))
119909 ⪯ 119910 iff 119909 (119905) ≪ 119910 (119905) for any 119905 isin [0 119879] (70)
Then (X ⪯) is a partially ordered set and (X sdot ) is acomplete metric space
Moreover for any increasing sequence 119909119899
inX converg-ing to 119909lowast isin X we have 119909
119899
(119905) ≪ 119909lowast(119905) for any 119905 isin [0 119879] Alsofor every 119909 119910 isin X there exists 119888 isin X which depends on 119909
and 119910 and is also comparable to 119909 and 119910 (cf [21])DefineT 119862([0 119879]) rarr 119862([0 119879]) by
T119906 (119905) = int119879
0
119870 (119905 119904 119906 (119904)) 119889119904 + 119892 (119905)
119905 isin [0 119879] 119906 isin 119862 ([0 119879])
(71)
Let us prove that
T119906 ⪯ T (T119906) forall119906 isin 119862 ([0 119879]) (72)
Let 119906 isin 119862([0 119879]) From (ii) for all 119905 isin [0 119879] we have
T119906 (119905)
= int119879
0
119870 (119905 119904 119906 (119904)) 119889119904 + 119892 (119905)
≪ int119879
0
119870(119905 119904 int119879
0
119870 (119904 120591 119906 (120591)) 119889120591 + 119892 (119904)) 119889119904 + 119892 (119905)
= int119879
0
119870 (119905 119904T119906 (119904)) 119889119904 + 119892 (119905)
= T (T119906) (119905)
(73)
Then (72) holdsNow for all 119909 119910 isin 119862([0 119879]) with 119910 ⪯ 119909 we have (by (iii))
1003816100381610038161003816T119909 (119905) minusT119910 (119905)1003816100381610038161003816
le int119879
0
1003816100381610038161003816119870 (119905 119904 119909 (119904)) minus 119870 (119905 119904 119910 (119904))1003816100381610038161003816 119889119904
le int119879
0
119901 (119905 119904) (1003816100381610038161003816119909 (119904) minus 119910 (119904)
1003816100381610038161003816) 119889119904
le1003817100381710038171003817119909 minus 119910
1003817100381710038171003817 int119879
0
119901 (119905 119904) 119889119904
(74)
Hence1003817100381710038171003817T119909 (119905) minusT119910 (119905)
1003817100381710038171003817
le1003817100381710038171003817119909 minus 119910
1003817100381710038171003817 sup119905isin[0119879]
int119879
0
119901 (119905 119904) 119889119904
le 1205731003817100381710038171003817119909 minus 119910
1003817100381710038171003817
(75)
8 Abstract and Applied Analysis
On the other hand it is demonstrated in [16] thatcondition of regularity is satisfied forX = 119862([0 119879]R119899)
Thus all the hypotheses of Theorem 13 are satisfied for120572 = 0 and then T has a fixed point 119906lowast isin 119862([0 119879]R119899) thatis 119906lowast is a solution of the integral equation (66)
Acknowledgments
The authors would like to thank the referees for careful read-ing and for providing valuable suggestions and comments forthis paper Also Poom Kumam would like to thank the KingMongkuts University of TechnologyThonburi (KMUTT) forthe financial support and Dhananjay Gopal is thankful toCSIR Government of India Grant no 25(0215)13EMR-II
References
[1] S Banach ldquoSur les operations dans les ensembles abstraits etleur application aux equations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922
[2] H K Nashine and I Altun ldquoFixed point theorems for gener-alized weakly contractive condition in ordered metric spacesrdquoFixed PointTheory and Applications vol 2011 Article ID 13236720 pages 2011
[3] R P Agarwal M A El-Gebeily and D OrsquoRegan ldquoGeneralizedcontractions in partially ordered metric spacesrdquo ApplicableAnalysis vol 87 no 1 pp 109ndash116 2008
[4] I Altun and H Simsek ldquoSome fixed point theorems onordered metric spaces and applicationrdquo Fixed Point Theory andApplications vol 2010 Article ID 621469 17 pages 2010
[5] A Amini-Harandi and H Emami ldquoA fixed point theoremfor contraction type maps in partially ordered metric spacesand application to ordinary differential equationsrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 5 pp2238ndash2242 2010
[6] I Beg and A R Butt ldquoFixed point for set-valued mappings sat-isfying an implicit relation in partially ordered metric spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no9 pp 3699ndash3704 2009
[7] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 65 no 7 pp 1379ndash1393 2006
[8] A Cabada and J J Nieto ldquoFixed points and approximatesolutions for nonlinear operator equationsrdquo Journal of Com-putational and Applied Mathematics vol 113 no 1-2 pp 17ndash25 2000 Fixed point theory with applications in nonlinearanalysis
[9] J Caballero J Harjani and K Sadarangani ldquoContractive-likemapping principles in ordered metric spaces and applicationto ordinary differential equationsrdquo Fixed Point Theory andApplications vol 2010 Article ID 916064 14 pages 2010
[10] L Ciric N Cakic M Rajovic and J S Ume ldquoMonotonegeneralized nonlinear contractions in partially ordered metricspacesrdquo Fixed Point Theory and Applications vol 2008 ArticleID 131294 11 pages 2008
[11] Lj B Ciric D Mihet and R Saadati ldquoMonotone generalizedcontractions in partially ordered probabilistic metric spacesrdquoTopology and its Applications vol 156 no 17 pp 2838ndash28442009
[12] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysis Theory Methods amp Applications vol 71 no 7-8 pp 3403ndash3410 2009
[13] J Harjani and K Sadarangani ldquoGeneralized contractions inpartially ordered metric spaces and applications to ordinarydifferential equationsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1188ndash1197 2010
[14] H K Nashine and B Samet ldquoFixed point results for map-pings satisfying (120595 120593)-weakly contractive condition in partiallyordered metric spacesrdquo Nonlinear Analysis Theory Methods ampApplications vol 74 no 6 pp 2201ndash2209 2011
[15] H K Nashine and W Shatanawi ldquoCoupled common fixedpoint theorems for a pair of commuting mappings in partiallyordered complete metric spacesrdquo Computers amp Mathematicswith Applications vol 62 no 4 pp 1984ndash1993 2011
[16] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005
[17] J J Nieto and R Rodrıguez-Lopez ldquoExistence and uniquenessof fixed point in partially ordered sets and applications toordinary differential equationsrdquo Acta Mathematica Sinica vol23 no 12 pp 2205ndash2212 2007
[18] R P Pant ldquoCommon fixed points of four mappingsrdquo Bulletin ofthe Calcutta Mathematical Society vol 90 no 4 pp 281ndash2861998
[19] R P Pant R K Bisht andDArora ldquoWeak reciprocal continuityand fixed point theoremsrdquo Annali dellrsquoUniversita di FerraraSezione VII ScienzeMatematiche vol 57 no 1 pp 181ndash190 2011
[20] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrixequationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004
[21] D OrsquoRegan and A Petrusel ldquoFixed point theorems for gen-eralized contractions in ordered metric spacesrdquo Journal ofMathematical Analysis andApplications vol 341 no 2 pp 1241ndash1252 2008
[22] B Samet ldquoCoupled fixed point theorems for a generalizedMeir-Keeler contraction in partially ordered metric spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 72 no12 pp 4508ndash4517 2010
[23] W Sintunavarat Y J Cho and P Kumam ldquoUrysohn integralequations approach by common fixed points in complex valuedmetric spacesrdquo Advances in Difference Equations vol 2013article 49 2013
[24] W Shatanawi ldquoPartially ordered cone metric spaces andcoupled fixed point resultsrdquo Computers amp Mathematics withApplications vol 60 no 8 pp 2508ndash2515 2010
[25] J Harjani B Lopez and K Sadarangani ldquoA fixed point theoremfor mappings satisfying a contractive condition of rationaltype on a partially ordered metric spacerdquo Abstract and AppliedAnalysis vol 2010 Article ID 190701 8 pages 2010
[26] D S Jaggi ldquoSome unique fixed point theoremsrdquo Indian Journalof Pure and AppliedMathematics vol 8 no 2 pp 223ndash230 1977
[27] B C Dhage ldquoCondensing mappings and applications to exis-tence theorems for common solution of differential equationsrdquoBulletin of the Korean Mathematical Society vol 36 no 3 pp565ndash578 1999
[28] B C Dhage D OrsquoRegan and R P Agarwal ldquoCommon fixedpoint theorems for a pair of countably condensing mappingsin ordered Banach spacesrdquo Journal of Applied Mathematics andStochastic Analysis vol 16 no 3 pp 243ndash248 2003
Abstract and Applied Analysis 9
[29] Lj Ciric B Samet C Vetro and M Abbas ldquoFixed point resultsfor weak contractive mappings in ordered 119870-metric spacesrdquoFixed Point Theory vol 13 no 1 pp 59ndash72 2012
[30] H K Nashine W Sintunavarat and P Kumam ldquoCyclic gener-alized contractions and fixed point results with applications toan integral equationrdquo Fixed Point Theory and Applications vol2012 article 217 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
Definition 3 Let (X ⪯) be a partially ordered set and letTR X rarr X be two mappings such that TX sube RXWe say that T is weakly increasing with respect to R if andonly if for all 119909 isin X one has
T119909 ⪯ T119910 forall119910 isin Rminus1
(T119909) (2)
Remark 4 If R X rarr X is the identity mapping on Xthen T is weakly increasing with respect to R if and only ifT(119909) ⪯ TT(119909) for all 119909 isin X
As mentioned earlier the notion of weakly increasingmappings was introduced in [4] which is presently in use (eg[27 28]) In what follows we furnish a relatively new exampleto demonstrate the preceding definition
Example 5 Consider X = [0infin) endowed with the naturalorder le Define two mappingsTR X rarr X as
T119909 = 1199092 + 1 0 le 119909 lt 1
1 1 le 119909
R119909 = 1199093
+ 1 0 le 119909 lt 1
1 1 le 119909
(3)
In order to show that the mappingT is weakly increasingwith respect to mappingR we distinguish three cases
Firstly we consider the case 119909 = 0 Let 119910 isin
Rminus1(T(0)) that isR(119910) = T(0) = 1 so that R(119910) = 1 andhenceforth 119910 = 0 or 119910 ge 1 By the definitions ofT we have
1 = T (0) le T(119910) = 1 (4)
Secondly we argue the case 119909 ge 1 Let 119910 isin Rminus1(T(119909)) thatisR(119910) = T(119909) = 1 which amounts to say (in view ofdefinition 119877) that 119910 = 0 or 119910 ge 1 implying thereby
1 = T (119909) le T(119910) = 1 (5)
Finally we need to consider the case 0 lt 119909 lt 1 Let 119910 isin
Rminus1(T(119909)) that isR(119910) = T(119909) By the definition ofT wehaveT(119909) = 1199092 + 1 so thatR(119910) = 1199092 + 1 Now in view ofdefinition ofR we haveR(119910) = 1199103+1 so that1199103+1 = 1199092+1yielding thereby 119910 = (119909)
23 Now we have
1199092
+ 1 = T(119909) le T (119910) = T((119909)23
) = (119909)43
+ 1 (6)
Thus we have shown thatT is weakly increasing with respecttoR
Definition 6 Let (X ⪯) be an ordered metric space We saythatX is regular if and only if the following hypothesis holds
if 119911119899
is a nondecreasing sequence inXwith respect to ⪯such that lim
119899rarrinfin
119911119899
= 119911 isin X then 119911119899
⪯ 119911 for all 119899 isin N
Definition 7 A pair (RT) of self-mappings of a met-ric space (X 119889) is said to be compatible if and only iflim119899rarr+infin
119889(TR119909119899
RT119909119899
) = 0whenever 119909119899
is a sequenceinX such that lim
119899rarr+infin
R119909119899
= lim119899rarr+infin
T119909119899
= 119911 isin X
Definition 8 (see [18]) A pair (RT) of self-mappings of ametric space (X 119889) is said to be reciprocally continuous ifand only if lim
119899rarrinfin
RT119909119899
= R119911 and lim119899rarrinfin
TR119909119899
= T119911
for every sequence 119909119899
inX satisfying
lim119899rarrinfin
R119909119899
= lim119899rarrinfin
T119909119899
= 119911 (7)
for some 119911 isin XNotice that a pair of continuous mappings is always
reciprocally continuous but not conversely as substantiatedby examples in [18]
Definition 9 (see [19]) A pair (RT) of self mappingsof a metric space (X 119889) is said to be weakly reciprocallycontinuous if and only if lim
119899rarrinfin
RT119909119899
= R119911 for everysequence 119909
119899
inX satisfying
lim119899rarrinfin
R119909119899
= lim119899rarrinfin
T119909119899
= 119911 (8)
for some 119911 isin XEvidently every pair of reciprocally continuousmappings
is always weakly reciprocally but not conversely as demon-strated in Pant et al [19]
2 Results
Themain result of this paper runs as follows
Theorem 10 Let (X ⪯) be a partially ordered set equippedwith a metric 119889 on X such that (X 119889) is a complete metricspace Let TR X rarr X be two mappings satisfying (forpairs (119909 119910) isin X timesX whereinR119909 andR119910 are comparable)
119889 (T119909T119910) le 120572119889 (R119909T119909) sdot 119889 (R119910T119910)
1 + 119889 (R119909R119910)
+ 120573119889 (R119909R119910)
(9)
where 120572 120573 are nonnegative real numbers with 120572 + 120573 lt 1Suppose that
(a) X is regular andT is weakly increasing withR
(b) the pair (RT) is commuting as well as weaklyreciprocally continuous
Then T and R have a coincidence point That is there exists119906 isin X such thatR119906 = T119906
Proof Let 1199090
be an arbitrary point in X Since TX sube RXone can inductively construct a sequence 119909
119899
in X definedby
R119909119899
= T119909119899minus1
forall119899 isin N (10)
As 1199091
isin Rminus1(T1199090
) and 1199092
isin Rminus1(T1199091
) using weakly in-creasing property ofT with respect toR we obtain
R1199091
= T1199090
⪯ T1199091
= R1199092
⪯ T1199092
= R1199093
(11)
Abstract and Applied Analysis 3
Continuing this process indefinitely we get
R1199091
⪯ R1199092
⪯ R1199093
⪯ sdot sdot sdot ⪯ R119909119899
⪯ R119909119899+1
⪯ sdot sdot sdot (12)
Now we proceed to show that for 119899 isin N
119889 (R (119909119899+1
) R (119909119899+2
)) le (120573
1 minus 120572)
119899
119889 (R (1199091
) R (1199092
))
(13)
AsR(1199092
) ⪰ R(1199091
) using (9) we have
119889 (R (1199092
) R (1199093
))
= 119889 (T (1199091
) T (1199092
))
le 120572119889 (R (119909
1
) T (1199091
)) sdot 119889 (R (1199092
) T (1199092
))
1 + 119889 (R (1199091
) R (1199092
))
+ 120573119889 (R (1199091
) R (1199092
))
(owing to 119889 (R (1199091
) R (1199092
))
le 1 + 119889 (R (1199091
) R (1199092
)))
le 120572119889 (R (1199092
) R (1199093
)) + 120573119889 (R (1199091
) R (1199092
))
(14)
which implies that
(1 minus 120572) 119889 (R (1199092
) R (1199093
)) le 120573119889 (R (1199091
) R (1199092
)) (15)
so that
119889 (R (1199092
) R (1199093
)) le (120573
1 minus 120572)119889 (R (119909
1
) R (1199092
)) (16)
Let (13) holds for all 119899 gt 0 As R(119909119899+1
) ⪰ R(119909119899
) using(9) we have
119889 (R (119909119899+2
) R (119909119899+3
)) = 119889 (T (119909119899+1
) T (119909119899+2
))
le 120572119889 (R (119909
119899+1
) T (119909119899+1
)) 119889 (R (119909119899+2
) T (119909119899+2
))
1 + 119889 (R (119909119899+1
) R (119909119899+2
))
+ 120573119889 (R (119909119899+1
) R (119909119899+2
))
(owing to 119889 (R (119909119899+1
) R (119909119899+2
)))
le 1 + 119889 (R (119909119899+1
) R (119909119899+2
))
le 120572119889 (R (119909119899+2
) R (119909119899+3
))
+ 120573119889 (R (119909119899+1
) R (119909119899+2
))
(17)
so that
119889 (R (119909119899+2
) R (119909119899+3
))
le (120573
1 minus 120572)119889 (R (119909
119899+1
) R (119909119899+2
))
le (120573
1 minus 120572)
119899+1
119889 (R1199091
R1199092
)
(18)
Now with ℎ = 120573(1 minus 120572) for any119898 gt 119899 we have
119889 (R (119909119899
) R (119909119898
))
le 119889 (R (119909119899
) R (119909119899+1
))
+ 119889 (R (119909119899+1
) R (119909119899+2
))
+ sdot sdot sdot 119889 (R (119909119898minus1
) R (119909119898
))
le [ℎ119899
+ ℎ119899+1
+ sdot sdot sdot ℎ119898minus1
] 119889 (R1199091
R1199092
)
le [ℎ119899
1 minus ℎ] 119889 (R119909
1
R1199092
)
(19)
As 0 lt ℎ = 120573(1minus120572)lt 1 we have lim119899rarr+infin
119889(R(119909119899
)R(119909119898
)) =
0 that is R119909119899
is Cauchy sequences in the complete metricspace (X 119889) and hence there exists some 119909 isin X such that
lim119899rarr+infin
T (119909119899
) = lim119899rarr+infin
R (119909119899
) = 119909 (20)
Owing to commutativity ofT withR and by using (10) (foreach 119899 ge 1) we have
R (R119909119899+1
) = R (T119909119899
) = T (R119909119899
)
that is Rminus1 (T (R119909119899
)) = R119909119899+1
(21)
SinceT is weakly increasing withR we can write
R (R119909119899+1
) = T (R119909119899
) ⪯ T (R119909119899+1
) = R (R119909119899+2
) (22)
so that R(R119909119899
) is nondecreasing As the maps R andT are weakly reciprocally continuous lim
119899rarrinfin
R(R119909119899
) =
lim119899rarrinfin
R(T119909119899minus1
) = R119909 which together with regularity ofX gives rise toR(R119909
119899
) ⪯ R119909 that isR(R119909119899
) andR119909 arecomparable On using condition (9) we have
119889 (R119909T119909)
le 119889 (R119909R (R119909119899+1
)) + 119889 (R (R119909119899+1
) T119909)
= 119889 (R119909R (R119909119899+1
)) + 119889 (R (T119909119899
) T119909)
= 119889 (R119909R (R119909119899+1
)) + 119889 (T (R119909119899
) T119909)
= 119889 (R119909R (R119909119899+1
)) + 119889 (T119909T (R119909119899
))
le 119889 (R119909R (R119909119899+1
))
+ 120572119889 (R119909T119909)119889 (R (R119909
119899
) T (R119909119899
))
1 + 119889 (R119909R (R119909119899
))
+ 120573119889 (R119909R (R119909119899
))
(23)
On making 119899 rarr infin in the preceeding inequality one gets
119889 (R119909T119909) = 0 (24)
so thatT119909 = R119909 Thus we have shown thatT andR havea coincidence point This completes the proof
Setting 120572 = 0 in Theorem 10 we deduce the following
4 Abstract and Applied Analysis
Corollary 11 Let (X ⪯) be a partially ordered set on whichthere is a metric 119889 on X such that (X 119889) is a complete metricspace Let TR X rarr X be given mappings satisfying (forpairs (119909 119910) isin X timesX whereinR119909 andR119910 are comparable)
119889 (T119909T119910) le 120573119889 (R119909R119910) (25)
where 120573 is a nonnegative real number with 120573 lt 1 Suppose that
(a) X is regular andT is weakly increasing withR(b) the pair (RT) is commuting as well as weakly
reciprocally continuous
Then T and R have a coincidence point That is there exists119906 isin X such thatR119906 = T119906
Theorem 12 Let (X ⪯) be a partially ordered set equippedwith a metric 119889 on X such that (X 119889) is a complete metricspace Let TR X rarr X be two mappings satisfying (forpairs (119909 119910) isin X timesX wherein R119909 and R119910 are comparable)
119889 (T119909T119910) le 120572119889 (R119909T119909) sdot 119889 (R119910T119910)
1 + 119889 (R119909R119910)
+ 120573119889 (R119909R119910)
(26)
where 120572 120573 are non-negative real numbers with 120572 + 120573 lt 1Suppose that
(a) T is weakly increasing withR(b) the pair TR is compatible and reciprocally contin-
uous
