Research ArticleSome Integral Type Fixed Point Theorems forNon-Self-Mappings Satisfying Generalized (120595 120593)-WeakContractive Conditions in Symmetric Spaces

Marwan Amin Kutbi1 Mohammad Imdad2 Sunny Chauhan3 and Wutiphol Sintunavarat4

1 Department of Mathematics King AbdulAziz University PO Box 80203 Jeddah 21589 Saudi Arabia2Department of Mathematics Aligarh Muslim Univeristy Aligarh 202 002 India3 Near Nehru Training Centre H No 274 Nai Basti B-14 Bijnor Uttar Pradesh 246701 India4Department of Mathematics and Statistics Faculty of Science and Technology Thammasat University Rangsit CenterPathumThani 12121 Thailand

Correspondence should be addressed to Wutiphol Sintunavarat wutipholmathstatscituacth

Received 10 January 2014 Accepted 10 March 2014 Published 7 April 2014

Academic Editor Wei-Shih Du

Copyright copy 2014 Marwan Amin Kutbi et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The aim of this paper is to obtain some new integral type fixed point theorems for nonself weakly compatible mappings insymmetric spaces satisfying generalized (120595 120593)-contractive conditions employing the common limit range property We furnishsome interesting examples which support our main theorems Our results generalize and extend some recent results contained inImdad et al (2013) to symmetric spaces Consequently a host of metrical common fixed theorems are generalized and improvedIn the process we also derive a fixed point theorem for four finite families of mappings which can be utilized to derive commonfixed point theorems involving any number of finite mappings

1 Introduction

The celebrated Banach contraction principle is indeed themost fundamental result ofmetrical fixed point theory whichstates that a contraction mapping of a complete metric spaceinto itself has a unique fixed point This theorem is very effec-tively utilized to establish the existence of solutions of non-linear Volterra integral equations Fredholm integral equa-tions and nonlinear integrodifferential equations in Banachspaces besides supporting the convergence of algorithmsin computational mathematics In [1] Hicks and Rhoadesproved some common fixed point theorems in symmetricspaces and showed that a general probabilistic structureadmits a compatible symmetric or semimetric

The study of common fixed points for noncompatiblemappings is equally interesting due to Pant [2] Jungck [3]generalized the idea of weakly commuting pair of mappingsdue to Sessa [4] by introducing the notion of compatiblemappings and showed that compatible pair of mappings

commutes on the set of coincidence points of the involvedmappings In 1998 Jungck and Rhoades [5] introduced thenotion of weakly compatible mappings in nonmetric spacesFor more details on systematic comparisons and illustrationsof these described notions we refer to Singh and Tomar[6] and Murthy [7] Afterwards Al-Thagafi and Shahzad [8]introduced an even weaker notion which they called occa-sionally weak compatibility Many authors (see eg [9ndash12])used this notion to obtain common fixed point resultsRecently ETHoric et al [13] showed that for single-valuedmappings the condition of occasionally weak compatibilityreduces to weak compatibility in the case of a unique point ofcoincidence (and a unique common fixed point) of the givenmappings However for hybrid pairs of maps this is not thecase

On the other hand in 2002 Aamri and El Moutawakil[14] introduced the notion of property (EA) which is aspecial case of tangential property due to Sastry and KrishnaMurthy [15] Later on Liu et al [16] initiated the notion of

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 519038 11 pageshttpdxdoiorg1011552014519038

2 Abstract and Applied Analysis

common property (EA) for hybrid pairs of mappings whichcontained property (EA) In this continuation Imdad et al[17] and Soliman et al [18] extended the results of Sastryand Krishna Murthy [15] and Pant [19] to symmetric spacesby utilizing the weak compatible property with commonproperty (EA) Since the notions of property (EA) andcommon property (EA) always require the completeness(or closedness) of underlying subspaces for the existence ofcommon fixed point hence Sintunavarat and Kumam [20]coined the idea of ldquocommon limit range propertyrdquo whichrelaxes the requirement of completeness (or closedness)of the underlying subspace Afterward Imdad et al [21]extended the notion of common limit range property to twopairs of self-mappings and proved some fixed point theoremsin Menger and metric spaces Most recently Karapınar et al[22] utilized the notion of common limit range property andshowed that the new notion buys certain typical conditionsutilized by Pant [19] up to a pair of mappings on the cast of arelatively more natural absorbing property due to Gopal et al[23]

The concept of weak contraction was introduced byAlber and Guerre-Delabriere [24] in 1997 wherein authorsintroduced the following notion for mappings defined on aHilbert space119867

Consider the following set of real functions

Φ = 120593 [0 +infin) 997888rarr [0 +infin) 120593 is lower

semi-continuous and 120593minus1

(0) = 0

(1)

Amapping119879 119883 rarr 119883 is called a120593-weak contraction if thereexists a function 120593 isin Φ such that

119889 (119879119909 119879119910) le 119889 (119909 119910) minus 120593 (119889 (119909 119910)) forall119909 119910 isin 119883 (2)

Alber and Guerre-Delabriere [24] also showed that each120593-weak contraction on a Hilbert space has a unique fixedpoint Thereafter Rhoades [25] showed that the results con-tained in [24] are also valid for any Banach space In partic-ular he generalized the Banach contraction principle whichfollows in case one chooses 120593(119905) = (1 minus 119896)119905

In 2002 Branciari [26] firstly studied the integral ana-logue of Banachrsquos contraction principle Some interestingresults can be easily seen in [27ndash32] Most recently Vetroet al [33] proved some integral type fixed point results formappings in metric spaces employing common limit rangeproperty Zhang and Song [34] proved a common fixed pointtheorem for twomappings by using 120593-weak contractionThisresult was extended by ETHoric [35] and Dutta and Choudhury[36] to a pair of (120595 120593)-weak contractive mappings Howeverthe main fixed point theorem for a self-mapping satisfying(120595 120593)-weak contractive condition contained in Dutta andChoudhury [36] is given below but before that we considerthe following set of real functions

Ψ = 120595 [0 +infin) 997888rarr [0 +infin) 120595 is continuous

nondecreasing and 120595minus1

(0) = 0

(3)

Theorem 1 Let (119883 119889) be a complete metric space and let 119879

119883 rarr 119883 be a self-mapping satisfying

120595 (119889 (119879119909 119879119910)) le 120595 (119889 (119909 119910)) minus 120593 (119889 (119909 119910)) (4)

for some 120595 isin Ψ and 120593 isin Φ and all 119909 119910 isin 119883 Then 119879 has aunique fixed point in119883

The object of this paper is to prove some integral typecommon fixed point theorems for two pairs of nonselfweakly compatible mappings satisfying generalized (120595 120593)-contractive conditions by using the common limit rangeproperty in symmetric spaces We give some illustrativeexamples to highlight the superiority of our results overseveral results existing in the literature As an extension ofour main result we state some fixed point theorems for sixmappings and four finite families of mappings in symmetricspaces by using the notion of the pairwise commutingmappings which is studied by Imdad et al [37]

2 Preliminaries

A common fixed point result generally involves conditionson commutativity continuity and contraction along witha suitable condition on the containment of range of onemapping into the range of the other Hence one is alwaysrequired to improve one or more of these conditions inorder to prove a new common fixed point theorem It can beobserved that in the case of two mappings 119860 119878 119883 rarr 119883where (119883 119889) is metric space (or symmetric space) one canconsider the following classes of mappings for the existenceand uniqueness of common fixed points

119889 (119860119909 119860119910) le 119865 (119898 (119909 119910)) (5)

where 119865 is some function and119898(119909 119910) is the maximum of oneof the sets Thus

1198725

119860119878

(119909 119910) = 119889 (119878119909 119878119910) 119889 (119878119909 119860119909) 119889 (119878119910 119860119910)

119889 (119878119909 119860119910) 119889 (119878119910 119860119909)

1198724

119860119878

(119909 119910) = 119889 (119878119909 119878119910) 119889 (119878119909 119860119909) 119889 (119878119910 119860119910)

1

2(119889 (119878119909 119860119910) + 119889 (119878119910 119860119909))

1198723

119860119878

(119909 119910) = 119889 (119878119909 119878119910) 1

2(119889 (119878119909 119860119909) + 119889 (119878119910 119860119910))

1

2(119889 (119878119909 119860119910) + 119889 (119878119910 119860119909))

(6)

A further possible generalization is to consider fourmappings instead of two and ascertain analogous commonfixed point theorems In the case of four mappings119860 119861 119878 119879

Abstract and Applied Analysis 3

119883 rarr 119883 where (119883 119889) is metric space (or symmetric space)the corresponding sets take the form

1198725

119860119861119878119879

(119909 119910) = 119889 (119878119909 119879119910) 119889 (119878119909 119860119909) 119889 (119879119910 119861119910)

119889 (119878119909 119861119910) 119889 (119879119910 119860119909)

1198724

119860119861119878119879

(119909 119910) = 119889 (119878119909 119879119910) 119889 (119878119909 119860119909) 119889 (119879119910 119861119910)

1

2(119889 (119878119909 119861119910) + 119889 (119879119910 119860119909))

1198723

119860119861119878119879

(119909 119910) = 119889 (119878119909 119879119910) 1

2(119889 (119878119909 119860119909) + 119889 (119879119910 119861119910))

1

2(119889 (119878119909 119861119910) + 119889 (119879119910 119860119909))

(7)

In this case (5) is usually replaced by

119889 (119860119909 119861119910) le 119865 (119898 (119909 119910)) (8)

where119898(119909 119910) is the maximum of one of the119872-setsSimilarly we can define the 119872-sets for six mappings

119860 119861119867 119877 119878 119879 119883 rarr 119883 where (119883 119889) is metric space (orsymmetric space) as

1198725

119860119861119867119877119878119879

(119909 119910) = 119889 (119878119877119909 119879119867119910) 119889 (119878119877119909 119860119909)

119889 (119879119867119910 119861119910)

119889 (119878119877119909 119861119910) 119889 (119879119867119910119860119909)

1198724

119860119861119867119877119878119879

(119909 119910) = 119889 (119878119877119909 119879119867119910)

119889 (119878119877119909 119860119909) 119889 (119879119867119910 119861119910)

1

2(119889 (119878119877119909 119861119910) + 119889 (119879119867119910119860119909))

1198723

119860119861119867119877119878119879

(119909 119910) = 119889 (119878119877119909 119879119867119910)

1

2(119889 (119878119877119909 119860119909) + 119889 (119879119867119910 119861119910))

1

2(119889 (119878119877119909 119861119910) + 119889 (119879119867119910119860119909))

(9)

and the contractive condition is again in the form (8)By using different arguments of control functions Rade-

novic et al [38] proved some common fixed point resultsfor two and three mappings by using (120595 120593)-weak contractiveconditions and improved several known metrical fixed pointtheorems Motivated by these results we prove some com-mon fixed point theorems for two pairs of weakly compatiblemappings with common limit range property satisfyinggeneralized (120595 120593)-weak contractive conditionsMany knownfixed point results are improved especially the ones provedin [38] and also contained in the references cited therein We

also obtain a fixed point theorem for four finite families ofself-mappings Some related results are also derived besidesfurnishing illustrative examples

The following definitions and results will be needed in thesequel

A symmetric on a set119883 is a function 119889 119883times119883 rarr [0infin)

satisfying the following conditions

(1) 119889(119909 119910) = 0 if and only if 119909 = 119910 for 119909 119910 isin 119883(2) 119889(119909 119910) = 119889(119910 119909) for all 119909 119910 isin 119883

Let 119889 be a symmetric on a set 119883 For 119909 isin 119883 and 120598 gt 0let B(119909 120598) = 119910 isin 119883 119889(119909 119910) lt 120598 A topology 120591(119889) on119883 is defined as follows 119880 isin 120591(119889) if and only if for each119909 isin 119880 there exists an 120598 gt 0 such that B(119909 120598) sub 119880 Asubset 119878 of 119883 is a neighbourhood of 119909 isin 119883 if there exists119880 isin 120591(119889) such that 119909 isin 119880 sub 119878 A symmetric 119889 is asemimetric if for each 119909 isin 119883 and each 120598 gt 0 B(119909 120598)

is a neighbourhood of 119909 in the topology 120591(119889) A symmet-ric (resp semimetric) space (119883 119889) is a topological spacewhose topology 120591(119889) on 119883 is induced by symmetric (respsemimetric) 119889 The difference of a symmetric and a metriccomes from the triangle inequality Since a symmetric spaceis not essentially Hausdorff therefore in order to provefixed point theorems some additional axioms are requiredThe following axioms which are available in Wilson [39]Aliouche [40] and Imdad et al [17] are relevant to thispresentation

From now on symmetric space will be denoted by (119883 119889)whereas a nonempty arbitrary set will be denoted by 119884

(1198823

) Given 119909119899

119909 and 119910 in 119883 lim119899rarrinfin

119889(119909119899

119909) = 0 andlim119899rarrinfin

119889(119909119899

119910) = 0 imply 119909 = 119910 [39](1198824

) Given 119909119899

119910119899

and 119909 in 119883 lim119899rarrinfin

119889(119909119899

119909) = 0

and lim119899rarrinfin

119889(119909119899

119910119899

) = 0 imply lim119899rarrinfin

119889(119910119899

119909) = 0

[39](119867119864) Given 119909

119899

119910119899

and 119909 in 119883 lim119899rarrinfin

119889(119909119899

119909) = 0

and lim119899rarrinfin

119889(119910119899

119909) = 0 imply lim119899rarrinfin

119889(119909119899

119910119899

) = 0

[40](1119862) A symmetric 119889 is said to be 1-continuous if

lim119899rarrinfin

119889(119909119899

119909) = 0 implies lim119899rarrinfin

119889(119909119899

119910) =

119889(119909 119910) where 119909119899

is a sequence in 119883 and 119909 119910 isin 119883

[41](119862119862) A symmetric 119889 is said to be continuous if

lim119899rarrinfin

119889(119909119899

119909) = 0 and lim119899rarrinfin

119889(119910119899

119910) = 0

imply lim119899rarrinfin

119889(119909119899

119910119899

) = 119889(119909 119910) where 119909119899

and119910119899

are sequences in119883 and 119909 119910 isin 119883 [41]

Here it is observed that (119862119862) rArr (1119862) (1198824

) rArr (1198823

)and (1119862) rArr (119882

3

) but the converse implications are nottrue In general all other possible implications amongst (119882

3

)(1119862) and (119867119864) are not true For detailed description we referan interesting note of Cho et al [42] which contained someillustrative examples However (119862119862) implies all the remain-ing four conditions namely (119882

3

) (1198824

) (119867119864) and (1119862)Employing these axioms several authors proved commonfixed point theorems in the framework of symmetric spaces(see [22 43ndash48])

4 Abstract and Applied Analysis

Definition 2 Let (119860 119878) be a pair of self-mappings defined ona nonempty set 119883 equipped with a symmetric 119889 Then themappings 119860 and 119878 are said to be

(1) commuting if 119860119878119909 = 119878119860119909 for all 119909 isin 119883(2) compatible [3] if lim

119899rarrinfin

119889(119860119878119909119899

119878119860119909119899

) = 0 foreach sequence 119909

119899

in 119884 such that lim119899rarrinfin

119860119909119899

=

lim119899rarrinfin

119878119909119899

(3) noncompatible [2] if there exists a sequence 119909

119899

in 119883 such that lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

butlim119899rarrinfin

119889(119860119878119909119899

119878119860119909119899

) is either nonzero or nonexis-tent

(4) weakly compatible [5] if they commute at their coinci-dence points that is 119860119878119909 = 119878119860119909 whenever 119860119909 = 119878119909for some 119909 isin 119883

(5) satisfied the property (EA) [14] if there exists asequence 119909

119899

in 119883 such that lim119899rarrinfin

119860119909119899

=

lim119899rarrinfin

119878119909119899

= 119911 for some 119911 isin 119883

Any pair of compatible as well as noncompatible self-mappings satisfies the property (EA) but a pair of mappingssatisfying the property (EA) needs not be noncompatible

Definition 3 (see [16]) Let 119884 be an arbitrary set and let 119883 bea nonempty set equipped with symmetric 119889 Then the pairs(119860 119878) and (119861 119879) of mappings from 119884 into119883 are said to sharethe common property (EA) if there exist two sequences 119909

119899

and 119910119899

in 119884 such that

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= lim119899rarrinfin

119861119910119899

= lim119899rarrinfin

119879119910119899

= 119911 (10)

for some 119911 isin 119883

Definition 4 (see [20]) Let 119884 be an arbitrary set and let 119883be a nonempty set equipped with symmetric 119889 Then the pair(119860 119878) of mappings from 119884 into119883 is said to have the commonlimit range property with respect to the mapping 119878 (denotedby (CLR

119878

)) if there exists a sequence 119909119899

in 119884 such that

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= 119911 (11)

where 119911 isin 119878(119884)

Definition 5 (see [21]) Let 119884 be an arbitrary set and let 119883 bea nonempty set equipped with symmetric 119889 Then the pairs(119860 119878) and (119861 119879) of mappings from 119884 into 119883 are said to havethe common limit range property with respect to mappings 119878and 119879 denoted by (CLR

119878119879

) if there exist two sequences 119909119899

and 119910119899

in 119884 such that

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= lim119899rarrinfin

119861119910119899

= lim119899rarrinfin

119879119910119899

= 119911 (12)

where 119911 isin 119878(119884) cap 119879(119884)

Remark 6 It is clear that (CLR119878119879

) property implies thecommon property (EA) but the converse is not true (see [49Example 1])

Definition 7 (see [37]) Two families of self-mappings 119860119894

119898

119894=1

and 119878119896

119899

119896=1

are said to be pairwise commuting if

(1) 119860119894

119860119895

= 119860119895

119860119894

for all 119894 119895 isin 1 2 119898(2) 119878119896

119878119897

= 119878119897

119878119896

for all 119896 119897 isin 1 2 119899(3) 119860119894

119878119896

= 119878119896

119860119894

for all 119894 isin 1 2 119898 and 119896 isin

1 2 119899

3 Results

Now we state and prove our main results for four mappingsemploying the common limit range property in symmetricspaces Firstly we prove the following lemma

Lemma 8 Let (119883 119889) be a symmetric space wherein 119889 satisfiesthe conditions (119862119862) whereas 119884 is an arbitrary nonempty setwith 119860 119861 119878 and 119879 119884 rarr 119883 Suppose that

(1) the pair (119860 119878) (or (119861 119879)) satisfies the (119862119871119877119878

) (or(119862119871119877119879

)) property(2) 119860(119884) sub 119879(119884) (or 119861(119884) sub 119878(119884))(3) 119879(119884) (or 119878(119884)) is a closed subset of119883(4) 119861119910

119899

converges for every sequence 119910119899

in 119884 whenever119879119910119899

converges (or 119860119909119899

converges for every sequence119909119899

in 119884 whenever 119878119909119899

converges)(5) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905) 119889119905) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(13)

where

119898(119909 119910) = max1198725119860119861119878119879

(119909 119910) (14)

and 120601 [0 +infin) rarr [0 +infin) is a Lebesgue-integrable mappingwhich is summable and nonnegative such that

int120598

0

120601 (119905) 119889119905 gt 0 (15)

for all 120598 gt 0Then the pairs (119860 119878) and (119861 119879) satisfy the (CLR

119878119879

)

property

Proof First we show that the conclusion of this theoremholds for first case Since the pair (119860 119878) enjoys the (CLR

119878

)

property therefore there exists a sequence 119909119899

in119884 such that

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= 119911 (16)

where 119911 isin 119878(119884) Since 119860(119884) sub 119879(119884) hence for each sequence119909119899

there exists a sequence 119910119899

in 119884 such that 119860119909119899

= 119879119910119899

Therefore by closedness of 119879(119884)

lim119899rarrinfin

119879119910119899

= lim119899rarrinfin

119860119909119899

= 119911 (17)

for 119911 isin 119879(119884) and in all 119911 isin 119878(119884) cap 119879(119884) Thus in all we have119860119909119899

rarr 119911 119878119909119899

rarr 119911 and 119879119910119899

rarr 119911 as 119899 rarr infin Since by

Abstract and Applied Analysis 5

(4) 119861119910119899

converges in all we need to show that 119861119910119899

rarr 119911

as 119899 rarr infin Assume this contrary we get 119861119910119899

rarr 119905( = 119911) as119899 rarr infin Now using inequality (13) with = 119909

119899

119910 = 119910119899

wehave

120595(int119889(119860119909

119899119861119910

119899)

0

120601 (119905) 119889119905)

le 120595(int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905) minus 120593(int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

(18)

where

119898(119909119899

119910119899

) = max 119889 (119878119909119899

119879119910119899

) 119889 (119878119909119899

119860119909119899

) 119889 (119879119910119899

119861119910119899

)

119889 (119878119909119899

119861119910119899

) 119889 (119879119910119899

119860119909119899

)

(19)

Taking limit as 119899 rarr infin and using property (119862119862) ininequality (18) we get

lim119899rarrinfin

120595(int119889(119860119909

119899119861119910

119899)

0

120601 (119905) 119889119905)

le lim119899rarrinfin

120595(int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

minus lim119899rarrinfin

120593(int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

(20)

that is

120595(int119889(119911119905)

0

120601 (119905) 119889119905)

= 120595( lim119899rarrinfin

int119889(119860119909

119899119861119910

119899)

0

120601 (119905) 119889119905)

le 120595( lim119899rarrinfin

int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

minus 120593( lim119899rarrinfin

int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

(21)

where

lim119899rarrinfin

119898(119909119899

119910119899

)

= max 119889 (119911 119911) 119889 (119911 119911) 119889 (119911 119905) 119889 (119911 119905) 119889 (119911 119911)

= max 0 0 119889 (119911 119905) 119889 (119911 119905) 0

= 119889 (119911 119905)

(22)

Hence inequality (21) implies

120595(int119889(119911119905)

0

120601 (119905) 119889119905)

le 120595(int119889(119911119905)

0

120601 (119905) 119889119905) minus 120593(int119889(119911119905)

0

120601 (119905) 119889119905)

(23)

that is 120593(int119889(119911119905)0

120601(119905)119889119905) le 0 and so 120593(int119889(119911119905)0

120601(119905)119889119905) = 0 andby the property of the function 120593 we have 119889(119911 119905) = 0 orequivalently 119911 = 119905 which contradicts the hypothesis 119905 = 119911Hence both the pairs (119860 119878) and (119861 119879) satisfy the (CLR

119878119879

)

propertyIn the second case it similar to the first case So in order

to avoid repetition the details of the proof are omitted Thiscompletes the proof

Theorem9 Let (119883 119889) be a symmetric spacewherein119889 satisfiesthe conditions (1119862) and (119867119864) whereas 119884 is an arbitrarynonempty set with 119860 119861 119878 119879 119884 rarr 119883 which satisfy theinequalities (13) and (15) of Lemma 8 Suppose that the pairs(119860 119878) and (119861 119879) satisfy the (119862119871119877

119878119879

) propertyThen (119860 119878) and(119861 119879) have a coincidence point each Moreover if 119884 = 119883 then119860 119861 119878 and119879 have a unique common fixed point provided boththe pairs (119860 119878) and (119861 119879) are weakly compatible

Proof Since the pairs (119860 119878) and (119861 119879) enjoy the (CLR119878119879

)

property there exist two sequences 119909119899

and 119910119899

in 119884 suchthat

lim119899rarrinfin

119889 (119860119909119899

119911) = lim119899rarrinfin

119889 (119878119909119899

119911) = lim119899rarrinfin

119889 (119861119910119899

119911)

= lim119899rarrinfin

119889 (119879119910119899

119911) = 0(24)

where 119911 isin 119878(119884) cap 119879(119884) It follows from 119911 isin 119878(119884) that thereexists a point119908 isin 119884 such that 119878119908 = 119911 We assert that119860119908 = 119911If not then using inequality (13) with 119909 = 119908 and 119910 = 119910

119899

oneobtains

120595(int119889(119860119908119861119910

119899)

0

120601 (119905) 119889119905)

le 120595(int119898(119908119910

119899)

0

120601 (119905) 119889119905) minus 120593(int119898(119908119910

119899)

0

120601 (119905) 119889119905)

(25)

where

119898(119908 119910119899

) =max 119889 (119878119908 119879119910119899

) 119889 (119878119908 119860119908) 119889 (119879119910119899

119861119910119899

)

119889 (119878119908 119861119910119899

) 119889 (119879119910119899

119860119908)

(26)

Taking limit as 119899 rarr infin and using properties (1119862) and (119867119864)in inequality (25) one obtains

lim119899rarrinfin

120595(int119889(119860119908119861119910

119899)

0

120601 (119905) 119889119905)

le lim119899rarrinfin

120595(int119898(119908119910

119899)

0

120601 (119905) 119889119905)

minus lim119899rarrinfin

120593(int119898(119908119910

119899)

0

120601 (119905) 119889119905)

(27)

6 Abstract and Applied Analysis

that is

120595(int119889(119860119908119911)

0

120601 (119905) 119889119905)

= 120595( lim119899rarrinfin

int119889(119860119908119861119910

119899)

0

120601 (119905) 119889119905)

le 120595( lim119899rarrinfin

int119898(119908119910

119899)

0

120601 (119905) 119889119905)

minus 120593( lim119899rarrinfin

int119898(119908119910

119899)

0

120601 (119905) 119889119905)

(28)

wherelim119899rarrinfin

119898(119908 119910119899

)

= max 119889 (119911 119911) 119889 (119911 119860119908) 119889 (119911 119911) 119889 (119911 119911) 119889 (119911 119860119908)

= max 0 119889 (119911 119860119908) 0 0 119889 (119911 119860119908)

= 119889 (119911 119860119908)

(29)

From inequality (28) one gets

120595(int119889(119860119908119911)

0

120601 (119905) 119889119905)

le 120595(int119889(119860119908119911)

0

120601 (119905)) minus 120593(int119889(119860119908119911)

0

120601 (119905))

(30)

so that 120593(int119889(119860119908119911)0

120601(119905)119889119905) = 0 that is 119889(119860119908 119911) = 0 Hence119860119908 = 119878119908 = 119911 which shows that 119908 is a coincidence point ofthe pair (119860 119878)

Also 119911 isin 119879(119884) there exists a point V isin 119884 such that119879V = 119911We assert that 119861V = 119911 If not then using inequality (13) with119909 = 119908 119910 = V we have

120595(int119889(119911119861V)

0

120601 (119905) 119889119905)

= 120595(int119889(119860119908119861V)

0

120601 (119905) 119889119905)

le 120595(int119898(119908V)

0

120601 (119905) 119889119905) minus 120593(int119898(119908V)

0

120601 (119905) 119889119905)

(31)

where119898(119908 V)

= max 119889 (119878119908 119879V) 119889 (119878119908 119860119908) 119889 (119879V 119861V)

119889 (119878119908 119861V) 119889 (119879V 119860119908)

= max 119889 (119911 119911) 119889 (119911 119911) 119889 (119911 119861V) 119889 (119911 119861V) 119889 (119911 119911)

= max 0 0 119889 (119911 119861V) 119889 (119911 119861V) 0

= 119889 (119911 119861V) (32)

Hence inequality (31) implies

120595(int119889(119911119861V)

0

120601 (119905) 119889119905)

le 120595(int119889(119911119861V)

0

120601 (119905) 119889119905) minus 120593(int119889(119911119861V)

0

120601 (119905) 119889119905)

(33)

so that 120593(int119889(119911119861V)0

120601(119905)119889119905) = 0 that is 119889(119911 119861V) = 0 Therefore119911 = 119861V = 119879V which shows that V is a coincidence point of thepair (119861 119879)

Consider 119884 = 119883 Since the pair (119860 119878) is weakly compat-ible and 119860119908 = 119878119908 hence 119860119911 = 119860119878119908 = 119878119860119908 = 119878119911 Nowwe assert that 119911 is a common fixed point of the pair (119860 119878) Toaccomplish this using inequality (13) with 119909 = 119911 and 119910 = Vone gets

120595(int119889(119860119911119911)

0

120601 (119905) 119889119905)

= 120595(int119889(119860119911119861V)

0

120601 (119905) 119889119905)

le 120595(int119898(119911V)

0

120601 (119905) 119889119905) minus 120593(int119898(119911V)

0

120601 (119905) 119889119905)

(34)

where119898(119911 V)

= max 119889 (119878119911 119879V) 119889 (119878119911 119860119911) 119889 (119879V 119861V)

119889 (119878119911 119861V) 119889 (119879V 119860119911)

= max 119889 (119860119911 119911) 119889 (119860119911 119860119911) 119889 (119911 119911)

119889 (119860119911 119911) 119889 (119911 119860119911)

= 119889 (119860119911 119911)

(35)

From inequality (34) we have

120595(int119889(119860119911119911)

0

120601 (119905) 119889119905)

le 120595(int119889(119860119911119911)

0

120601 (119905) 119889119905) minus 120593(int119889(119860119911119911)

0

120601 (119905) 119889119905)