Then T and R have a coincidence point That is there exists119906 isin X such thatR119906 = T119906
Proof Proceeding on the lines of the proof of Theorem 10one can furnish a sequence 119909
119899
such that
lim119899rarr+infin
T (119909119899
) = lim119899rarr+infin
R (119909119899
) = 119909 (27)
Now it remains to show that 119909 is a coincidence point of Tand R To accomplish this as the pair TR is compatibleas well as reciprocally continuous
lim119899rarrinfin
119889 (R (T (119909119899
)) T (R (119909119899
))) = 0 (28)
R (119909) = lim119899rarrinfin
R (T (119909119899
)) T (119909) = lim119899rarrinfin
T (R (119909119899
))
(29)
whenever
lim119899rarr+infin
T (119909119899
) = lim119899rarr+infin
R (119909119899
) = 119909 (30)
By using (29) in (28) we have 119889(T119909R119909) = 0 so thatT119909 =
R119909
By appealing Theorems 10 and 12 one can also have thefollowing natural theorem
Theorem 13 Let (X ⪯) be a partially ordered set on whichthere is a metric 119889 onX such that (X 119889) is a complete metricspace Let T X rarr X be nondecreasing mapping satisfying(for pairs (119909 119910) isin XtimesXwhereinR119909 andR119910 are comparable)
119889 (T119909T119910) le 120572119889 (119909T119909) sdot 119889 (119910T119910)
1 + 119889 (119909 119910)+ 120573119889 (119909 119910) (31)
where 120572 120573 are non-negative real numbers such that 120572 + 120573 lt 1Suppose that
(I) T119909 ⪯ T(T119909) for all 119909 isin X(II) eitherT is continuous orX is regular
ThenT has a fixed point
Proof The proof of this theorem can be outlined on the linesof the proof of Theorem 10 realizing R to be the identitymapping onX
Now we introduce the following property which will beutilized in our next theorem
Property (A) IfR(119909119899
) is a nondecreasing sequence in119883 suchthat lim
119899rarr+infin
R(119909119899
) = 119909 thenR(119909119899
) is comparable toR(119909)for all 119899 isin 119873
Theorem 14 Let (X ⪯) be a partially ordered set on whichthere exists a metric 119889 on X such that (X 119889) is a completemetric space Let TR X rarr X be given mappingssatisfying (for pairs (119909 119910) isin X timesX whereinR119909 andR119910 arecomparable)
119889 (T119909T119910) le 120572119889 (R119909T119909) sdot 119889 (R119910T119910)
1 + 119889 (R119909R119910)
+ 120573119889 (R119909R119910)
(32)
where 120572 120573 are non-negative real numbers with 120572 + 120573 lt 1Suppose that
(a) X is regular andT is weakly increasing withR(b) the pair (RT) is commuting as well as weakly
reciprocally continuous(c) R satisfies Property (A)
ThenR andT have a common fixed point
Proof Proceeding on the lines of the proof of Theorem 10one can inductively construct nondecreasing sequenceR119909119899
such that lim119899rarrinfin
T119909119899
= lim119899rarrinfin
R119909119899+1
= 119909 andR(119909) = T(119909) SinceR(119909
119899
) andR(119909) are comparable (for all119899 isin 119873) by using condition (32) we have
119889 (R119909R119909119899+1
) = 119889 (T119909T119909119899
)
le 120572119889 (R119909T119909)119889 (R119909
119899
T119909119899
)
1 + 119889 (R119909R119909119899
)
+ 120573119889 (R119909R119909119899
)
(33)
Abstract and Applied Analysis 5
Taking the limit as 119899 rarr infin one gets
119909 = R119909 = T119909 (34)
This completes the proof
Theorem 15 Let (X ⪯) be a partially ordered set on whichthere exists a metric 119889 on X such that (X 119889) is a completemetric space Let TR X rarr X be given mappingssatisfying (for pairs (119909 119910) isin X timesX whereinR119909 andR119910 arecomparable)
119889 (T119909T119910) le 120572119889 (R119909T119909) sdot 119889 (R119910T119910)
1 + 119889 (R119909R119910)
+ 120573119889 (R119909R119910)
(35)
where 120572 120573 are non-negative real numbers with 120572 + 120573 lt 1Suppose that
(a) T is weakly increasing withR(b) the pair TR is compatible and reciprocally contin-
uous(c) R satisfies Property (A)
ThenR andT have a common fixed point
Proof Proof is obvious in view of Theorems 12 and 14
3 Uniqueness Results
In what follows we investigate the conditions under whichTheorem 10 ensures the uniqueness of common fixed point
Theorem 16 If in addition to the hypotheses of Theorem 10every (119909 119909lowast) isin X timesX there exists a 119906 isin X such that T119906 isupper bound of T119909 119886119899119889 T119909lowast then T and R have a uniquecommon fixed point
Proof In view of Theorem 10 the set of coincidence pointsof the maps T and R is non-empty If 119909 and 119909lowast are twocoincidence points of the maps T and R (ie R119909 =
T119909 R119909lowast = T119909lowast) then we proceed to show that
R119909 = R119909lowast
(36)
In view of the additional hypothesis of this theorem thereexists 119906 isin X such thatT119906 is upper bound ofT119909 and T119909lowastPut 1199060
= 119906 and choose 1199061
isin X such thatR1199061
= T1199060
Nowproceeding on the lines of the proof of Theorem 10 one caninductively define sequence R119906
119899
such that
R119906119899
= T119906119899minus1
forall119899 (37)
wherein
R1199061
⪯ R1199062
⪯ R1199093
⪯ sdot sdot sdot ⪯ R119906119899
⪯ R119906119899+1
⪯ sdot sdot sdot (38)
Further setting 1199090
= 119909 and 119909lowast0
= 119909lowast one can also definethe sequences R119909
119899
and R119909lowast119899
such that
R119909119899
= T119909119899minus1
R119909lowast
119899
= T119909lowast
119899minus1
(39)
For every 119899 ge 1 we have
R119909119899
= T119909 R119909lowast
119899
= T119909lowast
forall119899 ge 1 (40)
SinceT119906 = R1199061
is upper bound ofT119909 = R1199091
andT119909lowast =
R119909lowast1
then
R1199091
= R119909 ⪯ R1199061
R119909lowast
1
= R119909lowast
⪯ R1199061
(41)
It is easy to show that R119909 ⪯ R119906119899
and R119909lowast ⪯ R119906119899
for all119899 ge 1 thenR119909 andR119909lowast are comparable withR119906
119899
On using(9) we have
119889 (R119909R119906119899+1
) = 119889 (T119909T119906119899
)
le 120572119889 (R119909T119909)119889 (R119906
119899
T119906119899
)
1 + 119889 (R119909R119906119899
)
+ 120573119889 (R119909R119906119899
)
(42)
or
119889 (R119909R119906119899+1
) le 120573119889 (R119909R119906119899
) (43)
Owing to (43) we can write
119889 (R119909R119906119899+1
) le 120573119889 (R119909R119906119899
)
le 1205732
119889 (R119909R119906119899minus1
) le sdot sdot sdot
le 120573119899+1
119889 (R119909R1199060
)
(44)
Taking the limit as 119899 rarr infin in (44) we get
lim119899rarrinfin
119889 (R119909R119906119899
) = 0 (45)
as 0 lt 120573 lt 1Similarly one can also show that
lim119899rarrinfin
119889 (R119909lowast
R119906119899
) = 0 (46)
On using (45) and (46) we can have
119889 (R119909R119909lowast
) le 119889 (R119909R119906119899+1
) + 119889 (R119906119899+1
R119909lowast
) (47)
so that lim119899rarrinfin
119889(R119909R119909lowast)0 that isR119909 = R119909lowast Thus wehave proved (36)
Since R119909 = T119909 owing to commutativity of T and Rone can write
R (R119909) = R (T119909) = T (R119909) (48)
which on insertingR119909 = 119911 gives rise to
R119911 = T119911 (49)
Thus 119911 is another coincidence point of the pair Now due toR119909 = R119909lowast (for every coincidence point119909 and119909lowast) and owingto the fact that 119911 is coincidence point of the pair RT itfollows thatR119911 = R119909 that is
R119911 = 119911 (50)
6 Abstract and Applied Analysis
On making use of (49) and (50) we can have
119911 = R119911 = T119911 (51)
which shows that 119911 is a common fixed point ofT andRTo prove the uniqueness let 119901 be another common fixed
point of the pair RT Since119901 and 119911 are coincidence pointsthenR119901 = R119911
Owing to that119901 and 119911 are commonfixed point of the pairsRT one can have
119901 = R119901 = R119911 = 119911 (52)
This completes the proof
The following simple example demonstrates Theorem 16
Example 17 ConsiderX = (1 0) (0 1) sub 1198772 equipped withnatural order (119909 119910) le (119911 119905) hArr 119909 le 119911 and 119910 le 119905 Thus (X le)is a partially ordered set wherein the two elements are notcomparable to each other Also (X le) is a complete metricspace under Euclideanmetric DefinedmappingsT and R
X rarr X as
T (119909 119910) = (1 0) (119909 119910) = (1 0)
(1 0) (119909 119910) = (0 1) (53)
R (119909 119910) = (1 0) (119909 119910) = (1 0)
(0 1) (119909 119910) = (0 1) (54)
As R(119909 119910) and R(119911 119905) in X are merely comparable tothemselves inequality (9) is vacuously satisfied for every 120572 120573
Notice that T(X) = (1 0) sub R(X) = X Also for(119909 119910) = (1 0)
R (119911 119905) = T (119909 119910) = T (1 0) = (1 0) 997904rArr (119911 119905) = (1 0)
(55)
so that
T (1 0) = T (119909 119910) le T (119911 119905) = T (1 0) (56)
Otherwise for (119911 119905) = (0 1)
R (119911 119905) = T (119909 119910) = T (0 1) = (1 0) 997904rArr (119911 119905) = (1 0)
(57)
so that
(1 0) = T (0 1) = T (119909 119910) le T (119911 119905) = T (1 0) = (1 0)
(58)
ThusT is weakly increasing with respect toRIf (119909119899
119910119899
) sub X is a nondecreasing sequence convergingto (119909 119910) isin X then necessarily (119909
119899
119910119899
) must be constantsequence that is (119909
119899
119910119899
) = (119909 119910) for all 119899 isin 119873 so that limit(119909 119910) is an upper bond for all the terms in the sequencewhichshows thatX is regular
By using definitions of the mapsT andR we have
(119909 119910) = R (119909 119910) = lim119899rarrinfin
T (119909119899
119910119899
) = lim119899rarrinfin
RT (119909119899
119910119899
)
(59)
wherein lim119899rarrinfin
R(119909119899
119910119899
) = lim119899rarrinfin
T(119909119899
119910119899
) = (119909 119910) sothat
R (119909 119910) = lim119899rarrinfin
RT (119909119899
119910119899
) (60)
which shows that the pair RT is weakly reciprocallycontinuous Also the pair RT is clearly commuting
Now we show that for every 119909 and 119909lowast in X there existsa 119906 isin X such that T119906 is upper bound of T119909 and T119909lowast If119909 = 119909lowast then choice of 119906 is obvious Otherwise if 119909 = (1 0)
and 119909lowast = (0 1) we can choose 119906 = (0 1) such that T119906 =
T(0 1) = (1 0) is upper bound of T119909 = (1 0) and T119909lowast =
(1 0)Thus we have shown that all the conditions ofTheorem 16
are satisfied and (119909 119910) = (1 0) is the unique common fixedpoint ofT andR
Theorem 18 If in addition to the hypotheses of Theorem 12for pairs (119909 119909lowast) isin XtimesX there exists a 119906 isin X such thatT119906 isupper bound of T119909 119886119899119889 T119909lowast then T and R have a uniquecommon fixed point
Proof Proof is obvious in view of Theorems 12 and 16
Corollary 19 In addition to the hypotheses of Theorem 16 (orTheorem 18) suppose that for every (119909 119909lowast) isin XtimesX there existsa 119906 isin X such thatT119906 is upper bound ofT119909 119886119899119889 T119909lowast ThenT has a unique fixed point that is there exists a unique 119909 isin Xsuch that 119909 = T119909
Proof In Theorem 16 ifR = 119868 (the identity mapping on 119883)we have the result
The following example demonstrates Theorem 18
Example 20 Consider X = [0infin) equipped with the usualmetric and natural order le Define two mappings TR
X rarr X by
T119909 = 1199093 + 2 0 le 119909 lt 1
3 1 le 119909
R119909 = 21199093 + 1 0 le 119909 lt 1
3 1 le 119909
(61)
Then evidently (X le) is a partially ordered set and the mapsT andR satisfy inequality (9) with 120573 = 12 and 120572 = 13 for119909 119910 isin [0infin)
To show that T is weakly increasing with respect to Rnotice thatT(X) = [2 3] sub [1 3] = R(X) Firstly we arguethe case 119909 = 0 Let 119910 isin Rminus1(1198790) that isR(119910) = T(0) = 2
so that 21199103 + 1 = 2 or 119910 = 1213 Using definitions ofT andR we have
2 = T (0) le T (119910) = T(1
213) =
1
2+ 2 (62)
Secondly if 119909 ge 1 then we haveR(119910) = T(119909) = 3 By usingdefinition ofR we have 119910 ge 1 so that
3 = T (119909) le T (119910) = 3 (63)
Abstract and Applied Analysis 7
Finally we consider the case 0 lt 119909 lt 1 Let 119910 isin
Rminus1(T(119909)) that isR(119910) = T(119909) In view of definition ofT we have T(119909) = 1199093 + 2 so that R(119910) = 1199093 + 2 Byusing the definition of R we have R(119910) = 21199103 + 1 so that21199103 + 1 = 1199093 + 2 or 119910 = ((119909
3
+ 1)2)13 Thus we have
1199093
+ 2 = T (119909) le T (119910) = T((1199093 + 1
2)
13
) =1199093 + 1
2+ 2
(64)
ThereforeT is weakly increasing with respect toRSince T and R are continuous therefore this pair of
maps is reciprocally continuousNow we show that maps T and R are compatible
If lim119899rarrinfin
119909119899
= 119896 and 119896 ge 1 then lim119899rarrinfin
T119909119899
=
lim119899rarrinfin
R119909119899
= 3 and henceforth
lim119899rarrinfin
TR119909119899
= lim119899rarrinfin
RT119909119899
= 3 (65)
implying thereby lim119899rarrinfin
119889(RT119909119899
TR119909119899
) = 0 so thatT and R are compatible
Thus we have shown that all the conditions ofTheorem 18are satisfied and 119909 = 3 is the unique fixed point ofT and R
4 An Application
In this section we present an application of Theorem 13 andused the idea of Ciric et al [29] to define a partial order andprove an existence theorem for the solution of an integralequation
Theorem 21 Consider the integral equation
119909 (119905) = int119879
0
119870 (119905 119904 119909 (119904)) 119889119904 + 119892 (119905) 119905 isin [0 119879] (66)
with 119879 gt 0 wherein
(i) the functions 119870 [0 119879] times [0 119879] times R119899 rarr R119899 and 119892
R119899 rarr R119899 are continuous(ii) for each 119905 119904 isin [0 119879]
119870 (119905 119904 119909 (119904)) ≪ 119870(119905 119904 int119879
0
119870 (119904 120591 119909 (120591)) 119889120591 + 119892 (119904)) (67)
where≪ denotes a partial order relation on R119899(iii) there exists a continuous function 119901 [0 119879]times[0 119879] rarr
R+
such that
|119870 (119905 119904 119906) minus 119870 (119905 119904 V)| le 119901 (119905 119904) |119906 minus V| (68)
for each 119905 119904 isin [0 119879] and is also comparable 119906 V isin R119899
(iv) sup119905isin[0119879]
int119879
0
119901(119905 119904)119889119904 le 120573 lt 1
Then the integral equation (66) has a unique solution 119909lowast in119862([0 119879]R119899)
Proof ConsiderX = 119862([0 119879]R119899)with the usual supremumnorm that is
119909 = max119905isin[0119879]
|119909 (119905)| (69)
for 119909 119910 isin 119862([0 119879]R119899) Define onX a partial order as follows(119909 119910 isin 119862([0 119879]R119899))
119909 ⪯ 119910 iff 119909 (119905) ≪ 119910 (119905) for any 119905 isin [0 119879] (70)
Then (X ⪯) is a partially ordered set and (X sdot ) is acomplete metric space
Moreover for any increasing sequence 119909119899
inX converg-ing to 119909lowast isin X we have 119909
119899
(119905) ≪ 119909lowast(119905) for any 119905 isin [0 119879] Alsofor every 119909 119910 isin X there exists 119888 isin X which depends on 119909
and 119910 and is also comparable to 119909 and 119910 (cf [21])DefineT 119862([0 119879]) rarr 119862([0 119879]) by
T119906 (119905) = int119879
0
119870 (119905 119904 119906 (119904)) 119889119904 + 119892 (119905)
119905 isin [0 119879] 119906 isin 119862 ([0 119879])
(71)
Let us prove that
T119906 ⪯ T (T119906) forall119906 isin 119862 ([0 119879]) (72)
Let 119906 isin 119862([0 119879]) From (ii) for all 119905 isin [0 119879] we have
T119906 (119905)
= int119879
0
119870 (119905 119904 119906 (119904)) 119889119904 + 119892 (119905)
≪ int119879
0
119870(119905 119904 int119879
0
119870 (119904 120591 119906 (120591)) 119889120591 + 119892 (119904)) 119889119904 + 119892 (119905)
= int119879
0
119870 (119905 119904T119906 (119904)) 119889119904 + 119892 (119905)
= T (T119906) (119905)
(73)
Then (72) holdsNow for all 119909 119910 isin 119862([0 119879]) with 119910 ⪯ 119909 we have (by (iii))
1003816100381610038161003816T119909 (119905) minusT119910 (119905)1003816100381610038161003816
le int119879
0
1003816100381610038161003816119870 (119905 119904 119909 (119904)) minus 119870 (119905 119904 119910 (119904))1003816100381610038161003816 119889119904
le int119879
0
119901 (119905 119904) (1003816100381610038161003816119909 (119904) minus 119910 (119904)
1003816100381610038161003816) 119889119904
le1003817100381710038171003817119909 minus 119910
1003817100381710038171003817 int119879
0
119901 (119905 119904) 119889119904
(74)
Hence1003817100381710038171003817T119909 (119905) minusT119910 (119905)
1003817100381710038171003817
le1003817100381710038171003817119909 minus 119910
1003817100381710038171003817 sup119905isin[0119879]
int119879
0
119901 (119905 119904) 119889119904
le 1205731003817100381710038171003817119909 minus 119910
1003817100381710038171003817
(75)
8 Abstract and Applied Analysis
On the other hand it is demonstrated in [16] thatcondition of regularity is satisfied forX = 119862([0 119879]R119899)
Thus all the hypotheses of Theorem 13 are satisfied for120572 = 0 and then T has a fixed point 119906lowast isin 119862([0 119879]R119899) thatis 119906lowast is a solution of the integral equation (66)
Acknowledgments
The authors would like to thank the referees for careful read-ing and for providing valuable suggestions and comments forthis paper Also Poom Kumam would like to thank the KingMongkuts University of TechnologyThonburi (KMUTT) forthe financial support and Dhananjay Gopal is thankful toCSIR Government of India Grant no 25(0215)13EMR-II
References
[1] S Banach ldquoSur les operations dans les ensembles abstraits etleur application aux equations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922
[2] H K Nashine and I Altun ldquoFixed point theorems for gener-alized weakly contractive condition in ordered metric spacesrdquoFixed PointTheory and Applications vol 2011 Article ID 13236720 pages 2011
[3] R P Agarwal M A El-Gebeily and D OrsquoRegan ldquoGeneralizedcontractions in partially ordered metric spacesrdquo ApplicableAnalysis vol 87 no 1 pp 109ndash116 2008
[4] I Altun and H Simsek ldquoSome fixed point theorems onordered metric spaces and applicationrdquo Fixed Point Theory andApplications vol 2010 Article ID 621469 17 pages 2010
[5] A Amini-Harandi and H Emami ldquoA fixed point theoremfor contraction type maps in partially ordered metric spacesand application to ordinary differential equationsrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 5 pp2238ndash2242 2010
[6] I Beg and A R Butt ldquoFixed point for set-valued mappings sat-isfying an implicit relation in partially ordered metric spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no9 pp 3699ndash3704 2009
[7] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 65 no 7 pp 1379ndash1393 2006