(36)

so that 120593(int119889(119860119911119911)0

120601(119905)119889119905) = 0 that is 119889(119860119911 119911) = 0 Hence wehave119860119911 = 119911 = 119878119911which shows that 119911 is a commonfixed pointof the pair (119860 119878)

Also the pair (119861 119879) is weakly compatible and 119861V = 119879Vhence 119861119911 = 119861119879V = 119879119861V = 119879119911 Using inequality (13) with119909 = 119908 and 119910 = 119911 we have

120595(int119889(119911119861119911)

0

120601 (119905) 119889119905)

= 120595(int119889(119860119908119861119911)

0

120601 (119905) 119889119905)

le 120595(int119898(119908119911)

0

120601 (119905) 119889119905) minus 120593(int119898(119908119911)

0

120601 (119905) 119889119905)

(37)

Abstract and Applied Analysis 7

where

119898(119908 119911) = max 119889 (119878119908 119879119911) 119889 (119878119908 119860119908) 119889 (119879119911 119861119911)

119889 (119878119908 119861119911) 119889 (119879119911 119860119908)

= max 119889 (119911 119861119911) 119889 (119911 119911) 119889 (119861119911 119861119911)

119889 (119911 119861119911) 119889 (119861119911 119911)

= 119889 (119911 119861119911)

(38)

Hence inequality (37) implies

120595(int119889(119911119861119911)

0

120601 (119905) 119889119905)

le 120595(int119889(119911119861119911)

0

120601 (119905) 119889119905) minus 120593(int119889(119911119861119911)

0

120601 (119905) 119889119905)

(39)

so that 120593(int119889(119911119861119911)0

120601(119905)119889119905) = 0 that is 119889(119911 119861119911) = 0 Therefore119861119911 = 119911 = 119879119911 which shows that 119911 is a common fixed pointof the pair (119861 119879) Hence 119911 is a common fixed point of thepairs (119860 119878) and (119861 119879) Uniqueness of common fixed point isan easy consequence of the inequality (13)This concludes theproof

Now we furnish an illustrative example which demon-strates the validity of the hypotheses and degree of generalityof Theorem 9

Example 10 Consider 119883 = 119884 = [2 11) equipped with thesymmetric 119889(119909 119910) = (119909 minus 119910)

2 for all 119909 119910 isin 119883 which alsosatisfies (1119862) and (119867119864) Define the mappings 119860 119861 119878 and 119879

by

119860119909 = 2 if 119909 isin 2 cup (5 11)

5 if 119909 isin (2 5]

119861119909 = 2 if 119909 isin 2 cup (5 11)

4 if 119909 isin (2 5]

119878119909 =

2 if 119909 = 2

6 if 119909 isin (2 5] 3119909 + 1

8 if 119909 isin (5 11)

119879119909 =

2 if 119909 = 2

8 if 119909 isin (2 5]

119909 minus 3 if 119909 isin (5 11)

(40)

Then 119860(119883) = 2 5 119861(119883) = 2 4 119879(119883) = [2 8] and 119878(119883) =[2 174) cup 6 Now consider the sequences 119909

119899

= 5 + 1119899

and 119910119899

= 2 Then

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= lim119899rarrinfin

119861119910119899

= lim119899rarrinfin

119879119910119899

= 2 isin 119878 (119883) cap 119879 (119883)

(41)

that is both the pairs (119860 119878) and (119861 119879) satisfy the (CLR119878119879

)

property

Let Lebesgue-integrable 120601 [0 +infin) rarr [0 +infin) definedby 120601(119905) = 119890

119905 Take 120595 isin Ψ and 120593 isin Φ given by 120595(119905) = 2119905 and120593(119905) = (27)119905 In order to check the contractive condition (13)consider the following nine cases

(i) 119909 = 119910 = 2

(ii) 119909 = 2 119910 isin (2 5]

(iii) 119909 = 2 119910 isin (5 11)

(iv) 119909 isin (2 5] 119910 = 2

(v) 119909 119910 isin (2 5]

(vi) 119909 isin (2 5] 119910 isin (5 11)

(vii) 119909 isin (5 11) 119910 = 2

(viii) 119909 isin (5 11) 119910 isin (2 5]

(ix) 119909 119910 isin (5 11)

In the cases (i) (iii) (vii) and (ix) we get that119889(119860119909 119861119910) =0 and (13) is trivially satisfied

In the cases (ii) and (viii) we have 119889(119860119909 119861119910) = 4 and119898(119909 119910) = 36 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int4

0

119890119905

119889119905)

= 120595 (1198904

minus 1)

= 2 (1198904

minus 1)

le12

7(11989036

minus 1)

= 2 (11989036

minus 1) minus2

7(11989036

minus 1)

= 120595(int36

0

119890119905

119889119905) minus 120593(int36

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(42)

Therefore we get the fact that (13) holds

8 Abstract and Applied Analysis

In the case (iv) we get 119889(119860119909 119861119910) = 9 and 119898(119909 119910) = 16Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int9

0

119890119905

119889119905)

= 120595 (1198909

minus 1)

= 2 (1198909

minus 1)

le12

7(11989016

minus 1)

= 2 (11989016

minus 1) minus2

7(11989016

minus 1)

= 120595(int16

0

119890119905

119889119905) minus 120593(int16

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(43)

This implies that (13) holdsIn the case (vi) we have 119889(119860119909 119861119910) = 9 and 119889(119878119909 119861119910) =

16 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int9

0

119890119905

119889119905)

= 120595 (1198909

minus 1)

= 2 (1198909

minus 1)

le12

7(11989016

minus 1)

=12

7(119890119889(119878119909119861119910)

minus 1)

le12

7(119890119898(119909119910)

minus 1)

= 2 (119890119898(119909119910)

minus 1) minus2

7(119890119898(119909119910)

minus 1)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(44)

This shows that (13) holds

Finally in the case (v) we obtain 119889(119860119909 119861119910) = 1 and119898(119909 119910) = 16 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int1

0

119890119905

119889119905)

= 120595 (119890 minus 1)

= 2 (119890 minus 1)

le12

7(11989016

minus 1)

= 2 (11989016

minus 1) minus2

7(11989016

minus 1)

= 120595(int16

0

119890119905

119889119905) minus 120593(int16

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(45)

that is the contractive condition (13) holdsHence all the conditions ofTheorem 9 are satisfied and 2

is a unique common fixed point of the pairs (119860 119878) and (119861 119879)which also remains a point of coincidence as well Here onemay notice that all the involved mappings are discontinuousat their unique common fixed point 2

Corollary 11 Let (119883 119889) be a symmetric space wherein 119889

satisfies the condition (119862119862)whereas119884 is an arbitrary nonemptyset with 119860 119861 119878 119879 119884 rarr 119883 satisfying all the hypothesesof Lemma 8 Then (119860 119878) and (119861 119879) have a coincidence pointeach Moreover if 119884 = 119883 then 119860 119861 119878 and 119879 have a uniquecommon fixed point provided both the pairs (119860 119878) and (119861 119879)are weakly compatible

Proof Owing to Lemma 8 it follows that the pairs (119860 119878) and(119861 119879) enjoy the (CLR

119878119879

) property Hence all the conditionsof Theorem 9 are satisfied and 119860 119861 119878 and 119879 have a uniquecommon fixed point provided both the pairs (119860 119878) and (119861 119879)are weakly compatible

Remark 12 The conclusions of Lemma 8 Theorem 9and Corollary 11 remain true if we choose 119898(119909 119910) =

max1198724119860119861119878119879

(119909 119910) or119898(119909 119910) = max1198723119860119861119878119879

(119909 119910)

Our next result shows the importance of common limitrange property over common property (EA)

Theorem 13 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861 119878 119879 119884 rarr 119883 which satisfy theinequalities (13) and (15) of Lemma 8 Suppose that

(1) the pairs (119860 119878) and (119861 119879) satisfy the common property(EA)

(2) 119878(119884) and 119879(119884) are closed subsets of119883

Abstract and Applied Analysis 9

Then (119860 119878) and (119861 119879) have a coincidence point eachMoreover if 119884 = 119883 then119860 119861 119878 and 119879 have a unique commonfixed point provided both the pairs (119860 119878) and (119861 119879) are weaklycompatible

Proof Since the pairs (119860 119878) and (119861 119879) satisfy the commonproperty (EA) there exist two sequences 119909

119899

and 119910119899

in 119884such that

lim119899rarrinfin

119889 (119860119909119899

119911) = lim119899rarrinfin

119889 (119878119909119899

119911) = lim119899rarrinfin

119889 (119861119910119899

119911)

= lim119899rarrinfin

119889 (119879119910119899

119911) = 0(46)

for some 119911 isin 119883 Since 119878(119884) and 119879(119884) are closed subsets of119883therefore 119911 isin 119878(119884)cap119879(119884) Since 119911 isin 119878(119884) there exists a point119908 isin 119884 such that 119878119908 = 119911 Also since 119911 isin 119879(119884) there exists apoint V isin 119883 such that 119879V = 119911 The rest of the proof runs onthe lines of the proof of Theorem 9

Remark 14 The conclusions of Theorem 13 remain true ifcondition (2) of Theorem 13 is replaced by one of thefollowing conditions

(2)1015840

119860(119884) sub 119879(119884) and 119861(119884) sub 119878(119884) where 119860(119884) and 119861(119884)denote the closure of ranges of themappings119860 and 119861

(2)10158401015840

119860(119884) and 119861(119884) are closed subsets of 119883 provided119860(119884) sub 119879(119884) and 119861(119884) sub 119878(119884)

By setting119860119861 119878 and119879 suitably we can deduce corollariesinvolving two and three self-mappings As a sample we candeduce the following corollary involving two self-mappings

Corollary 15 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119878 119884 rarr 119883 Suppose that

(1) the pair (119860 119878) satisfies the (119862119871119877119878

) property(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860119909119860119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905) 119889119905) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(47)

where 119898(119909 119910) = max119872119896119860119878

(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Then (119860 119878) has a coincidence point Moreover if 119884 = 119883then119860 and 119878 have a unique common fixed point in119883 providedthe pair (119860 119878) is weakly compatible

As an application of Theorem 9 we have the followingresult involving four finite families of self-mappings

Theorem 16 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860

119894

119898

119894=1

119861119895

119899

119895=1

119878119903

119901

119903=1

119879119897

119902

119897=1

119884 rarr

119883 satisfying the inequalities (13) and (15) of Lemma 8 where119860 = 119860

1

1198602

sdot sdot sdot 119860119898

119861 = 1198611

1198612

sdot sdot sdot 119861119899

119878 = 1198781

1198782

sdot sdot sdot 119878119901

and119879 = 119879

1

1198792

sdot sdot sdot 119879119902

Suppose that the pairs (119860 119878) and (119861 119879) satisfythe (119862119871119877

119878119879

) property Then (119860 119878) and (119861 119879) have a point ofcoincidence each

Moreover if 119884 = 119883 then 119860119894

119898

119894=1

119861119895

119899

119895=1

119878119903

119901

119903=1

and119879119897

119902

119897=1

have a unique common fixed point provided the families(119860119894

119878119903

) and (119861119895

119879119897

) commute pairwise where 119894 isin

1 2 119898 119903 isin 1 2 119901 119895 isin 1 2 119899 and 119897 isin

1 2 119902

Now we indicate that Theorem 16 can be utilized toderive common fixed point theorems for any finite number ofmappings As a sample we can derive a common fixed pointtheorem for six mappings by setting two families of twomembers while setting the rest two of single members

Corollary 17 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861119867 119877 119878 119879 119884 rarr 119883 Suppose that

(1) the pairs (119860 119878119877) and (119861 119879119867) share the (119862119871119877(119878119877)(119879119867)

)

property(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905)) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(48)

where 119898(119909 119910) = max119872119896119860119861119867119877119878119879

(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Then (119860 119878119877) and (119861 119879119867) have a coincidence point eachMoreover if 119884 = 119883 then 119860 119861 119867 119877 119878 and 119879 have a uniquecommon fixed point provided 119860119878 = 119878119860 119860119877 = 119877119860 119878119877 = 119877119878119861119879 = 119879119861 119861119867 = 119867119861 and 119879119867 = 119867119879

By setting 1198601

= 1198602

= sdot sdot sdot = 119860119898

= 119860 1198611

= 1198612

= sdot sdot sdot =

119861119899

= 119861 1198781

= 1198782

= sdot sdot sdot = 119878119901

= 119878 and 1198791

= 1198792

= sdot sdot sdot = 119879119902

= 119879

in Theorem 16 one gets the following corollary

Corollary 18 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861 119878 119879 119884 rarr 119883 Suppose that

(1) the pairs (119860119898 119878119901) and (119861119899 119879119902) share the (119862119871119877(119878

119901119879

119902)

)

property where119898 119899 119901 119902 are fixed positive integers(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860

119898

119909119861

119899

119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905) 119889119905) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(49)

10 Abstract and Applied Analysis

where 119898(119909 119910) = max119872119896119860

119898119861

119899119878

119901119879

119902(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Moreover if 119884 = 119883 then 119860 119861 119878 and 119879 have a uniquecommon fixed point provided 119860119878 = 119878119860 and 119861119879 = 119879119861

Remark 19 The above Corollary 18 is a slight but partial gen-eralization of Theorem 9 as the commutativity requirements(ie 119860119878 = 119878119860 and 119861119879 = 119879119861) in this corollary are relativelystronger as compared to weak compatibility in Theorem 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The first author gratefully acknowledges the support fromthe Deanship of Scientific Research (DSR) at King AbdulazizUniversity (KAU) during this research

References

[1] T L Hicks and B E Rhoades ldquoFixed point theory in symmetricspaces with applications to probabilistic spacesrdquo NonlinearAnalysis ATheory andMethods vol 36 no 3 pp 331ndash344 1999

[2] R P Pant ldquoNoncompatible mappings and common fixedpointsrdquo Soochow Journal of Mathematics vol 26 no 1 pp 29ndash35 2000

[3] G Jungck ldquoCompatible mappings and common fixed pointsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 9 no 4 pp 771ndash779 1986

[4] S Sessa ldquoOn a weak commutativity condition of mappings infixed point considerationsrdquo Institut Mathematique vol 32(46)pp 149ndash153 1982

[5] G Jungck and B E Rhoades ldquoFixed points for set valued func-tions without continuityrdquo Indian Journal of Pure and AppliedMathematics vol 29 no 3 pp 227ndash238 1998

[6] S L Singh and A Tomar ldquoWeaker forms of commuting mapsand existence of fixed pointsrdquo Journal of the Korea Society ofMathematical Education B vol 10 no 3 pp 145ndash161 2003

[7] P P Murthy ldquoImportant tools and possible applications ofmetric fixed point theoryrdquo Nonlinear Analysis Theory Methodsamp Applications vol 47 no 53479 3490 pages 2001

[8] M A Al-Thagafi andN Shahzad ldquoGeneralized 119868-nonexpansiveselfmaps and invariant approximationsrdquo Acta MathematicaSinica vol 24 no 5 pp 867ndash876 2008

[9] M Abbas and B E Rhoades ldquoCommon fixed point theoremsfor occasionally weakly compatible mappings satisfying a gen-eralized contractive conditionrdquoMathematical Communicationsvol 13 no 2 pp 295ndash301 2008

[10] A Aliouche and V Popa ldquoCommon fixed point theorems foroccasionally weakly compatible mappings via implicit rela-tionsrdquo Filomat vol 22 no 2 pp 99ndash107 2008

[11] A Bhatt H Chandra and D R Sahu ldquoCommon fixed pointtheorems for occasionally weakly compatible mappings underrelaxed conditionsrdquo Nonlinear Analysis A Theory and Methodsvol 73 no 1 pp 176ndash182 2010

[12] M Imdad S Chauhan Z Kadelburg andCVetro ldquoFixed pointtheorems for non-self mappings in symmetric spaces under 120593-weak contractive conditions and an application to functionalequations in dynamic programmingrdquo Applied Mathematics andComputation vol 227 pp 469ndash479 2014

[13] D ETHoric Z Kadelburg and S Radenovic ldquoA note on occasion-ally weakly compatible mappings and common fixed pointsrdquoFixed Point Theory vol 13 no 2 pp 475ndash479 2012

[14] M Aamri and D El Moutawakil ldquoSome new common fixedpoint theorems under strict contractive conditionsrdquo Journal ofMathematical Analysis and Applications vol 270 no 1 pp 181ndash188 2002

[15] K P R Sastry and I S R Krishna Murthy ldquoCommon fixedpoints of two partially commuting tangential selfmaps on amet-ric spacerdquo Journal of Mathematical Analysis and Applicationsvol 250 no 2 pp 731ndash734 2000

[16] Y Liu J Wu and Z Li ldquoCommon fixed points of single-valuedand multivalued mapsrdquo International Journal of Mathematicsand Mathematical Sciences no 19 pp 3045ndash3055 2005

[17] M Imdad S Chauhan and Z Kadelburg ldquoFixed point theoremfor mappings with common limit range property satisfyinggeneralized (120595 120593)-weak contractive conditionsrdquo MathematicalSciences vol 7 article 16 2013

[18] A H SolimanM Imdad andMHasan ldquoProving unified com-mon fixed point theorems via common property (EA) in sym-metric spacesrdquo Korean Mathematical Society vol 25 no 4 pp629ndash645 2010

[19] R P Pant ldquoCommon fixed points of Lipschitz type mappingpairsrdquo Journal of Mathematical Analysis and Applications vol240 no 1 pp 280ndash283 1999

[20] W Sintunavarat andPKumam ldquoCommonfixed point theoremsfor a pair of weakly compatible mappings in fuzzy metricspacesrdquo Journal of Applied Mathematics vol 2011 Article ID637958 14 pages 2011

[21] M Imdad B D Pant and S Chauhan ldquoFixed point theoremsinMenger spaces using the (119862119871119877

119878119879

) property and applicationsrdquoJournal of Nonlinear Analysis and Optimization Theory andApplications vol 3 no 2 pp 225ndash237 2012

[22] E Karapınar D K Patel M Imdad and D Gopal ldquoSomenonunique common fixed point theorems in symmetric spacesthrough 119862119871119877

(119878119879)

propertyrdquo International Journal of Mathemat-ics and Mathematical Sciences Article ID 753965 8 pages 2013

[23] D Gopal R P Pant and A S Ranadive ldquoCommon fixed pointof absorbingmapsrdquo BulletinMarathwadaMathematical Societyvol 9 no 1 pp 43ndash48 2008

[24] Ya I Alber and S Guerre-Delabriere ldquoPrinciple of weaklycontractive maps in Hilbert spacesrdquo in New Results in OperatorTheory and its Applications vol 98 ofOperatorTheory Advancesand Applications pp 7ndash22 1997

[25] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis vol 47 pp 2683ndash2693 2001

[26] A Branciari ldquoA fixed point theorem for mappings satisfyinga general contractive condition of integral typerdquo InternationalJournal of Mathematics and Mathematical Sciences vol 29 no9 pp 531ndash536 2002

[27] A Djoudi and A Aliouche ldquoCommon fixed point theoremsof Gregus type for weakly compatible mappings satisfyingcontractive conditions of integral typerdquo Journal ofMathematicalAnalysis and Applications vol 329 no 1 pp 31ndash45 2007

[28] Z Liu X Li S M Kang and S Y Cho ldquoFixed point theoremsfor mappings satisfying contractive conditions of integral type

Abstract and Applied Analysis 11

and applicationsrdquo Fixed Point Theory and Applications vol 64article 18 2011

[29] B Samet and C Vetro ldquoAn integral version of Cirics fixed pointtheoremrdquo Mediterranean Journal of Mathematics vol 9 no 1pp 225ndash238 2012

[30] W Sintunavarat and P Kumam ldquoGregus-type common fixedpoint theorems for tangential multivalued mappings of integraltype inmetric spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2011 Article ID 923458 12 pages2011

[31] W Sintunavarat and P Kumam ldquoGregus type fixed points for atangential multi-valued mappings satisfying contractive condi-tions of integral typerdquo Journal of Inequalities and Applicationsvol 2011 article 3 2011

[32] T Suzuki ldquoMeir-Keeler contractions of integral type are stillMeir-Keeler contractionsrdquo International Journal ofMathematicsand Mathematical Sciences vol 2007 Article ID 39281 6 pages2007

[33] C Vetro S Chauhan E Karapnar and W Shatanawi ldquoFixedpoints of weakly compatible mappings satisfying generalized120593-weak contractionsrdquo Bulletin of the Malaysian MathematicalSciences Society In press

[34] Q Zhang and Y Song ldquoFixed point theory for generalized 120593-weak contractionsrdquo Applied Mathematics Letters vol 22 no 1pp 75ndash78 2009

[35] D ETHoric ldquoCommon fixed point for generalized (120595 120593)-weakcontractionsrdquo Applied Mathematics Letters vol 22 no 12 pp1896ndash1900 2009

[36] P N Dutta and B S Choudhury ldquoA generalisation of con-traction principle in metric spacesrdquo Fixed Point Theory andApplications Article ID 406368 8 pages 2008

[37] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[38] S Radenovic Z Kadelburg D Jandrlic and A Jandrlic ldquoSomeresults on weakly contractive mapsrdquo Iranian MathematicalSociety Bulletin vol 38 no 3 pp 625ndash645 2012

[39] W A Wilson ldquoOn semi-metric spacesrdquo American Journal ofMathematics vol 53 no 2 pp 361ndash373 1931

[40] A Aliouche ldquoA common fixed point theorem for weakly com-patible mappings in symmetric spaces satisfying a contractivecondition of integral typerdquo Journal ofMathematical Analysis andApplications vol 322 no 2 pp 796ndash802 2006

[41] F Galvin and S D Shore ldquoCompleteness in semimetric spacesrdquoPacific Journal of Mathematics vol 113 no 1 pp 67ndash75 1984

[42] S-H Cho G-Y Lee and J-S Bae ldquoOn coincidence and fixed-point theorems in symmetric spacesrdquo Fixed Point Theory andApplications Article ID 562130 9 pages 2008

[43] M Aamri and D El Moutawakil ldquoCommon fixed points undercontractive conditions in symmetric spacesrdquo Applied Math-ematics E-Notes vol 3 pp 156ndash162 2003

[44] DGopalMHasan andM Imdad ldquoAbsorbing pairs facilitatingcommon fixed point theorems for Lipschitzian type mappingsin symmetric spacesrdquo Korean Mathematical Society Communi-cations vol 27 no 2 pp 385ndash397 2012

[45] D Gopal M Imdad and C Vetro ldquoCommon fixed pointtheorems for mappings satisfying common property (EA) insymmetric spacesrdquo Filomat vol 25 no 2 pp 59ndash78 2011

[46] M Imdad and J Ali ldquoCommon fixed point theorems in sym-metric spaces employing a new implicit function and common

property (EA)rdquo Bulletin of the Belgian Mathematical Societyvol 16 no 3 pp 421ndash433 2009

[47] M Imdad J Ali and L Khan ldquoCoincidence and fixed pointsin symmetric spaces under strict contractionsrdquo Journal ofMathematical Analysis and Applications vol 320 no 1 pp 352ndash360 2006

[48] D Turkoglu and I Altun ldquoA common fixed point theorem forweakly compatible mappings in symmetric spaces satisfying animplicit relationrdquo Sociedad Matematica Mexicana vol 13 no 1pp 195ndash205 2007

[49] M Imdad A Sharma Chauhan and S ldquoUnifying a multitudeof metrical fixed point theorems in symmetric spacesrdquo FilomatIn press

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

common property (EA) for hybrid pairs of mappings whichcontained property (EA) In this continuation Imdad et al[17] and Soliman et al [18] extended the results of Sastryand Krishna Murthy [15] and Pant [19] to symmetric spacesby utilizing the weak compatible property with commonproperty (EA) Since the notions of property (EA) andcommon property (EA) always require the completeness(or closedness) of underlying subspaces for the existence ofcommon fixed point hence Sintunavarat and Kumam [20]coined the idea of ldquocommon limit range propertyrdquo whichrelaxes the requirement of completeness (or closedness)of the underlying subspace Afterward Imdad et al [21]extended the notion of common limit range property to twopairs of self-mappings and proved some fixed point theoremsin Menger and metric spaces Most recently Karapınar et al[22] utilized the notion of common limit range property andshowed that the new notion buys certain typical conditionsutilized by Pant [19] up to a pair of mappings on the cast of arelatively more natural absorbing property due to Gopal et al[23]

The concept of weak contraction was introduced byAlber and Guerre-Delabriere [24] in 1997 wherein authorsintroduced the following notion for mappings defined on aHilbert space119867

Consider the following set of real functions

Φ = 120593 [0 +infin) 997888rarr [0 +infin) 120593 is lower

semi-continuous and 120593minus1

(0) = 0

(1)

Amapping119879 119883 rarr 119883 is called a120593-weak contraction if thereexists a function 120593 isin Φ such that

119889 (119879119909 119879119910) le 119889 (119909 119910) minus 120593 (119889 (119909 119910)) forall119909 119910 isin 119883 (2)

Alber and Guerre-Delabriere [24] also showed that each120593-weak contraction on a Hilbert space has a unique fixedpoint Thereafter Rhoades [25] showed that the results con-tained in [24] are also valid for any Banach space In partic-ular he generalized the Banach contraction principle whichfollows in case one chooses 120593(119905) = (1 minus 119896)119905

In 2002 Branciari [26] firstly studied the integral ana-logue of Banachrsquos contraction principle Some interestingresults can be easily seen in [27ndash32] Most recently Vetroet al [33] proved some integral type fixed point results formappings in metric spaces employing common limit rangeproperty Zhang and Song [34] proved a common fixed pointtheorem for twomappings by using 120593-weak contractionThisresult was extended by ETHoric [35] and Dutta and Choudhury[36] to a pair of (120595 120593)-weak contractive mappings Howeverthe main fixed point theorem for a self-mapping satisfying(120595 120593)-weak contractive condition contained in Dutta andChoudhury [36] is given below but before that we considerthe following set of real functions

Ψ = 120595 [0 +infin) 997888rarr [0 +infin) 120595 is continuous

nondecreasing and 120595minus1

(0) = 0

(3)

Theorem 1 Let (119883 119889) be a complete metric space and let 119879

119883 rarr 119883 be a self-mapping satisfying

120595 (119889 (119879119909 119879119910)) le 120595 (119889 (119909 119910)) minus 120593 (119889 (119909 119910)) (4)

for some 120595 isin Ψ and 120593 isin Φ and all 119909 119910 isin 119883 Then 119879 has aunique fixed point in119883

The object of this paper is to prove some integral typecommon fixed point theorems for two pairs of nonselfweakly compatible mappings satisfying generalized (120595 120593)-contractive conditions by using the common limit rangeproperty in symmetric spaces We give some illustrativeexamples to highlight the superiority of our results overseveral results existing in the literature As an extension ofour main result we state some fixed point theorems for sixmappings and four finite families of mappings in symmetricspaces by using the notion of the pairwise commutingmappings which is studied by Imdad et al [37]

2 Preliminaries

A common fixed point result generally involves conditionson commutativity continuity and contraction along witha suitable condition on the containment of range of onemapping into the range of the other Hence one is alwaysrequired to improve one or more of these conditions inorder to prove a new common fixed point theorem It can beobserved that in the case of two mappings 119860 119878 119883 rarr 119883where (119883 119889) is metric space (or symmetric space) one canconsider the following classes of mappings for the existenceand uniqueness of common fixed points

119889 (119860119909 119860119910) le 119865 (119898 (119909 119910)) (5)

where 119865 is some function and119898(119909 119910) is the maximum of oneof the sets Thus

1198725

119860119878

(119909 119910) = 119889 (119878119909 119878119910) 119889 (119878119909 119860119909) 119889 (119878119910 119860119910)

119889 (119878119909 119860119910) 119889 (119878119910 119860119909)

1198724

119860119878

(119909 119910) = 119889 (119878119909 119878119910) 119889 (119878119909 119860119909) 119889 (119878119910 119860119910)

1

2(119889 (119878119909 119860119910) + 119889 (119878119910 119860119909))

1198723

119860119878

(119909 119910) = 119889 (119878119909 119878119910) 1

2(119889 (119878119909 119860119909) + 119889 (119878119910 119860119910))

1

2(119889 (119878119909 119860119910) + 119889 (119878119910 119860119909))

(6)

A further possible generalization is to consider fourmappings instead of two and ascertain analogous commonfixed point theorems In the case of four mappings119860 119861 119878 119879

Abstract and Applied Analysis 3

119883 rarr 119883 where (119883 119889) is metric space (or symmetric space)the corresponding sets take the form

1198725

119860119861119878119879

(119909 119910) = 119889 (119878119909 119879119910) 119889 (119878119909 119860119909) 119889 (119879119910 119861119910)

119889 (119878119909 119861119910) 119889 (119879119910 119860119909)

1198724

119860119861119878119879

(119909 119910) = 119889 (119878119909 119879119910) 119889 (119878119909 119860119909) 119889 (119879119910 119861119910)

1

2(119889 (119878119909 119861119910) + 119889 (119879119910 119860119909))