[8] A Cabada and J J Nieto ldquoFixed points and approximatesolutions for nonlinear operator equationsrdquo Journal of Com-putational and Applied Mathematics vol 113 no 1-2 pp 17ndash25 2000 Fixed point theory with applications in nonlinearanalysis
[9] J Caballero J Harjani and K Sadarangani ldquoContractive-likemapping principles in ordered metric spaces and applicationto ordinary differential equationsrdquo Fixed Point Theory andApplications vol 2010 Article ID 916064 14 pages 2010
[10] L Ciric N Cakic M Rajovic and J S Ume ldquoMonotonegeneralized nonlinear contractions in partially ordered metricspacesrdquo Fixed Point Theory and Applications vol 2008 ArticleID 131294 11 pages 2008
[11] Lj B Ciric D Mihet and R Saadati ldquoMonotone generalizedcontractions in partially ordered probabilistic metric spacesrdquoTopology and its Applications vol 156 no 17 pp 2838ndash28442009
[12] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysis Theory Methods amp Applications vol 71 no 7-8 pp 3403ndash3410 2009
[13] J Harjani and K Sadarangani ldquoGeneralized contractions inpartially ordered metric spaces and applications to ordinarydifferential equationsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1188ndash1197 2010
[14] H K Nashine and B Samet ldquoFixed point results for map-pings satisfying (120595 120593)-weakly contractive condition in partiallyordered metric spacesrdquo Nonlinear Analysis Theory Methods ampApplications vol 74 no 6 pp 2201ndash2209 2011
[15] H K Nashine and W Shatanawi ldquoCoupled common fixedpoint theorems for a pair of commuting mappings in partiallyordered complete metric spacesrdquo Computers amp Mathematicswith Applications vol 62 no 4 pp 1984ndash1993 2011
[16] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005
[17] J J Nieto and R Rodrıguez-Lopez ldquoExistence and uniquenessof fixed point in partially ordered sets and applications toordinary differential equationsrdquo Acta Mathematica Sinica vol23 no 12 pp 2205ndash2212 2007
[18] R P Pant ldquoCommon fixed points of four mappingsrdquo Bulletin ofthe Calcutta Mathematical Society vol 90 no 4 pp 281ndash2861998
[19] R P Pant R K Bisht andDArora ldquoWeak reciprocal continuityand fixed point theoremsrdquo Annali dellrsquoUniversita di FerraraSezione VII ScienzeMatematiche vol 57 no 1 pp 181ndash190 2011
[20] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrixequationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004
[21] D OrsquoRegan and A Petrusel ldquoFixed point theorems for gen-eralized contractions in ordered metric spacesrdquo Journal ofMathematical Analysis andApplications vol 341 no 2 pp 1241ndash1252 2008
[22] B Samet ldquoCoupled fixed point theorems for a generalizedMeir-Keeler contraction in partially ordered metric spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 72 no12 pp 4508ndash4517 2010
[23] W Sintunavarat Y J Cho and P Kumam ldquoUrysohn integralequations approach by common fixed points in complex valuedmetric spacesrdquo Advances in Difference Equations vol 2013article 49 2013
[24] W Shatanawi ldquoPartially ordered cone metric spaces andcoupled fixed point resultsrdquo Computers amp Mathematics withApplications vol 60 no 8 pp 2508ndash2515 2010
[25] J Harjani B Lopez and K Sadarangani ldquoA fixed point theoremfor mappings satisfying a contractive condition of rationaltype on a partially ordered metric spacerdquo Abstract and AppliedAnalysis vol 2010 Article ID 190701 8 pages 2010
[26] D S Jaggi ldquoSome unique fixed point theoremsrdquo Indian Journalof Pure and AppliedMathematics vol 8 no 2 pp 223ndash230 1977
[27] B C Dhage ldquoCondensing mappings and applications to exis-tence theorems for common solution of differential equationsrdquoBulletin of the Korean Mathematical Society vol 36 no 3 pp565ndash578 1999
[28] B C Dhage D OrsquoRegan and R P Agarwal ldquoCommon fixedpoint theorems for a pair of countably condensing mappingsin ordered Banach spacesrdquo Journal of Applied Mathematics andStochastic Analysis vol 16 no 3 pp 243ndash248 2003
Abstract and Applied Analysis 9
[29] Lj Ciric B Samet C Vetro and M Abbas ldquoFixed point resultsfor weak contractive mappings in ordered 119870-metric spacesrdquoFixed Point Theory vol 13 no 1 pp 59ndash72 2012
[30] H K Nashine W Sintunavarat and P Kumam ldquoCyclic gener-alized contractions and fixed point results with applications toan integral equationrdquo Fixed Point Theory and Applications vol2012 article 217 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
Continuing this process indefinitely we get
R1199091
⪯ R1199092
⪯ R1199093
⪯ sdot sdot sdot ⪯ R119909119899
⪯ R119909119899+1
⪯ sdot sdot sdot (12)
Now we proceed to show that for 119899 isin N
119889 (R (119909119899+1
) R (119909119899+2
)) le (120573
1 minus 120572)
119899
119889 (R (1199091
) R (1199092
))
(13)
AsR(1199092
) ⪰ R(1199091
) using (9) we have
119889 (R (1199092
) R (1199093
))
= 119889 (T (1199091
) T (1199092
))
le 120572119889 (R (119909
1
) T (1199091
)) sdot 119889 (R (1199092
) T (1199092
))
1 + 119889 (R (1199091
) R (1199092
))
+ 120573119889 (R (1199091
) R (1199092
))
(owing to 119889 (R (1199091
) R (1199092
))
le 1 + 119889 (R (1199091
) R (1199092
)))
le 120572119889 (R (1199092
) R (1199093
)) + 120573119889 (R (1199091
) R (1199092
))
(14)
which implies that
(1 minus 120572) 119889 (R (1199092
) R (1199093
)) le 120573119889 (R (1199091
) R (1199092
)) (15)
so that
119889 (R (1199092
) R (1199093
)) le (120573
1 minus 120572)119889 (R (119909
1
) R (1199092
)) (16)
Let (13) holds for all 119899 gt 0 As R(119909119899+1
) ⪰ R(119909119899
) using(9) we have
119889 (R (119909119899+2
) R (119909119899+3
)) = 119889 (T (119909119899+1
) T (119909119899+2
))
le 120572119889 (R (119909
119899+1
) T (119909119899+1
)) 119889 (R (119909119899+2
) T (119909119899+2
))
1 + 119889 (R (119909119899+1
) R (119909119899+2
))
+ 120573119889 (R (119909119899+1
) R (119909119899+2
))
(owing to 119889 (R (119909119899+1
) R (119909119899+2
)))
le 1 + 119889 (R (119909119899+1
) R (119909119899+2
))
le 120572119889 (R (119909119899+2
) R (119909119899+3
))
+ 120573119889 (R (119909119899+1
) R (119909119899+2
))
(17)
so that
119889 (R (119909119899+2
) R (119909119899+3
))
le (120573
1 minus 120572)119889 (R (119909
119899+1
) R (119909119899+2
))
le (120573
1 minus 120572)
119899+1
119889 (R1199091
R1199092
)
(18)
Now with ℎ = 120573(1 minus 120572) for any119898 gt 119899 we have
119889 (R (119909119899
) R (119909119898
))
le 119889 (R (119909119899
) R (119909119899+1
))
+ 119889 (R (119909119899+1
) R (119909119899+2
))
+ sdot sdot sdot 119889 (R (119909119898minus1
) R (119909119898
))
le [ℎ119899
+ ℎ119899+1
+ sdot sdot sdot ℎ119898minus1
] 119889 (R1199091
R1199092
)
le [ℎ119899
1 minus ℎ] 119889 (R119909
1
R1199092
)
(19)
As 0 lt ℎ = 120573(1minus120572)lt 1 we have lim119899rarr+infin
119889(R(119909119899
)R(119909119898
)) =
0 that is R119909119899
is Cauchy sequences in the complete metricspace (X 119889) and hence there exists some 119909 isin X such that
lim119899rarr+infin
T (119909119899
) = lim119899rarr+infin
R (119909119899
) = 119909 (20)
Owing to commutativity ofT withR and by using (10) (foreach 119899 ge 1) we have
R (R119909119899+1
) = R (T119909119899
) = T (R119909119899
)
that is Rminus1 (T (R119909119899
)) = R119909119899+1
(21)
SinceT is weakly increasing withR we can write
R (R119909119899+1
) = T (R119909119899
) ⪯ T (R119909119899+1
) = R (R119909119899+2
) (22)
so that R(R119909119899
) is nondecreasing As the maps R andT are weakly reciprocally continuous lim
119899rarrinfin
R(R119909119899
) =
lim119899rarrinfin
R(T119909119899minus1
) = R119909 which together with regularity ofX gives rise toR(R119909
119899
) ⪯ R119909 that isR(R119909119899
) andR119909 arecomparable On using condition (9) we have
119889 (R119909T119909)
le 119889 (R119909R (R119909119899+1
)) + 119889 (R (R119909119899+1
) T119909)
= 119889 (R119909R (R119909119899+1
)) + 119889 (R (T119909119899
) T119909)
= 119889 (R119909R (R119909119899+1
)) + 119889 (T (R119909119899
) T119909)
= 119889 (R119909R (R119909119899+1
)) + 119889 (T119909T (R119909119899
))
le 119889 (R119909R (R119909119899+1
))
+ 120572119889 (R119909T119909)119889 (R (R119909
119899
) T (R119909119899
))
1 + 119889 (R119909R (R119909119899
))
+ 120573119889 (R119909R (R119909119899
))
(23)
On making 119899 rarr infin in the preceeding inequality one gets
119889 (R119909T119909) = 0 (24)
so thatT119909 = R119909 Thus we have shown thatT andR havea coincidence point This completes the proof
Setting 120572 = 0 in Theorem 10 we deduce the following
4 Abstract and Applied Analysis
Corollary 11 Let (X ⪯) be a partially ordered set on whichthere is a metric 119889 on X such that (X 119889) is a complete metricspace Let TR X rarr X be given mappings satisfying (forpairs (119909 119910) isin X timesX whereinR119909 andR119910 are comparable)
119889 (T119909T119910) le 120573119889 (R119909R119910) (25)
where 120573 is a nonnegative real number with 120573 lt 1 Suppose that
(a) X is regular andT is weakly increasing withR(b) the pair (RT) is commuting as well as weakly
reciprocally continuous
Then T and R have a coincidence point That is there exists119906 isin X such thatR119906 = T119906
Theorem 12 Let (X ⪯) be a partially ordered set equippedwith a metric 119889 on X such that (X 119889) is a complete metricspace Let TR X rarr X be two mappings satisfying (forpairs (119909 119910) isin X timesX wherein R119909 and R119910 are comparable)
119889 (T119909T119910) le 120572119889 (R119909T119909) sdot 119889 (R119910T119910)
1 + 119889 (R119909R119910)
+ 120573119889 (R119909R119910)
(26)
where 120572 120573 are non-negative real numbers with 120572 + 120573 lt 1Suppose that
(a) T is weakly increasing withR(b) the pair TR is compatible and reciprocally contin-
uous
Then T and R have a coincidence point That is there exists119906 isin X such thatR119906 = T119906
Proof Proceeding on the lines of the proof of Theorem 10one can furnish a sequence 119909
119899
such that
lim119899rarr+infin
T (119909119899
) = lim119899rarr+infin
R (119909119899
) = 119909 (27)
Now it remains to show that 119909 is a coincidence point of Tand R To accomplish this as the pair TR is compatibleas well as reciprocally continuous
lim119899rarrinfin
119889 (R (T (119909119899
)) T (R (119909119899
))) = 0 (28)
R (119909) = lim119899rarrinfin
R (T (119909119899
)) T (119909) = lim119899rarrinfin
T (R (119909119899
))
(29)
whenever
lim119899rarr+infin
T (119909119899
) = lim119899rarr+infin
R (119909119899
) = 119909 (30)
By using (29) in (28) we have 119889(T119909R119909) = 0 so thatT119909 =
R119909
By appealing Theorems 10 and 12 one can also have thefollowing natural theorem
Theorem 13 Let (X ⪯) be a partially ordered set on whichthere is a metric 119889 onX such that (X 119889) is a complete metricspace Let T X rarr X be nondecreasing mapping satisfying(for pairs (119909 119910) isin XtimesXwhereinR119909 andR119910 are comparable)
119889 (T119909T119910) le 120572119889 (119909T119909) sdot 119889 (119910T119910)
1 + 119889 (119909 119910)+ 120573119889 (119909 119910) (31)
where 120572 120573 are non-negative real numbers such that 120572 + 120573 lt 1Suppose that
(I) T119909 ⪯ T(T119909) for all 119909 isin X(II) eitherT is continuous orX is regular
ThenT has a fixed point
Proof The proof of this theorem can be outlined on the linesof the proof of Theorem 10 realizing R to be the identitymapping onX
Now we introduce the following property which will beutilized in our next theorem
Property (A) IfR(119909119899
) is a nondecreasing sequence in119883 suchthat lim
119899rarr+infin
R(119909119899
) = 119909 thenR(119909119899
) is comparable toR(119909)for all 119899 isin 119873
Theorem 14 Let (X ⪯) be a partially ordered set on whichthere exists a metric 119889 on X such that (X 119889) is a completemetric space Let TR X rarr X be given mappingssatisfying (for pairs (119909 119910) isin X timesX whereinR119909 andR119910 arecomparable)
119889 (T119909T119910) le 120572119889 (R119909T119909) sdot 119889 (R119910T119910)
1 + 119889 (R119909R119910)
+ 120573119889 (R119909R119910)
(32)
where 120572 120573 are non-negative real numbers with 120572 + 120573 lt 1Suppose that
(a) X is regular andT is weakly increasing withR(b) the pair (RT) is commuting as well as weakly
reciprocally continuous(c) R satisfies Property (A)
ThenR andT have a common fixed point
Proof Proceeding on the lines of the proof of Theorem 10one can inductively construct nondecreasing sequenceR119909119899
such that lim119899rarrinfin
T119909119899
= lim119899rarrinfin
R119909119899+1
= 119909 andR(119909) = T(119909) SinceR(119909
119899
) andR(119909) are comparable (for all119899 isin 119873) by using condition (32) we have
119889 (R119909R119909119899+1
) = 119889 (T119909T119909119899
)
le 120572119889 (R119909T119909)119889 (R119909
119899
T119909119899
)
1 + 119889 (R119909R119909119899
)
+ 120573119889 (R119909R119909119899
)
(33)
Abstract and Applied Analysis 5
Taking the limit as 119899 rarr infin one gets
119909 = R119909 = T119909 (34)
This completes the proof
Theorem 15 Let (X ⪯) be a partially ordered set on whichthere exists a metric 119889 on X such that (X 119889) is a completemetric space Let TR X rarr X be given mappingssatisfying (for pairs (119909 119910) isin X timesX whereinR119909 andR119910 arecomparable)
119889 (T119909T119910) le 120572119889 (R119909T119909) sdot 119889 (R119910T119910)
1 + 119889 (R119909R119910)
+ 120573119889 (R119909R119910)
(35)
where 120572 120573 are non-negative real numbers with 120572 + 120573 lt 1Suppose that
(a) T is weakly increasing withR(b) the pair TR is compatible and reciprocally contin-
uous(c) R satisfies Property (A)
ThenR andT have a common fixed point
Proof Proof is obvious in view of Theorems 12 and 14
3 Uniqueness Results
In what follows we investigate the conditions under whichTheorem 10 ensures the uniqueness of common fixed point
Theorem 16 If in addition to the hypotheses of Theorem 10every (119909 119909lowast) isin X timesX there exists a 119906 isin X such that T119906 isupper bound of T119909 119886119899119889 T119909lowast then T and R have a uniquecommon fixed point
Proof In view of Theorem 10 the set of coincidence pointsof the maps T and R is non-empty If 119909 and 119909lowast are twocoincidence points of the maps T and R (ie R119909 =
T119909 R119909lowast = T119909lowast) then we proceed to show that
R119909 = R119909lowast
(36)
In view of the additional hypothesis of this theorem thereexists 119906 isin X such thatT119906 is upper bound ofT119909 and T119909lowastPut 1199060
= 119906 and choose 1199061
isin X such thatR1199061
= T1199060
Nowproceeding on the lines of the proof of Theorem 10 one caninductively define sequence R119906
119899
such that
R119906119899
= T119906119899minus1
forall119899 (37)
wherein
R1199061
⪯ R1199062
⪯ R1199093
⪯ sdot sdot sdot ⪯ R119906119899
⪯ R119906119899+1
⪯ sdot sdot sdot (38)
Further setting 1199090
= 119909 and 119909lowast0
= 119909lowast one can also definethe sequences R119909
119899
and R119909lowast119899
such that
R119909119899
= T119909119899minus1
R119909lowast
119899
= T119909lowast
119899minus1
(39)
For every 119899 ge 1 we have
R119909119899
= T119909 R119909lowast
119899
= T119909lowast
forall119899 ge 1 (40)
SinceT119906 = R1199061
is upper bound ofT119909 = R1199091
andT119909lowast =
R119909lowast1
then
R1199091
= R119909 ⪯ R1199061
R119909lowast
1
= R119909lowast
⪯ R1199061
(41)
It is easy to show that R119909 ⪯ R119906119899
and R119909lowast ⪯ R119906119899
for all119899 ge 1 thenR119909 andR119909lowast are comparable withR119906
119899
On using(9) we have
119889 (R119909R119906119899+1
) = 119889 (T119909T119906119899
)
le 120572119889 (R119909T119909)119889 (R119906
119899
T119906119899
)
1 + 119889 (R119909R119906119899
)
+ 120573119889 (R119909R119906119899
)
(42)
or
119889 (R119909R119906119899+1
) le 120573119889 (R119909R119906119899
) (43)
Owing to (43) we can write
119889 (R119909R119906119899+1
) le 120573119889 (R119909R119906119899
)
le 1205732
119889 (R119909R119906119899minus1
) le sdot sdot sdot
le 120573119899+1
119889 (R119909R1199060
)
(44)
Taking the limit as 119899 rarr infin in (44) we get
lim119899rarrinfin
119889 (R119909R119906119899
) = 0 (45)
as 0 lt 120573 lt 1Similarly one can also show that
lim119899rarrinfin
119889 (R119909lowast
R119906119899
) = 0 (46)
On using (45) and (46) we can have
119889 (R119909R119909lowast
) le 119889 (R119909R119906119899+1
) + 119889 (R119906119899+1
R119909lowast
) (47)
so that lim119899rarrinfin
119889(R119909R119909lowast)0 that isR119909 = R119909lowast Thus wehave proved (36)
Since R119909 = T119909 owing to commutativity of T and Rone can write
R (R119909) = R (T119909) = T (R119909) (48)
which on insertingR119909 = 119911 gives rise to
R119911 = T119911 (49)
Thus 119911 is another coincidence point of the pair Now due toR119909 = R119909lowast (for every coincidence point119909 and119909lowast) and owingto the fact that 119911 is coincidence point of the pair RT itfollows thatR119911 = R119909 that is
R119911 = 119911 (50)
6 Abstract and Applied Analysis
On making use of (49) and (50) we can have
119911 = R119911 = T119911 (51)
which shows that 119911 is a common fixed point ofT andRTo prove the uniqueness let 119901 be another common fixed
point of the pair RT Since119901 and 119911 are coincidence pointsthenR119901 = R119911
Owing to that119901 and 119911 are commonfixed point of the pairsRT one can have
119901 = R119901 = R119911 = 119911 (52)
This completes the proof
The following simple example demonstrates Theorem 16
Example 17 ConsiderX = (1 0) (0 1) sub 1198772 equipped withnatural order (119909 119910) le (119911 119905) hArr 119909 le 119911 and 119910 le 119905 Thus (X le)is a partially ordered set wherein the two elements are notcomparable to each other Also (X le) is a complete metricspace under Euclideanmetric DefinedmappingsT and R
X rarr X as
T (119909 119910) = (1 0) (119909 119910) = (1 0)
(1 0) (119909 119910) = (0 1) (53)
R (119909 119910) = (1 0) (119909 119910) = (1 0)
(0 1) (119909 119910) = (0 1) (54)
As R(119909 119910) and R(119911 119905) in X are merely comparable tothemselves inequality (9) is vacuously satisfied for every 120572 120573
Notice that T(X) = (1 0) sub R(X) = X Also for(119909 119910) = (1 0)
R (119911 119905) = T (119909 119910) = T (1 0) = (1 0) 997904rArr (119911 119905) = (1 0)
(55)
so that
T (1 0) = T (119909 119910) le T (119911 119905) = T (1 0) (56)
Otherwise for (119911 119905) = (0 1)
R (119911 119905) = T (119909 119910) = T (0 1) = (1 0) 997904rArr (119911 119905) = (1 0)
(57)
so that
(1 0) = T (0 1) = T (119909 119910) le T (119911 119905) = T (1 0) = (1 0)
(58)
ThusT is weakly increasing with respect toRIf (119909119899
119910119899
) sub X is a nondecreasing sequence convergingto (119909 119910) isin X then necessarily (119909
119899
119910119899
) must be constantsequence that is (119909
119899
119910119899
) = (119909 119910) for all 119899 isin 119873 so that limit(119909 119910) is an upper bond for all the terms in the sequencewhichshows thatX is regular
By using definitions of the mapsT andR we have
(119909 119910) = R (119909 119910) = lim119899rarrinfin
T (119909119899
119910119899
) = lim119899rarrinfin
RT (119909119899
119910119899
)
(59)
wherein lim119899rarrinfin
R(119909119899
119910119899
) = lim119899rarrinfin
T(119909119899
119910119899
) = (119909 119910) sothat
R (119909 119910) = lim119899rarrinfin
RT (119909119899
119910119899
) (60)
which shows that the pair RT is weakly reciprocallycontinuous Also the pair RT is clearly commuting
Now we show that for every 119909 and 119909lowast in X there existsa 119906 isin X such that T119906 is upper bound of T119909 and T119909lowast If119909 = 119909lowast then choice of 119906 is obvious Otherwise if 119909 = (1 0)
and 119909lowast = (0 1) we can choose 119906 = (0 1) such that T119906 =
T(0 1) = (1 0) is upper bound of T119909 = (1 0) and T119909lowast =
(1 0)Thus we have shown that all the conditions ofTheorem 16
are satisfied and (119909 119910) = (1 0) is the unique common fixedpoint ofT andR
Theorem 18 If in addition to the hypotheses of Theorem 12for pairs (119909 119909lowast) isin XtimesX there exists a 119906 isin X such thatT119906 isupper bound of T119909 119886119899119889 T119909lowast then T and R have a uniquecommon fixed point
Proof Proof is obvious in view of Theorems 12 and 16
Corollary 19 In addition to the hypotheses of Theorem 16 (orTheorem 18) suppose that for every (119909 119909lowast) isin XtimesX there existsa 119906 isin X such thatT119906 is upper bound ofT119909 119886119899119889 T119909lowast ThenT has a unique fixed point that is there exists a unique 119909 isin Xsuch that 119909 = T119909
Proof In Theorem 16 ifR = 119868 (the identity mapping on 119883)we have the result
The following example demonstrates Theorem 18
Example 20 Consider X = [0infin) equipped with the usualmetric and natural order le Define two mappings TR
X rarr X by
T119909 = 1199093 + 2 0 le 119909 lt 1
3 1 le 119909
R119909 = 21199093 + 1 0 le 119909 lt 1
3 1 le 119909
(61)
Then evidently (X le) is a partially ordered set and the mapsT andR satisfy inequality (9) with 120573 = 12 and 120572 = 13 for119909 119910 isin [0infin)
To show that T is weakly increasing with respect to Rnotice thatT(X) = [2 3] sub [1 3] = R(X) Firstly we arguethe case 119909 = 0 Let 119910 isin Rminus1(1198790) that isR(119910) = T(0) = 2
so that 21199103 + 1 = 2 or 119910 = 1213 Using definitions ofT andR we have
2 = T (0) le T (119910) = T(1
213) =
1
2+ 2 (62)
Secondly if 119909 ge 1 then we haveR(119910) = T(119909) = 3 By usingdefinition ofR we have 119910 ge 1 so that
3 = T (119909) le T (119910) = 3 (63)
Abstract and Applied Analysis 7
Finally we consider the case 0 lt 119909 lt 1 Let 119910 isin
Rminus1(T(119909)) that isR(119910) = T(119909) In view of definition ofT we have T(119909) = 1199093 + 2 so that R(119910) = 1199093 + 2 Byusing the definition of R we have R(119910) = 21199103 + 1 so that21199103 + 1 = 1199093 + 2 or 119910 = ((119909
3
+ 1)2)13 Thus we have
1199093
+ 2 = T (119909) le T (119910) = T((1199093 + 1
2)
13
) =1199093 + 1
2+ 2
(64)
ThereforeT is weakly increasing with respect toRSince T and R are continuous therefore this pair of
maps is reciprocally continuousNow we show that maps T and R are compatible
If lim119899rarrinfin
119909119899
= 119896 and 119896 ge 1 then lim119899rarrinfin
T119909119899
=
lim119899rarrinfin
R119909119899
= 3 and henceforth
lim119899rarrinfin
TR119909119899
= lim119899rarrinfin
RT119909119899
= 3 (65)
implying thereby lim119899rarrinfin
119889(RT119909119899
TR119909119899
) = 0 so thatT and R are compatible
Thus we have shown that all the conditions ofTheorem 18are satisfied and 119909 = 3 is the unique fixed point ofT and R
4 An Application
In this section we present an application of Theorem 13 andused the idea of Ciric et al [29] to define a partial order andprove an existence theorem for the solution of an integralequation
Theorem 21 Consider the integral equation
119909 (119905) = int119879
0
119870 (119905 119904 119909 (119904)) 119889119904 + 119892 (119905) 119905 isin [0 119879] (66)
with 119879 gt 0 wherein
(i) the functions 119870 [0 119879] times [0 119879] times R119899 rarr R119899 and 119892
R119899 rarr R119899 are continuous(ii) for each 119905 119904 isin [0 119879]
119870 (119905 119904 119909 (119904)) ≪ 119870(119905 119904 int119879
0
119870 (119904 120591 119909 (120591)) 119889120591 + 119892 (119904)) (67)
where≪ denotes a partial order relation on R119899(iii) there exists a continuous function 119901 [0 119879]times[0 119879] rarr
R+
such that
|119870 (119905 119904 119906) minus 119870 (119905 119904 V)| le 119901 (119905 119904) |119906 minus V| (68)
for each 119905 119904 isin [0 119879] and is also comparable 119906 V isin R119899
(iv) sup119905isin[0119879]
int119879
0
119901(119905 119904)119889119904 le 120573 lt 1
Then the integral equation (66) has a unique solution 119909lowast in119862([0 119879]R119899)
Proof ConsiderX = 119862([0 119879]R119899)with the usual supremumnorm that is
119909 = max119905isin[0119879]
|119909 (119905)| (69)
for 119909 119910 isin 119862([0 119879]R119899) Define onX a partial order as follows(119909 119910 isin 119862([0 119879]R119899))
119909 ⪯ 119910 iff 119909 (119905) ≪ 119910 (119905) for any 119905 isin [0 119879] (70)
Then (X ⪯) is a partially ordered set and (X sdot ) is acomplete metric space
Moreover for any increasing sequence 119909119899
inX converg-ing to 119909lowast isin X we have 119909
119899
(119905) ≪ 119909lowast(119905) for any 119905 isin [0 119879] Alsofor every 119909 119910 isin X there exists 119888 isin X which depends on 119909
and 119910 and is also comparable to 119909 and 119910 (cf [21])DefineT 119862([0 119879]) rarr 119862([0 119879]) by
T119906 (119905) = int119879
0
119870 (119905 119904 119906 (119904)) 119889119904 + 119892 (119905)
119905 isin [0 119879] 119906 isin 119862 ([0 119879])
(71)
Let us prove that
T119906 ⪯ T (T119906) forall119906 isin 119862 ([0 119879]) (72)
Let 119906 isin 119862([0 119879]) From (ii) for all 119905 isin [0 119879] we have
T119906 (119905)
= int119879
0
119870 (119905 119904 119906 (119904)) 119889119904 + 119892 (119905)
≪ int119879
0
119870(119905 119904 int119879
0
119870 (119904 120591 119906 (120591)) 119889120591 + 119892 (119904)) 119889119904 + 119892 (119905)
= int119879
0
119870 (119905 119904T119906 (119904)) 119889119904 + 119892 (119905)
= T (T119906) (119905)
(73)
Then (72) holdsNow for all 119909 119910 isin 119862([0 119879]) with 119910 ⪯ 119909 we have (by (iii))
1003816100381610038161003816T119909 (119905) minusT119910 (119905)1003816100381610038161003816
le int119879
0
1003816100381610038161003816119870 (119905 119904 119909 (119904)) minus 119870 (119905 119904 119910 (119904))1003816100381610038161003816 119889119904
le int119879
0
119901 (119905 119904) (1003816100381610038161003816119909 (119904) minus 119910 (119904)
1003816100381610038161003816) 119889119904
le1003817100381710038171003817119909 minus 119910
1003817100381710038171003817 int119879
0
119901 (119905 119904) 119889119904
(74)
Hence1003817100381710038171003817T119909 (119905) minusT119910 (119905)
1003817100381710038171003817
le1003817100381710038171003817119909 minus 119910
1003817100381710038171003817 sup119905isin[0119879]
int119879
0
119901 (119905 119904) 119889119904
le 1205731003817100381710038171003817119909 minus 119910
1003817100381710038171003817
(75)
8 Abstract and Applied Analysis
On the other hand it is demonstrated in [16] thatcondition of regularity is satisfied forX = 119862([0 119879]R119899)
Thus all the hypotheses of Theorem 13 are satisfied for120572 = 0 and then T has a fixed point 119906lowast isin 119862([0 119879]R119899) thatis 119906lowast is a solution of the integral equation (66)
Acknowledgments
The authors would like to thank the referees for careful read-ing and for providing valuable suggestions and comments forthis paper Also Poom Kumam would like to thank the KingMongkuts University of TechnologyThonburi (KMUTT) forthe financial support and Dhananjay Gopal is thankful toCSIR Government of India Grant no 25(0215)13EMR-II
References
[1] S Banach ldquoSur les operations dans les ensembles abstraits etleur application aux equations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922
[2] H K Nashine and I Altun ldquoFixed point theorems for gener-alized weakly contractive condition in ordered metric spacesrdquoFixed PointTheory and Applications vol 2011 Article ID 13236720 pages 2011
[3] R P Agarwal M A El-Gebeily and D OrsquoRegan ldquoGeneralizedcontractions in partially ordered metric spacesrdquo ApplicableAnalysis vol 87 no 1 pp 109ndash116 2008
[4] I Altun and H Simsek ldquoSome fixed point theorems onordered metric spaces and applicationrdquo Fixed Point Theory andApplications vol 2010 Article ID 621469 17 pages 2010
[5] A Amini-Harandi and H Emami ldquoA fixed point theoremfor contraction type maps in partially ordered metric spacesand application to ordinary differential equationsrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 5 pp2238ndash2242 2010
[6] I Beg and A R Butt ldquoFixed point for set-valued mappings sat-isfying an implicit relation in partially ordered metric spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no9 pp 3699ndash3704 2009
[7] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 65 no 7 pp 1379ndash1393 2006
[8] A Cabada and J J Nieto ldquoFixed points and approximatesolutions for nonlinear operator equationsrdquo Journal of Com-putational and Applied Mathematics vol 113 no 1-2 pp 17ndash25 2000 Fixed point theory with applications in nonlinearanalysis
[9] J Caballero J Harjani and K Sadarangani ldquoContractive-likemapping principles in ordered metric spaces and applicationto ordinary differential equationsrdquo Fixed Point Theory andApplications vol 2010 Article ID 916064 14 pages 2010
[10] L Ciric N Cakic M Rajovic and J S Ume ldquoMonotonegeneralized nonlinear contractions in partially ordered metricspacesrdquo Fixed Point Theory and Applications vol 2008 ArticleID 131294 11 pages 2008
[11] Lj B Ciric D Mihet and R Saadati ldquoMonotone generalizedcontractions in partially ordered probabilistic metric spacesrdquoTopology and its Applications vol 156 no 17 pp 2838ndash28442009
[12] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysis Theory Methods amp Applications vol 71 no 7-8 pp 3403ndash3410 2009
[13] J Harjani and K Sadarangani ldquoGeneralized contractions inpartially ordered metric spaces and applications to ordinarydifferential equationsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1188ndash1197 2010
[14] H K Nashine and B Samet ldquoFixed point results for map-pings satisfying (120595 120593)-weakly contractive condition in partiallyordered metric spacesrdquo Nonlinear Analysis Theory Methods ampApplications vol 74 no 6 pp 2201ndash2209 2011
[15] H K Nashine and W Shatanawi ldquoCoupled common fixedpoint theorems for a pair of commuting mappings in partiallyordered complete metric spacesrdquo Computers amp Mathematicswith Applications vol 62 no 4 pp 1984ndash1993 2011
[16] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005
[17] J J Nieto and R Rodrıguez-Lopez ldquoExistence and uniquenessof fixed point in partially ordered sets and applications toordinary differential equationsrdquo Acta Mathematica Sinica vol23 no 12 pp 2205ndash2212 2007
[18] R P Pant ldquoCommon fixed points of four mappingsrdquo Bulletin ofthe Calcutta Mathematical Society vol 90 no 4 pp 281ndash2861998
[19] R P Pant R K Bisht andDArora ldquoWeak reciprocal continuityand fixed point theoremsrdquo Annali dellrsquoUniversita di FerraraSezione VII ScienzeMatematiche vol 57 no 1 pp 181ndash190 2011
[20] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrixequationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004
[21] D OrsquoRegan and A Petrusel ldquoFixed point theorems for gen-eralized contractions in ordered metric spacesrdquo Journal ofMathematical Analysis andApplications vol 341 no 2 pp 1241ndash1252 2008
[22] B Samet ldquoCoupled fixed point theorems for a generalizedMeir-Keeler contraction in partially ordered metric spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 72 no12 pp 4508ndash4517 2010
[23] W Sintunavarat Y J Cho and P Kumam ldquoUrysohn integralequations approach by common fixed points in complex valuedmetric spacesrdquo Advances in Difference Equations vol 2013article 49 2013
[24] W Shatanawi ldquoPartially ordered cone metric spaces andcoupled fixed point resultsrdquo Computers amp Mathematics withApplications vol 60 no 8 pp 2508ndash2515 2010
[25] J Harjani B Lopez and K Sadarangani ldquoA fixed point theoremfor mappings satisfying a contractive condition of rationaltype on a partially ordered metric spacerdquo Abstract and AppliedAnalysis vol 2010 Article ID 190701 8 pages 2010
[26] D S Jaggi ldquoSome unique fixed point theoremsrdquo Indian Journalof Pure and AppliedMathematics vol 8 no 2 pp 223ndash230 1977
[27] B C Dhage ldquoCondensing mappings and applications to exis-tence theorems for common solution of differential equationsrdquoBulletin of the Korean Mathematical Society vol 36 no 3 pp565ndash578 1999
[28] B C Dhage D OrsquoRegan and R P Agarwal ldquoCommon fixedpoint theorems for a pair of countably condensing mappingsin ordered Banach spacesrdquo Journal of Applied Mathematics andStochastic Analysis vol 16 no 3 pp 243ndash248 2003
Abstract and Applied Analysis 9
[29] Lj Ciric B Samet C Vetro and M Abbas ldquoFixed point resultsfor weak contractive mappings in ordered 119870-metric spacesrdquoFixed Point Theory vol 13 no 1 pp 59ndash72 2012
[30] H K Nashine W Sintunavarat and P Kumam ldquoCyclic gener-alized contractions and fixed point results with applications toan integral equationrdquo Fixed Point Theory and Applications vol2012 article 217 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
Corollary 11 Let (X ⪯) be a partially ordered set on whichthere is a metric 119889 on X such that (X 119889) is a complete metricspace Let TR X rarr X be given mappings satisfying (forpairs (119909 119910) isin X timesX whereinR119909 andR119910 are comparable)
119889 (T119909T119910) le 120573119889 (R119909R119910) (25)
where 120573 is a nonnegative real number with 120573 lt 1 Suppose that
(a) X is regular andT is weakly increasing withR(b) the pair (RT) is commuting as well as weakly
reciprocally continuous
Then T and R have a coincidence point That is there exists119906 isin X such thatR119906 = T119906
Theorem 12 Let (X ⪯) be a partially ordered set equippedwith a metric 119889 on X such that (X 119889) is a complete metricspace Let TR X rarr X be two mappings satisfying (forpairs (119909 119910) isin X timesX wherein R119909 and R119910 are comparable)
119889 (T119909T119910) le 120572119889 (R119909T119909) sdot 119889 (R119910T119910)
1 + 119889 (R119909R119910)
+ 120573119889 (R119909R119910)
(26)
where 120572 120573 are non-negative real numbers with 120572 + 120573 lt 1Suppose that
(a) T is weakly increasing withR(b) the pair TR is compatible and reciprocally contin-
uous
Then T and R have a coincidence point That is there exists119906 isin X such thatR119906 = T119906
Proof Proceeding on the lines of the proof of Theorem 10one can furnish a sequence 119909
119899
such that
lim119899rarr+infin
T (119909119899
) = lim119899rarr+infin
R (119909119899
) = 119909 (27)
Now it remains to show that 119909 is a coincidence point of Tand R To accomplish this as the pair TR is compatibleas well as reciprocally continuous
lim119899rarrinfin
119889 (R (T (119909119899
)) T (R (119909119899
))) = 0 (28)
R (119909) = lim119899rarrinfin
R (T (119909119899
)) T (119909) = lim119899rarrinfin
T (R (119909119899
))
(29)
whenever
lim119899rarr+infin
T (119909119899
) = lim119899rarr+infin
R (119909119899
) = 119909 (30)
By using (29) in (28) we have 119889(T119909R119909) = 0 so thatT119909 =
R119909
By appealing Theorems 10 and 12 one can also have thefollowing natural theorem
Theorem 13 Let (X ⪯) be a partially ordered set on whichthere is a metric 119889 onX such that (X 119889) is a complete metricspace Let T X rarr X be nondecreasing mapping satisfying(for pairs (119909 119910) isin XtimesXwhereinR119909 andR119910 are comparable)
119889 (T119909T119910) le 120572119889 (119909T119909) sdot 119889 (119910T119910)
1 + 119889 (119909 119910)+ 120573119889 (119909 119910) (31)
where 120572 120573 are non-negative real numbers such that 120572 + 120573 lt 1Suppose that
(I) T119909 ⪯ T(T119909) for all 119909 isin X(II) eitherT is continuous orX is regular
ThenT has a fixed point
Proof The proof of this theorem can be outlined on the linesof the proof of Theorem 10 realizing R to be the identitymapping onX
Now we introduce the following property which will beutilized in our next theorem
Property (A) IfR(119909119899
) is a nondecreasing sequence in119883 suchthat lim
119899rarr+infin
R(119909119899
) = 119909 thenR(119909119899
) is comparable toR(119909)for all 119899 isin 119873
Theorem 14 Let (X ⪯) be a partially ordered set on whichthere exists a metric 119889 on X such that (X 119889) is a completemetric space Let TR X rarr X be given mappingssatisfying (for pairs (119909 119910) isin X timesX whereinR119909 andR119910 arecomparable)
119889 (T119909T119910) le 120572119889 (R119909T119909) sdot 119889 (R119910T119910)
1 + 119889 (R119909R119910)
+ 120573119889 (R119909R119910)
(32)
where 120572 120573 are non-negative real numbers with 120572 + 120573 lt 1Suppose that
(a) X is regular andT is weakly increasing withR(b) the pair (RT) is commuting as well as weakly
reciprocally continuous(c) R satisfies Property (A)
ThenR andT have a common fixed point
Proof Proceeding on the lines of the proof of Theorem 10one can inductively construct nondecreasing sequenceR119909119899
such that lim119899rarrinfin
T119909119899
= lim119899rarrinfin
R119909119899+1
= 119909 andR(119909) = T(119909) SinceR(119909
119899
) andR(119909) are comparable (for all119899 isin 119873) by using condition (32) we have
119889 (R119909R119909119899+1
) = 119889 (T119909T119909119899
)
le 120572119889 (R119909T119909)119889 (R119909
119899
T119909119899
)