1198723

119860119861119878119879

(119909 119910) = 119889 (119878119909 119879119910) 1

2(119889 (119878119909 119860119909) + 119889 (119879119910 119861119910))

1

2(119889 (119878119909 119861119910) + 119889 (119879119910 119860119909))

(7)

In this case (5) is usually replaced by

119889 (119860119909 119861119910) le 119865 (119898 (119909 119910)) (8)

where119898(119909 119910) is the maximum of one of the119872-setsSimilarly we can define the 119872-sets for six mappings

119860 119861119867 119877 119878 119879 119883 rarr 119883 where (119883 119889) is metric space (orsymmetric space) as

1198725

119860119861119867119877119878119879

(119909 119910) = 119889 (119878119877119909 119879119867119910) 119889 (119878119877119909 119860119909)

119889 (119879119867119910 119861119910)

119889 (119878119877119909 119861119910) 119889 (119879119867119910119860119909)

1198724

119860119861119867119877119878119879

(119909 119910) = 119889 (119878119877119909 119879119867119910)

119889 (119878119877119909 119860119909) 119889 (119879119867119910 119861119910)

1

2(119889 (119878119877119909 119861119910) + 119889 (119879119867119910119860119909))

1198723

119860119861119867119877119878119879

(119909 119910) = 119889 (119878119877119909 119879119867119910)

1

2(119889 (119878119877119909 119860119909) + 119889 (119879119867119910 119861119910))

1

2(119889 (119878119877119909 119861119910) + 119889 (119879119867119910119860119909))

(9)

and the contractive condition is again in the form (8)By using different arguments of control functions Rade-

novic et al [38] proved some common fixed point resultsfor two and three mappings by using (120595 120593)-weak contractiveconditions and improved several known metrical fixed pointtheorems Motivated by these results we prove some com-mon fixed point theorems for two pairs of weakly compatiblemappings with common limit range property satisfyinggeneralized (120595 120593)-weak contractive conditionsMany knownfixed point results are improved especially the ones provedin [38] and also contained in the references cited therein We

also obtain a fixed point theorem for four finite families ofself-mappings Some related results are also derived besidesfurnishing illustrative examples

The following definitions and results will be needed in thesequel

A symmetric on a set119883 is a function 119889 119883times119883 rarr [0infin)

satisfying the following conditions

(1) 119889(119909 119910) = 0 if and only if 119909 = 119910 for 119909 119910 isin 119883(2) 119889(119909 119910) = 119889(119910 119909) for all 119909 119910 isin 119883

Let 119889 be a symmetric on a set 119883 For 119909 isin 119883 and 120598 gt 0let B(119909 120598) = 119910 isin 119883 119889(119909 119910) lt 120598 A topology 120591(119889) on119883 is defined as follows 119880 isin 120591(119889) if and only if for each119909 isin 119880 there exists an 120598 gt 0 such that B(119909 120598) sub 119880 Asubset 119878 of 119883 is a neighbourhood of 119909 isin 119883 if there exists119880 isin 120591(119889) such that 119909 isin 119880 sub 119878 A symmetric 119889 is asemimetric if for each 119909 isin 119883 and each 120598 gt 0 B(119909 120598)

is a neighbourhood of 119909 in the topology 120591(119889) A symmet-ric (resp semimetric) space (119883 119889) is a topological spacewhose topology 120591(119889) on 119883 is induced by symmetric (respsemimetric) 119889 The difference of a symmetric and a metriccomes from the triangle inequality Since a symmetric spaceis not essentially Hausdorff therefore in order to provefixed point theorems some additional axioms are requiredThe following axioms which are available in Wilson [39]Aliouche [40] and Imdad et al [17] are relevant to thispresentation

From now on symmetric space will be denoted by (119883 119889)whereas a nonempty arbitrary set will be denoted by 119884

(1198823

) Given 119909119899

119909 and 119910 in 119883 lim119899rarrinfin

119889(119909119899

119909) = 0 andlim119899rarrinfin

119889(119909119899

119910) = 0 imply 119909 = 119910 [39](1198824

) Given 119909119899

119910119899

and 119909 in 119883 lim119899rarrinfin

119889(119909119899

119909) = 0

and lim119899rarrinfin

119889(119909119899

119910119899

) = 0 imply lim119899rarrinfin

119889(119910119899

119909) = 0

[39](119867119864) Given 119909

119899

119910119899

and 119909 in 119883 lim119899rarrinfin

119889(119909119899

119909) = 0

and lim119899rarrinfin

119889(119910119899

119909) = 0 imply lim119899rarrinfin

119889(119909119899

119910119899

) = 0

[40](1119862) A symmetric 119889 is said to be 1-continuous if

lim119899rarrinfin

119889(119909119899

119909) = 0 implies lim119899rarrinfin

119889(119909119899

119910) =

119889(119909 119910) where 119909119899

is a sequence in 119883 and 119909 119910 isin 119883

[41](119862119862) A symmetric 119889 is said to be continuous if

lim119899rarrinfin

119889(119909119899

119909) = 0 and lim119899rarrinfin

119889(119910119899

119910) = 0

imply lim119899rarrinfin

119889(119909119899

119910119899

) = 119889(119909 119910) where 119909119899

and119910119899

are sequences in119883 and 119909 119910 isin 119883 [41]

Here it is observed that (119862119862) rArr (1119862) (1198824

) rArr (1198823

)and (1119862) rArr (119882

3

) but the converse implications are nottrue In general all other possible implications amongst (119882

3

)(1119862) and (119867119864) are not true For detailed description we referan interesting note of Cho et al [42] which contained someillustrative examples However (119862119862) implies all the remain-ing four conditions namely (119882

3

) (1198824

) (119867119864) and (1119862)Employing these axioms several authors proved commonfixed point theorems in the framework of symmetric spaces(see [22 43ndash48])

4 Abstract and Applied Analysis

Definition 2 Let (119860 119878) be a pair of self-mappings defined ona nonempty set 119883 equipped with a symmetric 119889 Then themappings 119860 and 119878 are said to be

(1) commuting if 119860119878119909 = 119878119860119909 for all 119909 isin 119883(2) compatible [3] if lim

119899rarrinfin

119889(119860119878119909119899

119878119860119909119899

) = 0 foreach sequence 119909

119899

in 119884 such that lim119899rarrinfin

119860119909119899

=

lim119899rarrinfin

119878119909119899

(3) noncompatible [2] if there exists a sequence 119909

119899

in 119883 such that lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

butlim119899rarrinfin

119889(119860119878119909119899

119878119860119909119899

) is either nonzero or nonexis-tent

(4) weakly compatible [5] if they commute at their coinci-dence points that is 119860119878119909 = 119878119860119909 whenever 119860119909 = 119878119909for some 119909 isin 119883

(5) satisfied the property (EA) [14] if there exists asequence 119909

119899

in 119883 such that lim119899rarrinfin

119860119909119899

=

lim119899rarrinfin

119878119909119899

= 119911 for some 119911 isin 119883

Any pair of compatible as well as noncompatible self-mappings satisfies the property (EA) but a pair of mappingssatisfying the property (EA) needs not be noncompatible

Definition 3 (see [16]) Let 119884 be an arbitrary set and let 119883 bea nonempty set equipped with symmetric 119889 Then the pairs(119860 119878) and (119861 119879) of mappings from 119884 into119883 are said to sharethe common property (EA) if there exist two sequences 119909

119899

and 119910119899

in 119884 such that

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= lim119899rarrinfin

119861119910119899

= lim119899rarrinfin

119879119910119899

= 119911 (10)

for some 119911 isin 119883

Definition 4 (see [20]) Let 119884 be an arbitrary set and let 119883be a nonempty set equipped with symmetric 119889 Then the pair(119860 119878) of mappings from 119884 into119883 is said to have the commonlimit range property with respect to the mapping 119878 (denotedby (CLR

119878

)) if there exists a sequence 119909119899

in 119884 such that

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= 119911 (11)

where 119911 isin 119878(119884)

Definition 5 (see [21]) Let 119884 be an arbitrary set and let 119883 bea nonempty set equipped with symmetric 119889 Then the pairs(119860 119878) and (119861 119879) of mappings from 119884 into 119883 are said to havethe common limit range property with respect to mappings 119878and 119879 denoted by (CLR

119878119879

) if there exist two sequences 119909119899

and 119910119899

in 119884 such that

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= lim119899rarrinfin

119861119910119899

= lim119899rarrinfin

119879119910119899

= 119911 (12)

where 119911 isin 119878(119884) cap 119879(119884)

Remark 6 It is clear that (CLR119878119879

) property implies thecommon property (EA) but the converse is not true (see [49Example 1])

Definition 7 (see [37]) Two families of self-mappings 119860119894

119898

119894=1

and 119878119896

119899

119896=1

are said to be pairwise commuting if

(1) 119860119894

119860119895

= 119860119895

119860119894

for all 119894 119895 isin 1 2 119898(2) 119878119896

119878119897

= 119878119897

119878119896

for all 119896 119897 isin 1 2 119899(3) 119860119894

119878119896

= 119878119896

119860119894

for all 119894 isin 1 2 119898 and 119896 isin

1 2 119899

3 Results

Now we state and prove our main results for four mappingsemploying the common limit range property in symmetricspaces Firstly we prove the following lemma

Lemma 8 Let (119883 119889) be a symmetric space wherein 119889 satisfiesthe conditions (119862119862) whereas 119884 is an arbitrary nonempty setwith 119860 119861 119878 and 119879 119884 rarr 119883 Suppose that

(1) the pair (119860 119878) (or (119861 119879)) satisfies the (119862119871119877119878

) (or(119862119871119877119879

)) property(2) 119860(119884) sub 119879(119884) (or 119861(119884) sub 119878(119884))(3) 119879(119884) (or 119878(119884)) is a closed subset of119883(4) 119861119910

119899

converges for every sequence 119910119899

in 119884 whenever119879119910119899

converges (or 119860119909119899

converges for every sequence119909119899

in 119884 whenever 119878119909119899

converges)(5) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905) 119889119905) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(13)

where

119898(119909 119910) = max1198725119860119861119878119879

(119909 119910) (14)

and 120601 [0 +infin) rarr [0 +infin) is a Lebesgue-integrable mappingwhich is summable and nonnegative such that

int120598

0

120601 (119905) 119889119905 gt 0 (15)

for all 120598 gt 0Then the pairs (119860 119878) and (119861 119879) satisfy the (CLR

119878119879

)

property

Proof First we show that the conclusion of this theoremholds for first case Since the pair (119860 119878) enjoys the (CLR

119878

)

property therefore there exists a sequence 119909119899

in119884 such that

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= 119911 (16)

where 119911 isin 119878(119884) Since 119860(119884) sub 119879(119884) hence for each sequence119909119899

there exists a sequence 119910119899

in 119884 such that 119860119909119899

= 119879119910119899

Therefore by closedness of 119879(119884)

lim119899rarrinfin

119879119910119899

= lim119899rarrinfin

119860119909119899

= 119911 (17)

for 119911 isin 119879(119884) and in all 119911 isin 119878(119884) cap 119879(119884) Thus in all we have119860119909119899

rarr 119911 119878119909119899

rarr 119911 and 119879119910119899

rarr 119911 as 119899 rarr infin Since by

Abstract and Applied Analysis 5

(4) 119861119910119899

converges in all we need to show that 119861119910119899

rarr 119911

as 119899 rarr infin Assume this contrary we get 119861119910119899

rarr 119905( = 119911) as119899 rarr infin Now using inequality (13) with = 119909

119899

119910 = 119910119899

wehave

120595(int119889(119860119909

119899119861119910

119899)

0

120601 (119905) 119889119905)

le 120595(int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905) minus 120593(int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

(18)

where

119898(119909119899

119910119899

) = max 119889 (119878119909119899

119879119910119899

) 119889 (119878119909119899

119860119909119899

) 119889 (119879119910119899

119861119910119899

)

119889 (119878119909119899

119861119910119899

) 119889 (119879119910119899

119860119909119899

)

(19)

Taking limit as 119899 rarr infin and using property (119862119862) ininequality (18) we get

lim119899rarrinfin

120595(int119889(119860119909

119899119861119910

119899)

0

120601 (119905) 119889119905)

le lim119899rarrinfin

120595(int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

minus lim119899rarrinfin

120593(int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

(20)

that is

120595(int119889(119911119905)

0

120601 (119905) 119889119905)

= 120595( lim119899rarrinfin

int119889(119860119909

119899119861119910

119899)

0

120601 (119905) 119889119905)

le 120595( lim119899rarrinfin

int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

minus 120593( lim119899rarrinfin

int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

(21)

where

lim119899rarrinfin

119898(119909119899

119910119899

)

= max 119889 (119911 119911) 119889 (119911 119911) 119889 (119911 119905) 119889 (119911 119905) 119889 (119911 119911)

= max 0 0 119889 (119911 119905) 119889 (119911 119905) 0

= 119889 (119911 119905)

(22)

Hence inequality (21) implies

120595(int119889(119911119905)

0

120601 (119905) 119889119905)

le 120595(int119889(119911119905)

0

120601 (119905) 119889119905) minus 120593(int119889(119911119905)

0

120601 (119905) 119889119905)

(23)

that is 120593(int119889(119911119905)0

120601(119905)119889119905) le 0 and so 120593(int119889(119911119905)0

120601(119905)119889119905) = 0 andby the property of the function 120593 we have 119889(119911 119905) = 0 orequivalently 119911 = 119905 which contradicts the hypothesis 119905 = 119911Hence both the pairs (119860 119878) and (119861 119879) satisfy the (CLR

119878119879

)

propertyIn the second case it similar to the first case So in order

to avoid repetition the details of the proof are omitted Thiscompletes the proof

Theorem9 Let (119883 119889) be a symmetric spacewherein119889 satisfiesthe conditions (1119862) and (119867119864) whereas 119884 is an arbitrarynonempty set with 119860 119861 119878 119879 119884 rarr 119883 which satisfy theinequalities (13) and (15) of Lemma 8 Suppose that the pairs(119860 119878) and (119861 119879) satisfy the (119862119871119877

119878119879

) propertyThen (119860 119878) and(119861 119879) have a coincidence point each Moreover if 119884 = 119883 then119860 119861 119878 and119879 have a unique common fixed point provided boththe pairs (119860 119878) and (119861 119879) are weakly compatible

Proof Since the pairs (119860 119878) and (119861 119879) enjoy the (CLR119878119879

)

property there exist two sequences 119909119899

and 119910119899

in 119884 suchthat

lim119899rarrinfin

119889 (119860119909119899

119911) = lim119899rarrinfin

119889 (119878119909119899

119911) = lim119899rarrinfin

119889 (119861119910119899

119911)

= lim119899rarrinfin

119889 (119879119910119899

119911) = 0(24)

where 119911 isin 119878(119884) cap 119879(119884) It follows from 119911 isin 119878(119884) that thereexists a point119908 isin 119884 such that 119878119908 = 119911 We assert that119860119908 = 119911If not then using inequality (13) with 119909 = 119908 and 119910 = 119910

119899

oneobtains

120595(int119889(119860119908119861119910

119899)

0

120601 (119905) 119889119905)

le 120595(int119898(119908119910

119899)

0

120601 (119905) 119889119905) minus 120593(int119898(119908119910

119899)

0

120601 (119905) 119889119905)

(25)

where

119898(119908 119910119899

) =max 119889 (119878119908 119879119910119899

) 119889 (119878119908 119860119908) 119889 (119879119910119899

119861119910119899

)

119889 (119878119908 119861119910119899

) 119889 (119879119910119899

119860119908)

(26)

Taking limit as 119899 rarr infin and using properties (1119862) and (119867119864)in inequality (25) one obtains

lim119899rarrinfin

120595(int119889(119860119908119861119910

119899)

0

120601 (119905) 119889119905)

le lim119899rarrinfin

120595(int119898(119908119910

119899)

0

120601 (119905) 119889119905)

minus lim119899rarrinfin

120593(int119898(119908119910

119899)

0

120601 (119905) 119889119905)

(27)

6 Abstract and Applied Analysis

that is

120595(int119889(119860119908119911)

0

120601 (119905) 119889119905)

= 120595( lim119899rarrinfin

int119889(119860119908119861119910

119899)

0

120601 (119905) 119889119905)

le 120595( lim119899rarrinfin

int119898(119908119910

119899)

0

120601 (119905) 119889119905)

minus 120593( lim119899rarrinfin

int119898(119908119910

119899)

0

120601 (119905) 119889119905)

(28)

wherelim119899rarrinfin

119898(119908 119910119899

)

= max 119889 (119911 119911) 119889 (119911 119860119908) 119889 (119911 119911) 119889 (119911 119911) 119889 (119911 119860119908)

= max 0 119889 (119911 119860119908) 0 0 119889 (119911 119860119908)

= 119889 (119911 119860119908)

(29)

From inequality (28) one gets

120595(int119889(119860119908119911)

0

120601 (119905) 119889119905)

le 120595(int119889(119860119908119911)

0

120601 (119905)) minus 120593(int119889(119860119908119911)

0

120601 (119905))

(30)

so that 120593(int119889(119860119908119911)0

120601(119905)119889119905) = 0 that is 119889(119860119908 119911) = 0 Hence119860119908 = 119878119908 = 119911 which shows that 119908 is a coincidence point ofthe pair (119860 119878)

Also 119911 isin 119879(119884) there exists a point V isin 119884 such that119879V = 119911We assert that 119861V = 119911 If not then using inequality (13) with119909 = 119908 119910 = V we have

120595(int119889(119911119861V)

0

120601 (119905) 119889119905)

= 120595(int119889(119860119908119861V)

0

120601 (119905) 119889119905)

le 120595(int119898(119908V)

0

120601 (119905) 119889119905) minus 120593(int119898(119908V)

0

120601 (119905) 119889119905)

(31)

where119898(119908 V)

= max 119889 (119878119908 119879V) 119889 (119878119908 119860119908) 119889 (119879V 119861V)

119889 (119878119908 119861V) 119889 (119879V 119860119908)

= max 119889 (119911 119911) 119889 (119911 119911) 119889 (119911 119861V) 119889 (119911 119861V) 119889 (119911 119911)

= max 0 0 119889 (119911 119861V) 119889 (119911 119861V) 0

= 119889 (119911 119861V) (32)

Hence inequality (31) implies

120595(int119889(119911119861V)

0

120601 (119905) 119889119905)

le 120595(int119889(119911119861V)

0

120601 (119905) 119889119905) minus 120593(int119889(119911119861V)

0

120601 (119905) 119889119905)

(33)

so that 120593(int119889(119911119861V)0

120601(119905)119889119905) = 0 that is 119889(119911 119861V) = 0 Therefore119911 = 119861V = 119879V which shows that V is a coincidence point of thepair (119861 119879)

Consider 119884 = 119883 Since the pair (119860 119878) is weakly compat-ible and 119860119908 = 119878119908 hence 119860119911 = 119860119878119908 = 119878119860119908 = 119878119911 Nowwe assert that 119911 is a common fixed point of the pair (119860 119878) Toaccomplish this using inequality (13) with 119909 = 119911 and 119910 = Vone gets

120595(int119889(119860119911119911)

0

120601 (119905) 119889119905)

= 120595(int119889(119860119911119861V)

0

120601 (119905) 119889119905)

le 120595(int119898(119911V)

0

120601 (119905) 119889119905) minus 120593(int119898(119911V)

0

120601 (119905) 119889119905)

(34)

where119898(119911 V)

= max 119889 (119878119911 119879V) 119889 (119878119911 119860119911) 119889 (119879V 119861V)

119889 (119878119911 119861V) 119889 (119879V 119860119911)

= max 119889 (119860119911 119911) 119889 (119860119911 119860119911) 119889 (119911 119911)

119889 (119860119911 119911) 119889 (119911 119860119911)

= 119889 (119860119911 119911)

(35)

From inequality (34) we have

120595(int119889(119860119911119911)

0

120601 (119905) 119889119905)

le 120595(int119889(119860119911119911)

0

120601 (119905) 119889119905) minus 120593(int119889(119860119911119911)

0

120601 (119905) 119889119905)

(36)

so that 120593(int119889(119860119911119911)0

120601(119905)119889119905) = 0 that is 119889(119860119911 119911) = 0 Hence wehave119860119911 = 119911 = 119878119911which shows that 119911 is a commonfixed pointof the pair (119860 119878)

Also the pair (119861 119879) is weakly compatible and 119861V = 119879Vhence 119861119911 = 119861119879V = 119879119861V = 119879119911 Using inequality (13) with119909 = 119908 and 119910 = 119911 we have

120595(int119889(119911119861119911)

0

120601 (119905) 119889119905)

= 120595(int119889(119860119908119861119911)

0

120601 (119905) 119889119905)

le 120595(int119898(119908119911)

0

120601 (119905) 119889119905) minus 120593(int119898(119908119911)

0

120601 (119905) 119889119905)

(37)

Abstract and Applied Analysis 7

where

119898(119908 119911) = max 119889 (119878119908 119879119911) 119889 (119878119908 119860119908) 119889 (119879119911 119861119911)

119889 (119878119908 119861119911) 119889 (119879119911 119860119908)

= max 119889 (119911 119861119911) 119889 (119911 119911) 119889 (119861119911 119861119911)

119889 (119911 119861119911) 119889 (119861119911 119911)

= 119889 (119911 119861119911)

(38)

Hence inequality (37) implies

120595(int119889(119911119861119911)

0

120601 (119905) 119889119905)

le 120595(int119889(119911119861119911)

0

120601 (119905) 119889119905) minus 120593(int119889(119911119861119911)

0

120601 (119905) 119889119905)

(39)

so that 120593(int119889(119911119861119911)0

120601(119905)119889119905) = 0 that is 119889(119911 119861119911) = 0 Therefore119861119911 = 119911 = 119879119911 which shows that 119911 is a common fixed pointof the pair (119861 119879) Hence 119911 is a common fixed point of thepairs (119860 119878) and (119861 119879) Uniqueness of common fixed point isan easy consequence of the inequality (13)This concludes theproof

Now we furnish an illustrative example which demon-strates the validity of the hypotheses and degree of generalityof Theorem 9

Example 10 Consider 119883 = 119884 = [2 11) equipped with thesymmetric 119889(119909 119910) = (119909 minus 119910)

2 for all 119909 119910 isin 119883 which alsosatisfies (1119862) and (119867119864) Define the mappings 119860 119861 119878 and 119879

by

119860119909 = 2 if 119909 isin 2 cup (5 11)

5 if 119909 isin (2 5]

119861119909 = 2 if 119909 isin 2 cup (5 11)

4 if 119909 isin (2 5]

119878119909 =

2 if 119909 = 2

6 if 119909 isin (2 5] 3119909 + 1

8 if 119909 isin (5 11)

119879119909 =

2 if 119909 = 2

8 if 119909 isin (2 5]

119909 minus 3 if 119909 isin (5 11)

(40)

Then 119860(119883) = 2 5 119861(119883) = 2 4 119879(119883) = [2 8] and 119878(119883) =[2 174) cup 6 Now consider the sequences 119909

119899

= 5 + 1119899

and 119910119899

= 2 Then

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= lim119899rarrinfin

119861119910119899

= lim119899rarrinfin

119879119910119899

= 2 isin 119878 (119883) cap 119879 (119883)

(41)

that is both the pairs (119860 119878) and (119861 119879) satisfy the (CLR119878119879

)

property

Let Lebesgue-integrable 120601 [0 +infin) rarr [0 +infin) definedby 120601(119905) = 119890

119905 Take 120595 isin Ψ and 120593 isin Φ given by 120595(119905) = 2119905 and120593(119905) = (27)119905 In order to check the contractive condition (13)consider the following nine cases

(i) 119909 = 119910 = 2

(ii) 119909 = 2 119910 isin (2 5]

(iii) 119909 = 2 119910 isin (5 11)

(iv) 119909 isin (2 5] 119910 = 2

(v) 119909 119910 isin (2 5]

(vi) 119909 isin (2 5] 119910 isin (5 11)

(vii) 119909 isin (5 11) 119910 = 2

(viii) 119909 isin (5 11) 119910 isin (2 5]

(ix) 119909 119910 isin (5 11)

In the cases (i) (iii) (vii) and (ix) we get that119889(119860119909 119861119910) =0 and (13) is trivially satisfied

In the cases (ii) and (viii) we have 119889(119860119909 119861119910) = 4 and119898(119909 119910) = 36 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int4

0

119890119905

119889119905)

= 120595 (1198904

minus 1)

= 2 (1198904

minus 1)

le12

7(11989036

minus 1)

= 2 (11989036

minus 1) minus2

7(11989036

minus 1)

= 120595(int36

0

119890119905

119889119905) minus 120593(int36

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(42)

Therefore we get the fact that (13) holds

8 Abstract and Applied Analysis

In the case (iv) we get 119889(119860119909 119861119910) = 9 and 119898(119909 119910) = 16Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int9

0

119890119905

119889119905)

= 120595 (1198909

minus 1)

= 2 (1198909

minus 1)

le12

7(11989016

minus 1)

= 2 (11989016

minus 1) minus2

7(11989016

minus 1)

= 120595(int16

0

119890119905

119889119905) minus 120593(int16

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(43)

This implies that (13) holdsIn the case (vi) we have 119889(119860119909 119861119910) = 9 and 119889(119878119909 119861119910) =

16 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int9

0

119890119905

119889119905)

= 120595 (1198909

minus 1)

= 2 (1198909

minus 1)

le12

7(11989016

minus 1)

=12

7(119890119889(119878119909119861119910)

minus 1)

le12

7(119890119898(119909119910)

minus 1)

= 2 (119890119898(119909119910)

minus 1) minus2

7(119890119898(119909119910)

minus 1)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(44)

This shows that (13) holds

Finally in the case (v) we obtain 119889(119860119909 119861119910) = 1 and119898(119909 119910) = 16 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int1

0

119890119905

119889119905)

= 120595 (119890 minus 1)

= 2 (119890 minus 1)

le12

7(11989016

minus 1)

= 2 (11989016

minus 1) minus2

7(11989016

minus 1)

= 120595(int16

0

119890119905

119889119905) minus 120593(int16

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(45)

that is the contractive condition (13) holdsHence all the conditions ofTheorem 9 are satisfied and 2

is a unique common fixed point of the pairs (119860 119878) and (119861 119879)which also remains a point of coincidence as well Here onemay notice that all the involved mappings are discontinuousat their unique common fixed point 2

Corollary 11 Let (119883 119889) be a symmetric space wherein 119889

satisfies the condition (119862119862)whereas119884 is an arbitrary nonemptyset with 119860 119861 119878 119879 119884 rarr 119883 satisfying all the hypothesesof Lemma 8 Then (119860 119878) and (119861 119879) have a coincidence pointeach Moreover if 119884 = 119883 then 119860 119861 119878 and 119879 have a uniquecommon fixed point provided both the pairs (119860 119878) and (119861 119879)are weakly compatible

Proof Owing to Lemma 8 it follows that the pairs (119860 119878) and(119861 119879) enjoy the (CLR

119878119879

) property Hence all the conditionsof Theorem 9 are satisfied and 119860 119861 119878 and 119879 have a uniquecommon fixed point provided both the pairs (119860 119878) and (119861 119879)are weakly compatible

Remark 12 The conclusions of Lemma 8 Theorem 9and Corollary 11 remain true if we choose 119898(119909 119910) =

max1198724119860119861119878119879

(119909 119910) or119898(119909 119910) = max1198723119860119861119878119879

(119909 119910)

Our next result shows the importance of common limitrange property over common property (EA)

Theorem 13 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861 119878 119879 119884 rarr 119883 which satisfy theinequalities (13) and (15) of Lemma 8 Suppose that

(1) the pairs (119860 119878) and (119861 119879) satisfy the common property(EA)

(2) 119878(119884) and 119879(119884) are closed subsets of119883

Abstract and Applied Analysis 9

Then (119860 119878) and (119861 119879) have a coincidence point eachMoreover if 119884 = 119883 then119860 119861 119878 and 119879 have a unique commonfixed point provided both the pairs (119860 119878) and (119861 119879) are weaklycompatible

Proof Since the pairs (119860 119878) and (119861 119879) satisfy the commonproperty (EA) there exist two sequences 119909

119899

and 119910119899

in 119884such that

lim119899rarrinfin

119889 (119860119909119899

119911) = lim119899rarrinfin

119889 (119878119909119899

119911) = lim119899rarrinfin

119889 (119861119910119899

119911)

= lim119899rarrinfin

119889 (119879119910119899

119911) = 0(46)

for some 119911 isin 119883 Since 119878(119884) and 119879(119884) are closed subsets of119883therefore 119911 isin 119878(119884)cap119879(119884) Since 119911 isin 119878(119884) there exists a point119908 isin 119884 such that 119878119908 = 119911 Also since 119911 isin 119879(119884) there exists apoint V isin 119883 such that 119879V = 119911 The rest of the proof runs onthe lines of the proof of Theorem 9