1 + 119889 (R119909R119909119899
)
+ 120573119889 (R119909R119909119899
)
(33)
Abstract and Applied Analysis 5
Taking the limit as 119899 rarr infin one gets
119909 = R119909 = T119909 (34)
This completes the proof
Theorem 15 Let (X ⪯) be a partially ordered set on whichthere exists a metric 119889 on X such that (X 119889) is a completemetric space Let TR X rarr X be given mappingssatisfying (for pairs (119909 119910) isin X timesX whereinR119909 andR119910 arecomparable)
119889 (T119909T119910) le 120572119889 (R119909T119909) sdot 119889 (R119910T119910)
1 + 119889 (R119909R119910)
+ 120573119889 (R119909R119910)
(35)
where 120572 120573 are non-negative real numbers with 120572 + 120573 lt 1Suppose that
(a) T is weakly increasing withR(b) the pair TR is compatible and reciprocally contin-
uous(c) R satisfies Property (A)
ThenR andT have a common fixed point
Proof Proof is obvious in view of Theorems 12 and 14
3 Uniqueness Results
In what follows we investigate the conditions under whichTheorem 10 ensures the uniqueness of common fixed point
Theorem 16 If in addition to the hypotheses of Theorem 10every (119909 119909lowast) isin X timesX there exists a 119906 isin X such that T119906 isupper bound of T119909 119886119899119889 T119909lowast then T and R have a uniquecommon fixed point
Proof In view of Theorem 10 the set of coincidence pointsof the maps T and R is non-empty If 119909 and 119909lowast are twocoincidence points of the maps T and R (ie R119909 =
T119909 R119909lowast = T119909lowast) then we proceed to show that
R119909 = R119909lowast
(36)
In view of the additional hypothesis of this theorem thereexists 119906 isin X such thatT119906 is upper bound ofT119909 and T119909lowastPut 1199060
= 119906 and choose 1199061
isin X such thatR1199061
= T1199060
Nowproceeding on the lines of the proof of Theorem 10 one caninductively define sequence R119906
119899
such that
R119906119899
= T119906119899minus1
forall119899 (37)
wherein
R1199061
⪯ R1199062
⪯ R1199093
⪯ sdot sdot sdot ⪯ R119906119899
⪯ R119906119899+1
⪯ sdot sdot sdot (38)
Further setting 1199090
= 119909 and 119909lowast0
= 119909lowast one can also definethe sequences R119909
119899
and R119909lowast119899
such that
R119909119899
= T119909119899minus1
R119909lowast
119899
= T119909lowast
119899minus1
(39)
For every 119899 ge 1 we have
R119909119899
= T119909 R119909lowast
119899
= T119909lowast
forall119899 ge 1 (40)
SinceT119906 = R1199061
is upper bound ofT119909 = R1199091
andT119909lowast =
R119909lowast1
then
R1199091
= R119909 ⪯ R1199061
R119909lowast
1
= R119909lowast
⪯ R1199061
(41)
It is easy to show that R119909 ⪯ R119906119899
and R119909lowast ⪯ R119906119899
for all119899 ge 1 thenR119909 andR119909lowast are comparable withR119906
119899
On using(9) we have
119889 (R119909R119906119899+1
) = 119889 (T119909T119906119899
)
le 120572119889 (R119909T119909)119889 (R119906
119899
T119906119899
)
1 + 119889 (R119909R119906119899
)
+ 120573119889 (R119909R119906119899
)
(42)
or
119889 (R119909R119906119899+1
) le 120573119889 (R119909R119906119899
) (43)
Owing to (43) we can write
119889 (R119909R119906119899+1
) le 120573119889 (R119909R119906119899
)
le 1205732
119889 (R119909R119906119899minus1
) le sdot sdot sdot
le 120573119899+1
119889 (R119909R1199060
)
(44)
Taking the limit as 119899 rarr infin in (44) we get
lim119899rarrinfin
119889 (R119909R119906119899
) = 0 (45)
as 0 lt 120573 lt 1Similarly one can also show that
lim119899rarrinfin
119889 (R119909lowast
R119906119899
) = 0 (46)
On using (45) and (46) we can have
119889 (R119909R119909lowast
) le 119889 (R119909R119906119899+1
) + 119889 (R119906119899+1
R119909lowast
) (47)
so that lim119899rarrinfin
119889(R119909R119909lowast)0 that isR119909 = R119909lowast Thus wehave proved (36)
Since R119909 = T119909 owing to commutativity of T and Rone can write
R (R119909) = R (T119909) = T (R119909) (48)
which on insertingR119909 = 119911 gives rise to
R119911 = T119911 (49)
Thus 119911 is another coincidence point of the pair Now due toR119909 = R119909lowast (for every coincidence point119909 and119909lowast) and owingto the fact that 119911 is coincidence point of the pair RT itfollows thatR119911 = R119909 that is
R119911 = 119911 (50)
6 Abstract and Applied Analysis
On making use of (49) and (50) we can have
119911 = R119911 = T119911 (51)
which shows that 119911 is a common fixed point ofT andRTo prove the uniqueness let 119901 be another common fixed
point of the pair RT Since119901 and 119911 are coincidence pointsthenR119901 = R119911
Owing to that119901 and 119911 are commonfixed point of the pairsRT one can have
119901 = R119901 = R119911 = 119911 (52)
This completes the proof
The following simple example demonstrates Theorem 16
Example 17 ConsiderX = (1 0) (0 1) sub 1198772 equipped withnatural order (119909 119910) le (119911 119905) hArr 119909 le 119911 and 119910 le 119905 Thus (X le)is a partially ordered set wherein the two elements are notcomparable to each other Also (X le) is a complete metricspace under Euclideanmetric DefinedmappingsT and R
X rarr X as
T (119909 119910) = (1 0) (119909 119910) = (1 0)
(1 0) (119909 119910) = (0 1) (53)
R (119909 119910) = (1 0) (119909 119910) = (1 0)
(0 1) (119909 119910) = (0 1) (54)
As R(119909 119910) and R(119911 119905) in X are merely comparable tothemselves inequality (9) is vacuously satisfied for every 120572 120573
Notice that T(X) = (1 0) sub R(X) = X Also for(119909 119910) = (1 0)
R (119911 119905) = T (119909 119910) = T (1 0) = (1 0) 997904rArr (119911 119905) = (1 0)
(55)
so that
T (1 0) = T (119909 119910) le T (119911 119905) = T (1 0) (56)
Otherwise for (119911 119905) = (0 1)
R (119911 119905) = T (119909 119910) = T (0 1) = (1 0) 997904rArr (119911 119905) = (1 0)
(57)
so that
(1 0) = T (0 1) = T (119909 119910) le T (119911 119905) = T (1 0) = (1 0)
(58)
ThusT is weakly increasing with respect toRIf (119909119899
119910119899
) sub X is a nondecreasing sequence convergingto (119909 119910) isin X then necessarily (119909
119899
119910119899
) must be constantsequence that is (119909
119899
119910119899
) = (119909 119910) for all 119899 isin 119873 so that limit(119909 119910) is an upper bond for all the terms in the sequencewhichshows thatX is regular
By using definitions of the mapsT andR we have
(119909 119910) = R (119909 119910) = lim119899rarrinfin
T (119909119899
119910119899
) = lim119899rarrinfin
RT (119909119899
119910119899
)
(59)
wherein lim119899rarrinfin
R(119909119899
119910119899
) = lim119899rarrinfin
T(119909119899
119910119899
) = (119909 119910) sothat
R (119909 119910) = lim119899rarrinfin
RT (119909119899
119910119899
) (60)
which shows that the pair RT is weakly reciprocallycontinuous Also the pair RT is clearly commuting
Now we show that for every 119909 and 119909lowast in X there existsa 119906 isin X such that T119906 is upper bound of T119909 and T119909lowast If119909 = 119909lowast then choice of 119906 is obvious Otherwise if 119909 = (1 0)
and 119909lowast = (0 1) we can choose 119906 = (0 1) such that T119906 =
T(0 1) = (1 0) is upper bound of T119909 = (1 0) and T119909lowast =
(1 0)Thus we have shown that all the conditions ofTheorem 16
are satisfied and (119909 119910) = (1 0) is the unique common fixedpoint ofT andR
Theorem 18 If in addition to the hypotheses of Theorem 12for pairs (119909 119909lowast) isin XtimesX there exists a 119906 isin X such thatT119906 isupper bound of T119909 119886119899119889 T119909lowast then T and R have a uniquecommon fixed point
Proof Proof is obvious in view of Theorems 12 and 16
Corollary 19 In addition to the hypotheses of Theorem 16 (orTheorem 18) suppose that for every (119909 119909lowast) isin XtimesX there existsa 119906 isin X such thatT119906 is upper bound ofT119909 119886119899119889 T119909lowast ThenT has a unique fixed point that is there exists a unique 119909 isin Xsuch that 119909 = T119909
Proof In Theorem 16 ifR = 119868 (the identity mapping on 119883)we have the result
The following example demonstrates Theorem 18
Example 20 Consider X = [0infin) equipped with the usualmetric and natural order le Define two mappings TR
X rarr X by
T119909 = 1199093 + 2 0 le 119909 lt 1
3 1 le 119909
R119909 = 21199093 + 1 0 le 119909 lt 1
3 1 le 119909
(61)
Then evidently (X le) is a partially ordered set and the mapsT andR satisfy inequality (9) with 120573 = 12 and 120572 = 13 for119909 119910 isin [0infin)
To show that T is weakly increasing with respect to Rnotice thatT(X) = [2 3] sub [1 3] = R(X) Firstly we arguethe case 119909 = 0 Let 119910 isin Rminus1(1198790) that isR(119910) = T(0) = 2
so that 21199103 + 1 = 2 or 119910 = 1213 Using definitions ofT andR we have
2 = T (0) le T (119910) = T(1
213) =
1
2+ 2 (62)
Secondly if 119909 ge 1 then we haveR(119910) = T(119909) = 3 By usingdefinition ofR we have 119910 ge 1 so that
3 = T (119909) le T (119910) = 3 (63)
Abstract and Applied Analysis 7
Finally we consider the case 0 lt 119909 lt 1 Let 119910 isin
Rminus1(T(119909)) that isR(119910) = T(119909) In view of definition ofT we have T(119909) = 1199093 + 2 so that R(119910) = 1199093 + 2 Byusing the definition of R we have R(119910) = 21199103 + 1 so that21199103 + 1 = 1199093 + 2 or 119910 = ((119909
3
+ 1)2)13 Thus we have
1199093
+ 2 = T (119909) le T (119910) = T((1199093 + 1
2)
13
) =1199093 + 1
2+ 2
(64)
ThereforeT is weakly increasing with respect toRSince T and R are continuous therefore this pair of
maps is reciprocally continuousNow we show that maps T and R are compatible
If lim119899rarrinfin
119909119899
= 119896 and 119896 ge 1 then lim119899rarrinfin
T119909119899
=
lim119899rarrinfin
R119909119899
= 3 and henceforth
lim119899rarrinfin
TR119909119899
= lim119899rarrinfin
RT119909119899
= 3 (65)
implying thereby lim119899rarrinfin
119889(RT119909119899
TR119909119899
) = 0 so thatT and R are compatible
Thus we have shown that all the conditions ofTheorem 18are satisfied and 119909 = 3 is the unique fixed point ofT and R
4 An Application
In this section we present an application of Theorem 13 andused the idea of Ciric et al [29] to define a partial order andprove an existence theorem for the solution of an integralequation
Theorem 21 Consider the integral equation
119909 (119905) = int119879
0
119870 (119905 119904 119909 (119904)) 119889119904 + 119892 (119905) 119905 isin [0 119879] (66)
with 119879 gt 0 wherein
(i) the functions 119870 [0 119879] times [0 119879] times R119899 rarr R119899 and 119892
R119899 rarr R119899 are continuous(ii) for each 119905 119904 isin [0 119879]
119870 (119905 119904 119909 (119904)) ≪ 119870(119905 119904 int119879
0
119870 (119904 120591 119909 (120591)) 119889120591 + 119892 (119904)) (67)
where≪ denotes a partial order relation on R119899(iii) there exists a continuous function 119901 [0 119879]times[0 119879] rarr
R+
such that
|119870 (119905 119904 119906) minus 119870 (119905 119904 V)| le 119901 (119905 119904) |119906 minus V| (68)
for each 119905 119904 isin [0 119879] and is also comparable 119906 V isin R119899
(iv) sup119905isin[0119879]
int119879
0
119901(119905 119904)119889119904 le 120573 lt 1
Then the integral equation (66) has a unique solution 119909lowast in119862([0 119879]R119899)
Proof ConsiderX = 119862([0 119879]R119899)with the usual supremumnorm that is
119909 = max119905isin[0119879]
|119909 (119905)| (69)
for 119909 119910 isin 119862([0 119879]R119899) Define onX a partial order as follows(119909 119910 isin 119862([0 119879]R119899))
119909 ⪯ 119910 iff 119909 (119905) ≪ 119910 (119905) for any 119905 isin [0 119879] (70)
Then (X ⪯) is a partially ordered set and (X sdot ) is acomplete metric space
Moreover for any increasing sequence 119909119899
inX converg-ing to 119909lowast isin X we have 119909
119899
(119905) ≪ 119909lowast(119905) for any 119905 isin [0 119879] Alsofor every 119909 119910 isin X there exists 119888 isin X which depends on 119909
and 119910 and is also comparable to 119909 and 119910 (cf [21])DefineT 119862([0 119879]) rarr 119862([0 119879]) by
T119906 (119905) = int119879
0
119870 (119905 119904 119906 (119904)) 119889119904 + 119892 (119905)
119905 isin [0 119879] 119906 isin 119862 ([0 119879])
(71)
Let us prove that
T119906 ⪯ T (T119906) forall119906 isin 119862 ([0 119879]) (72)
Let 119906 isin 119862([0 119879]) From (ii) for all 119905 isin [0 119879] we have
T119906 (119905)
= int119879
0
119870 (119905 119904 119906 (119904)) 119889119904 + 119892 (119905)
≪ int119879
0
119870(119905 119904 int119879
0
119870 (119904 120591 119906 (120591)) 119889120591 + 119892 (119904)) 119889119904 + 119892 (119905)
= int119879
0
119870 (119905 119904T119906 (119904)) 119889119904 + 119892 (119905)
= T (T119906) (119905)
(73)
Then (72) holdsNow for all 119909 119910 isin 119862([0 119879]) with 119910 ⪯ 119909 we have (by (iii))
1003816100381610038161003816T119909 (119905) minusT119910 (119905)1003816100381610038161003816
le int119879
0
1003816100381610038161003816119870 (119905 119904 119909 (119904)) minus 119870 (119905 119904 119910 (119904))1003816100381610038161003816 119889119904
le int119879
0
119901 (119905 119904) (1003816100381610038161003816119909 (119904) minus 119910 (119904)
1003816100381610038161003816) 119889119904
le1003817100381710038171003817119909 minus 119910
1003817100381710038171003817 int119879
0
119901 (119905 119904) 119889119904
(74)
Hence1003817100381710038171003817T119909 (119905) minusT119910 (119905)
1003817100381710038171003817
le1003817100381710038171003817119909 minus 119910
1003817100381710038171003817 sup119905isin[0119879]
int119879
0
119901 (119905 119904) 119889119904
le 1205731003817100381710038171003817119909 minus 119910
1003817100381710038171003817
(75)
8 Abstract and Applied Analysis
On the other hand it is demonstrated in [16] thatcondition of regularity is satisfied forX = 119862([0 119879]R119899)
Thus all the hypotheses of Theorem 13 are satisfied for120572 = 0 and then T has a fixed point 119906lowast isin 119862([0 119879]R119899) thatis 119906lowast is a solution of the integral equation (66)
Acknowledgments
The authors would like to thank the referees for careful read-ing and for providing valuable suggestions and comments forthis paper Also Poom Kumam would like to thank the KingMongkuts University of TechnologyThonburi (KMUTT) forthe financial support and Dhananjay Gopal is thankful toCSIR Government of India Grant no 25(0215)13EMR-II
References
[1] S Banach ldquoSur les operations dans les ensembles abstraits etleur application aux equations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922
[2] H K Nashine and I Altun ldquoFixed point theorems for gener-alized weakly contractive condition in ordered metric spacesrdquoFixed PointTheory and Applications vol 2011 Article ID 13236720 pages 2011
[3] R P Agarwal M A El-Gebeily and D OrsquoRegan ldquoGeneralizedcontractions in partially ordered metric spacesrdquo ApplicableAnalysis vol 87 no 1 pp 109ndash116 2008
[4] I Altun and H Simsek ldquoSome fixed point theorems onordered metric spaces and applicationrdquo Fixed Point Theory andApplications vol 2010 Article ID 621469 17 pages 2010
[5] A Amini-Harandi and H Emami ldquoA fixed point theoremfor contraction type maps in partially ordered metric spacesand application to ordinary differential equationsrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 5 pp2238ndash2242 2010
[6] I Beg and A R Butt ldquoFixed point for set-valued mappings sat-isfying an implicit relation in partially ordered metric spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no9 pp 3699ndash3704 2009
[7] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 65 no 7 pp 1379ndash1393 2006
[8] A Cabada and J J Nieto ldquoFixed points and approximatesolutions for nonlinear operator equationsrdquo Journal of Com-putational and Applied Mathematics vol 113 no 1-2 pp 17ndash25 2000 Fixed point theory with applications in nonlinearanalysis
[9] J Caballero J Harjani and K Sadarangani ldquoContractive-likemapping principles in ordered metric spaces and applicationto ordinary differential equationsrdquo Fixed Point Theory andApplications vol 2010 Article ID 916064 14 pages 2010
[10] L Ciric N Cakic M Rajovic and J S Ume ldquoMonotonegeneralized nonlinear contractions in partially ordered metricspacesrdquo Fixed Point Theory and Applications vol 2008 ArticleID 131294 11 pages 2008
[11] Lj B Ciric D Mihet and R Saadati ldquoMonotone generalizedcontractions in partially ordered probabilistic metric spacesrdquoTopology and its Applications vol 156 no 17 pp 2838ndash28442009
[12] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysis Theory Methods amp Applications vol 71 no 7-8 pp 3403ndash3410 2009
[13] J Harjani and K Sadarangani ldquoGeneralized contractions inpartially ordered metric spaces and applications to ordinarydifferential equationsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1188ndash1197 2010
[14] H K Nashine and B Samet ldquoFixed point results for map-pings satisfying (120595 120593)-weakly contractive condition in partiallyordered metric spacesrdquo Nonlinear Analysis Theory Methods ampApplications vol 74 no 6 pp 2201ndash2209 2011
[15] H K Nashine and W Shatanawi ldquoCoupled common fixedpoint theorems for a pair of commuting mappings in partiallyordered complete metric spacesrdquo Computers amp Mathematicswith Applications vol 62 no 4 pp 1984ndash1993 2011
[16] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005
[17] J J Nieto and R Rodrıguez-Lopez ldquoExistence and uniquenessof fixed point in partially ordered sets and applications toordinary differential equationsrdquo Acta Mathematica Sinica vol23 no 12 pp 2205ndash2212 2007
[18] R P Pant ldquoCommon fixed points of four mappingsrdquo Bulletin ofthe Calcutta Mathematical Society vol 90 no 4 pp 281ndash2861998
[19] R P Pant R K Bisht andDArora ldquoWeak reciprocal continuityand fixed point theoremsrdquo Annali dellrsquoUniversita di FerraraSezione VII ScienzeMatematiche vol 57 no 1 pp 181ndash190 2011
[20] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrixequationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004
[21] D OrsquoRegan and A Petrusel ldquoFixed point theorems for gen-eralized contractions in ordered metric spacesrdquo Journal ofMathematical Analysis andApplications vol 341 no 2 pp 1241ndash1252 2008
[22] B Samet ldquoCoupled fixed point theorems for a generalizedMeir-Keeler contraction in partially ordered metric spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 72 no12 pp 4508ndash4517 2010
[23] W Sintunavarat Y J Cho and P Kumam ldquoUrysohn integralequations approach by common fixed points in complex valuedmetric spacesrdquo Advances in Difference Equations vol 2013article 49 2013
[24] W Shatanawi ldquoPartially ordered cone metric spaces andcoupled fixed point resultsrdquo Computers amp Mathematics withApplications vol 60 no 8 pp 2508ndash2515 2010
[25] J Harjani B Lopez and K Sadarangani ldquoA fixed point theoremfor mappings satisfying a contractive condition of rationaltype on a partially ordered metric spacerdquo Abstract and AppliedAnalysis vol 2010 Article ID 190701 8 pages 2010
[26] D S Jaggi ldquoSome unique fixed point theoremsrdquo Indian Journalof Pure and AppliedMathematics vol 8 no 2 pp 223ndash230 1977
[27] B C Dhage ldquoCondensing mappings and applications to exis-tence theorems for common solution of differential equationsrdquoBulletin of the Korean Mathematical Society vol 36 no 3 pp565ndash578 1999
[28] B C Dhage D OrsquoRegan and R P Agarwal ldquoCommon fixedpoint theorems for a pair of countably condensing mappingsin ordered Banach spacesrdquo Journal of Applied Mathematics andStochastic Analysis vol 16 no 3 pp 243ndash248 2003
Abstract and Applied Analysis 9
[29] Lj Ciric B Samet C Vetro and M Abbas ldquoFixed point resultsfor weak contractive mappings in ordered 119870-metric spacesrdquoFixed Point Theory vol 13 no 1 pp 59ndash72 2012
[30] H K Nashine W Sintunavarat and P Kumam ldquoCyclic gener-alized contractions and fixed point results with applications toan integral equationrdquo Fixed Point Theory and Applications vol2012 article 217 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
Taking the limit as 119899 rarr infin one gets
119909 = R119909 = T119909 (34)
This completes the proof
Theorem 15 Let (X ⪯) be a partially ordered set on whichthere exists a metric 119889 on X such that (X 119889) is a completemetric space Let TR X rarr X be given mappingssatisfying (for pairs (119909 119910) isin X timesX whereinR119909 andR119910 arecomparable)
119889 (T119909T119910) le 120572119889 (R119909T119909) sdot 119889 (R119910T119910)
1 + 119889 (R119909R119910)
+ 120573119889 (R119909R119910)
(35)
where 120572 120573 are non-negative real numbers with 120572 + 120573 lt 1Suppose that
(a) T is weakly increasing withR(b) the pair TR is compatible and reciprocally contin-
uous(c) R satisfies Property (A)
ThenR andT have a common fixed point
Proof Proof is obvious in view of Theorems 12 and 14
3 Uniqueness Results
In what follows we investigate the conditions under whichTheorem 10 ensures the uniqueness of common fixed point
Theorem 16 If in addition to the hypotheses of Theorem 10every (119909 119909lowast) isin X timesX there exists a 119906 isin X such that T119906 isupper bound of T119909 119886119899119889 T119909lowast then T and R have a uniquecommon fixed point
Proof In view of Theorem 10 the set of coincidence pointsof the maps T and R is non-empty If 119909 and 119909lowast are twocoincidence points of the maps T and R (ie R119909 =
T119909 R119909lowast = T119909lowast) then we proceed to show that
R119909 = R119909lowast
(36)
In view of the additional hypothesis of this theorem thereexists 119906 isin X such thatT119906 is upper bound ofT119909 and T119909lowastPut 1199060
= 119906 and choose 1199061
isin X such thatR1199061
= T1199060
Nowproceeding on the lines of the proof of Theorem 10 one caninductively define sequence R119906
119899
such that
R119906119899
= T119906119899minus1
forall119899 (37)
wherein
R1199061
⪯ R1199062
⪯ R1199093
⪯ sdot sdot sdot ⪯ R119906119899
⪯ R119906119899+1
⪯ sdot sdot sdot (38)
Further setting 1199090
= 119909 and 119909lowast0
= 119909lowast one can also definethe sequences R119909
119899
and R119909lowast119899
such that
R119909119899
= T119909119899minus1
R119909lowast
119899
= T119909lowast
119899minus1
(39)
For every 119899 ge 1 we have
R119909119899
= T119909 R119909lowast
119899
= T119909lowast
forall119899 ge 1 (40)
SinceT119906 = R1199061
is upper bound ofT119909 = R1199091
andT119909lowast =
R119909lowast1
then
R1199091
= R119909 ⪯ R1199061
R119909lowast
1
= R119909lowast
⪯ R1199061
(41)
It is easy to show that R119909 ⪯ R119906119899
and R119909lowast ⪯ R119906119899
for all119899 ge 1 thenR119909 andR119909lowast are comparable withR119906
119899
On using(9) we have
119889 (R119909R119906119899+1
) = 119889 (T119909T119906119899
)
le 120572119889 (R119909T119909)119889 (R119906
119899
T119906119899
)
1 + 119889 (R119909R119906119899
)
+ 120573119889 (R119909R119906119899
)
(42)
or
119889 (R119909R119906119899+1
) le 120573119889 (R119909R119906119899
) (43)
Owing to (43) we can write
119889 (R119909R119906119899+1
) le 120573119889 (R119909R119906119899
)
le 1205732
119889 (R119909R119906119899minus1
) le sdot sdot sdot
le 120573119899+1
119889 (R119909R1199060
)
(44)
Taking the limit as 119899 rarr infin in (44) we get
lim119899rarrinfin
119889 (R119909R119906119899
) = 0 (45)
as 0 lt 120573 lt 1Similarly one can also show that
lim119899rarrinfin
119889 (R119909lowast
R119906119899
) = 0 (46)
On using (45) and (46) we can have
119889 (R119909R119909lowast
) le 119889 (R119909R119906119899+1
) + 119889 (R119906119899+1
R119909lowast
) (47)
so that lim119899rarrinfin
119889(R119909R119909lowast)0 that isR119909 = R119909lowast Thus wehave proved (36)
Since R119909 = T119909 owing to commutativity of T and Rone can write
R (R119909) = R (T119909) = T (R119909) (48)
which on insertingR119909 = 119911 gives rise to
R119911 = T119911 (49)
Thus 119911 is another coincidence point of the pair Now due toR119909 = R119909lowast (for every coincidence point119909 and119909lowast) and owingto the fact that 119911 is coincidence point of the pair RT itfollows thatR119911 = R119909 that is
R119911 = 119911 (50)
6 Abstract and Applied Analysis
On making use of (49) and (50) we can have
119911 = R119911 = T119911 (51)
which shows that 119911 is a common fixed point ofT andRTo prove the uniqueness let 119901 be another common fixed
point of the pair RT Since119901 and 119911 are coincidence pointsthenR119901 = R119911
Owing to that119901 and 119911 are commonfixed point of the pairsRT one can have
119901 = R119901 = R119911 = 119911 (52)
This completes the proof
The following simple example demonstrates Theorem 16
Example 17 ConsiderX = (1 0) (0 1) sub 1198772 equipped withnatural order (119909 119910) le (119911 119905) hArr 119909 le 119911 and 119910 le 119905 Thus (X le)is a partially ordered set wherein the two elements are notcomparable to each other Also (X le) is a complete metricspace under Euclideanmetric DefinedmappingsT and R
X rarr X as
T (119909 119910) = (1 0) (119909 119910) = (1 0)
(1 0) (119909 119910) = (0 1) (53)
R (119909 119910) = (1 0) (119909 119910) = (1 0)
(0 1) (119909 119910) = (0 1) (54)
As R(119909 119910) and R(119911 119905) in X are merely comparable tothemselves inequality (9) is vacuously satisfied for every 120572 120573
Notice that T(X) = (1 0) sub R(X) = X Also for(119909 119910) = (1 0)
R (119911 119905) = T (119909 119910) = T (1 0) = (1 0) 997904rArr (119911 119905) = (1 0)
(55)
so that
T (1 0) = T (119909 119910) le T (119911 119905) = T (1 0) (56)
Otherwise for (119911 119905) = (0 1)
R (119911 119905) = T (119909 119910) = T (0 1) = (1 0) 997904rArr (119911 119905) = (1 0)
(57)
so that
(1 0) = T (0 1) = T (119909 119910) le T (119911 119905) = T (1 0) = (1 0)
(58)
ThusT is weakly increasing with respect toRIf (119909119899
119910119899
) sub X is a nondecreasing sequence convergingto (119909 119910) isin X then necessarily (119909
119899
119910119899
) must be constantsequence that is (119909
119899
119910119899
) = (119909 119910) for all 119899 isin 119873 so that limit(119909 119910) is an upper bond for all the terms in the sequencewhichshows thatX is regular
By using definitions of the mapsT andR we have
(119909 119910) = R (119909 119910) = lim119899rarrinfin
T (119909119899
119910119899
) = lim119899rarrinfin
RT (119909119899
119910119899
)
(59)
wherein lim119899rarrinfin
R(119909119899
119910119899
) = lim119899rarrinfin
T(119909119899
119910119899
) = (119909 119910) sothat
R (119909 119910) = lim119899rarrinfin
RT (119909119899
119910119899
) (60)
which shows that the pair RT is weakly reciprocallycontinuous Also the pair RT is clearly commuting
Now we show that for every 119909 and 119909lowast in X there existsa 119906 isin X such that T119906 is upper bound of T119909 and T119909lowast If119909 = 119909lowast then choice of 119906 is obvious Otherwise if 119909 = (1 0)
and 119909lowast = (0 1) we can choose 119906 = (0 1) such that T119906 =
T(0 1) = (1 0) is upper bound of T119909 = (1 0) and T119909lowast =
(1 0)Thus we have shown that all the conditions ofTheorem 16
are satisfied and (119909 119910) = (1 0) is the unique common fixedpoint ofT andR
Theorem 18 If in addition to the hypotheses of Theorem 12for pairs (119909 119909lowast) isin XtimesX there exists a 119906 isin X such thatT119906 isupper bound of T119909 119886119899119889 T119909lowast then T and R have a uniquecommon fixed point
Proof Proof is obvious in view of Theorems 12 and 16
Corollary 19 In addition to the hypotheses of Theorem 16 (orTheorem 18) suppose that for every (119909 119909lowast) isin XtimesX there existsa 119906 isin X such thatT119906 is upper bound ofT119909 119886119899119889 T119909lowast ThenT has a unique fixed point that is there exists a unique 119909 isin Xsuch that 119909 = T119909
Proof In Theorem 16 ifR = 119868 (the identity mapping on 119883)we have the result
The following example demonstrates Theorem 18
Example 20 Consider X = [0infin) equipped with the usualmetric and natural order le Define two mappings TR
X rarr X by
T119909 = 1199093 + 2 0 le 119909 lt 1
3 1 le 119909
R119909 = 21199093 + 1 0 le 119909 lt 1
3 1 le 119909
(61)
Then evidently (X le) is a partially ordered set and the mapsT andR satisfy inequality (9) with 120573 = 12 and 120572 = 13 for119909 119910 isin [0infin)
To show that T is weakly increasing with respect to Rnotice thatT(X) = [2 3] sub [1 3] = R(X) Firstly we arguethe case 119909 = 0 Let 119910 isin Rminus1(1198790) that isR(119910) = T(0) = 2
so that 21199103 + 1 = 2 or 119910 = 1213 Using definitions ofT andR we have
2 = T (0) le T (119910) = T(1
213) =
1
2+ 2 (62)
Secondly if 119909 ge 1 then we haveR(119910) = T(119909) = 3 By usingdefinition ofR we have 119910 ge 1 so that
3 = T (119909) le T (119910) = 3 (63)
Abstract and Applied Analysis 7
Finally we consider the case 0 lt 119909 lt 1 Let 119910 isin
Rminus1(T(119909)) that isR(119910) = T(119909) In view of definition ofT we have T(119909) = 1199093 + 2 so that R(119910) = 1199093 + 2 Byusing the definition of R we have R(119910) = 21199103 + 1 so that21199103 + 1 = 1199093 + 2 or 119910 = ((119909
3
+ 1)2)13 Thus we have
1199093
+ 2 = T (119909) le T (119910) = T((1199093 + 1
2)
13
) =1199093 + 1
2+ 2
(64)
ThereforeT is weakly increasing with respect toRSince T and R are continuous therefore this pair of
maps is reciprocally continuousNow we show that maps T and R are compatible
If lim119899rarrinfin
119909119899
= 119896 and 119896 ge 1 then lim119899rarrinfin
T119909119899
=
lim119899rarrinfin
R119909119899
= 3 and henceforth
lim119899rarrinfin
TR119909119899
= lim119899rarrinfin
RT119909119899
= 3 (65)
implying thereby lim119899rarrinfin
119889(RT119909119899
TR119909119899
) = 0 so thatT and R are compatible
Thus we have shown that all the conditions ofTheorem 18are satisfied and 119909 = 3 is the unique fixed point ofT and R
4 An Application
In this section we present an application of Theorem 13 andused the idea of Ciric et al [29] to define a partial order andprove an existence theorem for the solution of an integralequation
Theorem 21 Consider the integral equation
119909 (119905) = int119879
0
119870 (119905 119904 119909 (119904)) 119889119904 + 119892 (119905) 119905 isin [0 119879] (66)
with 119879 gt 0 wherein
(i) the functions 119870 [0 119879] times [0 119879] times R119899 rarr R119899 and 119892
R119899 rarr R119899 are continuous(ii) for each 119905 119904 isin [0 119879]
119870 (119905 119904 119909 (119904)) ≪ 119870(119905 119904 int119879
0
119870 (119904 120591 119909 (120591)) 119889120591 + 119892 (119904)) (67)
where≪ denotes a partial order relation on R119899(iii) there exists a continuous function 119901 [0 119879]times[0 119879] rarr
R+
such that
|119870 (119905 119904 119906) minus 119870 (119905 119904 V)| le 119901 (119905 119904) |119906 minus V| (68)
for each 119905 119904 isin [0 119879] and is also comparable 119906 V isin R119899
(iv) sup119905isin[0119879]
int119879
0
119901(119905 119904)119889119904 le 120573 lt 1
Then the integral equation (66) has a unique solution 119909lowast in119862([0 119879]R119899)
Proof ConsiderX = 119862([0 119879]R119899)with the usual supremumnorm that is
119909 = max119905isin[0119879]
|119909 (119905)| (69)
for 119909 119910 isin 119862([0 119879]R119899) Define onX a partial order as follows(119909 119910 isin 119862([0 119879]R119899))
119909 ⪯ 119910 iff 119909 (119905) ≪ 119910 (119905) for any 119905 isin [0 119879] (70)
Then (X ⪯) is a partially ordered set and (X sdot ) is acomplete metric space
Moreover for any increasing sequence 119909119899
inX converg-ing to 119909lowast isin X we have 119909
119899
(119905) ≪ 119909lowast(119905) for any 119905 isin [0 119879] Alsofor every 119909 119910 isin X there exists 119888 isin X which depends on 119909
and 119910 and is also comparable to 119909 and 119910 (cf [21])DefineT 119862([0 119879]) rarr 119862([0 119879]) by
T119906 (119905) = int119879
0
119870 (119905 119904 119906 (119904)) 119889119904 + 119892 (119905)
119905 isin [0 119879] 119906 isin 119862 ([0 119879])
(71)
Let us prove that
T119906 ⪯ T (T119906) forall119906 isin 119862 ([0 119879]) (72)
Let 119906 isin 119862([0 119879]) From (ii) for all 119905 isin [0 119879] we have
T119906 (119905)
= int119879
0
119870 (119905 119904 119906 (119904)) 119889119904 + 119892 (119905)
≪ int119879
0
119870(119905 119904 int119879
0
119870 (119904 120591 119906 (120591)) 119889120591 + 119892 (119904)) 119889119904 + 119892 (119905)
= int119879
0
119870 (119905 119904T119906 (119904)) 119889119904 + 119892 (119905)
= T (T119906) (119905)
(73)
Then (72) holdsNow for all 119909 119910 isin 119862([0 119879]) with 119910 ⪯ 119909 we have (by (iii))
1003816100381610038161003816T119909 (119905) minusT119910 (119905)1003816100381610038161003816
le int119879
0
1003816100381610038161003816119870 (119905 119904 119909 (119904)) minus 119870 (119905 119904 119910 (119904))1003816100381610038161003816 119889119904
le int119879
0
119901 (119905 119904) (1003816100381610038161003816119909 (119904) minus 119910 (119904)
1003816100381610038161003816) 119889119904
le1003817100381710038171003817119909 minus 119910
1003817100381710038171003817 int119879
0
119901 (119905 119904) 119889119904
(74)
Hence1003817100381710038171003817T119909 (119905) minusT119910 (119905)
1003817100381710038171003817
le1003817100381710038171003817119909 minus 119910
1003817100381710038171003817 sup119905isin[0119879]
int119879
0
119901 (119905 119904) 119889119904
le 1205731003817100381710038171003817119909 minus 119910
1003817100381710038171003817
(75)
8 Abstract and Applied Analysis
On the other hand it is demonstrated in [16] thatcondition of regularity is satisfied forX = 119862([0 119879]R119899)
Thus all the hypotheses of Theorem 13 are satisfied for120572 = 0 and then T has a fixed point 119906lowast isin 119862([0 119879]R119899) thatis 119906lowast is a solution of the integral equation (66)
Acknowledgments
The authors would like to thank the referees for careful read-ing and for providing valuable suggestions and comments forthis paper Also Poom Kumam would like to thank the KingMongkuts University of TechnologyThonburi (KMUTT) forthe financial support and Dhananjay Gopal is thankful toCSIR Government of India Grant no 25(0215)13EMR-II
References
[1] S Banach ldquoSur les operations dans les ensembles abstraits etleur application aux equations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922
[2] H K Nashine and I Altun ldquoFixed point theorems for gener-alized weakly contractive condition in ordered metric spacesrdquoFixed PointTheory and Applications vol 2011 Article ID 13236720 pages 2011
[3] R P Agarwal M A El-Gebeily and D OrsquoRegan ldquoGeneralizedcontractions in partially ordered metric spacesrdquo ApplicableAnalysis vol 87 no 1 pp 109ndash116 2008
[4] I Altun and H Simsek ldquoSome fixed point theorems onordered metric spaces and applicationrdquo Fixed Point Theory andApplications vol 2010 Article ID 621469 17 pages 2010
[5] A Amini-Harandi and H Emami ldquoA fixed point theoremfor contraction type maps in partially ordered metric spacesand application to ordinary differential equationsrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 5 pp2238ndash2242 2010
[6] I Beg and A R Butt ldquoFixed point for set-valued mappings sat-isfying an implicit relation in partially ordered metric spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no9 pp 3699ndash3704 2009
[7] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 65 no 7 pp 1379ndash1393 2006
[8] A Cabada and J J Nieto ldquoFixed points and approximatesolutions for nonlinear operator equationsrdquo Journal of Com-putational and Applied Mathematics vol 113 no 1-2 pp 17ndash25 2000 Fixed point theory with applications in nonlinearanalysis
[9] J Caballero J Harjani and K Sadarangani ldquoContractive-likemapping principles in ordered metric spaces and applicationto ordinary differential equationsrdquo Fixed Point Theory andApplications vol 2010 Article ID 916064 14 pages 2010
[10] L Ciric N Cakic M Rajovic and J S Ume ldquoMonotonegeneralized nonlinear contractions in partially ordered metricspacesrdquo Fixed Point Theory and Applications vol 2008 ArticleID 131294 11 pages 2008
[11] Lj B Ciric D Mihet and R Saadati ldquoMonotone generalizedcontractions in partially ordered probabilistic metric spacesrdquoTopology and its Applications vol 156 no 17 pp 2838ndash28442009
[12] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysis Theory Methods amp Applications vol 71 no 7-8 pp 3403ndash3410 2009
[13] J Harjani and K Sadarangani ldquoGeneralized contractions inpartially ordered metric spaces and applications to ordinarydifferential equationsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1188ndash1197 2010
[14] H K Nashine and B Samet ldquoFixed point results for map-pings satisfying (120595 120593)-weakly contractive condition in partiallyordered metric spacesrdquo Nonlinear Analysis Theory Methods ampApplications vol 74 no 6 pp 2201ndash2209 2011
[15] H K Nashine and W Shatanawi ldquoCoupled common fixedpoint theorems for a pair of commuting mappings in partiallyordered complete metric spacesrdquo Computers amp Mathematicswith Applications vol 62 no 4 pp 1984ndash1993 2011