Remark 14 The conclusions of Theorem 13 remain true ifcondition (2) of Theorem 13 is replaced by one of thefollowing conditions

(2)1015840

119860(119884) sub 119879(119884) and 119861(119884) sub 119878(119884) where 119860(119884) and 119861(119884)denote the closure of ranges of themappings119860 and 119861

(2)10158401015840

119860(119884) and 119861(119884) are closed subsets of 119883 provided119860(119884) sub 119879(119884) and 119861(119884) sub 119878(119884)

By setting119860119861 119878 and119879 suitably we can deduce corollariesinvolving two and three self-mappings As a sample we candeduce the following corollary involving two self-mappings

Corollary 15 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119878 119884 rarr 119883 Suppose that

(1) the pair (119860 119878) satisfies the (119862119871119877119878

) property(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860119909119860119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905) 119889119905) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(47)

where 119898(119909 119910) = max119872119896119860119878

(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Then (119860 119878) has a coincidence point Moreover if 119884 = 119883then119860 and 119878 have a unique common fixed point in119883 providedthe pair (119860 119878) is weakly compatible

As an application of Theorem 9 we have the followingresult involving four finite families of self-mappings

Theorem 16 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860

119894

119898

119894=1

119861119895

119899

119895=1

119878119903

119901

119903=1

119879119897

119902

119897=1

119884 rarr

119883 satisfying the inequalities (13) and (15) of Lemma 8 where119860 = 119860

1

1198602

sdot sdot sdot 119860119898

119861 = 1198611

1198612

sdot sdot sdot 119861119899

119878 = 1198781

1198782

sdot sdot sdot 119878119901

and119879 = 119879

1

1198792

sdot sdot sdot 119879119902

Suppose that the pairs (119860 119878) and (119861 119879) satisfythe (119862119871119877

119878119879

) property Then (119860 119878) and (119861 119879) have a point ofcoincidence each

Moreover if 119884 = 119883 then 119860119894

119898

119894=1

119861119895

119899

119895=1

119878119903

119901

119903=1

and119879119897

119902

119897=1

have a unique common fixed point provided the families(119860119894

119878119903

) and (119861119895

119879119897

) commute pairwise where 119894 isin

1 2 119898 119903 isin 1 2 119901 119895 isin 1 2 119899 and 119897 isin

1 2 119902

Now we indicate that Theorem 16 can be utilized toderive common fixed point theorems for any finite number ofmappings As a sample we can derive a common fixed pointtheorem for six mappings by setting two families of twomembers while setting the rest two of single members

Corollary 17 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861119867 119877 119878 119879 119884 rarr 119883 Suppose that

(1) the pairs (119860 119878119877) and (119861 119879119867) share the (119862119871119877(119878119877)(119879119867)

)

property(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905)) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(48)

where 119898(119909 119910) = max119872119896119860119861119867119877119878119879

(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Then (119860 119878119877) and (119861 119879119867) have a coincidence point eachMoreover if 119884 = 119883 then 119860 119861 119867 119877 119878 and 119879 have a uniquecommon fixed point provided 119860119878 = 119878119860 119860119877 = 119877119860 119878119877 = 119877119878119861119879 = 119879119861 119861119867 = 119867119861 and 119879119867 = 119867119879

By setting 1198601

= 1198602

= sdot sdot sdot = 119860119898

= 119860 1198611

= 1198612

= sdot sdot sdot =

119861119899

= 119861 1198781

= 1198782

= sdot sdot sdot = 119878119901

= 119878 and 1198791

= 1198792

= sdot sdot sdot = 119879119902

= 119879

in Theorem 16 one gets the following corollary

Corollary 18 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861 119878 119879 119884 rarr 119883 Suppose that

(1) the pairs (119860119898 119878119901) and (119861119899 119879119902) share the (119862119871119877(119878

119901119879

119902)

)

property where119898 119899 119901 119902 are fixed positive integers(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860

119898

119909119861

119899

119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905) 119889119905) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(49)

10 Abstract and Applied Analysis

where 119898(119909 119910) = max119872119896119860

119898119861

119899119878

119901119879

119902(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Moreover if 119884 = 119883 then 119860 119861 119878 and 119879 have a uniquecommon fixed point provided 119860119878 = 119878119860 and 119861119879 = 119879119861

Remark 19 The above Corollary 18 is a slight but partial gen-eralization of Theorem 9 as the commutativity requirements(ie 119860119878 = 119878119860 and 119861119879 = 119879119861) in this corollary are relativelystronger as compared to weak compatibility in Theorem 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The first author gratefully acknowledges the support fromthe Deanship of Scientific Research (DSR) at King AbdulazizUniversity (KAU) during this research

References

[1] T L Hicks and B E Rhoades ldquoFixed point theory in symmetricspaces with applications to probabilistic spacesrdquo NonlinearAnalysis ATheory andMethods vol 36 no 3 pp 331ndash344 1999

[2] R P Pant ldquoNoncompatible mappings and common fixedpointsrdquo Soochow Journal of Mathematics vol 26 no 1 pp 29ndash35 2000

[3] G Jungck ldquoCompatible mappings and common fixed pointsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 9 no 4 pp 771ndash779 1986

[4] S Sessa ldquoOn a weak commutativity condition of mappings infixed point considerationsrdquo Institut Mathematique vol 32(46)pp 149ndash153 1982

[5] G Jungck and B E Rhoades ldquoFixed points for set valued func-tions without continuityrdquo Indian Journal of Pure and AppliedMathematics vol 29 no 3 pp 227ndash238 1998

[6] S L Singh and A Tomar ldquoWeaker forms of commuting mapsand existence of fixed pointsrdquo Journal of the Korea Society ofMathematical Education B vol 10 no 3 pp 145ndash161 2003

[7] P P Murthy ldquoImportant tools and possible applications ofmetric fixed point theoryrdquo Nonlinear Analysis Theory Methodsamp Applications vol 47 no 53479 3490 pages 2001

[8] M A Al-Thagafi andN Shahzad ldquoGeneralized 119868-nonexpansiveselfmaps and invariant approximationsrdquo Acta MathematicaSinica vol 24 no 5 pp 867ndash876 2008

[9] M Abbas and B E Rhoades ldquoCommon fixed point theoremsfor occasionally weakly compatible mappings satisfying a gen-eralized contractive conditionrdquoMathematical Communicationsvol 13 no 2 pp 295ndash301 2008

[10] A Aliouche and V Popa ldquoCommon fixed point theorems foroccasionally weakly compatible mappings via implicit rela-tionsrdquo Filomat vol 22 no 2 pp 99ndash107 2008

[11] A Bhatt H Chandra and D R Sahu ldquoCommon fixed pointtheorems for occasionally weakly compatible mappings underrelaxed conditionsrdquo Nonlinear Analysis A Theory and Methodsvol 73 no 1 pp 176ndash182 2010

[12] M Imdad S Chauhan Z Kadelburg andCVetro ldquoFixed pointtheorems for non-self mappings in symmetric spaces under 120593-weak contractive conditions and an application to functionalequations in dynamic programmingrdquo Applied Mathematics andComputation vol 227 pp 469ndash479 2014

[13] D ETHoric Z Kadelburg and S Radenovic ldquoA note on occasion-ally weakly compatible mappings and common fixed pointsrdquoFixed Point Theory vol 13 no 2 pp 475ndash479 2012

[14] M Aamri and D El Moutawakil ldquoSome new common fixedpoint theorems under strict contractive conditionsrdquo Journal ofMathematical Analysis and Applications vol 270 no 1 pp 181ndash188 2002

[15] K P R Sastry and I S R Krishna Murthy ldquoCommon fixedpoints of two partially commuting tangential selfmaps on amet-ric spacerdquo Journal of Mathematical Analysis and Applicationsvol 250 no 2 pp 731ndash734 2000

[16] Y Liu J Wu and Z Li ldquoCommon fixed points of single-valuedand multivalued mapsrdquo International Journal of Mathematicsand Mathematical Sciences no 19 pp 3045ndash3055 2005

[17] M Imdad S Chauhan and Z Kadelburg ldquoFixed point theoremfor mappings with common limit range property satisfyinggeneralized (120595 120593)-weak contractive conditionsrdquo MathematicalSciences vol 7 article 16 2013

[18] A H SolimanM Imdad andMHasan ldquoProving unified com-mon fixed point theorems via common property (EA) in sym-metric spacesrdquo Korean Mathematical Society vol 25 no 4 pp629ndash645 2010

[19] R P Pant ldquoCommon fixed points of Lipschitz type mappingpairsrdquo Journal of Mathematical Analysis and Applications vol240 no 1 pp 280ndash283 1999

[20] W Sintunavarat andPKumam ldquoCommonfixed point theoremsfor a pair of weakly compatible mappings in fuzzy metricspacesrdquo Journal of Applied Mathematics vol 2011 Article ID637958 14 pages 2011

[21] M Imdad B D Pant and S Chauhan ldquoFixed point theoremsinMenger spaces using the (119862119871119877

119878119879

) property and applicationsrdquoJournal of Nonlinear Analysis and Optimization Theory andApplications vol 3 no 2 pp 225ndash237 2012

[22] E Karapınar D K Patel M Imdad and D Gopal ldquoSomenonunique common fixed point theorems in symmetric spacesthrough 119862119871119877

(119878119879)

propertyrdquo International Journal of Mathemat-ics and Mathematical Sciences Article ID 753965 8 pages 2013

[23] D Gopal R P Pant and A S Ranadive ldquoCommon fixed pointof absorbingmapsrdquo BulletinMarathwadaMathematical Societyvol 9 no 1 pp 43ndash48 2008

[24] Ya I Alber and S Guerre-Delabriere ldquoPrinciple of weaklycontractive maps in Hilbert spacesrdquo in New Results in OperatorTheory and its Applications vol 98 ofOperatorTheory Advancesand Applications pp 7ndash22 1997

[25] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis vol 47 pp 2683ndash2693 2001

[26] A Branciari ldquoA fixed point theorem for mappings satisfyinga general contractive condition of integral typerdquo InternationalJournal of Mathematics and Mathematical Sciences vol 29 no9 pp 531ndash536 2002

[27] A Djoudi and A Aliouche ldquoCommon fixed point theoremsof Gregus type for weakly compatible mappings satisfyingcontractive conditions of integral typerdquo Journal ofMathematicalAnalysis and Applications vol 329 no 1 pp 31ndash45 2007

[28] Z Liu X Li S M Kang and S Y Cho ldquoFixed point theoremsfor mappings satisfying contractive conditions of integral type

Abstract and Applied Analysis 11

and applicationsrdquo Fixed Point Theory and Applications vol 64article 18 2011

[29] B Samet and C Vetro ldquoAn integral version of Cirics fixed pointtheoremrdquo Mediterranean Journal of Mathematics vol 9 no 1pp 225ndash238 2012

[30] W Sintunavarat and P Kumam ldquoGregus-type common fixedpoint theorems for tangential multivalued mappings of integraltype inmetric spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2011 Article ID 923458 12 pages2011

[31] W Sintunavarat and P Kumam ldquoGregus type fixed points for atangential multi-valued mappings satisfying contractive condi-tions of integral typerdquo Journal of Inequalities and Applicationsvol 2011 article 3 2011

[32] T Suzuki ldquoMeir-Keeler contractions of integral type are stillMeir-Keeler contractionsrdquo International Journal ofMathematicsand Mathematical Sciences vol 2007 Article ID 39281 6 pages2007

[33] C Vetro S Chauhan E Karapnar and W Shatanawi ldquoFixedpoints of weakly compatible mappings satisfying generalized120593-weak contractionsrdquo Bulletin of the Malaysian MathematicalSciences Society In press

[34] Q Zhang and Y Song ldquoFixed point theory for generalized 120593-weak contractionsrdquo Applied Mathematics Letters vol 22 no 1pp 75ndash78 2009

[35] D ETHoric ldquoCommon fixed point for generalized (120595 120593)-weakcontractionsrdquo Applied Mathematics Letters vol 22 no 12 pp1896ndash1900 2009

[36] P N Dutta and B S Choudhury ldquoA generalisation of con-traction principle in metric spacesrdquo Fixed Point Theory andApplications Article ID 406368 8 pages 2008

[37] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[38] S Radenovic Z Kadelburg D Jandrlic and A Jandrlic ldquoSomeresults on weakly contractive mapsrdquo Iranian MathematicalSociety Bulletin vol 38 no 3 pp 625ndash645 2012

[39] W A Wilson ldquoOn semi-metric spacesrdquo American Journal ofMathematics vol 53 no 2 pp 361ndash373 1931

[40] A Aliouche ldquoA common fixed point theorem for weakly com-patible mappings in symmetric spaces satisfying a contractivecondition of integral typerdquo Journal ofMathematical Analysis andApplications vol 322 no 2 pp 796ndash802 2006

[41] F Galvin and S D Shore ldquoCompleteness in semimetric spacesrdquoPacific Journal of Mathematics vol 113 no 1 pp 67ndash75 1984

[42] S-H Cho G-Y Lee and J-S Bae ldquoOn coincidence and fixed-point theorems in symmetric spacesrdquo Fixed Point Theory andApplications Article ID 562130 9 pages 2008

[43] M Aamri and D El Moutawakil ldquoCommon fixed points undercontractive conditions in symmetric spacesrdquo Applied Math-ematics E-Notes vol 3 pp 156ndash162 2003

[44] DGopalMHasan andM Imdad ldquoAbsorbing pairs facilitatingcommon fixed point theorems for Lipschitzian type mappingsin symmetric spacesrdquo Korean Mathematical Society Communi-cations vol 27 no 2 pp 385ndash397 2012

[45] D Gopal M Imdad and C Vetro ldquoCommon fixed pointtheorems for mappings satisfying common property (EA) insymmetric spacesrdquo Filomat vol 25 no 2 pp 59ndash78 2011

[46] M Imdad and J Ali ldquoCommon fixed point theorems in sym-metric spaces employing a new implicit function and common

property (EA)rdquo Bulletin of the Belgian Mathematical Societyvol 16 no 3 pp 421ndash433 2009

[47] M Imdad J Ali and L Khan ldquoCoincidence and fixed pointsin symmetric spaces under strict contractionsrdquo Journal ofMathematical Analysis and Applications vol 320 no 1 pp 352ndash360 2006

[48] D Turkoglu and I Altun ldquoA common fixed point theorem forweakly compatible mappings in symmetric spaces satisfying animplicit relationrdquo Sociedad Matematica Mexicana vol 13 no 1pp 195ndash205 2007

[49] M Imdad A Sharma Chauhan and S ldquoUnifying a multitudeof metrical fixed point theorems in symmetric spacesrdquo FilomatIn press

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

119883 rarr 119883 where (119883 119889) is metric space (or symmetric space)the corresponding sets take the form

1198725

119860119861119878119879

(119909 119910) = 119889 (119878119909 119879119910) 119889 (119878119909 119860119909) 119889 (119879119910 119861119910)

119889 (119878119909 119861119910) 119889 (119879119910 119860119909)

1198724

119860119861119878119879

(119909 119910) = 119889 (119878119909 119879119910) 119889 (119878119909 119860119909) 119889 (119879119910 119861119910)

1

2(119889 (119878119909 119861119910) + 119889 (119879119910 119860119909))

1198723

119860119861119878119879

(119909 119910) = 119889 (119878119909 119879119910) 1

2(119889 (119878119909 119860119909) + 119889 (119879119910 119861119910))

1

2(119889 (119878119909 119861119910) + 119889 (119879119910 119860119909))

(7)

In this case (5) is usually replaced by

119889 (119860119909 119861119910) le 119865 (119898 (119909 119910)) (8)

where119898(119909 119910) is the maximum of one of the119872-setsSimilarly we can define the 119872-sets for six mappings

119860 119861119867 119877 119878 119879 119883 rarr 119883 where (119883 119889) is metric space (orsymmetric space) as

1198725

119860119861119867119877119878119879

(119909 119910) = 119889 (119878119877119909 119879119867119910) 119889 (119878119877119909 119860119909)

119889 (119879119867119910 119861119910)

119889 (119878119877119909 119861119910) 119889 (119879119867119910119860119909)

1198724

119860119861119867119877119878119879

(119909 119910) = 119889 (119878119877119909 119879119867119910)

119889 (119878119877119909 119860119909) 119889 (119879119867119910 119861119910)

1

2(119889 (119878119877119909 119861119910) + 119889 (119879119867119910119860119909))

1198723

119860119861119867119877119878119879

(119909 119910) = 119889 (119878119877119909 119879119867119910)

1

2(119889 (119878119877119909 119860119909) + 119889 (119879119867119910 119861119910))

1

2(119889 (119878119877119909 119861119910) + 119889 (119879119867119910119860119909))

(9)

and the contractive condition is again in the form (8)By using different arguments of control functions Rade-

novic et al [38] proved some common fixed point resultsfor two and three mappings by using (120595 120593)-weak contractiveconditions and improved several known metrical fixed pointtheorems Motivated by these results we prove some com-mon fixed point theorems for two pairs of weakly compatiblemappings with common limit range property satisfyinggeneralized (120595 120593)-weak contractive conditionsMany knownfixed point results are improved especially the ones provedin [38] and also contained in the references cited therein We

also obtain a fixed point theorem for four finite families ofself-mappings Some related results are also derived besidesfurnishing illustrative examples

The following definitions and results will be needed in thesequel

A symmetric on a set119883 is a function 119889 119883times119883 rarr [0infin)

satisfying the following conditions

(1) 119889(119909 119910) = 0 if and only if 119909 = 119910 for 119909 119910 isin 119883(2) 119889(119909 119910) = 119889(119910 119909) for all 119909 119910 isin 119883

Let 119889 be a symmetric on a set 119883 For 119909 isin 119883 and 120598 gt 0let B(119909 120598) = 119910 isin 119883 119889(119909 119910) lt 120598 A topology 120591(119889) on119883 is defined as follows 119880 isin 120591(119889) if and only if for each119909 isin 119880 there exists an 120598 gt 0 such that B(119909 120598) sub 119880 Asubset 119878 of 119883 is a neighbourhood of 119909 isin 119883 if there exists119880 isin 120591(119889) such that 119909 isin 119880 sub 119878 A symmetric 119889 is asemimetric if for each 119909 isin 119883 and each 120598 gt 0 B(119909 120598)

is a neighbourhood of 119909 in the topology 120591(119889) A symmet-ric (resp semimetric) space (119883 119889) is a topological spacewhose topology 120591(119889) on 119883 is induced by symmetric (respsemimetric) 119889 The difference of a symmetric and a metriccomes from the triangle inequality Since a symmetric spaceis not essentially Hausdorff therefore in order to provefixed point theorems some additional axioms are requiredThe following axioms which are available in Wilson [39]Aliouche [40] and Imdad et al [17] are relevant to thispresentation

From now on symmetric space will be denoted by (119883 119889)whereas a nonempty arbitrary set will be denoted by 119884

(1198823

) Given 119909119899

119909 and 119910 in 119883 lim119899rarrinfin

119889(119909119899

119909) = 0 andlim119899rarrinfin

119889(119909119899

119910) = 0 imply 119909 = 119910 [39](1198824

) Given 119909119899

119910119899

and 119909 in 119883 lim119899rarrinfin

119889(119909119899

119909) = 0

and lim119899rarrinfin

119889(119909119899

119910119899

) = 0 imply lim119899rarrinfin

119889(119910119899

119909) = 0

[39](119867119864) Given 119909

119899

119910119899

and 119909 in 119883 lim119899rarrinfin

119889(119909119899

119909) = 0

and lim119899rarrinfin

119889(119910119899

119909) = 0 imply lim119899rarrinfin

119889(119909119899

119910119899

) = 0

[40](1119862) A symmetric 119889 is said to be 1-continuous if

lim119899rarrinfin

119889(119909119899

119909) = 0 implies lim119899rarrinfin

119889(119909119899

119910) =

119889(119909 119910) where 119909119899

is a sequence in 119883 and 119909 119910 isin 119883

[41](119862119862) A symmetric 119889 is said to be continuous if

lim119899rarrinfin

119889(119909119899

119909) = 0 and lim119899rarrinfin

119889(119910119899

119910) = 0

imply lim119899rarrinfin

119889(119909119899

119910119899

) = 119889(119909 119910) where 119909119899

and119910119899

are sequences in119883 and 119909 119910 isin 119883 [41]

Here it is observed that (119862119862) rArr (1119862) (1198824

) rArr (1198823

)and (1119862) rArr (119882

3

) but the converse implications are nottrue In general all other possible implications amongst (119882

3

)(1119862) and (119867119864) are not true For detailed description we referan interesting note of Cho et al [42] which contained someillustrative examples However (119862119862) implies all the remain-ing four conditions namely (119882

3

) (1198824

) (119867119864) and (1119862)Employing these axioms several authors proved commonfixed point theorems in the framework of symmetric spaces(see [22 43ndash48])

4 Abstract and Applied Analysis

Definition 2 Let (119860 119878) be a pair of self-mappings defined ona nonempty set 119883 equipped with a symmetric 119889 Then themappings 119860 and 119878 are said to be

(1) commuting if 119860119878119909 = 119878119860119909 for all 119909 isin 119883(2) compatible [3] if lim

119899rarrinfin

119889(119860119878119909119899

119878119860119909119899

) = 0 foreach sequence 119909

119899

in 119884 such that lim119899rarrinfin

119860119909119899

=

lim119899rarrinfin

119878119909119899

(3) noncompatible [2] if there exists a sequence 119909

119899

in 119883 such that lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

butlim119899rarrinfin

119889(119860119878119909119899

119878119860119909119899

) is either nonzero or nonexis-tent

(4) weakly compatible [5] if they commute at their coinci-dence points that is 119860119878119909 = 119878119860119909 whenever 119860119909 = 119878119909for some 119909 isin 119883

(5) satisfied the property (EA) [14] if there exists asequence 119909

119899

in 119883 such that lim119899rarrinfin

119860119909119899

=

lim119899rarrinfin

119878119909119899

= 119911 for some 119911 isin 119883

Any pair of compatible as well as noncompatible self-mappings satisfies the property (EA) but a pair of mappingssatisfying the property (EA) needs not be noncompatible

Definition 3 (see [16]) Let 119884 be an arbitrary set and let 119883 bea nonempty set equipped with symmetric 119889 Then the pairs(119860 119878) and (119861 119879) of mappings from 119884 into119883 are said to sharethe common property (EA) if there exist two sequences 119909

119899

and 119910119899

in 119884 such that

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= lim119899rarrinfin

119861119910119899

= lim119899rarrinfin

119879119910119899

= 119911 (10)

for some 119911 isin 119883

Definition 4 (see [20]) Let 119884 be an arbitrary set and let 119883be a nonempty set equipped with symmetric 119889 Then the pair(119860 119878) of mappings from 119884 into119883 is said to have the commonlimit range property with respect to the mapping 119878 (denotedby (CLR

119878

)) if there exists a sequence 119909119899

in 119884 such that

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= 119911 (11)

where 119911 isin 119878(119884)

Definition 5 (see [21]) Let 119884 be an arbitrary set and let 119883 bea nonempty set equipped with symmetric 119889 Then the pairs(119860 119878) and (119861 119879) of mappings from 119884 into 119883 are said to havethe common limit range property with respect to mappings 119878and 119879 denoted by (CLR

119878119879

) if there exist two sequences 119909119899

and 119910119899

in 119884 such that

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= lim119899rarrinfin

119861119910119899

= lim119899rarrinfin

119879119910119899

= 119911 (12)

where 119911 isin 119878(119884) cap 119879(119884)

Remark 6 It is clear that (CLR119878119879

) property implies thecommon property (EA) but the converse is not true (see [49Example 1])

Definition 7 (see [37]) Two families of self-mappings 119860119894

119898

119894=1

and 119878119896

119899

119896=1

are said to be pairwise commuting if

(1) 119860119894

119860119895

= 119860119895

119860119894

for all 119894 119895 isin 1 2 119898(2) 119878119896

119878119897

= 119878119897

119878119896

for all 119896 119897 isin 1 2 119899(3) 119860119894

119878119896

= 119878119896

119860119894

for all 119894 isin 1 2 119898 and 119896 isin

1 2 119899

3 Results

Now we state and prove our main results for four mappingsemploying the common limit range property in symmetricspaces Firstly we prove the following lemma

Lemma 8 Let (119883 119889) be a symmetric space wherein 119889 satisfiesthe conditions (119862119862) whereas 119884 is an arbitrary nonempty setwith 119860 119861 119878 and 119879 119884 rarr 119883 Suppose that

(1) the pair (119860 119878) (or (119861 119879)) satisfies the (119862119871119877119878

) (or(119862119871119877119879

)) property(2) 119860(119884) sub 119879(119884) (or 119861(119884) sub 119878(119884))(3) 119879(119884) (or 119878(119884)) is a closed subset of119883(4) 119861119910

119899

converges for every sequence 119910119899

in 119884 whenever119879119910119899

converges (or 119860119909119899

converges for every sequence119909119899

in 119884 whenever 119878119909119899

converges)(5) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905) 119889119905) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(13)

where

119898(119909 119910) = max1198725119860119861119878119879

(119909 119910) (14)

and 120601 [0 +infin) rarr [0 +infin) is a Lebesgue-integrable mappingwhich is summable and nonnegative such that

int120598

0

120601 (119905) 119889119905 gt 0 (15)

for all 120598 gt 0Then the pairs (119860 119878) and (119861 119879) satisfy the (CLR

119878119879

)

property

Proof First we show that the conclusion of this theoremholds for first case Since the pair (119860 119878) enjoys the (CLR

119878

)

property therefore there exists a sequence 119909119899

in119884 such that

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= 119911 (16)

where 119911 isin 119878(119884) Since 119860(119884) sub 119879(119884) hence for each sequence119909119899

there exists a sequence 119910119899

in 119884 such that 119860119909119899

= 119879119910119899

Therefore by closedness of 119879(119884)

lim119899rarrinfin

119879119910119899

= lim119899rarrinfin

119860119909119899

= 119911 (17)

for 119911 isin 119879(119884) and in all 119911 isin 119878(119884) cap 119879(119884) Thus in all we have119860119909119899

rarr 119911 119878119909119899

rarr 119911 and 119879119910119899

rarr 119911 as 119899 rarr infin Since by

Abstract and Applied Analysis 5

(4) 119861119910119899

converges in all we need to show that 119861119910119899

rarr 119911

as 119899 rarr infin Assume this contrary we get 119861119910119899

rarr 119905( = 119911) as119899 rarr infin Now using inequality (13) with = 119909

119899

119910 = 119910119899

wehave

120595(int119889(119860119909

119899119861119910

119899)

0

120601 (119905) 119889119905)

le 120595(int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905) minus 120593(int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

(18)

where

119898(119909119899

119910119899

) = max 119889 (119878119909119899

119879119910119899

) 119889 (119878119909119899

119860119909119899

) 119889 (119879119910119899

119861119910119899

)

119889 (119878119909119899

119861119910119899

) 119889 (119879119910119899

119860119909119899

)

(19)

Taking limit as 119899 rarr infin and using property (119862119862) ininequality (18) we get

lim119899rarrinfin

120595(int119889(119860119909

119899119861119910

119899)

0

120601 (119905) 119889119905)

le lim119899rarrinfin

120595(int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

minus lim119899rarrinfin

120593(int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

(20)

that is

120595(int119889(119911119905)

0

120601 (119905) 119889119905)

= 120595( lim119899rarrinfin

int119889(119860119909

119899119861119910

119899)

0

120601 (119905) 119889119905)

le 120595( lim119899rarrinfin

int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

minus 120593( lim119899rarrinfin

int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

(21)

where

lim119899rarrinfin

119898(119909119899

119910119899

)

= max 119889 (119911 119911) 119889 (119911 119911) 119889 (119911 119905) 119889 (119911 119905) 119889 (119911 119911)

= max 0 0 119889 (119911 119905) 119889 (119911 119905) 0

= 119889 (119911 119905)

(22)

Hence inequality (21) implies

120595(int119889(119911119905)

0

120601 (119905) 119889119905)

le 120595(int119889(119911119905)

0

120601 (119905) 119889119905) minus 120593(int119889(119911119905)

0

120601 (119905) 119889119905)

(23)

that is 120593(int119889(119911119905)0

120601(119905)119889119905) le 0 and so 120593(int119889(119911119905)0

120601(119905)119889119905) = 0 andby the property of the function 120593 we have 119889(119911 119905) = 0 orequivalently 119911 = 119905 which contradicts the hypothesis 119905 = 119911Hence both the pairs (119860 119878) and (119861 119879) satisfy the (CLR

119878119879

)

propertyIn the second case it similar to the first case So in order

to avoid repetition the details of the proof are omitted Thiscompletes the proof

Theorem9 Let (119883 119889) be a symmetric spacewherein119889 satisfiesthe conditions (1119862) and (119867119864) whereas 119884 is an arbitrarynonempty set with 119860 119861 119878 119879 119884 rarr 119883 which satisfy theinequalities (13) and (15) of Lemma 8 Suppose that the pairs(119860 119878) and (119861 119879) satisfy the (119862119871119877