[16] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005
[17] J J Nieto and R Rodrıguez-Lopez ldquoExistence and uniquenessof fixed point in partially ordered sets and applications toordinary differential equationsrdquo Acta Mathematica Sinica vol23 no 12 pp 2205ndash2212 2007
[18] R P Pant ldquoCommon fixed points of four mappingsrdquo Bulletin ofthe Calcutta Mathematical Society vol 90 no 4 pp 281ndash2861998
[19] R P Pant R K Bisht andDArora ldquoWeak reciprocal continuityand fixed point theoremsrdquo Annali dellrsquoUniversita di FerraraSezione VII ScienzeMatematiche vol 57 no 1 pp 181ndash190 2011
[20] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrixequationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004
[21] D OrsquoRegan and A Petrusel ldquoFixed point theorems for gen-eralized contractions in ordered metric spacesrdquo Journal ofMathematical Analysis andApplications vol 341 no 2 pp 1241ndash1252 2008
[22] B Samet ldquoCoupled fixed point theorems for a generalizedMeir-Keeler contraction in partially ordered metric spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 72 no12 pp 4508ndash4517 2010
[23] W Sintunavarat Y J Cho and P Kumam ldquoUrysohn integralequations approach by common fixed points in complex valuedmetric spacesrdquo Advances in Difference Equations vol 2013article 49 2013
[24] W Shatanawi ldquoPartially ordered cone metric spaces andcoupled fixed point resultsrdquo Computers amp Mathematics withApplications vol 60 no 8 pp 2508ndash2515 2010
[25] J Harjani B Lopez and K Sadarangani ldquoA fixed point theoremfor mappings satisfying a contractive condition of rationaltype on a partially ordered metric spacerdquo Abstract and AppliedAnalysis vol 2010 Article ID 190701 8 pages 2010
[26] D S Jaggi ldquoSome unique fixed point theoremsrdquo Indian Journalof Pure and AppliedMathematics vol 8 no 2 pp 223ndash230 1977
[27] B C Dhage ldquoCondensing mappings and applications to exis-tence theorems for common solution of differential equationsrdquoBulletin of the Korean Mathematical Society vol 36 no 3 pp565ndash578 1999
[28] B C Dhage D OrsquoRegan and R P Agarwal ldquoCommon fixedpoint theorems for a pair of countably condensing mappingsin ordered Banach spacesrdquo Journal of Applied Mathematics andStochastic Analysis vol 16 no 3 pp 243ndash248 2003
Abstract and Applied Analysis 9
[29] Lj Ciric B Samet C Vetro and M Abbas ldquoFixed point resultsfor weak contractive mappings in ordered 119870-metric spacesrdquoFixed Point Theory vol 13 no 1 pp 59ndash72 2012
[30] H K Nashine W Sintunavarat and P Kumam ldquoCyclic gener-alized contractions and fixed point results with applications toan integral equationrdquo Fixed Point Theory and Applications vol2012 article 217 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
On making use of (49) and (50) we can have
119911 = R119911 = T119911 (51)
which shows that 119911 is a common fixed point ofT andRTo prove the uniqueness let 119901 be another common fixed
point of the pair RT Since119901 and 119911 are coincidence pointsthenR119901 = R119911
Owing to that119901 and 119911 are commonfixed point of the pairsRT one can have
119901 = R119901 = R119911 = 119911 (52)
This completes the proof
The following simple example demonstrates Theorem 16
Example 17 ConsiderX = (1 0) (0 1) sub 1198772 equipped withnatural order (119909 119910) le (119911 119905) hArr 119909 le 119911 and 119910 le 119905 Thus (X le)is a partially ordered set wherein the two elements are notcomparable to each other Also (X le) is a complete metricspace under Euclideanmetric DefinedmappingsT and R
X rarr X as
T (119909 119910) = (1 0) (119909 119910) = (1 0)
(1 0) (119909 119910) = (0 1) (53)
R (119909 119910) = (1 0) (119909 119910) = (1 0)
(0 1) (119909 119910) = (0 1) (54)
As R(119909 119910) and R(119911 119905) in X are merely comparable tothemselves inequality (9) is vacuously satisfied for every 120572 120573
Notice that T(X) = (1 0) sub R(X) = X Also for(119909 119910) = (1 0)
R (119911 119905) = T (119909 119910) = T (1 0) = (1 0) 997904rArr (119911 119905) = (1 0)
(55)
so that
T (1 0) = T (119909 119910) le T (119911 119905) = T (1 0) (56)
Otherwise for (119911 119905) = (0 1)
R (119911 119905) = T (119909 119910) = T (0 1) = (1 0) 997904rArr (119911 119905) = (1 0)
(57)
so that
(1 0) = T (0 1) = T (119909 119910) le T (119911 119905) = T (1 0) = (1 0)
(58)
ThusT is weakly increasing with respect toRIf (119909119899
119910119899
) sub X is a nondecreasing sequence convergingto (119909 119910) isin X then necessarily (119909
119899
119910119899
) must be constantsequence that is (119909
119899
119910119899
) = (119909 119910) for all 119899 isin 119873 so that limit(119909 119910) is an upper bond for all the terms in the sequencewhichshows thatX is regular
By using definitions of the mapsT andR we have
(119909 119910) = R (119909 119910) = lim119899rarrinfin
T (119909119899
119910119899
) = lim119899rarrinfin
RT (119909119899
119910119899
)
(59)
wherein lim119899rarrinfin
R(119909119899
119910119899
) = lim119899rarrinfin
T(119909119899
119910119899
) = (119909 119910) sothat
R (119909 119910) = lim119899rarrinfin
RT (119909119899
119910119899
) (60)
which shows that the pair RT is weakly reciprocallycontinuous Also the pair RT is clearly commuting
Now we show that for every 119909 and 119909lowast in X there existsa 119906 isin X such that T119906 is upper bound of T119909 and T119909lowast If119909 = 119909lowast then choice of 119906 is obvious Otherwise if 119909 = (1 0)
and 119909lowast = (0 1) we can choose 119906 = (0 1) such that T119906 =
T(0 1) = (1 0) is upper bound of T119909 = (1 0) and T119909lowast =
(1 0)Thus we have shown that all the conditions ofTheorem 16
are satisfied and (119909 119910) = (1 0) is the unique common fixedpoint ofT andR
Theorem 18 If in addition to the hypotheses of Theorem 12for pairs (119909 119909lowast) isin XtimesX there exists a 119906 isin X such thatT119906 isupper bound of T119909 119886119899119889 T119909lowast then T and R have a uniquecommon fixed point
Proof Proof is obvious in view of Theorems 12 and 16
Corollary 19 In addition to the hypotheses of Theorem 16 (orTheorem 18) suppose that for every (119909 119909lowast) isin XtimesX there existsa 119906 isin X such thatT119906 is upper bound ofT119909 119886119899119889 T119909lowast ThenT has a unique fixed point that is there exists a unique 119909 isin Xsuch that 119909 = T119909
Proof In Theorem 16 ifR = 119868 (the identity mapping on 119883)we have the result
The following example demonstrates Theorem 18
Example 20 Consider X = [0infin) equipped with the usualmetric and natural order le Define two mappings TR
X rarr X by
T119909 = 1199093 + 2 0 le 119909 lt 1
3 1 le 119909
R119909 = 21199093 + 1 0 le 119909 lt 1
3 1 le 119909
(61)
Then evidently (X le) is a partially ordered set and the mapsT andR satisfy inequality (9) with 120573 = 12 and 120572 = 13 for119909 119910 isin [0infin)
To show that T is weakly increasing with respect to Rnotice thatT(X) = [2 3] sub [1 3] = R(X) Firstly we arguethe case 119909 = 0 Let 119910 isin Rminus1(1198790) that isR(119910) = T(0) = 2
so that 21199103 + 1 = 2 or 119910 = 1213 Using definitions ofT andR we have
2 = T (0) le T (119910) = T(1
213) =
1
2+ 2 (62)
Secondly if 119909 ge 1 then we haveR(119910) = T(119909) = 3 By usingdefinition ofR we have 119910 ge 1 so that
3 = T (119909) le T (119910) = 3 (63)
Abstract and Applied Analysis 7
Finally we consider the case 0 lt 119909 lt 1 Let 119910 isin
Rminus1(T(119909)) that isR(119910) = T(119909) In view of definition ofT we have T(119909) = 1199093 + 2 so that R(119910) = 1199093 + 2 Byusing the definition of R we have R(119910) = 21199103 + 1 so that21199103 + 1 = 1199093 + 2 or 119910 = ((119909
3
+ 1)2)13 Thus we have
1199093
+ 2 = T (119909) le T (119910) = T((1199093 + 1
2)
13
) =1199093 + 1
2+ 2
(64)
ThereforeT is weakly increasing with respect toRSince T and R are continuous therefore this pair of
maps is reciprocally continuousNow we show that maps T and R are compatible
If lim119899rarrinfin
119909119899
= 119896 and 119896 ge 1 then lim119899rarrinfin
T119909119899
=
lim119899rarrinfin
R119909119899
= 3 and henceforth
lim119899rarrinfin
TR119909119899
= lim119899rarrinfin
RT119909119899
= 3 (65)
implying thereby lim119899rarrinfin
119889(RT119909119899
TR119909119899
) = 0 so thatT and R are compatible
Thus we have shown that all the conditions ofTheorem 18are satisfied and 119909 = 3 is the unique fixed point ofT and R
4 An Application
In this section we present an application of Theorem 13 andused the idea of Ciric et al [29] to define a partial order andprove an existence theorem for the solution of an integralequation
Theorem 21 Consider the integral equation
119909 (119905) = int119879
0
119870 (119905 119904 119909 (119904)) 119889119904 + 119892 (119905) 119905 isin [0 119879] (66)
with 119879 gt 0 wherein
(i) the functions 119870 [0 119879] times [0 119879] times R119899 rarr R119899 and 119892
R119899 rarr R119899 are continuous(ii) for each 119905 119904 isin [0 119879]
119870 (119905 119904 119909 (119904)) ≪ 119870(119905 119904 int119879
0
119870 (119904 120591 119909 (120591)) 119889120591 + 119892 (119904)) (67)
where≪ denotes a partial order relation on R119899(iii) there exists a continuous function 119901 [0 119879]times[0 119879] rarr
R+
such that
|119870 (119905 119904 119906) minus 119870 (119905 119904 V)| le 119901 (119905 119904) |119906 minus V| (68)
for each 119905 119904 isin [0 119879] and is also comparable 119906 V isin R119899
(iv) sup119905isin[0119879]
int119879
0
119901(119905 119904)119889119904 le 120573 lt 1
Then the integral equation (66) has a unique solution 119909lowast in119862([0 119879]R119899)
Proof ConsiderX = 119862([0 119879]R119899)with the usual supremumnorm that is
119909 = max119905isin[0119879]
|119909 (119905)| (69)
for 119909 119910 isin 119862([0 119879]R119899) Define onX a partial order as follows(119909 119910 isin 119862([0 119879]R119899))
119909 ⪯ 119910 iff 119909 (119905) ≪ 119910 (119905) for any 119905 isin [0 119879] (70)
Then (X ⪯) is a partially ordered set and (X sdot ) is acomplete metric space
Moreover for any increasing sequence 119909119899
inX converg-ing to 119909lowast isin X we have 119909
119899
(119905) ≪ 119909lowast(119905) for any 119905 isin [0 119879] Alsofor every 119909 119910 isin X there exists 119888 isin X which depends on 119909
and 119910 and is also comparable to 119909 and 119910 (cf [21])DefineT 119862([0 119879]) rarr 119862([0 119879]) by
T119906 (119905) = int119879
0
119870 (119905 119904 119906 (119904)) 119889119904 + 119892 (119905)
119905 isin [0 119879] 119906 isin 119862 ([0 119879])
(71)
Let us prove that
T119906 ⪯ T (T119906) forall119906 isin 119862 ([0 119879]) (72)
Let 119906 isin 119862([0 119879]) From (ii) for all 119905 isin [0 119879] we have
T119906 (119905)
= int119879
0
119870 (119905 119904 119906 (119904)) 119889119904 + 119892 (119905)
≪ int119879
0
119870(119905 119904 int119879
0
119870 (119904 120591 119906 (120591)) 119889120591 + 119892 (119904)) 119889119904 + 119892 (119905)
= int119879
0
119870 (119905 119904T119906 (119904)) 119889119904 + 119892 (119905)
= T (T119906) (119905)
(73)
Then (72) holdsNow for all 119909 119910 isin 119862([0 119879]) with 119910 ⪯ 119909 we have (by (iii))
1003816100381610038161003816T119909 (119905) minusT119910 (119905)1003816100381610038161003816
le int119879
0
1003816100381610038161003816119870 (119905 119904 119909 (119904)) minus 119870 (119905 119904 119910 (119904))1003816100381610038161003816 119889119904
le int119879
0
119901 (119905 119904) (1003816100381610038161003816119909 (119904) minus 119910 (119904)
1003816100381610038161003816) 119889119904
le1003817100381710038171003817119909 minus 119910
1003817100381710038171003817 int119879
0
119901 (119905 119904) 119889119904
(74)
Hence1003817100381710038171003817T119909 (119905) minusT119910 (119905)
1003817100381710038171003817
le1003817100381710038171003817119909 minus 119910
1003817100381710038171003817 sup119905isin[0119879]
int119879
0
119901 (119905 119904) 119889119904
le 1205731003817100381710038171003817119909 minus 119910
1003817100381710038171003817
(75)
8 Abstract and Applied Analysis
On the other hand it is demonstrated in [16] thatcondition of regularity is satisfied forX = 119862([0 119879]R119899)
Thus all the hypotheses of Theorem 13 are satisfied for120572 = 0 and then T has a fixed point 119906lowast isin 119862([0 119879]R119899) thatis 119906lowast is a solution of the integral equation (66)
Acknowledgments
The authors would like to thank the referees for careful read-ing and for providing valuable suggestions and comments forthis paper Also Poom Kumam would like to thank the KingMongkuts University of TechnologyThonburi (KMUTT) forthe financial support and Dhananjay Gopal is thankful toCSIR Government of India Grant no 25(0215)13EMR-II
References
[1] S Banach ldquoSur les operations dans les ensembles abstraits etleur application aux equations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922
[2] H K Nashine and I Altun ldquoFixed point theorems for gener-alized weakly contractive condition in ordered metric spacesrdquoFixed PointTheory and Applications vol 2011 Article ID 13236720 pages 2011
[3] R P Agarwal M A El-Gebeily and D OrsquoRegan ldquoGeneralizedcontractions in partially ordered metric spacesrdquo ApplicableAnalysis vol 87 no 1 pp 109ndash116 2008
[4] I Altun and H Simsek ldquoSome fixed point theorems onordered metric spaces and applicationrdquo Fixed Point Theory andApplications vol 2010 Article ID 621469 17 pages 2010
[5] A Amini-Harandi and H Emami ldquoA fixed point theoremfor contraction type maps in partially ordered metric spacesand application to ordinary differential equationsrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 5 pp2238ndash2242 2010
[6] I Beg and A R Butt ldquoFixed point for set-valued mappings sat-isfying an implicit relation in partially ordered metric spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no9 pp 3699ndash3704 2009
[7] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 65 no 7 pp 1379ndash1393 2006
[8] A Cabada and J J Nieto ldquoFixed points and approximatesolutions for nonlinear operator equationsrdquo Journal of Com-putational and Applied Mathematics vol 113 no 1-2 pp 17ndash25 2000 Fixed point theory with applications in nonlinearanalysis
[9] J Caballero J Harjani and K Sadarangani ldquoContractive-likemapping principles in ordered metric spaces and applicationto ordinary differential equationsrdquo Fixed Point Theory andApplications vol 2010 Article ID 916064 14 pages 2010
[10] L Ciric N Cakic M Rajovic and J S Ume ldquoMonotonegeneralized nonlinear contractions in partially ordered metricspacesrdquo Fixed Point Theory and Applications vol 2008 ArticleID 131294 11 pages 2008
[11] Lj B Ciric D Mihet and R Saadati ldquoMonotone generalizedcontractions in partially ordered probabilistic metric spacesrdquoTopology and its Applications vol 156 no 17 pp 2838ndash28442009
[12] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysis Theory Methods amp Applications vol 71 no 7-8 pp 3403ndash3410 2009
[13] J Harjani and K Sadarangani ldquoGeneralized contractions inpartially ordered metric spaces and applications to ordinarydifferential equationsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1188ndash1197 2010
[14] H K Nashine and B Samet ldquoFixed point results for map-pings satisfying (120595 120593)-weakly contractive condition in partiallyordered metric spacesrdquo Nonlinear Analysis Theory Methods ampApplications vol 74 no 6 pp 2201ndash2209 2011
[15] H K Nashine and W Shatanawi ldquoCoupled common fixedpoint theorems for a pair of commuting mappings in partiallyordered complete metric spacesrdquo Computers amp Mathematicswith Applications vol 62 no 4 pp 1984ndash1993 2011
[16] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005
[17] J J Nieto and R Rodrıguez-Lopez ldquoExistence and uniquenessof fixed point in partially ordered sets and applications toordinary differential equationsrdquo Acta Mathematica Sinica vol23 no 12 pp 2205ndash2212 2007
[18] R P Pant ldquoCommon fixed points of four mappingsrdquo Bulletin ofthe Calcutta Mathematical Society vol 90 no 4 pp 281ndash2861998
[19] R P Pant R K Bisht andDArora ldquoWeak reciprocal continuityand fixed point theoremsrdquo Annali dellrsquoUniversita di FerraraSezione VII ScienzeMatematiche vol 57 no 1 pp 181ndash190 2011
[20] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrixequationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004
[21] D OrsquoRegan and A Petrusel ldquoFixed point theorems for gen-eralized contractions in ordered metric spacesrdquo Journal ofMathematical Analysis andApplications vol 341 no 2 pp 1241ndash1252 2008
[22] B Samet ldquoCoupled fixed point theorems for a generalizedMeir-Keeler contraction in partially ordered metric spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 72 no12 pp 4508ndash4517 2010
[23] W Sintunavarat Y J Cho and P Kumam ldquoUrysohn integralequations approach by common fixed points in complex valuedmetric spacesrdquo Advances in Difference Equations vol 2013article 49 2013
[24] W Shatanawi ldquoPartially ordered cone metric spaces andcoupled fixed point resultsrdquo Computers amp Mathematics withApplications vol 60 no 8 pp 2508ndash2515 2010
[25] J Harjani B Lopez and K Sadarangani ldquoA fixed point theoremfor mappings satisfying a contractive condition of rationaltype on a partially ordered metric spacerdquo Abstract and AppliedAnalysis vol 2010 Article ID 190701 8 pages 2010
[26] D S Jaggi ldquoSome unique fixed point theoremsrdquo Indian Journalof Pure and AppliedMathematics vol 8 no 2 pp 223ndash230 1977
[27] B C Dhage ldquoCondensing mappings and applications to exis-tence theorems for common solution of differential equationsrdquoBulletin of the Korean Mathematical Society vol 36 no 3 pp565ndash578 1999
[28] B C Dhage D OrsquoRegan and R P Agarwal ldquoCommon fixedpoint theorems for a pair of countably condensing mappingsin ordered Banach spacesrdquo Journal of Applied Mathematics andStochastic Analysis vol 16 no 3 pp 243ndash248 2003
Abstract and Applied Analysis 9
[29] Lj Ciric B Samet C Vetro and M Abbas ldquoFixed point resultsfor weak contractive mappings in ordered 119870-metric spacesrdquoFixed Point Theory vol 13 no 1 pp 59ndash72 2012
[30] H K Nashine W Sintunavarat and P Kumam ldquoCyclic gener-alized contractions and fixed point results with applications toan integral equationrdquo Fixed Point Theory and Applications vol2012 article 217 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