119878119879

) propertyThen (119860 119878) and(119861 119879) have a coincidence point each Moreover if 119884 = 119883 then119860 119861 119878 and119879 have a unique common fixed point provided boththe pairs (119860 119878) and (119861 119879) are weakly compatible

Proof Since the pairs (119860 119878) and (119861 119879) enjoy the (CLR119878119879

)

property there exist two sequences 119909119899

and 119910119899

in 119884 suchthat

lim119899rarrinfin

119889 (119860119909119899

119911) = lim119899rarrinfin

119889 (119878119909119899

119911) = lim119899rarrinfin

119889 (119861119910119899

119911)

= lim119899rarrinfin

119889 (119879119910119899

119911) = 0(24)

where 119911 isin 119878(119884) cap 119879(119884) It follows from 119911 isin 119878(119884) that thereexists a point119908 isin 119884 such that 119878119908 = 119911 We assert that119860119908 = 119911If not then using inequality (13) with 119909 = 119908 and 119910 = 119910

119899

oneobtains

120595(int119889(119860119908119861119910

119899)

0

120601 (119905) 119889119905)

le 120595(int119898(119908119910

119899)

0

120601 (119905) 119889119905) minus 120593(int119898(119908119910

119899)

0

120601 (119905) 119889119905)

(25)

where

119898(119908 119910119899

) =max 119889 (119878119908 119879119910119899

) 119889 (119878119908 119860119908) 119889 (119879119910119899

119861119910119899

)

119889 (119878119908 119861119910119899

) 119889 (119879119910119899

119860119908)

(26)

Taking limit as 119899 rarr infin and using properties (1119862) and (119867119864)in inequality (25) one obtains

lim119899rarrinfin

120595(int119889(119860119908119861119910

119899)

0

120601 (119905) 119889119905)

le lim119899rarrinfin

120595(int119898(119908119910

119899)

0

120601 (119905) 119889119905)

minus lim119899rarrinfin

120593(int119898(119908119910

119899)

0

120601 (119905) 119889119905)

(27)

6 Abstract and Applied Analysis

that is

120595(int119889(119860119908119911)

0

120601 (119905) 119889119905)

= 120595( lim119899rarrinfin

int119889(119860119908119861119910

119899)

0

120601 (119905) 119889119905)

le 120595( lim119899rarrinfin

int119898(119908119910

119899)

0

120601 (119905) 119889119905)

minus 120593( lim119899rarrinfin

int119898(119908119910

119899)

0

120601 (119905) 119889119905)

(28)

wherelim119899rarrinfin

119898(119908 119910119899

)

= max 119889 (119911 119911) 119889 (119911 119860119908) 119889 (119911 119911) 119889 (119911 119911) 119889 (119911 119860119908)

= max 0 119889 (119911 119860119908) 0 0 119889 (119911 119860119908)

= 119889 (119911 119860119908)

(29)

From inequality (28) one gets

120595(int119889(119860119908119911)

0

120601 (119905) 119889119905)

le 120595(int119889(119860119908119911)

0

120601 (119905)) minus 120593(int119889(119860119908119911)

0

120601 (119905))

(30)

so that 120593(int119889(119860119908119911)0

120601(119905)119889119905) = 0 that is 119889(119860119908 119911) = 0 Hence119860119908 = 119878119908 = 119911 which shows that 119908 is a coincidence point ofthe pair (119860 119878)

Also 119911 isin 119879(119884) there exists a point V isin 119884 such that119879V = 119911We assert that 119861V = 119911 If not then using inequality (13) with119909 = 119908 119910 = V we have

120595(int119889(119911119861V)

0

120601 (119905) 119889119905)

= 120595(int119889(119860119908119861V)

0

120601 (119905) 119889119905)

le 120595(int119898(119908V)

0

120601 (119905) 119889119905) minus 120593(int119898(119908V)

0

120601 (119905) 119889119905)

(31)

where119898(119908 V)

= max 119889 (119878119908 119879V) 119889 (119878119908 119860119908) 119889 (119879V 119861V)

119889 (119878119908 119861V) 119889 (119879V 119860119908)

= max 119889 (119911 119911) 119889 (119911 119911) 119889 (119911 119861V) 119889 (119911 119861V) 119889 (119911 119911)

= max 0 0 119889 (119911 119861V) 119889 (119911 119861V) 0

= 119889 (119911 119861V) (32)

Hence inequality (31) implies

120595(int119889(119911119861V)

0

120601 (119905) 119889119905)

le 120595(int119889(119911119861V)

0

120601 (119905) 119889119905) minus 120593(int119889(119911119861V)

0

120601 (119905) 119889119905)

(33)

so that 120593(int119889(119911119861V)0

120601(119905)119889119905) = 0 that is 119889(119911 119861V) = 0 Therefore119911 = 119861V = 119879V which shows that V is a coincidence point of thepair (119861 119879)

Consider 119884 = 119883 Since the pair (119860 119878) is weakly compat-ible and 119860119908 = 119878119908 hence 119860119911 = 119860119878119908 = 119878119860119908 = 119878119911 Nowwe assert that 119911 is a common fixed point of the pair (119860 119878) Toaccomplish this using inequality (13) with 119909 = 119911 and 119910 = Vone gets

120595(int119889(119860119911119911)

0

120601 (119905) 119889119905)

= 120595(int119889(119860119911119861V)

0

120601 (119905) 119889119905)

le 120595(int119898(119911V)

0

120601 (119905) 119889119905) minus 120593(int119898(119911V)

0

120601 (119905) 119889119905)

(34)

where119898(119911 V)

= max 119889 (119878119911 119879V) 119889 (119878119911 119860119911) 119889 (119879V 119861V)

119889 (119878119911 119861V) 119889 (119879V 119860119911)

= max 119889 (119860119911 119911) 119889 (119860119911 119860119911) 119889 (119911 119911)

119889 (119860119911 119911) 119889 (119911 119860119911)

= 119889 (119860119911 119911)

(35)

From inequality (34) we have

120595(int119889(119860119911119911)

0

120601 (119905) 119889119905)

le 120595(int119889(119860119911119911)

0

120601 (119905) 119889119905) minus 120593(int119889(119860119911119911)

0

120601 (119905) 119889119905)

(36)

so that 120593(int119889(119860119911119911)0

120601(119905)119889119905) = 0 that is 119889(119860119911 119911) = 0 Hence wehave119860119911 = 119911 = 119878119911which shows that 119911 is a commonfixed pointof the pair (119860 119878)

Also the pair (119861 119879) is weakly compatible and 119861V = 119879Vhence 119861119911 = 119861119879V = 119879119861V = 119879119911 Using inequality (13) with119909 = 119908 and 119910 = 119911 we have

120595(int119889(119911119861119911)

0

120601 (119905) 119889119905)

= 120595(int119889(119860119908119861119911)

0

120601 (119905) 119889119905)

le 120595(int119898(119908119911)

0

120601 (119905) 119889119905) minus 120593(int119898(119908119911)

0

120601 (119905) 119889119905)

(37)

Abstract and Applied Analysis 7

where

119898(119908 119911) = max 119889 (119878119908 119879119911) 119889 (119878119908 119860119908) 119889 (119879119911 119861119911)

119889 (119878119908 119861119911) 119889 (119879119911 119860119908)

= max 119889 (119911 119861119911) 119889 (119911 119911) 119889 (119861119911 119861119911)

119889 (119911 119861119911) 119889 (119861119911 119911)

= 119889 (119911 119861119911)

(38)

Hence inequality (37) implies

120595(int119889(119911119861119911)

0

120601 (119905) 119889119905)

le 120595(int119889(119911119861119911)

0

120601 (119905) 119889119905) minus 120593(int119889(119911119861119911)

0

120601 (119905) 119889119905)

(39)

so that 120593(int119889(119911119861119911)0

120601(119905)119889119905) = 0 that is 119889(119911 119861119911) = 0 Therefore119861119911 = 119911 = 119879119911 which shows that 119911 is a common fixed pointof the pair (119861 119879) Hence 119911 is a common fixed point of thepairs (119860 119878) and (119861 119879) Uniqueness of common fixed point isan easy consequence of the inequality (13)This concludes theproof

Now we furnish an illustrative example which demon-strates the validity of the hypotheses and degree of generalityof Theorem 9

Example 10 Consider 119883 = 119884 = [2 11) equipped with thesymmetric 119889(119909 119910) = (119909 minus 119910)

2 for all 119909 119910 isin 119883 which alsosatisfies (1119862) and (119867119864) Define the mappings 119860 119861 119878 and 119879

by

119860119909 = 2 if 119909 isin 2 cup (5 11)

5 if 119909 isin (2 5]

119861119909 = 2 if 119909 isin 2 cup (5 11)

4 if 119909 isin (2 5]

119878119909 =

2 if 119909 = 2

6 if 119909 isin (2 5] 3119909 + 1

8 if 119909 isin (5 11)

119879119909 =

2 if 119909 = 2

8 if 119909 isin (2 5]

119909 minus 3 if 119909 isin (5 11)

(40)

Then 119860(119883) = 2 5 119861(119883) = 2 4 119879(119883) = [2 8] and 119878(119883) =[2 174) cup 6 Now consider the sequences 119909

119899

= 5 + 1119899

and 119910119899

= 2 Then

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= lim119899rarrinfin

119861119910119899

= lim119899rarrinfin

119879119910119899

= 2 isin 119878 (119883) cap 119879 (119883)

(41)

that is both the pairs (119860 119878) and (119861 119879) satisfy the (CLR119878119879

)

property

Let Lebesgue-integrable 120601 [0 +infin) rarr [0 +infin) definedby 120601(119905) = 119890

119905 Take 120595 isin Ψ and 120593 isin Φ given by 120595(119905) = 2119905 and120593(119905) = (27)119905 In order to check the contractive condition (13)consider the following nine cases

(i) 119909 = 119910 = 2

(ii) 119909 = 2 119910 isin (2 5]

(iii) 119909 = 2 119910 isin (5 11)

(iv) 119909 isin (2 5] 119910 = 2

(v) 119909 119910 isin (2 5]

(vi) 119909 isin (2 5] 119910 isin (5 11)

(vii) 119909 isin (5 11) 119910 = 2

(viii) 119909 isin (5 11) 119910 isin (2 5]

(ix) 119909 119910 isin (5 11)

In the cases (i) (iii) (vii) and (ix) we get that119889(119860119909 119861119910) =0 and (13) is trivially satisfied

In the cases (ii) and (viii) we have 119889(119860119909 119861119910) = 4 and119898(119909 119910) = 36 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int4

0

119890119905

119889119905)

= 120595 (1198904

minus 1)

= 2 (1198904

minus 1)

le12

7(11989036

minus 1)

= 2 (11989036

minus 1) minus2

7(11989036

minus 1)

= 120595(int36

0

119890119905

119889119905) minus 120593(int36

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(42)

Therefore we get the fact that (13) holds

8 Abstract and Applied Analysis

In the case (iv) we get 119889(119860119909 119861119910) = 9 and 119898(119909 119910) = 16Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int9

0

119890119905

119889119905)

= 120595 (1198909

minus 1)

= 2 (1198909

minus 1)

le12

7(11989016

minus 1)

= 2 (11989016

minus 1) minus2

7(11989016

minus 1)

= 120595(int16

0

119890119905

119889119905) minus 120593(int16

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(43)

This implies that (13) holdsIn the case (vi) we have 119889(119860119909 119861119910) = 9 and 119889(119878119909 119861119910) =

16 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int9

0

119890119905

119889119905)

= 120595 (1198909

minus 1)

= 2 (1198909

minus 1)

le12

7(11989016

minus 1)

=12

7(119890119889(119878119909119861119910)

minus 1)

le12

7(119890119898(119909119910)

minus 1)

= 2 (119890119898(119909119910)

minus 1) minus2

7(119890119898(119909119910)

minus 1)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(44)

This shows that (13) holds

Finally in the case (v) we obtain 119889(119860119909 119861119910) = 1 and119898(119909 119910) = 16 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int1

0

119890119905

119889119905)

= 120595 (119890 minus 1)

= 2 (119890 minus 1)

le12

7(11989016

minus 1)

= 2 (11989016

minus 1) minus2

7(11989016

minus 1)

= 120595(int16

0

119890119905

119889119905) minus 120593(int16

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(45)

that is the contractive condition (13) holdsHence all the conditions ofTheorem 9 are satisfied and 2

is a unique common fixed point of the pairs (119860 119878) and (119861 119879)which also remains a point of coincidence as well Here onemay notice that all the involved mappings are discontinuousat their unique common fixed point 2

Corollary 11 Let (119883 119889) be a symmetric space wherein 119889

satisfies the condition (119862119862)whereas119884 is an arbitrary nonemptyset with 119860 119861 119878 119879 119884 rarr 119883 satisfying all the hypothesesof Lemma 8 Then (119860 119878) and (119861 119879) have a coincidence pointeach Moreover if 119884 = 119883 then 119860 119861 119878 and 119879 have a uniquecommon fixed point provided both the pairs (119860 119878) and (119861 119879)are weakly compatible

Proof Owing to Lemma 8 it follows that the pairs (119860 119878) and(119861 119879) enjoy the (CLR

119878119879

) property Hence all the conditionsof Theorem 9 are satisfied and 119860 119861 119878 and 119879 have a uniquecommon fixed point provided both the pairs (119860 119878) and (119861 119879)are weakly compatible

Remark 12 The conclusions of Lemma 8 Theorem 9and Corollary 11 remain true if we choose 119898(119909 119910) =

max1198724119860119861119878119879

(119909 119910) or119898(119909 119910) = max1198723119860119861119878119879

(119909 119910)

Our next result shows the importance of common limitrange property over common property (EA)

Theorem 13 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861 119878 119879 119884 rarr 119883 which satisfy theinequalities (13) and (15) of Lemma 8 Suppose that

(1) the pairs (119860 119878) and (119861 119879) satisfy the common property(EA)

(2) 119878(119884) and 119879(119884) are closed subsets of119883

Abstract and Applied Analysis 9

Then (119860 119878) and (119861 119879) have a coincidence point eachMoreover if 119884 = 119883 then119860 119861 119878 and 119879 have a unique commonfixed point provided both the pairs (119860 119878) and (119861 119879) are weaklycompatible

Proof Since the pairs (119860 119878) and (119861 119879) satisfy the commonproperty (EA) there exist two sequences 119909

119899

and 119910119899

in 119884such that

lim119899rarrinfin

119889 (119860119909119899

119911) = lim119899rarrinfin

119889 (119878119909119899

119911) = lim119899rarrinfin

119889 (119861119910119899

119911)

= lim119899rarrinfin

119889 (119879119910119899

119911) = 0(46)

for some 119911 isin 119883 Since 119878(119884) and 119879(119884) are closed subsets of119883therefore 119911 isin 119878(119884)cap119879(119884) Since 119911 isin 119878(119884) there exists a point119908 isin 119884 such that 119878119908 = 119911 Also since 119911 isin 119879(119884) there exists apoint V isin 119883 such that 119879V = 119911 The rest of the proof runs onthe lines of the proof of Theorem 9

Remark 14 The conclusions of Theorem 13 remain true ifcondition (2) of Theorem 13 is replaced by one of thefollowing conditions

(2)1015840

119860(119884) sub 119879(119884) and 119861(119884) sub 119878(119884) where 119860(119884) and 119861(119884)denote the closure of ranges of themappings119860 and 119861

(2)10158401015840

119860(119884) and 119861(119884) are closed subsets of 119883 provided119860(119884) sub 119879(119884) and 119861(119884) sub 119878(119884)

By setting119860119861 119878 and119879 suitably we can deduce corollariesinvolving two and three self-mappings As a sample we candeduce the following corollary involving two self-mappings

Corollary 15 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119878 119884 rarr 119883 Suppose that

(1) the pair (119860 119878) satisfies the (119862119871119877119878

) property(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860119909119860119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905) 119889119905) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(47)

where 119898(119909 119910) = max119872119896119860119878

(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Then (119860 119878) has a coincidence point Moreover if 119884 = 119883then119860 and 119878 have a unique common fixed point in119883 providedthe pair (119860 119878) is weakly compatible

As an application of Theorem 9 we have the followingresult involving four finite families of self-mappings

Theorem 16 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860

119894

119898

119894=1

119861119895

119899

119895=1

119878119903

119901

119903=1

119879119897

119902

119897=1

119884 rarr

119883 satisfying the inequalities (13) and (15) of Lemma 8 where119860 = 119860

1

1198602

sdot sdot sdot 119860119898

119861 = 1198611

1198612

sdot sdot sdot 119861119899

119878 = 1198781

1198782

sdot sdot sdot 119878119901

and119879 = 119879

1

1198792

sdot sdot sdot 119879119902

Suppose that the pairs (119860 119878) and (119861 119879) satisfythe (119862119871119877

119878119879

) property Then (119860 119878) and (119861 119879) have a point ofcoincidence each

Moreover if 119884 = 119883 then 119860119894

119898

119894=1

119861119895

119899

119895=1

119878119903

119901

119903=1

and119879119897

119902

119897=1

have a unique common fixed point provided the families(119860119894

119878119903

) and (119861119895

119879119897

) commute pairwise where 119894 isin

1 2 119898 119903 isin 1 2 119901 119895 isin 1 2 119899 and 119897 isin

1 2 119902

Now we indicate that Theorem 16 can be utilized toderive common fixed point theorems for any finite number ofmappings As a sample we can derive a common fixed pointtheorem for six mappings by setting two families of twomembers while setting the rest two of single members

Corollary 17 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861119867 119877 119878 119879 119884 rarr 119883 Suppose that

(1) the pairs (119860 119878119877) and (119861 119879119867) share the (119862119871119877(119878119877)(119879119867)

)

property(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905)) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(48)

where 119898(119909 119910) = max119872119896119860119861119867119877119878119879

(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Then (119860 119878119877) and (119861 119879119867) have a coincidence point eachMoreover if 119884 = 119883 then 119860 119861 119867 119877 119878 and 119879 have a uniquecommon fixed point provided 119860119878 = 119878119860 119860119877 = 119877119860 119878119877 = 119877119878119861119879 = 119879119861 119861119867 = 119867119861 and 119879119867 = 119867119879

By setting 1198601

= 1198602

= sdot sdot sdot = 119860119898

= 119860 1198611

= 1198612

= sdot sdot sdot =

119861119899

= 119861 1198781

= 1198782

= sdot sdot sdot = 119878119901

= 119878 and 1198791

= 1198792

= sdot sdot sdot = 119879119902

= 119879

in Theorem 16 one gets the following corollary

Corollary 18 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861 119878 119879 119884 rarr 119883 Suppose that

(1) the pairs (119860119898 119878119901) and (119861119899 119879119902) share the (119862119871119877(119878

119901119879

119902)

)

property where119898 119899 119901 119902 are fixed positive integers(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860

119898

119909119861

119899

119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905) 119889119905) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(49)

10 Abstract and Applied Analysis

where 119898(119909 119910) = max119872119896119860

119898119861

119899119878

119901119879

119902(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Moreover if 119884 = 119883 then 119860 119861 119878 and 119879 have a uniquecommon fixed point provided 119860119878 = 119878119860 and 119861119879 = 119879119861

Remark 19 The above Corollary 18 is a slight but partial gen-eralization of Theorem 9 as the commutativity requirements(ie 119860119878 = 119878119860 and 119861119879 = 119879119861) in this corollary are relativelystronger as compared to weak compatibility in Theorem 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The first author gratefully acknowledges the support fromthe Deanship of Scientific Research (DSR) at King AbdulazizUniversity (KAU) during this research

References

[1] T L Hicks and B E Rhoades ldquoFixed point theory in symmetricspaces with applications to probabilistic spacesrdquo NonlinearAnalysis ATheory andMethods vol 36 no 3 pp 331ndash344 1999

[2] R P Pant ldquoNoncompatible mappings and common fixedpointsrdquo Soochow Journal of Mathematics vol 26 no 1 pp 29ndash35 2000

[3] G Jungck ldquoCompatible mappings and common fixed pointsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 9 no 4 pp 771ndash779 1986

[4] S Sessa ldquoOn a weak commutativity condition of mappings infixed point considerationsrdquo Institut Mathematique vol 32(46)pp 149ndash153 1982

[5] G Jungck and B E Rhoades ldquoFixed points for set valued func-tions without continuityrdquo Indian Journal of Pure and AppliedMathematics vol 29 no 3 pp 227ndash238 1998

[6] S L Singh and A Tomar ldquoWeaker forms of commuting mapsand existence of fixed pointsrdquo Journal of the Korea Society ofMathematical Education B vol 10 no 3 pp 145ndash161 2003

[7] P P Murthy ldquoImportant tools and possible applications ofmetric fixed point theoryrdquo Nonlinear Analysis Theory Methodsamp Applications vol 47 no 53479 3490 pages 2001

[8] M A Al-Thagafi andN Shahzad ldquoGeneralized 119868-nonexpansiveselfmaps and invariant approximationsrdquo Acta MathematicaSinica vol 24 no 5 pp 867ndash876 2008

[9] M Abbas and B E Rhoades ldquoCommon fixed point theoremsfor occasionally weakly compatible mappings satisfying a gen-eralized contractive conditionrdquoMathematical Communicationsvol 13 no 2 pp 295ndash301 2008

[10] A Aliouche and V Popa ldquoCommon fixed point theorems foroccasionally weakly compatible mappings via implicit rela-tionsrdquo Filomat vol 22 no 2 pp 99ndash107 2008

[11] A Bhatt H Chandra and D R Sahu ldquoCommon fixed pointtheorems for occasionally weakly compatible mappings underrelaxed conditionsrdquo Nonlinear Analysis A Theory and Methodsvol 73 no 1 pp 176ndash182 2010

[12] M Imdad S Chauhan Z Kadelburg andCVetro ldquoFixed pointtheorems for non-self mappings in symmetric spaces under 120593-weak contractive conditions and an application to functionalequations in dynamic programmingrdquo Applied Mathematics andComputation vol 227 pp 469ndash479 2014

[13] D ETHoric Z Kadelburg and S Radenovic ldquoA note on occasion-ally weakly compatible mappings and common fixed pointsrdquoFixed Point Theory vol 13 no 2 pp 475ndash479 2012

[14] M Aamri and D El Moutawakil ldquoSome new common fixedpoint theorems under strict contractive conditionsrdquo Journal ofMathematical Analysis and Applications vol 270 no 1 pp 181ndash188 2002

[15] K P R Sastry and I S R Krishna Murthy ldquoCommon fixedpoints of two partially commuting tangential selfmaps on amet-ric spacerdquo Journal of Mathematical Analysis and Applicationsvol 250 no 2 pp 731ndash734 2000

[16] Y Liu J Wu and Z Li ldquoCommon fixed points of single-valuedand multivalued mapsrdquo International Journal of Mathematicsand Mathematical Sciences no 19 pp 3045ndash3055 2005

[17] M Imdad S Chauhan and Z Kadelburg ldquoFixed point theoremfor mappings with common limit range property satisfyinggeneralized (120595 120593)-weak contractive conditionsrdquo MathematicalSciences vol 7 article 16 2013

[18] A H SolimanM Imdad andMHasan ldquoProving unified com-mon fixed point theorems via common property (EA) in sym-metric spacesrdquo Korean Mathematical Society vol 25 no 4 pp629ndash645 2010

[19] R P Pant ldquoCommon fixed points of Lipschitz type mappingpairsrdquo Journal of Mathematical Analysis and Applications vol240 no 1 pp 280ndash283 1999

[20] W Sintunavarat andPKumam ldquoCommonfixed point theoremsfor a pair of weakly compatible mappings in fuzzy metricspacesrdquo Journal of Applied Mathematics vol 2011 Article ID637958 14 pages 2011

[21] M Imdad B D Pant and S Chauhan ldquoFixed point theoremsinMenger spaces using the (119862119871119877

119878119879

) property and applicationsrdquoJournal of Nonlinear Analysis and Optimization Theory andApplications vol 3 no 2 pp 225ndash237 2012

[22] E Karapınar D K Patel M Imdad and D Gopal ldquoSomenonunique common fixed point theorems in symmetric spacesthrough 119862119871119877

(119878119879)

propertyrdquo International Journal of Mathemat-ics and Mathematical Sciences Article ID 753965 8 pages 2013

[23] D Gopal R P Pant and A S Ranadive ldquoCommon fixed pointof absorbingmapsrdquo BulletinMarathwadaMathematical Societyvol 9 no 1 pp 43ndash48 2008

[24] Ya I Alber and S Guerre-Delabriere ldquoPrinciple of weaklycontractive maps in Hilbert spacesrdquo in New Results in OperatorTheory and its Applications vol 98 ofOperatorTheory Advancesand Applications pp 7ndash22 1997

[25] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis vol 47 pp 2683ndash2693 2001

[26] A Branciari ldquoA fixed point theorem for mappings satisfyinga general contractive condition of integral typerdquo InternationalJournal of Mathematics and Mathematical Sciences vol 29 no9 pp 531ndash536 2002

[27] A Djoudi and A Aliouche ldquoCommon fixed point theoremsof Gregus type for weakly compatible mappings satisfyingcontractive conditions of integral typerdquo Journal ofMathematicalAnalysis and Applications vol 329 no 1 pp 31ndash45 2007

[28] Z Liu X Li S M Kang and S Y Cho ldquoFixed point theoremsfor mappings satisfying contractive conditions of integral type

Abstract and Applied Analysis 11

and applicationsrdquo Fixed Point Theory and Applications vol 64article 18 2011

[29] B Samet and C Vetro ldquoAn integral version of Cirics fixed pointtheoremrdquo Mediterranean Journal of Mathematics vol 9 no 1pp 225ndash238 2012

[30] W Sintunavarat and P Kumam ldquoGregus-type common fixedpoint theorems for tangential multivalued mappings of integraltype inmetric spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2011 Article ID 923458 12 pages2011

[31] W Sintunavarat and P Kumam ldquoGregus type fixed points for atangential multi-valued mappings satisfying contractive condi-tions of integral typerdquo Journal of Inequalities and Applicationsvol 2011 article 3 2011

[32] T Suzuki ldquoMeir-Keeler contractions of integral type are stillMeir-Keeler contractionsrdquo International Journal ofMathematicsand Mathematical Sciences vol 2007 Article ID 39281 6 pages2007

[33] C Vetro S Chauhan E Karapnar and W Shatanawi ldquoFixedpoints of weakly compatible mappings satisfying generalized120593-weak contractionsrdquo Bulletin of the Malaysian MathematicalSciences Society In press

[34] Q Zhang and Y Song ldquoFixed point theory for generalized 120593-weak contractionsrdquo Applied Mathematics Letters vol 22 no 1pp 75ndash78 2009

[35] D ETHoric ldquoCommon fixed point for generalized (120595 120593)-weakcontractionsrdquo Applied Mathematics Letters vol 22 no 12 pp1896ndash1900 2009

[36] P N Dutta and B S Choudhury ldquoA generalisation of con-traction principle in metric spacesrdquo Fixed Point Theory andApplications Article ID 406368 8 pages 2008

[37] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[38] S Radenovic Z Kadelburg D Jandrlic and A Jandrlic ldquoSomeresults on weakly contractive mapsrdquo Iranian MathematicalSociety Bulletin vol 38 no 3 pp 625ndash645 2012

[39] W A Wilson ldquoOn semi-metric spacesrdquo American Journal ofMathematics vol 53 no 2 pp 361ndash373 1931

[40] A Aliouche ldquoA common fixed point theorem for weakly com-patible mappings in symmetric spaces satisfying a contractivecondition of integral typerdquo Journal ofMathematical Analysis andApplications vol 322 no 2 pp 796ndash802 2006

[41] F Galvin and S D Shore ldquoCompleteness in semimetric spacesrdquoPacific Journal of Mathematics vol 113 no 1 pp 67ndash75 1984

[42] S-H Cho G-Y Lee and J-S Bae ldquoOn coincidence and fixed-point theorems in symmetric spacesrdquo Fixed Point Theory andApplications Article ID 562130 9 pages 2008

[43] M Aamri and D El Moutawakil ldquoCommon fixed points undercontractive conditions in symmetric spacesrdquo Applied Math-ematics E-Notes vol 3 pp 156ndash162 2003

[44] DGopalMHasan andM Imdad ldquoAbsorbing pairs facilitatingcommon fixed point theorems for Lipschitzian type mappingsin symmetric spacesrdquo Korean Mathematical Society Communi-cations vol 27 no 2 pp 385ndash397 2012

[45] D Gopal M Imdad and C Vetro ldquoCommon fixed pointtheorems for mappings satisfying common property (EA) insymmetric spacesrdquo Filomat vol 25 no 2 pp 59ndash78 2011

[46] M Imdad and J Ali ldquoCommon fixed point theorems in sym-metric spaces employing a new implicit function and common

property (EA)rdquo Bulletin of the Belgian Mathematical Societyvol 16 no 3 pp 421ndash433 2009

[47] M Imdad J Ali and L Khan ldquoCoincidence and fixed pointsin symmetric spaces under strict contractionsrdquo Journal ofMathematical Analysis and Applications vol 320 no 1 pp 352ndash360 2006

[48] D Turkoglu and I Altun ldquoA common fixed point theorem forweakly compatible mappings in symmetric spaces satisfying animplicit relationrdquo Sociedad Matematica Mexicana vol 13 no 1pp 195ndash205 2007

[49] M Imdad A Sharma Chauhan and S ldquoUnifying a multitudeof metrical fixed point theorems in symmetric spacesrdquo FilomatIn press

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

Definition 2 Let (119860 119878) be a pair of self-mappings defined ona nonempty set 119883 equipped with a symmetric 119889 Then themappings 119860 and 119878 are said to be

(1) commuting if 119860119878119909 = 119878119860119909 for all 119909 isin 119883(2) compatible [3] if lim

119899rarrinfin

119889(119860119878119909119899

119878119860119909119899

) = 0 foreach sequence 119909

119899

in 119884 such that lim119899rarrinfin

119860119909119899

=

lim119899rarrinfin

119878119909119899

(3) noncompatible [2] if there exists a sequence 119909

119899

in 119883 such that lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

butlim119899rarrinfin

119889(119860119878119909119899

119878119860119909119899

) is either nonzero or nonexis-tent

(4) weakly compatible [5] if they commute at their coinci-dence points that is 119860119878119909 = 119878119860119909 whenever 119860119909 = 119878119909for some 119909 isin 119883

(5) satisfied the property (EA) [14] if there exists asequence 119909

119899

in 119883 such that lim119899rarrinfin

119860119909119899

=

lim119899rarrinfin

119878119909119899

= 119911 for some 119911 isin 119883

Any pair of compatible as well as noncompatible self-mappings satisfies the property (EA) but a pair of mappingssatisfying the property (EA) needs not be noncompatible

Definition 3 (see [16]) Let 119884 be an arbitrary set and let 119883 bea nonempty set equipped with symmetric 119889 Then the pairs(119860 119878) and (119861 119879) of mappings from 119884 into119883 are said to sharethe common property (EA) if there exist two sequences 119909

119899

and 119910119899

in 119884 such that

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= lim119899rarrinfin

119861119910119899

= lim119899rarrinfin

119879119910119899

= 119911 (10)

for some 119911 isin 119883

Definition 4 (see [20]) Let 119884 be an arbitrary set and let 119883be a nonempty set equipped with symmetric 119889 Then the pair(119860 119878) of mappings from 119884 into119883 is said to have the commonlimit range property with respect to the mapping 119878 (denotedby (CLR

119878

)) if there exists a sequence 119909119899

in 119884 such that

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= 119911 (11)

where 119911 isin 119878(119884)

Definition 5 (see [21]) Let 119884 be an arbitrary set and let 119883 bea nonempty set equipped with symmetric 119889 Then the pairs(119860 119878) and (119861 119879) of mappings from 119884 into 119883 are said to havethe common limit range property with respect to mappings 119878and 119879 denoted by (CLR

119878119879

) if there exist two sequences 119909119899

and 119910119899

in 119884 such that

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= lim119899rarrinfin

119861119910119899

= lim119899rarrinfin

119879119910119899

= 119911 (12)

where 119911 isin 119878(119884) cap 119879(119884)

Remark 6 It is clear that (CLR119878119879

) property implies thecommon property (EA) but the converse is not true (see [49Example 1])

Definition 7 (see [37]) Two families of self-mappings 119860119894

119898

119894=1

and 119878119896

119899

119896=1

are said to be pairwise commuting if

(1) 119860119894

119860119895

= 119860119895

119860119894

for all 119894 119895 isin 1 2 119898(2) 119878119896

119878119897

= 119878119897

119878119896

for all 119896 119897 isin 1 2 119899(3) 119860119894

119878119896

= 119878119896

119860119894

for all 119894 isin 1 2 119898 and 119896 isin

1 2 119899

3 Results

Now we state and prove our main results for four mappingsemploying the common limit range property in symmetricspaces Firstly we prove the following lemma

Lemma 8 Let (119883 119889) be a symmetric space wherein 119889 satisfiesthe conditions (119862119862) whereas 119884 is an arbitrary nonempty setwith 119860 119861 119878 and 119879 119884 rarr 119883 Suppose that

(1) the pair (119860 119878) (or (119861 119879)) satisfies the (119862119871119877119878

) (or(119862119871119877119879

)) property(2) 119860(119884) sub 119879(119884) (or 119861(119884) sub 119878(119884))(3) 119879(119884) (or 119878(119884)) is a closed subset of119883(4) 119861119910

119899

converges for every sequence 119910119899

in 119884 whenever119879119910119899

converges (or 119860119909119899

converges for every sequence119909119899

in 119884 whenever 119878119909119899

converges)(5) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905) 119889119905) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(13)

where

119898(119909 119910) = max1198725119860119861119878119879

(119909 119910) (14)

and 120601 [0 +infin) rarr [0 +infin) is a Lebesgue-integrable mappingwhich is summable and nonnegative such that

int120598

0

120601 (119905) 119889119905 gt 0 (15)

for all 120598 gt 0Then the pairs (119860 119878) and (119861 119879) satisfy the (CLR

119878119879

)

property

Proof First we show that the conclusion of this theoremholds for first case Since the pair (119860 119878) enjoys the (CLR

119878

)

property therefore there exists a sequence 119909119899

in119884 such that

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= 119911 (16)

where 119911 isin 119878(119884) Since 119860(119884) sub 119879(119884) hence for each sequence119909119899

there exists a sequence 119910119899

in 119884 such that 119860119909119899

= 119879119910119899

Therefore by closedness of 119879(119884)

lim119899rarrinfin

119879119910119899

= lim119899rarrinfin

119860119909119899

= 119911 (17)

for 119911 isin 119879(119884) and in all 119911 isin 119878(119884) cap 119879(119884) Thus in all we have119860119909119899

rarr 119911 119878119909119899

rarr 119911 and 119879119910119899

rarr 119911 as 119899 rarr infin Since by

Abstract and Applied Analysis 5

(4) 119861119910119899

converges in all we need to show that 119861119910119899

rarr 119911

as 119899 rarr infin Assume this contrary we get 119861119910119899

rarr 119905( = 119911) as119899 rarr infin Now using inequality (13) with = 119909

119899

119910 = 119910119899

wehave

120595(int119889(119860119909

119899119861119910

119899)

0

120601 (119905) 119889119905)

le 120595(int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905) minus 120593(int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

(18)

where

119898(119909119899

119910119899

) = max 119889 (119878119909119899

119879119910119899

) 119889 (119878119909119899

119860119909119899

) 119889 (119879119910119899

119861119910119899

)

119889 (119878119909119899

119861119910119899

) 119889 (119879119910119899

119860119909119899

)

(19)

Taking limit as 119899 rarr infin and using property (119862119862) ininequality (18) we get

lim119899rarrinfin

120595(int119889(119860119909

119899119861119910

119899)

0

120601 (119905) 119889119905)

le lim119899rarrinfin

120595(int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

minus lim119899rarrinfin

120593(int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

(20)

that is

120595(int119889(119911119905)

0

120601 (119905) 119889119905)

= 120595( lim119899rarrinfin

int119889(119860119909

119899119861119910

119899)

0

120601 (119905) 119889119905)

le 120595( lim119899rarrinfin

int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

minus 120593( lim119899rarrinfin

int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

(21)

where

lim119899rarrinfin

119898(119909119899

119910119899

)

= max 119889 (119911 119911) 119889 (119911 119911) 119889 (119911 119905) 119889 (119911 119905) 119889 (119911 119911)

= max 0 0 119889 (119911 119905) 119889 (119911 119905) 0

= 119889 (119911 119905)

(22)

Hence inequality (21) implies

120595(int119889(119911119905)

0

120601 (119905) 119889119905)

le 120595(int119889(119911119905)

0

120601 (119905) 119889119905) minus 120593(int119889(119911119905)

0

120601 (119905) 119889119905)

(23)

that is 120593(int119889(119911119905)0

120601(119905)119889119905) le 0 and so 120593(int119889(119911119905)0

120601(119905)119889119905) = 0 andby the property of the function 120593 we have 119889(119911 119905) = 0 orequivalently 119911 = 119905 which contradicts the hypothesis 119905 = 119911Hence both the pairs (119860 119878) and (119861 119879) satisfy the (CLR

119878119879

)

propertyIn the second case it similar to the first case So in order

to avoid repetition the details of the proof are omitted Thiscompletes the proof

Theorem9 Let (119883 119889) be a symmetric spacewherein119889 satisfiesthe conditions (1119862) and (119867119864) whereas 119884 is an arbitrarynonempty set with 119860 119861 119878 119879 119884 rarr 119883 which satisfy theinequalities (13) and (15) of Lemma 8 Suppose that the pairs(119860 119878) and (119861 119879) satisfy the (119862119871119877

119878119879

) propertyThen (119860 119878) and(119861 119879) have a coincidence point each Moreover if 119884 = 119883 then119860 119861 119878 and119879 have a unique common fixed point provided boththe pairs (119860 119878) and (119861 119879) are weakly compatible

Proof Since the pairs (119860 119878) and (119861 119879) enjoy the (CLR119878119879

)

property there exist two sequences 119909119899

and 119910119899

in 119884 suchthat

lim119899rarrinfin

119889 (119860119909119899

119911) = lim119899rarrinfin

119889 (119878119909119899

119911) = lim119899rarrinfin

119889 (119861119910119899

119911)

= lim119899rarrinfin

119889 (119879119910119899

119911) = 0(24)

where 119911 isin 119878(119884) cap 119879(119884) It follows from 119911 isin 119878(119884) that thereexists a point119908 isin 119884 such that 119878119908 = 119911 We assert that119860119908 = 119911If not then using inequality (13) with 119909 = 119908 and 119910 = 119910

119899

oneobtains

120595(int119889(119860119908119861119910

119899)

0

120601 (119905) 119889119905)

le 120595(int119898(119908119910

119899)

0

120601 (119905) 119889119905) minus 120593(int119898(119908119910

119899)

0

120601 (119905) 119889119905)

(25)

where

119898(119908 119910119899

) =max 119889 (119878119908 119879119910119899

) 119889 (119878119908 119860119908) 119889 (119879119910119899

119861119910119899

)

119889 (119878119908 119861119910119899

) 119889 (119879119910119899

119860119908)

(26)

Taking limit as 119899 rarr infin and using properties (1119862) and (119867119864)in inequality (25) one obtains

lim119899rarrinfin

120595(int119889(119860119908119861119910

119899)

0

120601 (119905) 119889119905)

le lim119899rarrinfin

120595(int119898(119908119910

119899)

0

120601 (119905) 119889119905)

minus lim119899rarrinfin

120593(int119898(119908119910

119899)

0

120601 (119905) 119889119905)

(27)

6 Abstract and Applied Analysis

that is

120595(int119889(119860119908119911)

0

120601 (119905) 119889119905)

= 120595( lim119899rarrinfin

int119889(119860119908119861119910

119899)

0

120601 (119905) 119889119905)

le 120595( lim119899rarrinfin

int119898(119908119910

119899)

0

120601 (119905) 119889119905)

minus 120593( lim119899rarrinfin

int119898(119908119910

119899)

0

120601 (119905) 119889119905)

(28)

wherelim119899rarrinfin

119898(119908 119910119899

)

= max 119889 (119911 119911) 119889 (119911 119860119908) 119889 (119911 119911) 119889 (119911 119911) 119889 (119911 119860119908)

= max 0 119889 (119911 119860119908) 0 0 119889 (119911 119860119908)

= 119889 (119911 119860119908)

(29)

From inequality (28) one gets

120595(int119889(119860119908119911)

0

120601 (119905) 119889119905)

le 120595(int119889(119860119908119911)

0

120601 (119905)) minus 120593(int119889(119860119908119911)

0

120601 (119905))

(30)

so that 120593(int119889(119860119908119911)0

120601(119905)119889119905) = 0 that is 119889(119860119908 119911) = 0 Hence119860119908 = 119878119908 = 119911 which shows that 119908 is a coincidence point ofthe pair (119860 119878)

Also 119911 isin 119879(119884) there exists a point V isin 119884 such that119879V = 119911We assert that 119861V = 119911 If not then using inequality (13) with119909 = 119908 119910 = V we have

120595(int119889(119911119861V)

0

120601 (119905) 119889119905)

= 120595(int119889(119860119908119861V)

0

120601 (119905) 119889119905)

le 120595(int119898(119908V)

0

120601 (119905) 119889119905) minus 120593(int119898(119908V)

0

120601 (119905) 119889119905)

(31)

where119898(119908 V)

= max 119889 (119878119908 119879V) 119889 (119878119908 119860119908) 119889 (119879V 119861V)

119889 (119878119908 119861V) 119889 (119879V 119860119908)

= max 119889 (119911 119911) 119889 (119911 119911) 119889 (119911 119861V) 119889 (119911 119861V) 119889 (119911 119911)

= max 0 0 119889 (119911 119861V) 119889 (119911 119861V) 0

= 119889 (119911 119861V) (32)

Hence inequality (31) implies

120595(int119889(119911119861V)

0

120601 (119905) 119889119905)

le 120595(int119889(119911119861V)

0

120601 (119905) 119889119905) minus 120593(int119889(119911119861V)

0

120601 (119905) 119889119905)

(33)

so that 120593(int119889(119911119861V)0

120601(119905)119889119905) = 0 that is 119889(119911 119861V) = 0 Therefore119911 = 119861V = 119879V which shows that V is a coincidence point of thepair (119861 119879)

Consider 119884 = 119883 Since the pair (119860 119878) is weakly compat-ible and 119860119908 = 119878119908 hence 119860119911 = 119860119878119908 = 119878119860119908 = 119878119911 Nowwe assert that 119911 is a common fixed point of the pair (119860 119878) Toaccomplish this using inequality (13) with 119909 = 119911 and 119910 = Vone gets

120595(int119889(119860119911119911)

0

120601 (119905) 119889119905)

= 120595(int119889(119860119911119861V)

0

120601 (119905) 119889119905)

le 120595(int119898(119911V)

0

120601 (119905) 119889119905) minus 120593(int119898(119911V)

0

120601 (119905) 119889119905)

(34)

where119898(119911 V)

= max 119889 (119878119911 119879V) 119889 (119878119911 119860119911) 119889 (119879V 119861V)

119889 (119878119911 119861V) 119889 (119879V 119860119911)

= max 119889 (119860119911 119911) 119889 (119860119911 119860119911) 119889 (119911 119911)

119889 (119860119911 119911) 119889 (119911 119860119911)

= 119889 (119860119911 119911)

(35)

From inequality (34) we have

120595(int119889(119860119911119911)

0

120601 (119905) 119889119905)

le 120595(int119889(119860119911119911)

0

120601 (119905) 119889119905) minus 120593(int119889(119860119911119911)

0

120601 (119905) 119889119905)

(36)

so that 120593(int119889(119860119911119911)0

120601(119905)119889119905) = 0 that is 119889(119860119911 119911) = 0 Hence wehave119860119911 = 119911 = 119878119911which shows that 119911 is a commonfixed pointof the pair (119860 119878)

Also the pair (119861 119879) is weakly compatible and 119861V = 119879Vhence 119861119911 = 119861119879V = 119879119861V = 119879119911 Using inequality (13) with119909 = 119908 and 119910 = 119911 we have

120595(int119889(119911119861119911)

0

120601 (119905) 119889119905)

= 120595(int119889(119860119908119861119911)

0

120601 (119905) 119889119905)

le 120595(int119898(119908119911)

0

120601 (119905) 119889119905) minus 120593(int119898(119908119911)

0

120601 (119905) 119889119905)

(37)

Abstract and Applied Analysis 7

where

119898(119908 119911) = max 119889 (119878119908 119879119911) 119889 (119878119908 119860119908) 119889 (119879119911 119861119911)

119889 (119878119908 119861119911) 119889 (119879119911 119860119908)

= max 119889 (119911 119861119911) 119889 (119911 119911) 119889 (119861119911 119861119911)

119889 (119911 119861119911) 119889 (119861119911 119911)

= 119889 (119911 119861119911)

(38)

Hence inequality (37) implies

120595(int119889(119911119861119911)

0

120601 (119905) 119889119905)

le 120595(int119889(119911119861119911)

0

120601 (119905) 119889119905) minus 120593(int119889(119911119861119911)

0

120601 (119905) 119889119905)

(39)

so that 120593(int119889(119911119861119911)0

120601(119905)119889119905) = 0 that is 119889(119911 119861119911) = 0 Therefore119861119911 = 119911 = 119879119911 which shows that 119911 is a common fixed pointof the pair (119861 119879) Hence 119911 is a common fixed point of thepairs (119860 119878) and (119861 119879) Uniqueness of common fixed point isan easy consequence of the inequality (13)This concludes theproof

Now we furnish an illustrative example which demon-strates the validity of the hypotheses and degree of generalityof Theorem 9

Example 10 Consider 119883 = 119884 = [2 11) equipped with thesymmetric 119889(119909 119910) = (119909 minus 119910)

2 for all 119909 119910 isin 119883 which alsosatisfies (1119862) and (119867119864) Define the mappings 119860 119861 119878 and 119879

by

119860119909 = 2 if 119909 isin 2 cup (5 11)

5 if 119909 isin (2 5]

119861119909 = 2 if 119909 isin 2 cup (5 11)

4 if 119909 isin (2 5]

119878119909 =

2 if 119909 = 2

6 if 119909 isin (2 5] 3119909 + 1

8 if 119909 isin (5 11)

119879119909 =

2 if 119909 = 2

8 if 119909 isin (2 5]

119909 minus 3 if 119909 isin (5 11)

(40)

Then 119860(119883) = 2 5 119861(119883) = 2 4 119879(119883) = [2 8] and 119878(119883) =[2 174) cup 6 Now consider the sequences 119909

119899

= 5 + 1119899

and 119910119899

= 2 Then

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= lim119899rarrinfin

119861119910119899

= lim119899rarrinfin

119879119910119899

= 2 isin 119878 (119883) cap 119879 (119883)

(41)

that is both the pairs (119860 119878) and (119861 119879) satisfy the (CLR119878119879

)

property

Let Lebesgue-integrable 120601 [0 +infin) rarr [0 +infin) definedby 120601(119905) = 119890

119905 Take 120595 isin Ψ and 120593 isin Φ given by 120595(119905) = 2119905 and120593(119905) = (27)119905 In order to check the contractive condition (13)consider the following nine cases

(i) 119909 = 119910 = 2

(ii) 119909 = 2 119910 isin (2 5]

(iii) 119909 = 2 119910 isin (5 11)

(iv) 119909 isin (2 5] 119910 = 2

(v) 119909 119910 isin (2 5]

(vi) 119909 isin (2 5] 119910 isin (5 11)

(vii) 119909 isin (5 11) 119910 = 2

(viii) 119909 isin (5 11) 119910 isin (2 5]

(ix) 119909 119910 isin (5 11)

In the cases (i) (iii) (vii) and (ix) we get that119889(119860119909 119861119910) =0 and (13) is trivially satisfied

In the cases (ii) and (viii) we have 119889(119860119909 119861119910) = 4 and119898(119909 119910) = 36 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int4

0

119890119905

119889119905)

= 120595 (1198904

minus 1)

= 2 (1198904

minus 1)

le12

7(11989036

minus 1)

= 2 (11989036

minus 1) minus2

7(11989036

minus 1)

= 120595(int36

0

119890119905

119889119905) minus 120593(int36

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(42)

Therefore we get the fact that (13) holds

8 Abstract and Applied Analysis

In the case (iv) we get 119889(119860119909 119861119910) = 9 and 119898(119909 119910) = 16Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int9

0

119890119905

119889119905)

= 120595 (1198909

minus 1)

= 2 (1198909

minus 1)

le12

7(11989016

minus 1)

= 2 (11989016

minus 1) minus2

7(11989016

minus 1)

= 120595(int16

0

119890119905

119889119905) minus 120593(int16

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(43)

This implies that (13) holdsIn the case (vi) we have 119889(119860119909 119861119910) = 9 and 119889(119878119909 119861119910) =

16 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int9

0

119890119905

119889119905)

= 120595 (1198909

minus 1)

= 2 (1198909

minus 1)

le12

7(11989016

minus 1)

=12

7(119890119889(119878119909119861119910)

minus 1)

le12

7(119890119898(119909119910)

minus 1)

= 2 (119890119898(119909119910)

minus 1) minus2

7(119890119898(119909119910)

minus 1)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(44)

This shows that (13) holds

Finally in the case (v) we obtain 119889(119860119909 119861119910) = 1 and119898(119909 119910) = 16 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int1

0

119890119905

119889119905)

= 120595 (119890 minus 1)

= 2 (119890 minus 1)

le12

7(11989016

minus 1)

= 2 (11989016

minus 1) minus2

7(11989016

minus 1)

= 120595(int16

0

119890119905

119889119905) minus 120593(int16

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(45)

that is the contractive condition (13) holdsHence all the conditions ofTheorem 9 are satisfied and 2

is a unique common fixed point of the pairs (119860 119878) and (119861 119879)which also remains a point of coincidence as well Here onemay notice that all the involved mappings are discontinuousat their unique common fixed point 2

Corollary 11 Let (119883 119889) be a symmetric space wherein 119889

satisfies the condition (119862119862)whereas119884 is an arbitrary nonemptyset with 119860 119861 119878 119879 119884 rarr 119883 satisfying all the hypothesesof Lemma 8 Then (119860 119878) and (119861 119879) have a coincidence pointeach Moreover if 119884 = 119883 then 119860 119861 119878 and 119879 have a uniquecommon fixed point provided both the pairs (119860 119878) and (119861 119879)are weakly compatible

Proof Owing to Lemma 8 it follows that the pairs (119860 119878) and(119861 119879) enjoy the (CLR

119878119879

) property Hence all the conditionsof Theorem 9 are satisfied and 119860 119861 119878 and 119879 have a uniquecommon fixed point provided both the pairs (119860 119878) and (119861 119879)are weakly compatible

Remark 12 The conclusions of Lemma 8 Theorem 9and Corollary 11 remain true if we choose 119898(119909 119910) =

max1198724119860119861119878119879

(119909 119910) or119898(119909 119910) = max1198723119860119861119878119879

(119909 119910)

Our next result shows the importance of common limitrange property over common property (EA)

Theorem 13 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861 119878 119879 119884 rarr 119883 which satisfy theinequalities (13) and (15) of Lemma 8 Suppose that

(1) the pairs (119860 119878) and (119861 119879) satisfy the common property(EA)

(2) 119878(119884) and 119879(119884) are closed subsets of119883

Abstract and Applied Analysis 9

Then (119860 119878) and (119861 119879) have a coincidence point eachMoreover if 119884 = 119883 then119860 119861 119878 and 119879 have a unique commonfixed point provided both the pairs (119860 119878) and (119861 119879) are weaklycompatible

Proof Since the pairs (119860 119878) and (119861 119879) satisfy the commonproperty (EA) there exist two sequences 119909

119899

and 119910119899

in 119884such that

lim119899rarrinfin

119889 (119860119909119899

119911) = lim119899rarrinfin

119889 (119878119909119899

119911) = lim119899rarrinfin

119889 (119861119910119899

119911)

= lim119899rarrinfin

119889 (119879119910119899

119911) = 0(46)

for some 119911 isin 119883 Since 119878(119884) and 119879(119884) are closed subsets of119883therefore 119911 isin 119878(119884)cap119879(119884) Since 119911 isin 119878(119884) there exists a point119908 isin 119884 such that 119878119908 = 119911 Also since 119911 isin 119879(119884) there exists apoint V isin 119883 such that 119879V = 119911 The rest of the proof runs onthe lines of the proof of Theorem 9

Remark 14 The conclusions of Theorem 13 remain true ifcondition (2) of Theorem 13 is replaced by one of thefollowing conditions

(2)1015840

119860(119884) sub 119879(119884) and 119861(119884) sub 119878(119884) where 119860(119884) and 119861(119884)denote the closure of ranges of themappings119860 and 119861

(2)10158401015840

119860(119884) and 119861(119884) are closed subsets of 119883 provided119860(119884) sub 119879(119884) and 119861(119884) sub 119878(119884)

By setting119860119861 119878 and119879 suitably we can deduce corollariesinvolving two and three self-mappings As a sample we candeduce the following corollary involving two self-mappings

Corollary 15 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119878 119884 rarr 119883 Suppose that

(1) the pair (119860 119878) satisfies the (119862119871119877119878

) property(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860119909119860119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905) 119889119905) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(47)

where 119898(119909 119910) = max119872119896119860119878

(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Then (119860 119878) has a coincidence point Moreover if 119884 = 119883then119860 and 119878 have a unique common fixed point in119883 providedthe pair (119860 119878) is weakly compatible

As an application of Theorem 9 we have the followingresult involving four finite families of self-mappings

Theorem 16 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860

119894

119898

119894=1

119861119895

119899

119895=1

119878119903

119901

119903=1

119879119897

119902

119897=1

119884 rarr

119883 satisfying the inequalities (13) and (15) of Lemma 8 where119860 = 119860

1

1198602

sdot sdot sdot 119860119898

119861 = 1198611

1198612

sdot sdot sdot 119861119899

119878 = 1198781

1198782

sdot sdot sdot 119878119901

and119879 = 119879

1

1198792

sdot sdot sdot 119879119902

Suppose that the pairs (119860 119878) and (119861 119879) satisfythe (119862119871119877

119878119879

) property Then (119860 119878) and (119861 119879) have a point ofcoincidence each

Moreover if 119884 = 119883 then 119860119894

119898

119894=1

119861119895

119899

119895=1

119878119903

119901

119903=1

and119879119897

119902

119897=1

have a unique common fixed point provided the families(119860119894

119878119903

) and (119861119895

119879119897

) commute pairwise where 119894 isin

1 2 119898 119903 isin 1 2 119901 119895 isin 1 2 119899 and 119897 isin

1 2 119902

Now we indicate that Theorem 16 can be utilized toderive common fixed point theorems for any finite number ofmappings As a sample we can derive a common fixed pointtheorem for six mappings by setting two families of twomembers while setting the rest two of single members

Corollary 17 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861119867 119877 119878 119879 119884 rarr 119883 Suppose that

(1) the pairs (119860 119878119877) and (119861 119879119867) share the (119862119871119877(119878119877)(119879119867)

)

property(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905)) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(48)

where 119898(119909 119910) = max119872119896119860119861119867119877119878119879

(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Then (119860 119878119877) and (119861 119879119867) have a coincidence point eachMoreover if 119884 = 119883 then 119860 119861 119867 119877 119878 and 119879 have a uniquecommon fixed point provided 119860119878 = 119878119860 119860119877 = 119877119860 119878119877 = 119877119878119861119879 = 119879119861 119861119867 = 119867119861 and 119879119867 = 119867119879

By setting 1198601

= 1198602

= sdot sdot sdot = 119860119898

= 119860 1198611

= 1198612

= sdot sdot sdot =

119861119899

= 119861 1198781

= 1198782

= sdot sdot sdot = 119878119901

= 119878 and 1198791

= 1198792

= sdot sdot sdot = 119879119902

= 119879

in Theorem 16 one gets the following corollary

Corollary 18 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861 119878 119879 119884 rarr 119883 Suppose that

(1) the pairs (119860119898 119878119901) and (119861119899 119879119902) share the (119862119871119877(119878

119901119879

119902)

)

property where119898 119899 119901 119902 are fixed positive integers(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860

119898

119909119861

119899

119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905) 119889119905) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(49)

10 Abstract and Applied Analysis

where 119898(119909 119910) = max119872119896119860

119898119861

119899119878

119901119879

119902(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Moreover if 119884 = 119883 then 119860 119861 119878 and 119879 have a uniquecommon fixed point provided 119860119878 = 119878119860 and 119861119879 = 119879119861

Remark 19 The above Corollary 18 is a slight but partial gen-eralization of Theorem 9 as the commutativity requirements(ie 119860119878 = 119878119860 and 119861119879 = 119879119861) in this corollary are relativelystronger as compared to weak compatibility in Theorem 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The first author gratefully acknowledges the support fromthe Deanship of Scientific Research (DSR) at King AbdulazizUniversity (KAU) during this research

References

[1] T L Hicks and B E Rhoades ldquoFixed point theory in symmetricspaces with applications to probabilistic spacesrdquo NonlinearAnalysis ATheory andMethods vol 36 no 3 pp 331ndash344 1999

[2] R P Pant ldquoNoncompatible mappings and common fixedpointsrdquo Soochow Journal of Mathematics vol 26 no 1 pp 29ndash35 2000

[3] G Jungck ldquoCompatible mappings and common fixed pointsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 9 no 4 pp 771ndash779 1986

[4] S Sessa ldquoOn a weak commutativity condition of mappings infixed point considerationsrdquo Institut Mathematique vol 32(46)pp 149ndash153 1982