Finally we consider the case 0 lt 119909 lt 1 Let 119910 isin
Rminus1(T(119909)) that isR(119910) = T(119909) In view of definition ofT we have T(119909) = 1199093 + 2 so that R(119910) = 1199093 + 2 Byusing the definition of R we have R(119910) = 21199103 + 1 so that21199103 + 1 = 1199093 + 2 or 119910 = ((119909
3
+ 1)2)13 Thus we have
1199093
+ 2 = T (119909) le T (119910) = T((1199093 + 1
2)
13
) =1199093 + 1
2+ 2
(64)
ThereforeT is weakly increasing with respect toRSince T and R are continuous therefore this pair of
maps is reciprocally continuousNow we show that maps T and R are compatible
If lim119899rarrinfin
119909119899
= 119896 and 119896 ge 1 then lim119899rarrinfin
T119909119899
=
lim119899rarrinfin
R119909119899
= 3 and henceforth
lim119899rarrinfin
TR119909119899
= lim119899rarrinfin
RT119909119899
= 3 (65)
implying thereby lim119899rarrinfin
119889(RT119909119899
TR119909119899
) = 0 so thatT and R are compatible
Thus we have shown that all the conditions ofTheorem 18are satisfied and 119909 = 3 is the unique fixed point ofT and R
4 An Application
In this section we present an application of Theorem 13 andused the idea of Ciric et al [29] to define a partial order andprove an existence theorem for the solution of an integralequation
Theorem 21 Consider the integral equation
119909 (119905) = int119879
0
119870 (119905 119904 119909 (119904)) 119889119904 + 119892 (119905) 119905 isin [0 119879] (66)
with 119879 gt 0 wherein
(i) the functions 119870 [0 119879] times [0 119879] times R119899 rarr R119899 and 119892
R119899 rarr R119899 are continuous(ii) for each 119905 119904 isin [0 119879]
119870 (119905 119904 119909 (119904)) ≪ 119870(119905 119904 int119879
0
119870 (119904 120591 119909 (120591)) 119889120591 + 119892 (119904)) (67)
where≪ denotes a partial order relation on R119899(iii) there exists a continuous function 119901 [0 119879]times[0 119879] rarr
R+
such that
|119870 (119905 119904 119906) minus 119870 (119905 119904 V)| le 119901 (119905 119904) |119906 minus V| (68)
for each 119905 119904 isin [0 119879] and is also comparable 119906 V isin R119899
(iv) sup119905isin[0119879]
int119879
0
119901(119905 119904)119889119904 le 120573 lt 1
Then the integral equation (66) has a unique solution 119909lowast in119862([0 119879]R119899)
Proof ConsiderX = 119862([0 119879]R119899)with the usual supremumnorm that is
119909 = max119905isin[0119879]
|119909 (119905)| (69)
for 119909 119910 isin 119862([0 119879]R119899) Define onX a partial order as follows(119909 119910 isin 119862([0 119879]R119899))
119909 ⪯ 119910 iff 119909 (119905) ≪ 119910 (119905) for any 119905 isin [0 119879] (70)
Then (X ⪯) is a partially ordered set and (X sdot ) is acomplete metric space
Moreover for any increasing sequence 119909119899
inX converg-ing to 119909lowast isin X we have 119909
119899
(119905) ≪ 119909lowast(119905) for any 119905 isin [0 119879] Alsofor every 119909 119910 isin X there exists 119888 isin X which depends on 119909
and 119910 and is also comparable to 119909 and 119910 (cf [21])DefineT 119862([0 119879]) rarr 119862([0 119879]) by
T119906 (119905) = int119879
0
119870 (119905 119904 119906 (119904)) 119889119904 + 119892 (119905)
119905 isin [0 119879] 119906 isin 119862 ([0 119879])
(71)
Let us prove that
T119906 ⪯ T (T119906) forall119906 isin 119862 ([0 119879]) (72)
Let 119906 isin 119862([0 119879]) From (ii) for all 119905 isin [0 119879] we have
T119906 (119905)
= int119879
0
119870 (119905 119904 119906 (119904)) 119889119904 + 119892 (119905)
≪ int119879
0
119870(119905 119904 int119879
0
119870 (119904 120591 119906 (120591)) 119889120591 + 119892 (119904)) 119889119904 + 119892 (119905)
= int119879
0
119870 (119905 119904T119906 (119904)) 119889119904 + 119892 (119905)
= T (T119906) (119905)
(73)
Then (72) holdsNow for all 119909 119910 isin 119862([0 119879]) with 119910 ⪯ 119909 we have (by (iii))
1003816100381610038161003816T119909 (119905) minusT119910 (119905)1003816100381610038161003816
le int119879
0
1003816100381610038161003816119870 (119905 119904 119909 (119904)) minus 119870 (119905 119904 119910 (119904))1003816100381610038161003816 119889119904
le int119879
0
119901 (119905 119904) (1003816100381610038161003816119909 (119904) minus 119910 (119904)
1003816100381610038161003816) 119889119904
le1003817100381710038171003817119909 minus 119910
1003817100381710038171003817 int119879
0
119901 (119905 119904) 119889119904
(74)
Hence1003817100381710038171003817T119909 (119905) minusT119910 (119905)
1003817100381710038171003817
le1003817100381710038171003817119909 minus 119910
1003817100381710038171003817 sup119905isin[0119879]
int119879
0
119901 (119905 119904) 119889119904
le 1205731003817100381710038171003817119909 minus 119910
1003817100381710038171003817
(75)
8 Abstract and Applied Analysis
On the other hand it is demonstrated in [16] thatcondition of regularity is satisfied forX = 119862([0 119879]R119899)
Thus all the hypotheses of Theorem 13 are satisfied for120572 = 0 and then T has a fixed point 119906lowast isin 119862([0 119879]R119899) thatis 119906lowast is a solution of the integral equation (66)
Acknowledgments
The authors would like to thank the referees for careful read-ing and for providing valuable suggestions and comments forthis paper Also Poom Kumam would like to thank the KingMongkuts University of TechnologyThonburi (KMUTT) forthe financial support and Dhananjay Gopal is thankful toCSIR Government of India Grant no 25(0215)13EMR-II
References
[1] S Banach ldquoSur les operations dans les ensembles abstraits etleur application aux equations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922
[2] H K Nashine and I Altun ldquoFixed point theorems for gener-alized weakly contractive condition in ordered metric spacesrdquoFixed PointTheory and Applications vol 2011 Article ID 13236720 pages 2011
[3] R P Agarwal M A El-Gebeily and D OrsquoRegan ldquoGeneralizedcontractions in partially ordered metric spacesrdquo ApplicableAnalysis vol 87 no 1 pp 109ndash116 2008
[4] I Altun and H Simsek ldquoSome fixed point theorems onordered metric spaces and applicationrdquo Fixed Point Theory andApplications vol 2010 Article ID 621469 17 pages 2010
[5] A Amini-Harandi and H Emami ldquoA fixed point theoremfor contraction type maps in partially ordered metric spacesand application to ordinary differential equationsrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 5 pp2238ndash2242 2010
[6] I Beg and A R Butt ldquoFixed point for set-valued mappings sat-isfying an implicit relation in partially ordered metric spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no9 pp 3699ndash3704 2009
[7] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 65 no 7 pp 1379ndash1393 2006
[8] A Cabada and J J Nieto ldquoFixed points and approximatesolutions for nonlinear operator equationsrdquo Journal of Com-putational and Applied Mathematics vol 113 no 1-2 pp 17ndash25 2000 Fixed point theory with applications in nonlinearanalysis
[9] J Caballero J Harjani and K Sadarangani ldquoContractive-likemapping principles in ordered metric spaces and applicationto ordinary differential equationsrdquo Fixed Point Theory andApplications vol 2010 Article ID 916064 14 pages 2010
[10] L Ciric N Cakic M Rajovic and J S Ume ldquoMonotonegeneralized nonlinear contractions in partially ordered metricspacesrdquo Fixed Point Theory and Applications vol 2008 ArticleID 131294 11 pages 2008
[11] Lj B Ciric D Mihet and R Saadati ldquoMonotone generalizedcontractions in partially ordered probabilistic metric spacesrdquoTopology and its Applications vol 156 no 17 pp 2838ndash28442009
[12] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysis Theory Methods amp Applications vol 71 no 7-8 pp 3403ndash3410 2009
[13] J Harjani and K Sadarangani ldquoGeneralized contractions inpartially ordered metric spaces and applications to ordinarydifferential equationsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1188ndash1197 2010
[14] H K Nashine and B Samet ldquoFixed point results for map-pings satisfying (120595 120593)-weakly contractive condition in partiallyordered metric spacesrdquo Nonlinear Analysis Theory Methods ampApplications vol 74 no 6 pp 2201ndash2209 2011
[15] H K Nashine and W Shatanawi ldquoCoupled common fixedpoint theorems for a pair of commuting mappings in partiallyordered complete metric spacesrdquo Computers amp Mathematicswith Applications vol 62 no 4 pp 1984ndash1993 2011
[16] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005
[17] J J Nieto and R Rodrıguez-Lopez ldquoExistence and uniquenessof fixed point in partially ordered sets and applications toordinary differential equationsrdquo Acta Mathematica Sinica vol23 no 12 pp 2205ndash2212 2007
[18] R P Pant ldquoCommon fixed points of four mappingsrdquo Bulletin ofthe Calcutta Mathematical Society vol 90 no 4 pp 281ndash2861998
[19] R P Pant R K Bisht andDArora ldquoWeak reciprocal continuityand fixed point theoremsrdquo Annali dellrsquoUniversita di FerraraSezione VII ScienzeMatematiche vol 57 no 1 pp 181ndash190 2011
[20] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrixequationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004
[21] D OrsquoRegan and A Petrusel ldquoFixed point theorems for gen-eralized contractions in ordered metric spacesrdquo Journal ofMathematical Analysis andApplications vol 341 no 2 pp 1241ndash1252 2008
[22] B Samet ldquoCoupled fixed point theorems for a generalizedMeir-Keeler contraction in partially ordered metric spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 72 no12 pp 4508ndash4517 2010
[23] W Sintunavarat Y J Cho and P Kumam ldquoUrysohn integralequations approach by common fixed points in complex valuedmetric spacesrdquo Advances in Difference Equations vol 2013article 49 2013
[24] W Shatanawi ldquoPartially ordered cone metric spaces andcoupled fixed point resultsrdquo Computers amp Mathematics withApplications vol 60 no 8 pp 2508ndash2515 2010
[25] J Harjani B Lopez and K Sadarangani ldquoA fixed point theoremfor mappings satisfying a contractive condition of rationaltype on a partially ordered metric spacerdquo Abstract and AppliedAnalysis vol 2010 Article ID 190701 8 pages 2010
[26] D S Jaggi ldquoSome unique fixed point theoremsrdquo Indian Journalof Pure and AppliedMathematics vol 8 no 2 pp 223ndash230 1977
[27] B C Dhage ldquoCondensing mappings and applications to exis-tence theorems for common solution of differential equationsrdquoBulletin of the Korean Mathematical Society vol 36 no 3 pp565ndash578 1999
[28] B C Dhage D OrsquoRegan and R P Agarwal ldquoCommon fixedpoint theorems for a pair of countably condensing mappingsin ordered Banach spacesrdquo Journal of Applied Mathematics andStochastic Analysis vol 16 no 3 pp 243ndash248 2003
Abstract and Applied Analysis 9
[29] Lj Ciric B Samet C Vetro and M Abbas ldquoFixed point resultsfor weak contractive mappings in ordered 119870-metric spacesrdquoFixed Point Theory vol 13 no 1 pp 59ndash72 2012
[30] H K Nashine W Sintunavarat and P Kumam ldquoCyclic gener-alized contractions and fixed point results with applications toan integral equationrdquo Fixed Point Theory and Applications vol2012 article 217 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Abstract and Applied Analysis
On the other hand it is demonstrated in [16] thatcondition of regularity is satisfied forX = 119862([0 119879]R119899)
Thus all the hypotheses of Theorem 13 are satisfied for120572 = 0 and then T has a fixed point 119906lowast isin 119862([0 119879]R119899) thatis 119906lowast is a solution of the integral equation (66)
Acknowledgments
The authors would like to thank the referees for careful read-ing and for providing valuable suggestions and comments forthis paper Also Poom Kumam would like to thank the KingMongkuts University of TechnologyThonburi (KMUTT) forthe financial support and Dhananjay Gopal is thankful toCSIR Government of India Grant no 25(0215)13EMR-II
References
[1] S Banach ldquoSur les operations dans les ensembles abstraits etleur application aux equations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922
[2] H K Nashine and I Altun ldquoFixed point theorems for gener-alized weakly contractive condition in ordered metric spacesrdquoFixed PointTheory and Applications vol 2011 Article ID 13236720 pages 2011
[3] R P Agarwal M A El-Gebeily and D OrsquoRegan ldquoGeneralizedcontractions in partially ordered metric spacesrdquo ApplicableAnalysis vol 87 no 1 pp 109ndash116 2008
[4] I Altun and H Simsek ldquoSome fixed point theorems onordered metric spaces and applicationrdquo Fixed Point Theory andApplications vol 2010 Article ID 621469 17 pages 2010
[5] A Amini-Harandi and H Emami ldquoA fixed point theoremfor contraction type maps in partially ordered metric spacesand application to ordinary differential equationsrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 5 pp2238ndash2242 2010
[6] I Beg and A R Butt ldquoFixed point for set-valued mappings sat-isfying an implicit relation in partially ordered metric spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no9 pp 3699ndash3704 2009
[7] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 65 no 7 pp 1379ndash1393 2006
[8] A Cabada and J J Nieto ldquoFixed points and approximatesolutions for nonlinear operator equationsrdquo Journal of Com-putational and Applied Mathematics vol 113 no 1-2 pp 17ndash25 2000 Fixed point theory with applications in nonlinearanalysis
[9] J Caballero J Harjani and K Sadarangani ldquoContractive-likemapping principles in ordered metric spaces and applicationto ordinary differential equationsrdquo Fixed Point Theory andApplications vol 2010 Article ID 916064 14 pages 2010
[10] L Ciric N Cakic M Rajovic and J S Ume ldquoMonotonegeneralized nonlinear contractions in partially ordered metricspacesrdquo Fixed Point Theory and Applications vol 2008 ArticleID 131294 11 pages 2008
[11] Lj B Ciric D Mihet and R Saadati ldquoMonotone generalizedcontractions in partially ordered probabilistic metric spacesrdquoTopology and its Applications vol 156 no 17 pp 2838ndash28442009
[12] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysis Theory Methods amp Applications vol 71 no 7-8 pp 3403ndash3410 2009
[13] J Harjani and K Sadarangani ldquoGeneralized contractions inpartially ordered metric spaces and applications to ordinarydifferential equationsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1188ndash1197 2010
[14] H K Nashine and B Samet ldquoFixed point results for map-pings satisfying (120595 120593)-weakly contractive condition in partiallyordered metric spacesrdquo Nonlinear Analysis Theory Methods ampApplications vol 74 no 6 pp 2201ndash2209 2011
[15] H K Nashine and W Shatanawi ldquoCoupled common fixedpoint theorems for a pair of commuting mappings in partiallyordered complete metric spacesrdquo Computers amp Mathematicswith Applications vol 62 no 4 pp 1984ndash1993 2011
[16] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005
[17] J J Nieto and R Rodrıguez-Lopez ldquoExistence and uniquenessof fixed point in partially ordered sets and applications toordinary differential equationsrdquo Acta Mathematica Sinica vol23 no 12 pp 2205ndash2212 2007
[18] R P Pant ldquoCommon fixed points of four mappingsrdquo Bulletin ofthe Calcutta Mathematical Society vol 90 no 4 pp 281ndash2861998
[19] R P Pant R K Bisht andDArora ldquoWeak reciprocal continuityand fixed point theoremsrdquo Annali dellrsquoUniversita di FerraraSezione VII ScienzeMatematiche vol 57 no 1 pp 181ndash190 2011
[20] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrixequationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004
[21] D OrsquoRegan and A Petrusel ldquoFixed point theorems for gen-eralized contractions in ordered metric spacesrdquo Journal ofMathematical Analysis andApplications vol 341 no 2 pp 1241ndash1252 2008
[22] B Samet ldquoCoupled fixed point theorems for a generalizedMeir-Keeler contraction in partially ordered metric spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 72 no12 pp 4508ndash4517 2010
[23] W Sintunavarat Y J Cho and P Kumam ldquoUrysohn integralequations approach by common fixed points in complex valuedmetric spacesrdquo Advances in Difference Equations vol 2013article 49 2013
[24] W Shatanawi ldquoPartially ordered cone metric spaces andcoupled fixed point resultsrdquo Computers amp Mathematics withApplications vol 60 no 8 pp 2508ndash2515 2010
[25] J Harjani B Lopez and K Sadarangani ldquoA fixed point theoremfor mappings satisfying a contractive condition of rationaltype on a partially ordered metric spacerdquo Abstract and AppliedAnalysis vol 2010 Article ID 190701 8 pages 2010
[26] D S Jaggi ldquoSome unique fixed point theoremsrdquo Indian Journalof Pure and AppliedMathematics vol 8 no 2 pp 223ndash230 1977
[27] B C Dhage ldquoCondensing mappings and applications to exis-tence theorems for common solution of differential equationsrdquoBulletin of the Korean Mathematical Society vol 36 no 3 pp565ndash578 1999
[28] B C Dhage D OrsquoRegan and R P Agarwal ldquoCommon fixedpoint theorems for a pair of countably condensing mappingsin ordered Banach spacesrdquo Journal of Applied Mathematics andStochastic Analysis vol 16 no 3 pp 243ndash248 2003
Abstract and Applied Analysis 9
[29] Lj Ciric B Samet C Vetro and M Abbas ldquoFixed point resultsfor weak contractive mappings in ordered 119870-metric spacesrdquoFixed Point Theory vol 13 no 1 pp 59ndash72 2012
[30] H K Nashine W Sintunavarat and P Kumam ldquoCyclic gener-alized contractions and fixed point results with applications toan integral equationrdquo Fixed Point Theory and Applications vol2012 article 217 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 9
[29] Lj Ciric B Samet C Vetro and M Abbas ldquoFixed point resultsfor weak contractive mappings in ordered 119870-metric spacesrdquoFixed Point Theory vol 13 no 1 pp 59ndash72 2012
[30] H K Nashine W Sintunavarat and P Kumam ldquoCyclic gener-alized contractions and fixed point results with applications toan integral equationrdquo Fixed Point Theory and Applications vol2012 article 217 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Related Documents
![11.[76-90]Coupled Fixed Point Theorems in Partially Ordered Metric Space](https://static.cupdf.com/doc/110x72/577d1e5a1a28ab4e1e8e558d/1176-90coupled-fixed-point-theorems-in-partially-ordered-metric-space.jpg)
![arxiv.org · arXiv:1601.02512v1 [math.FA] 7 Jan 2016 UNIFIED MULTI-TUPLED FIXED POINT THEOREMS INVOLVING MONOTONE PROPERTY IN ORDERED METRIC SPACES AFTAB ALAM1, MOHAMMAD …](https://static.cupdf.com/doc/110x72/6035acab22caba551d1d479f/arxivorg-arxiv160102512v1-mathfa-7-jan-2016-unified-multi-tupled-fixed-point.jpg)