[5] G Jungck and B E Rhoades ldquoFixed points for set valued func-tions without continuityrdquo Indian Journal of Pure and AppliedMathematics vol 29 no 3 pp 227ndash238 1998

[6] S L Singh and A Tomar ldquoWeaker forms of commuting mapsand existence of fixed pointsrdquo Journal of the Korea Society ofMathematical Education B vol 10 no 3 pp 145ndash161 2003

[7] P P Murthy ldquoImportant tools and possible applications ofmetric fixed point theoryrdquo Nonlinear Analysis Theory Methodsamp Applications vol 47 no 53479 3490 pages 2001

[8] M A Al-Thagafi andN Shahzad ldquoGeneralized 119868-nonexpansiveselfmaps and invariant approximationsrdquo Acta MathematicaSinica vol 24 no 5 pp 867ndash876 2008

[9] M Abbas and B E Rhoades ldquoCommon fixed point theoremsfor occasionally weakly compatible mappings satisfying a gen-eralized contractive conditionrdquoMathematical Communicationsvol 13 no 2 pp 295ndash301 2008

[10] A Aliouche and V Popa ldquoCommon fixed point theorems foroccasionally weakly compatible mappings via implicit rela-tionsrdquo Filomat vol 22 no 2 pp 99ndash107 2008

[11] A Bhatt H Chandra and D R Sahu ldquoCommon fixed pointtheorems for occasionally weakly compatible mappings underrelaxed conditionsrdquo Nonlinear Analysis A Theory and Methodsvol 73 no 1 pp 176ndash182 2010

[12] M Imdad S Chauhan Z Kadelburg andCVetro ldquoFixed pointtheorems for non-self mappings in symmetric spaces under 120593-weak contractive conditions and an application to functionalequations in dynamic programmingrdquo Applied Mathematics andComputation vol 227 pp 469ndash479 2014

[13] D ETHoric Z Kadelburg and S Radenovic ldquoA note on occasion-ally weakly compatible mappings and common fixed pointsrdquoFixed Point Theory vol 13 no 2 pp 475ndash479 2012

[14] M Aamri and D El Moutawakil ldquoSome new common fixedpoint theorems under strict contractive conditionsrdquo Journal ofMathematical Analysis and Applications vol 270 no 1 pp 181ndash188 2002

[15] K P R Sastry and I S R Krishna Murthy ldquoCommon fixedpoints of two partially commuting tangential selfmaps on amet-ric spacerdquo Journal of Mathematical Analysis and Applicationsvol 250 no 2 pp 731ndash734 2000

[16] Y Liu J Wu and Z Li ldquoCommon fixed points of single-valuedand multivalued mapsrdquo International Journal of Mathematicsand Mathematical Sciences no 19 pp 3045ndash3055 2005

[17] M Imdad S Chauhan and Z Kadelburg ldquoFixed point theoremfor mappings with common limit range property satisfyinggeneralized (120595 120593)-weak contractive conditionsrdquo MathematicalSciences vol 7 article 16 2013

[18] A H SolimanM Imdad andMHasan ldquoProving unified com-mon fixed point theorems via common property (EA) in sym-metric spacesrdquo Korean Mathematical Society vol 25 no 4 pp629ndash645 2010

[19] R P Pant ldquoCommon fixed points of Lipschitz type mappingpairsrdquo Journal of Mathematical Analysis and Applications vol240 no 1 pp 280ndash283 1999

[20] W Sintunavarat andPKumam ldquoCommonfixed point theoremsfor a pair of weakly compatible mappings in fuzzy metricspacesrdquo Journal of Applied Mathematics vol 2011 Article ID637958 14 pages 2011

[21] M Imdad B D Pant and S Chauhan ldquoFixed point theoremsinMenger spaces using the (119862119871119877

119878119879

) property and applicationsrdquoJournal of Nonlinear Analysis and Optimization Theory andApplications vol 3 no 2 pp 225ndash237 2012

[22] E Karapınar D K Patel M Imdad and D Gopal ldquoSomenonunique common fixed point theorems in symmetric spacesthrough 119862119871119877

(119878119879)

propertyrdquo International Journal of Mathemat-ics and Mathematical Sciences Article ID 753965 8 pages 2013

[23] D Gopal R P Pant and A S Ranadive ldquoCommon fixed pointof absorbingmapsrdquo BulletinMarathwadaMathematical Societyvol 9 no 1 pp 43ndash48 2008

[24] Ya I Alber and S Guerre-Delabriere ldquoPrinciple of weaklycontractive maps in Hilbert spacesrdquo in New Results in OperatorTheory and its Applications vol 98 ofOperatorTheory Advancesand Applications pp 7ndash22 1997

[25] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis vol 47 pp 2683ndash2693 2001

[26] A Branciari ldquoA fixed point theorem for mappings satisfyinga general contractive condition of integral typerdquo InternationalJournal of Mathematics and Mathematical Sciences vol 29 no9 pp 531ndash536 2002

[27] A Djoudi and A Aliouche ldquoCommon fixed point theoremsof Gregus type for weakly compatible mappings satisfyingcontractive conditions of integral typerdquo Journal ofMathematicalAnalysis and Applications vol 329 no 1 pp 31ndash45 2007

[28] Z Liu X Li S M Kang and S Y Cho ldquoFixed point theoremsfor mappings satisfying contractive conditions of integral type

Abstract and Applied Analysis 11

and applicationsrdquo Fixed Point Theory and Applications vol 64article 18 2011

[29] B Samet and C Vetro ldquoAn integral version of Cirics fixed pointtheoremrdquo Mediterranean Journal of Mathematics vol 9 no 1pp 225ndash238 2012

[30] W Sintunavarat and P Kumam ldquoGregus-type common fixedpoint theorems for tangential multivalued mappings of integraltype inmetric spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2011 Article ID 923458 12 pages2011

[31] W Sintunavarat and P Kumam ldquoGregus type fixed points for atangential multi-valued mappings satisfying contractive condi-tions of integral typerdquo Journal of Inequalities and Applicationsvol 2011 article 3 2011

[32] T Suzuki ldquoMeir-Keeler contractions of integral type are stillMeir-Keeler contractionsrdquo International Journal ofMathematicsand Mathematical Sciences vol 2007 Article ID 39281 6 pages2007

[33] C Vetro S Chauhan E Karapnar and W Shatanawi ldquoFixedpoints of weakly compatible mappings satisfying generalized120593-weak contractionsrdquo Bulletin of the Malaysian MathematicalSciences Society In press

[34] Q Zhang and Y Song ldquoFixed point theory for generalized 120593-weak contractionsrdquo Applied Mathematics Letters vol 22 no 1pp 75ndash78 2009

[35] D ETHoric ldquoCommon fixed point for generalized (120595 120593)-weakcontractionsrdquo Applied Mathematics Letters vol 22 no 12 pp1896ndash1900 2009

[36] P N Dutta and B S Choudhury ldquoA generalisation of con-traction principle in metric spacesrdquo Fixed Point Theory andApplications Article ID 406368 8 pages 2008

[37] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[38] S Radenovic Z Kadelburg D Jandrlic and A Jandrlic ldquoSomeresults on weakly contractive mapsrdquo Iranian MathematicalSociety Bulletin vol 38 no 3 pp 625ndash645 2012

[39] W A Wilson ldquoOn semi-metric spacesrdquo American Journal ofMathematics vol 53 no 2 pp 361ndash373 1931

[40] A Aliouche ldquoA common fixed point theorem for weakly com-patible mappings in symmetric spaces satisfying a contractivecondition of integral typerdquo Journal ofMathematical Analysis andApplications vol 322 no 2 pp 796ndash802 2006

[41] F Galvin and S D Shore ldquoCompleteness in semimetric spacesrdquoPacific Journal of Mathematics vol 113 no 1 pp 67ndash75 1984

[42] S-H Cho G-Y Lee and J-S Bae ldquoOn coincidence and fixed-point theorems in symmetric spacesrdquo Fixed Point Theory andApplications Article ID 562130 9 pages 2008

[43] M Aamri and D El Moutawakil ldquoCommon fixed points undercontractive conditions in symmetric spacesrdquo Applied Math-ematics E-Notes vol 3 pp 156ndash162 2003

[44] DGopalMHasan andM Imdad ldquoAbsorbing pairs facilitatingcommon fixed point theorems for Lipschitzian type mappingsin symmetric spacesrdquo Korean Mathematical Society Communi-cations vol 27 no 2 pp 385ndash397 2012

[45] D Gopal M Imdad and C Vetro ldquoCommon fixed pointtheorems for mappings satisfying common property (EA) insymmetric spacesrdquo Filomat vol 25 no 2 pp 59ndash78 2011

[46] M Imdad and J Ali ldquoCommon fixed point theorems in sym-metric spaces employing a new implicit function and common

property (EA)rdquo Bulletin of the Belgian Mathematical Societyvol 16 no 3 pp 421ndash433 2009

[47] M Imdad J Ali and L Khan ldquoCoincidence and fixed pointsin symmetric spaces under strict contractionsrdquo Journal ofMathematical Analysis and Applications vol 320 no 1 pp 352ndash360 2006

[48] D Turkoglu and I Altun ldquoA common fixed point theorem forweakly compatible mappings in symmetric spaces satisfying animplicit relationrdquo Sociedad Matematica Mexicana vol 13 no 1pp 195ndash205 2007

[49] M Imdad A Sharma Chauhan and S ldquoUnifying a multitudeof metrical fixed point theorems in symmetric spacesrdquo FilomatIn press

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

(4) 119861119910119899

converges in all we need to show that 119861119910119899

rarr 119911

as 119899 rarr infin Assume this contrary we get 119861119910119899

rarr 119905( = 119911) as119899 rarr infin Now using inequality (13) with = 119909

119899

119910 = 119910119899

wehave

120595(int119889(119860119909

119899119861119910

119899)

0

120601 (119905) 119889119905)

le 120595(int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905) minus 120593(int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

(18)

where

119898(119909119899

119910119899

) = max 119889 (119878119909119899

119879119910119899

) 119889 (119878119909119899

119860119909119899

) 119889 (119879119910119899

119861119910119899

)

119889 (119878119909119899

119861119910119899

) 119889 (119879119910119899

119860119909119899

)

(19)

Taking limit as 119899 rarr infin and using property (119862119862) ininequality (18) we get

lim119899rarrinfin

120595(int119889(119860119909

119899119861119910

119899)

0

120601 (119905) 119889119905)

le lim119899rarrinfin

120595(int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

minus lim119899rarrinfin

120593(int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

(20)

that is

120595(int119889(119911119905)

0

120601 (119905) 119889119905)

= 120595( lim119899rarrinfin

int119889(119860119909

119899119861119910

119899)

0

120601 (119905) 119889119905)

le 120595( lim119899rarrinfin

int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

minus 120593( lim119899rarrinfin

int119898(119909

119899119910

119899)

0

120601 (119905) 119889119905)

(21)

where

lim119899rarrinfin

119898(119909119899

119910119899

)

= max 119889 (119911 119911) 119889 (119911 119911) 119889 (119911 119905) 119889 (119911 119905) 119889 (119911 119911)

= max 0 0 119889 (119911 119905) 119889 (119911 119905) 0

= 119889 (119911 119905)

(22)

Hence inequality (21) implies

120595(int119889(119911119905)

0

120601 (119905) 119889119905)

le 120595(int119889(119911119905)

0

120601 (119905) 119889119905) minus 120593(int119889(119911119905)

0

120601 (119905) 119889119905)

(23)

that is 120593(int119889(119911119905)0

120601(119905)119889119905) le 0 and so 120593(int119889(119911119905)0

120601(119905)119889119905) = 0 andby the property of the function 120593 we have 119889(119911 119905) = 0 orequivalently 119911 = 119905 which contradicts the hypothesis 119905 = 119911Hence both the pairs (119860 119878) and (119861 119879) satisfy the (CLR

119878119879

)

propertyIn the second case it similar to the first case So in order

to avoid repetition the details of the proof are omitted Thiscompletes the proof

Theorem9 Let (119883 119889) be a symmetric spacewherein119889 satisfiesthe conditions (1119862) and (119867119864) whereas 119884 is an arbitrarynonempty set with 119860 119861 119878 119879 119884 rarr 119883 which satisfy theinequalities (13) and (15) of Lemma 8 Suppose that the pairs(119860 119878) and (119861 119879) satisfy the (119862119871119877

119878119879

) propertyThen (119860 119878) and(119861 119879) have a coincidence point each Moreover if 119884 = 119883 then119860 119861 119878 and119879 have a unique common fixed point provided boththe pairs (119860 119878) and (119861 119879) are weakly compatible

Proof Since the pairs (119860 119878) and (119861 119879) enjoy the (CLR119878119879

)

property there exist two sequences 119909119899

and 119910119899

in 119884 suchthat

lim119899rarrinfin

119889 (119860119909119899

119911) = lim119899rarrinfin

119889 (119878119909119899

119911) = lim119899rarrinfin

119889 (119861119910119899

119911)

= lim119899rarrinfin

119889 (119879119910119899

119911) = 0(24)

where 119911 isin 119878(119884) cap 119879(119884) It follows from 119911 isin 119878(119884) that thereexists a point119908 isin 119884 such that 119878119908 = 119911 We assert that119860119908 = 119911If not then using inequality (13) with 119909 = 119908 and 119910 = 119910

119899

oneobtains

120595(int119889(119860119908119861119910

119899)

0

120601 (119905) 119889119905)

le 120595(int119898(119908119910

119899)

0

120601 (119905) 119889119905) minus 120593(int119898(119908119910

119899)

0

120601 (119905) 119889119905)

(25)

where

119898(119908 119910119899

) =max 119889 (119878119908 119879119910119899

) 119889 (119878119908 119860119908) 119889 (119879119910119899

119861119910119899

)

119889 (119878119908 119861119910119899

) 119889 (119879119910119899

119860119908)

(26)

Taking limit as 119899 rarr infin and using properties (1119862) and (119867119864)in inequality (25) one obtains

lim119899rarrinfin

120595(int119889(119860119908119861119910

119899)

0

120601 (119905) 119889119905)

le lim119899rarrinfin

120595(int119898(119908119910

119899)

0

120601 (119905) 119889119905)

minus lim119899rarrinfin

120593(int119898(119908119910

119899)

0

120601 (119905) 119889119905)

(27)

6 Abstract and Applied Analysis

that is

120595(int119889(119860119908119911)

0

120601 (119905) 119889119905)

= 120595( lim119899rarrinfin

int119889(119860119908119861119910

119899)

0

120601 (119905) 119889119905)

le 120595( lim119899rarrinfin

int119898(119908119910

119899)

0

120601 (119905) 119889119905)

minus 120593( lim119899rarrinfin

int119898(119908119910

119899)

0

120601 (119905) 119889119905)

(28)

wherelim119899rarrinfin

119898(119908 119910119899

)

= max 119889 (119911 119911) 119889 (119911 119860119908) 119889 (119911 119911) 119889 (119911 119911) 119889 (119911 119860119908)

= max 0 119889 (119911 119860119908) 0 0 119889 (119911 119860119908)

= 119889 (119911 119860119908)

(29)

From inequality (28) one gets

120595(int119889(119860119908119911)

0

120601 (119905) 119889119905)

le 120595(int119889(119860119908119911)

0

120601 (119905)) minus 120593(int119889(119860119908119911)

0

120601 (119905))

(30)

so that 120593(int119889(119860119908119911)0

120601(119905)119889119905) = 0 that is 119889(119860119908 119911) = 0 Hence119860119908 = 119878119908 = 119911 which shows that 119908 is a coincidence point ofthe pair (119860 119878)

Also 119911 isin 119879(119884) there exists a point V isin 119884 such that119879V = 119911We assert that 119861V = 119911 If not then using inequality (13) with119909 = 119908 119910 = V we have

120595(int119889(119911119861V)

0

120601 (119905) 119889119905)

= 120595(int119889(119860119908119861V)

0

120601 (119905) 119889119905)

le 120595(int119898(119908V)

0

120601 (119905) 119889119905) minus 120593(int119898(119908V)

0

120601 (119905) 119889119905)

(31)

where119898(119908 V)

= max 119889 (119878119908 119879V) 119889 (119878119908 119860119908) 119889 (119879V 119861V)

119889 (119878119908 119861V) 119889 (119879V 119860119908)

= max 119889 (119911 119911) 119889 (119911 119911) 119889 (119911 119861V) 119889 (119911 119861V) 119889 (119911 119911)

= max 0 0 119889 (119911 119861V) 119889 (119911 119861V) 0

= 119889 (119911 119861V) (32)

Hence inequality (31) implies

120595(int119889(119911119861V)

0

120601 (119905) 119889119905)

le 120595(int119889(119911119861V)

0

120601 (119905) 119889119905) minus 120593(int119889(119911119861V)

0

120601 (119905) 119889119905)

(33)

so that 120593(int119889(119911119861V)0

120601(119905)119889119905) = 0 that is 119889(119911 119861V) = 0 Therefore119911 = 119861V = 119879V which shows that V is a coincidence point of thepair (119861 119879)

Consider 119884 = 119883 Since the pair (119860 119878) is weakly compat-ible and 119860119908 = 119878119908 hence 119860119911 = 119860119878119908 = 119878119860119908 = 119878119911 Nowwe assert that 119911 is a common fixed point of the pair (119860 119878) Toaccomplish this using inequality (13) with 119909 = 119911 and 119910 = Vone gets

120595(int119889(119860119911119911)

0

120601 (119905) 119889119905)

= 120595(int119889(119860119911119861V)

0

120601 (119905) 119889119905)

le 120595(int119898(119911V)

0

120601 (119905) 119889119905) minus 120593(int119898(119911V)

0

120601 (119905) 119889119905)

(34)

where119898(119911 V)

= max 119889 (119878119911 119879V) 119889 (119878119911 119860119911) 119889 (119879V 119861V)

119889 (119878119911 119861V) 119889 (119879V 119860119911)

= max 119889 (119860119911 119911) 119889 (119860119911 119860119911) 119889 (119911 119911)

119889 (119860119911 119911) 119889 (119911 119860119911)

= 119889 (119860119911 119911)

(35)

From inequality (34) we have

120595(int119889(119860119911119911)

0

120601 (119905) 119889119905)

le 120595(int119889(119860119911119911)

0

120601 (119905) 119889119905) minus 120593(int119889(119860119911119911)

0

120601 (119905) 119889119905)

(36)

so that 120593(int119889(119860119911119911)0

120601(119905)119889119905) = 0 that is 119889(119860119911 119911) = 0 Hence wehave119860119911 = 119911 = 119878119911which shows that 119911 is a commonfixed pointof the pair (119860 119878)

Also the pair (119861 119879) is weakly compatible and 119861V = 119879Vhence 119861119911 = 119861119879V = 119879119861V = 119879119911 Using inequality (13) with119909 = 119908 and 119910 = 119911 we have

120595(int119889(119911119861119911)

0

120601 (119905) 119889119905)

= 120595(int119889(119860119908119861119911)

0

120601 (119905) 119889119905)

le 120595(int119898(119908119911)

0

120601 (119905) 119889119905) minus 120593(int119898(119908119911)

0

120601 (119905) 119889119905)

(37)

Abstract and Applied Analysis 7

where

119898(119908 119911) = max 119889 (119878119908 119879119911) 119889 (119878119908 119860119908) 119889 (119879119911 119861119911)

119889 (119878119908 119861119911) 119889 (119879119911 119860119908)

= max 119889 (119911 119861119911) 119889 (119911 119911) 119889 (119861119911 119861119911)

119889 (119911 119861119911) 119889 (119861119911 119911)

= 119889 (119911 119861119911)

(38)

Hence inequality (37) implies

120595(int119889(119911119861119911)

0

120601 (119905) 119889119905)

le 120595(int119889(119911119861119911)

0

120601 (119905) 119889119905) minus 120593(int119889(119911119861119911)

0

120601 (119905) 119889119905)

(39)

so that 120593(int119889(119911119861119911)0

120601(119905)119889119905) = 0 that is 119889(119911 119861119911) = 0 Therefore119861119911 = 119911 = 119879119911 which shows that 119911 is a common fixed pointof the pair (119861 119879) Hence 119911 is a common fixed point of thepairs (119860 119878) and (119861 119879) Uniqueness of common fixed point isan easy consequence of the inequality (13)This concludes theproof

Now we furnish an illustrative example which demon-strates the validity of the hypotheses and degree of generalityof Theorem 9

Example 10 Consider 119883 = 119884 = [2 11) equipped with thesymmetric 119889(119909 119910) = (119909 minus 119910)

2 for all 119909 119910 isin 119883 which alsosatisfies (1119862) and (119867119864) Define the mappings 119860 119861 119878 and 119879

by

119860119909 = 2 if 119909 isin 2 cup (5 11)

5 if 119909 isin (2 5]

119861119909 = 2 if 119909 isin 2 cup (5 11)

4 if 119909 isin (2 5]

119878119909 =

2 if 119909 = 2

6 if 119909 isin (2 5] 3119909 + 1

8 if 119909 isin (5 11)

119879119909 =

2 if 119909 = 2

8 if 119909 isin (2 5]

119909 minus 3 if 119909 isin (5 11)

(40)

Then 119860(119883) = 2 5 119861(119883) = 2 4 119879(119883) = [2 8] and 119878(119883) =[2 174) cup 6 Now consider the sequences 119909

119899

= 5 + 1119899

and 119910119899

= 2 Then

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= lim119899rarrinfin

119861119910119899

= lim119899rarrinfin

119879119910119899

= 2 isin 119878 (119883) cap 119879 (119883)

(41)

that is both the pairs (119860 119878) and (119861 119879) satisfy the (CLR119878119879

)

property

Let Lebesgue-integrable 120601 [0 +infin) rarr [0 +infin) definedby 120601(119905) = 119890

119905 Take 120595 isin Ψ and 120593 isin Φ given by 120595(119905) = 2119905 and120593(119905) = (27)119905 In order to check the contractive condition (13)consider the following nine cases

(i) 119909 = 119910 = 2

(ii) 119909 = 2 119910 isin (2 5]

(iii) 119909 = 2 119910 isin (5 11)

(iv) 119909 isin (2 5] 119910 = 2

(v) 119909 119910 isin (2 5]

(vi) 119909 isin (2 5] 119910 isin (5 11)

(vii) 119909 isin (5 11) 119910 = 2

(viii) 119909 isin (5 11) 119910 isin (2 5]

(ix) 119909 119910 isin (5 11)

In the cases (i) (iii) (vii) and (ix) we get that119889(119860119909 119861119910) =0 and (13) is trivially satisfied

In the cases (ii) and (viii) we have 119889(119860119909 119861119910) = 4 and119898(119909 119910) = 36 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int4

0

119890119905

119889119905)

= 120595 (1198904

minus 1)

= 2 (1198904

minus 1)

le12

7(11989036

minus 1)

= 2 (11989036

minus 1) minus2

7(11989036

minus 1)

= 120595(int36

0

119890119905

119889119905) minus 120593(int36

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(42)

Therefore we get the fact that (13) holds

8 Abstract and Applied Analysis

In the case (iv) we get 119889(119860119909 119861119910) = 9 and 119898(119909 119910) = 16Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int9

0

119890119905

119889119905)

= 120595 (1198909

minus 1)

= 2 (1198909

minus 1)

le12

7(11989016

minus 1)

= 2 (11989016

minus 1) minus2

7(11989016

minus 1)

= 120595(int16

0

119890119905

119889119905) minus 120593(int16

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(43)

This implies that (13) holdsIn the case (vi) we have 119889(119860119909 119861119910) = 9 and 119889(119878119909 119861119910) =

16 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int9

0

119890119905

119889119905)

= 120595 (1198909

minus 1)

= 2 (1198909

minus 1)

le12

7(11989016

minus 1)

=12

7(119890119889(119878119909119861119910)

minus 1)

le12

7(119890119898(119909119910)

minus 1)

= 2 (119890119898(119909119910)

minus 1) minus2

7(119890119898(119909119910)

minus 1)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(44)

This shows that (13) holds

Finally in the case (v) we obtain 119889(119860119909 119861119910) = 1 and119898(119909 119910) = 16 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int1

0

119890119905

119889119905)

= 120595 (119890 minus 1)

= 2 (119890 minus 1)

le12

7(11989016

minus 1)

= 2 (11989016

minus 1) minus2

7(11989016

minus 1)

= 120595(int16

0

119890119905

119889119905) minus 120593(int16

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(45)

that is the contractive condition (13) holdsHence all the conditions ofTheorem 9 are satisfied and 2

is a unique common fixed point of the pairs (119860 119878) and (119861 119879)which also remains a point of coincidence as well Here onemay notice that all the involved mappings are discontinuousat their unique common fixed point 2

Corollary 11 Let (119883 119889) be a symmetric space wherein 119889

satisfies the condition (119862119862)whereas119884 is an arbitrary nonemptyset with 119860 119861 119878 119879 119884 rarr 119883 satisfying all the hypothesesof Lemma 8 Then (119860 119878) and (119861 119879) have a coincidence pointeach Moreover if 119884 = 119883 then 119860 119861 119878 and 119879 have a uniquecommon fixed point provided both the pairs (119860 119878) and (119861 119879)are weakly compatible

Proof Owing to Lemma 8 it follows that the pairs (119860 119878) and(119861 119879) enjoy the (CLR

119878119879

) property Hence all the conditionsof Theorem 9 are satisfied and 119860 119861 119878 and 119879 have a uniquecommon fixed point provided both the pairs (119860 119878) and (119861 119879)are weakly compatible

Remark 12 The conclusions of Lemma 8 Theorem 9and Corollary 11 remain true if we choose 119898(119909 119910) =

max1198724119860119861119878119879

(119909 119910) or119898(119909 119910) = max1198723119860119861119878119879

(119909 119910)

Our next result shows the importance of common limitrange property over common property (EA)

Theorem 13 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861 119878 119879 119884 rarr 119883 which satisfy theinequalities (13) and (15) of Lemma 8 Suppose that

(1) the pairs (119860 119878) and (119861 119879) satisfy the common property(EA)

(2) 119878(119884) and 119879(119884) are closed subsets of119883

Abstract and Applied Analysis 9

Then (119860 119878) and (119861 119879) have a coincidence point eachMoreover if 119884 = 119883 then119860 119861 119878 and 119879 have a unique commonfixed point provided both the pairs (119860 119878) and (119861 119879) are weaklycompatible

Proof Since the pairs (119860 119878) and (119861 119879) satisfy the commonproperty (EA) there exist two sequences 119909

119899

and 119910119899

in 119884such that

lim119899rarrinfin

119889 (119860119909119899

119911) = lim119899rarrinfin

119889 (119878119909119899

119911) = lim119899rarrinfin

119889 (119861119910119899

119911)

= lim119899rarrinfin

119889 (119879119910119899

119911) = 0(46)

for some 119911 isin 119883 Since 119878(119884) and 119879(119884) are closed subsets of119883therefore 119911 isin 119878(119884)cap119879(119884) Since 119911 isin 119878(119884) there exists a point119908 isin 119884 such that 119878119908 = 119911 Also since 119911 isin 119879(119884) there exists apoint V isin 119883 such that 119879V = 119911 The rest of the proof runs onthe lines of the proof of Theorem 9

Remark 14 The conclusions of Theorem 13 remain true ifcondition (2) of Theorem 13 is replaced by one of thefollowing conditions

(2)1015840

119860(119884) sub 119879(119884) and 119861(119884) sub 119878(119884) where 119860(119884) and 119861(119884)denote the closure of ranges of themappings119860 and 119861

(2)10158401015840

119860(119884) and 119861(119884) are closed subsets of 119883 provided119860(119884) sub 119879(119884) and 119861(119884) sub 119878(119884)

By setting119860119861 119878 and119879 suitably we can deduce corollariesinvolving two and three self-mappings As a sample we candeduce the following corollary involving two self-mappings

Corollary 15 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119878 119884 rarr 119883 Suppose that

(1) the pair (119860 119878) satisfies the (119862119871119877119878

) property(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860119909119860119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905) 119889119905) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(47)

where 119898(119909 119910) = max119872119896119860119878

(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Then (119860 119878) has a coincidence point Moreover if 119884 = 119883then119860 and 119878 have a unique common fixed point in119883 providedthe pair (119860 119878) is weakly compatible

As an application of Theorem 9 we have the followingresult involving four finite families of self-mappings

Theorem 16 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860

119894

119898

119894=1

119861119895

119899

119895=1

119878119903

119901

119903=1

119879119897

119902

119897=1

119884 rarr

119883 satisfying the inequalities (13) and (15) of Lemma 8 where119860 = 119860

1

1198602

sdot sdot sdot 119860119898

119861 = 1198611

1198612

sdot sdot sdot 119861119899

119878 = 1198781

1198782

sdot sdot sdot 119878119901

and119879 = 119879

1

1198792

sdot sdot sdot 119879119902

Suppose that the pairs (119860 119878) and (119861 119879) satisfythe (119862119871119877

119878119879

) property Then (119860 119878) and (119861 119879) have a point ofcoincidence each

Moreover if 119884 = 119883 then 119860119894

119898

119894=1

119861119895

119899

119895=1

119878119903

119901

119903=1

and119879119897

119902

119897=1

have a unique common fixed point provided the families(119860119894

119878119903

) and (119861119895

119879119897

) commute pairwise where 119894 isin

1 2 119898 119903 isin 1 2 119901 119895 isin 1 2 119899 and 119897 isin

1 2 119902

Now we indicate that Theorem 16 can be utilized toderive common fixed point theorems for any finite number ofmappings As a sample we can derive a common fixed pointtheorem for six mappings by setting two families of twomembers while setting the rest two of single members

Corollary 17 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861119867 119877 119878 119879 119884 rarr 119883 Suppose that

(1) the pairs (119860 119878119877) and (119861 119879119867) share the (119862119871119877(119878119877)(119879119867)

)

property(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905)) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(48)

where 119898(119909 119910) = max119872119896119860119861119867119877119878119879

(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Then (119860 119878119877) and (119861 119879119867) have a coincidence point eachMoreover if 119884 = 119883 then 119860 119861 119867 119877 119878 and 119879 have a uniquecommon fixed point provided 119860119878 = 119878119860 119860119877 = 119877119860 119878119877 = 119877119878119861119879 = 119879119861 119861119867 = 119867119861 and 119879119867 = 119867119879

By setting 1198601

= 1198602

= sdot sdot sdot = 119860119898

= 119860 1198611

= 1198612

= sdot sdot sdot =

119861119899

= 119861 1198781

= 1198782

= sdot sdot sdot = 119878119901

= 119878 and 1198791

= 1198792

= sdot sdot sdot = 119879119902

= 119879

in Theorem 16 one gets the following corollary

Corollary 18 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861 119878 119879 119884 rarr 119883 Suppose that

(1) the pairs (119860119898 119878119901) and (119861119899 119879119902) share the (119862119871119877(119878

119901119879

119902)

)

property where119898 119899 119901 119902 are fixed positive integers(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860

119898

119909119861

119899

119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905) 119889119905) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(49)

10 Abstract and Applied Analysis

where 119898(119909 119910) = max119872119896119860

119898119861

119899119878

119901119879

119902(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Moreover if 119884 = 119883 then 119860 119861 119878 and 119879 have a uniquecommon fixed point provided 119860119878 = 119878119860 and 119861119879 = 119879119861

Remark 19 The above Corollary 18 is a slight but partial gen-eralization of Theorem 9 as the commutativity requirements(ie 119860119878 = 119878119860 and 119861119879 = 119879119861) in this corollary are relativelystronger as compared to weak compatibility in Theorem 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The first author gratefully acknowledges the support fromthe Deanship of Scientific Research (DSR) at King AbdulazizUniversity (KAU) during this research

References

[1] T L Hicks and B E Rhoades ldquoFixed point theory in symmetricspaces with applications to probabilistic spacesrdquo NonlinearAnalysis ATheory andMethods vol 36 no 3 pp 331ndash344 1999

[2] R P Pant ldquoNoncompatible mappings and common fixedpointsrdquo Soochow Journal of Mathematics vol 26 no 1 pp 29ndash35 2000

[3] G Jungck ldquoCompatible mappings and common fixed pointsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 9 no 4 pp 771ndash779 1986

[4] S Sessa ldquoOn a weak commutativity condition of mappings infixed point considerationsrdquo Institut Mathematique vol 32(46)pp 149ndash153 1982

[5] G Jungck and B E Rhoades ldquoFixed points for set valued func-tions without continuityrdquo Indian Journal of Pure and AppliedMathematics vol 29 no 3 pp 227ndash238 1998

[6] S L Singh and A Tomar ldquoWeaker forms of commuting mapsand existence of fixed pointsrdquo Journal of the Korea Society ofMathematical Education B vol 10 no 3 pp 145ndash161 2003

[7] P P Murthy ldquoImportant tools and possible applications ofmetric fixed point theoryrdquo Nonlinear Analysis Theory Methodsamp Applications vol 47 no 53479 3490 pages 2001

[8] M A Al-Thagafi andN Shahzad ldquoGeneralized 119868-nonexpansiveselfmaps and invariant approximationsrdquo Acta MathematicaSinica vol 24 no 5 pp 867ndash876 2008

[9] M Abbas and B E Rhoades ldquoCommon fixed point theoremsfor occasionally weakly compatible mappings satisfying a gen-eralized contractive conditionrdquoMathematical Communicationsvol 13 no 2 pp 295ndash301 2008

[10] A Aliouche and V Popa ldquoCommon fixed point theorems foroccasionally weakly compatible mappings via implicit rela-tionsrdquo Filomat vol 22 no 2 pp 99ndash107 2008

[11] A Bhatt H Chandra and D R Sahu ldquoCommon fixed pointtheorems for occasionally weakly compatible mappings underrelaxed conditionsrdquo Nonlinear Analysis A Theory and Methodsvol 73 no 1 pp 176ndash182 2010

[12] M Imdad S Chauhan Z Kadelburg andCVetro ldquoFixed pointtheorems for non-self mappings in symmetric spaces under 120593-weak contractive conditions and an application to functionalequations in dynamic programmingrdquo Applied Mathematics andComputation vol 227 pp 469ndash479 2014

[13] D ETHoric Z Kadelburg and S Radenovic ldquoA note on occasion-ally weakly compatible mappings and common fixed pointsrdquoFixed Point Theory vol 13 no 2 pp 475ndash479 2012

[14] M Aamri and D El Moutawakil ldquoSome new common fixedpoint theorems under strict contractive conditionsrdquo Journal ofMathematical Analysis and Applications vol 270 no 1 pp 181ndash188 2002

[15] K P R Sastry and I S R Krishna Murthy ldquoCommon fixedpoints of two partially commuting tangential selfmaps on amet-ric spacerdquo Journal of Mathematical Analysis and Applicationsvol 250 no 2 pp 731ndash734 2000

[16] Y Liu J Wu and Z Li ldquoCommon fixed points of single-valuedand multivalued mapsrdquo International Journal of Mathematicsand Mathematical Sciences no 19 pp 3045ndash3055 2005

[17] M Imdad S Chauhan and Z Kadelburg ldquoFixed point theoremfor mappings with common limit range property satisfyinggeneralized (120595 120593)-weak contractive conditionsrdquo MathematicalSciences vol 7 article 16 2013

[18] A H SolimanM Imdad andMHasan ldquoProving unified com-mon fixed point theorems via common property (EA) in sym-metric spacesrdquo Korean Mathematical Society vol 25 no 4 pp629ndash645 2010

[19] R P Pant ldquoCommon fixed points of Lipschitz type mappingpairsrdquo Journal of Mathematical Analysis and Applications vol240 no 1 pp 280ndash283 1999

[20] W Sintunavarat andPKumam ldquoCommonfixed point theoremsfor a pair of weakly compatible mappings in fuzzy metricspacesrdquo Journal of Applied Mathematics vol 2011 Article ID637958 14 pages 2011

[21] M Imdad B D Pant and S Chauhan ldquoFixed point theoremsinMenger spaces using the (119862119871119877

119878119879

) property and applicationsrdquoJournal of Nonlinear Analysis and Optimization Theory andApplications vol 3 no 2 pp 225ndash237 2012

[22] E Karapınar D K Patel M Imdad and D Gopal ldquoSomenonunique common fixed point theorems in symmetric spacesthrough 119862119871119877

(119878119879)

propertyrdquo International Journal of Mathemat-ics and Mathematical Sciences Article ID 753965 8 pages 2013

[23] D Gopal R P Pant and A S Ranadive ldquoCommon fixed pointof absorbingmapsrdquo BulletinMarathwadaMathematical Societyvol 9 no 1 pp 43ndash48 2008

[24] Ya I Alber and S Guerre-Delabriere ldquoPrinciple of weaklycontractive maps in Hilbert spacesrdquo in New Results in OperatorTheory and its Applications vol 98 ofOperatorTheory Advancesand Applications pp 7ndash22 1997

[25] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis vol 47 pp 2683ndash2693 2001

[26] A Branciari ldquoA fixed point theorem for mappings satisfyinga general contractive condition of integral typerdquo InternationalJournal of Mathematics and Mathematical Sciences vol 29 no9 pp 531ndash536 2002

[27] A Djoudi and A Aliouche ldquoCommon fixed point theoremsof Gregus type for weakly compatible mappings satisfyingcontractive conditions of integral typerdquo Journal ofMathematicalAnalysis and Applications vol 329 no 1 pp 31ndash45 2007

[28] Z Liu X Li S M Kang and S Y Cho ldquoFixed point theoremsfor mappings satisfying contractive conditions of integral type

Abstract and Applied Analysis 11

and applicationsrdquo Fixed Point Theory and Applications vol 64article 18 2011

[29] B Samet and C Vetro ldquoAn integral version of Cirics fixed pointtheoremrdquo Mediterranean Journal of Mathematics vol 9 no 1pp 225ndash238 2012

[30] W Sintunavarat and P Kumam ldquoGregus-type common fixedpoint theorems for tangential multivalued mappings of integraltype inmetric spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2011 Article ID 923458 12 pages2011

[31] W Sintunavarat and P Kumam ldquoGregus type fixed points for atangential multi-valued mappings satisfying contractive condi-tions of integral typerdquo Journal of Inequalities and Applicationsvol 2011 article 3 2011

[32] T Suzuki ldquoMeir-Keeler contractions of integral type are stillMeir-Keeler contractionsrdquo International Journal ofMathematicsand Mathematical Sciences vol 2007 Article ID 39281 6 pages2007

[33] C Vetro S Chauhan E Karapnar and W Shatanawi ldquoFixedpoints of weakly compatible mappings satisfying generalized120593-weak contractionsrdquo Bulletin of the Malaysian MathematicalSciences Society In press

[34] Q Zhang and Y Song ldquoFixed point theory for generalized 120593-weak contractionsrdquo Applied Mathematics Letters vol 22 no 1pp 75ndash78 2009

[35] D ETHoric ldquoCommon fixed point for generalized (120595 120593)-weakcontractionsrdquo Applied Mathematics Letters vol 22 no 12 pp1896ndash1900 2009

[36] P N Dutta and B S Choudhury ldquoA generalisation of con-traction principle in metric spacesrdquo Fixed Point Theory andApplications Article ID 406368 8 pages 2008

[37] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[38] S Radenovic Z Kadelburg D Jandrlic and A Jandrlic ldquoSomeresults on weakly contractive mapsrdquo Iranian MathematicalSociety Bulletin vol 38 no 3 pp 625ndash645 2012

[39] W A Wilson ldquoOn semi-metric spacesrdquo American Journal ofMathematics vol 53 no 2 pp 361ndash373 1931

[40] A Aliouche ldquoA common fixed point theorem for weakly com-patible mappings in symmetric spaces satisfying a contractivecondition of integral typerdquo Journal ofMathematical Analysis andApplications vol 322 no 2 pp 796ndash802 2006

[41] F Galvin and S D Shore ldquoCompleteness in semimetric spacesrdquoPacific Journal of Mathematics vol 113 no 1 pp 67ndash75 1984

[42] S-H Cho G-Y Lee and J-S Bae ldquoOn coincidence and fixed-point theorems in symmetric spacesrdquo Fixed Point Theory andApplications Article ID 562130 9 pages 2008

[43] M Aamri and D El Moutawakil ldquoCommon fixed points undercontractive conditions in symmetric spacesrdquo Applied Math-ematics E-Notes vol 3 pp 156ndash162 2003

[44] DGopalMHasan andM Imdad ldquoAbsorbing pairs facilitatingcommon fixed point theorems for Lipschitzian type mappingsin symmetric spacesrdquo Korean Mathematical Society Communi-cations vol 27 no 2 pp 385ndash397 2012

[45] D Gopal M Imdad and C Vetro ldquoCommon fixed pointtheorems for mappings satisfying common property (EA) insymmetric spacesrdquo Filomat vol 25 no 2 pp 59ndash78 2011

[46] M Imdad and J Ali ldquoCommon fixed point theorems in sym-metric spaces employing a new implicit function and common

property (EA)rdquo Bulletin of the Belgian Mathematical Societyvol 16 no 3 pp 421ndash433 2009

[47] M Imdad J Ali and L Khan ldquoCoincidence and fixed pointsin symmetric spaces under strict contractionsrdquo Journal ofMathematical Analysis and Applications vol 320 no 1 pp 352ndash360 2006

[48] D Turkoglu and I Altun ldquoA common fixed point theorem forweakly compatible mappings in symmetric spaces satisfying animplicit relationrdquo Sociedad Matematica Mexicana vol 13 no 1pp 195ndash205 2007

[49] M Imdad A Sharma Chauhan and S ldquoUnifying a multitudeof metrical fixed point theorems in symmetric spacesrdquo FilomatIn press

Submit your manuscripts athttpwwwhindawicom

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

that is

120595(int119889(119860119908119911)

0

120601 (119905) 119889119905)

= 120595( lim119899rarrinfin

int119889(119860119908119861119910

119899)

0

120601 (119905) 119889119905)

le 120595( lim119899rarrinfin

int119898(119908119910

119899)

0

120601 (119905) 119889119905)

minus 120593( lim119899rarrinfin

int119898(119908119910

119899)

0

120601 (119905) 119889119905)

(28)

wherelim119899rarrinfin

119898(119908 119910119899

)

= max 119889 (119911 119911) 119889 (119911 119860119908) 119889 (119911 119911) 119889 (119911 119911) 119889 (119911 119860119908)

= max 0 119889 (119911 119860119908) 0 0 119889 (119911 119860119908)

= 119889 (119911 119860119908)

(29)

From inequality (28) one gets

120595(int119889(119860119908119911)

0

120601 (119905) 119889119905)

le 120595(int119889(119860119908119911)

0

120601 (119905)) minus 120593(int119889(119860119908119911)

0

120601 (119905))

(30)

so that 120593(int119889(119860119908119911)0

120601(119905)119889119905) = 0 that is 119889(119860119908 119911) = 0 Hence119860119908 = 119878119908 = 119911 which shows that 119908 is a coincidence point ofthe pair (119860 119878)

Also 119911 isin 119879(119884) there exists a point V isin 119884 such that119879V = 119911We assert that 119861V = 119911 If not then using inequality (13) with119909 = 119908 119910 = V we have

120595(int119889(119911119861V)

0

120601 (119905) 119889119905)

= 120595(int119889(119860119908119861V)

0

120601 (119905) 119889119905)

le 120595(int119898(119908V)

0

120601 (119905) 119889119905) minus 120593(int119898(119908V)

0

120601 (119905) 119889119905)

(31)

where119898(119908 V)

= max 119889 (119878119908 119879V) 119889 (119878119908 119860119908) 119889 (119879V 119861V)

119889 (119878119908 119861V) 119889 (119879V 119860119908)

= max 119889 (119911 119911) 119889 (119911 119911) 119889 (119911 119861V) 119889 (119911 119861V) 119889 (119911 119911)

= max 0 0 119889 (119911 119861V) 119889 (119911 119861V) 0

= 119889 (119911 119861V) (32)

Hence inequality (31) implies

120595(int119889(119911119861V)

0

120601 (119905) 119889119905)

le 120595(int119889(119911119861V)

0

120601 (119905) 119889119905) minus 120593(int119889(119911119861V)

0

120601 (119905) 119889119905)

(33)

so that 120593(int119889(119911119861V)0

120601(119905)119889119905) = 0 that is 119889(119911 119861V) = 0 Therefore119911 = 119861V = 119879V which shows that V is a coincidence point of thepair (119861 119879)

Consider 119884 = 119883 Since the pair (119860 119878) is weakly compat-ible and 119860119908 = 119878119908 hence 119860119911 = 119860119878119908 = 119878119860119908 = 119878119911 Nowwe assert that 119911 is a common fixed point of the pair (119860 119878) Toaccomplish this using inequality (13) with 119909 = 119911 and 119910 = Vone gets

120595(int119889(119860119911119911)

0

120601 (119905) 119889119905)

= 120595(int119889(119860119911119861V)

0

120601 (119905) 119889119905)

le 120595(int119898(119911V)

0

120601 (119905) 119889119905) minus 120593(int119898(119911V)

0

120601 (119905) 119889119905)

(34)

where119898(119911 V)

= max 119889 (119878119911 119879V) 119889 (119878119911 119860119911) 119889 (119879V 119861V)

119889 (119878119911 119861V) 119889 (119879V 119860119911)

= max 119889 (119860119911 119911) 119889 (119860119911 119860119911) 119889 (119911 119911)

119889 (119860119911 119911) 119889 (119911 119860119911)

= 119889 (119860119911 119911)

(35)

From inequality (34) we have

120595(int119889(119860119911119911)

0

120601 (119905) 119889119905)

le 120595(int119889(119860119911119911)

0

120601 (119905) 119889119905) minus 120593(int119889(119860119911119911)

0

120601 (119905) 119889119905)

(36)

so that 120593(int119889(119860119911119911)0

120601(119905)119889119905) = 0 that is 119889(119860119911 119911) = 0 Hence wehave119860119911 = 119911 = 119878119911which shows that 119911 is a commonfixed pointof the pair (119860 119878)

Also the pair (119861 119879) is weakly compatible and 119861V = 119879Vhence 119861119911 = 119861119879V = 119879119861V = 119879119911 Using inequality (13) with119909 = 119908 and 119910 = 119911 we have

120595(int119889(119911119861119911)

0

120601 (119905) 119889119905)

= 120595(int119889(119860119908119861119911)

0

120601 (119905) 119889119905)

le 120595(int119898(119908119911)

0

120601 (119905) 119889119905) minus 120593(int119898(119908119911)

0

120601 (119905) 119889119905)

(37)

Abstract and Applied Analysis 7

where

119898(119908 119911) = max 119889 (119878119908 119879119911) 119889 (119878119908 119860119908) 119889 (119879119911 119861119911)

119889 (119878119908 119861119911) 119889 (119879119911 119860119908)

= max 119889 (119911 119861119911) 119889 (119911 119911) 119889 (119861119911 119861119911)

119889 (119911 119861119911) 119889 (119861119911 119911)

= 119889 (119911 119861119911)

(38)

Hence inequality (37) implies

120595(int119889(119911119861119911)

0

120601 (119905) 119889119905)

le 120595(int119889(119911119861119911)

0

120601 (119905) 119889119905) minus 120593(int119889(119911119861119911)

0

120601 (119905) 119889119905)

(39)

so that 120593(int119889(119911119861119911)0

120601(119905)119889119905) = 0 that is 119889(119911 119861119911) = 0 Therefore119861119911 = 119911 = 119879119911 which shows that 119911 is a common fixed pointof the pair (119861 119879) Hence 119911 is a common fixed point of thepairs (119860 119878) and (119861 119879) Uniqueness of common fixed point isan easy consequence of the inequality (13)This concludes theproof

Now we furnish an illustrative example which demon-strates the validity of the hypotheses and degree of generalityof Theorem 9

Example 10 Consider 119883 = 119884 = [2 11) equipped with thesymmetric 119889(119909 119910) = (119909 minus 119910)

2 for all 119909 119910 isin 119883 which alsosatisfies (1119862) and (119867119864) Define the mappings 119860 119861 119878 and 119879

by

119860119909 = 2 if 119909 isin 2 cup (5 11)

5 if 119909 isin (2 5]

119861119909 = 2 if 119909 isin 2 cup (5 11)

4 if 119909 isin (2 5]

119878119909 =

2 if 119909 = 2

6 if 119909 isin (2 5] 3119909 + 1

8 if 119909 isin (5 11)

119879119909 =

2 if 119909 = 2

8 if 119909 isin (2 5]

119909 minus 3 if 119909 isin (5 11)

(40)

Then 119860(119883) = 2 5 119861(119883) = 2 4 119879(119883) = [2 8] and 119878(119883) =[2 174) cup 6 Now consider the sequences 119909

119899

= 5 + 1119899

and 119910119899

= 2 Then

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= lim119899rarrinfin

119861119910119899

= lim119899rarrinfin

119879119910119899

= 2 isin 119878 (119883) cap 119879 (119883)

(41)

that is both the pairs (119860 119878) and (119861 119879) satisfy the (CLR119878119879

)

property

Let Lebesgue-integrable 120601 [0 +infin) rarr [0 +infin) definedby 120601(119905) = 119890

119905 Take 120595 isin Ψ and 120593 isin Φ given by 120595(119905) = 2119905 and120593(119905) = (27)119905 In order to check the contractive condition (13)consider the following nine cases

(i) 119909 = 119910 = 2

(ii) 119909 = 2 119910 isin (2 5]

(iii) 119909 = 2 119910 isin (5 11)

(iv) 119909 isin (2 5] 119910 = 2

(v) 119909 119910 isin (2 5]

(vi) 119909 isin (2 5] 119910 isin (5 11)

(vii) 119909 isin (5 11) 119910 = 2

(viii) 119909 isin (5 11) 119910 isin (2 5]

(ix) 119909 119910 isin (5 11)

In the cases (i) (iii) (vii) and (ix) we get that119889(119860119909 119861119910) =0 and (13) is trivially satisfied

In the cases (ii) and (viii) we have 119889(119860119909 119861119910) = 4 and119898(119909 119910) = 36 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int4

0

119890119905

119889119905)

= 120595 (1198904

minus 1)

= 2 (1198904

minus 1)

le12

7(11989036

minus 1)

= 2 (11989036

minus 1) minus2

7(11989036

minus 1)

= 120595(int36

0

119890119905

119889119905) minus 120593(int36

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(42)

Therefore we get the fact that (13) holds

8 Abstract and Applied Analysis

In the case (iv) we get 119889(119860119909 119861119910) = 9 and 119898(119909 119910) = 16Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int9

0

119890119905

119889119905)

= 120595 (1198909

minus 1)

= 2 (1198909

minus 1)

le12

7(11989016

minus 1)

= 2 (11989016

minus 1) minus2

7(11989016

minus 1)

= 120595(int16

0

119890119905

119889119905) minus 120593(int16

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(43)

This implies that (13) holdsIn the case (vi) we have 119889(119860119909 119861119910) = 9 and 119889(119878119909 119861119910) =

16 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int9

0

119890119905

119889119905)

= 120595 (1198909

minus 1)

= 2 (1198909

minus 1)

le12

7(11989016

minus 1)

=12

7(119890119889(119878119909119861119910)

minus 1)

le12

7(119890119898(119909119910)

minus 1)

= 2 (119890119898(119909119910)

minus 1) minus2

7(119890119898(119909119910)

minus 1)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(44)

This shows that (13) holds

Finally in the case (v) we obtain 119889(119860119909 119861119910) = 1 and119898(119909 119910) = 16 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int1

0

119890119905

119889119905)

= 120595 (119890 minus 1)

= 2 (119890 minus 1)

le12

7(11989016

minus 1)

= 2 (11989016

minus 1) minus2

7(11989016

minus 1)

= 120595(int16

0

119890119905

119889119905) minus 120593(int16

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(45)

that is the contractive condition (13) holdsHence all the conditions ofTheorem 9 are satisfied and 2

is a unique common fixed point of the pairs (119860 119878) and (119861 119879)which also remains a point of coincidence as well Here onemay notice that all the involved mappings are discontinuousat their unique common fixed point 2

Corollary 11 Let (119883 119889) be a symmetric space wherein 119889

satisfies the condition (119862119862)whereas119884 is an arbitrary nonemptyset with 119860 119861 119878 119879 119884 rarr 119883 satisfying all the hypothesesof Lemma 8 Then (119860 119878) and (119861 119879) have a coincidence pointeach Moreover if 119884 = 119883 then 119860 119861 119878 and 119879 have a uniquecommon fixed point provided both the pairs (119860 119878) and (119861 119879)are weakly compatible

Proof Owing to Lemma 8 it follows that the pairs (119860 119878) and(119861 119879) enjoy the (CLR

119878119879

) property Hence all the conditionsof Theorem 9 are satisfied and 119860 119861 119878 and 119879 have a uniquecommon fixed point provided both the pairs (119860 119878) and (119861 119879)are weakly compatible

Remark 12 The conclusions of Lemma 8 Theorem 9and Corollary 11 remain true if we choose 119898(119909 119910) =

max1198724119860119861119878119879

(119909 119910) or119898(119909 119910) = max1198723119860119861119878119879

(119909 119910)

Our next result shows the importance of common limitrange property over common property (EA)

Theorem 13 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861 119878 119879 119884 rarr 119883 which satisfy theinequalities (13) and (15) of Lemma 8 Suppose that

(1) the pairs (119860 119878) and (119861 119879) satisfy the common property(EA)

(2) 119878(119884) and 119879(119884) are closed subsets of119883

Abstract and Applied Analysis 9

Then (119860 119878) and (119861 119879) have a coincidence point eachMoreover if 119884 = 119883 then119860 119861 119878 and 119879 have a unique commonfixed point provided both the pairs (119860 119878) and (119861 119879) are weaklycompatible

Proof Since the pairs (119860 119878) and (119861 119879) satisfy the commonproperty (EA) there exist two sequences 119909

119899

and 119910119899

in 119884such that

lim119899rarrinfin

119889 (119860119909119899

119911) = lim119899rarrinfin

119889 (119878119909119899

119911) = lim119899rarrinfin

119889 (119861119910119899

119911)

= lim119899rarrinfin

119889 (119879119910119899

119911) = 0(46)

for some 119911 isin 119883 Since 119878(119884) and 119879(119884) are closed subsets of119883therefore 119911 isin 119878(119884)cap119879(119884) Since 119911 isin 119878(119884) there exists a point119908 isin 119884 such that 119878119908 = 119911 Also since 119911 isin 119879(119884) there exists apoint V isin 119883 such that 119879V = 119911 The rest of the proof runs onthe lines of the proof of Theorem 9

Remark 14 The conclusions of Theorem 13 remain true ifcondition (2) of Theorem 13 is replaced by one of thefollowing conditions

(2)1015840

119860(119884) sub 119879(119884) and 119861(119884) sub 119878(119884) where 119860(119884) and 119861(119884)denote the closure of ranges of themappings119860 and 119861

(2)10158401015840

119860(119884) and 119861(119884) are closed subsets of 119883 provided119860(119884) sub 119879(119884) and 119861(119884) sub 119878(119884)

By setting119860119861 119878 and119879 suitably we can deduce corollariesinvolving two and three self-mappings As a sample we candeduce the following corollary involving two self-mappings

Corollary 15 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119878 119884 rarr 119883 Suppose that

(1) the pair (119860 119878) satisfies the (119862119871119877119878

) property(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860119909119860119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905) 119889119905) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(47)

where 119898(119909 119910) = max119872119896119860119878

(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Then (119860 119878) has a coincidence point Moreover if 119884 = 119883then119860 and 119878 have a unique common fixed point in119883 providedthe pair (119860 119878) is weakly compatible

As an application of Theorem 9 we have the followingresult involving four finite families of self-mappings

Theorem 16 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860

119894

119898

119894=1

119861119895

119899

119895=1

119878119903

119901

119903=1

119879119897

119902

119897=1

119884 rarr

119883 satisfying the inequalities (13) and (15) of Lemma 8 where119860 = 119860

1

1198602

sdot sdot sdot 119860119898

119861 = 1198611

1198612

sdot sdot sdot 119861119899

119878 = 1198781

1198782

sdot sdot sdot 119878119901

and119879 = 119879

1

1198792

sdot sdot sdot 119879119902

Suppose that the pairs (119860 119878) and (119861 119879) satisfythe (119862119871119877

119878119879

) property Then (119860 119878) and (119861 119879) have a point ofcoincidence each

Moreover if 119884 = 119883 then 119860119894

119898

119894=1

119861119895

119899

119895=1

119878119903

119901

119903=1

and119879119897

119902

119897=1

have a unique common fixed point provided the families(119860119894

119878119903

) and (119861119895

119879119897

) commute pairwise where 119894 isin

1 2 119898 119903 isin 1 2 119901 119895 isin 1 2 119899 and 119897 isin

1 2 119902

Now we indicate that Theorem 16 can be utilized toderive common fixed point theorems for any finite number ofmappings As a sample we can derive a common fixed pointtheorem for six mappings by setting two families of twomembers while setting the rest two of single members

Corollary 17 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861119867 119877 119878 119879 119884 rarr 119883 Suppose that

(1) the pairs (119860 119878119877) and (119861 119879119867) share the (119862119871119877(119878119877)(119879119867)

)

property(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905)) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(48)

where 119898(119909 119910) = max119872119896119860119861119867119877119878119879

(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Then (119860 119878119877) and (119861 119879119867) have a coincidence point eachMoreover if 119884 = 119883 then 119860 119861 119867 119877 119878 and 119879 have a uniquecommon fixed point provided 119860119878 = 119878119860 119860119877 = 119877119860 119878119877 = 119877119878119861119879 = 119879119861 119861119867 = 119867119861 and 119879119867 = 119867119879

By setting 1198601

= 1198602

= sdot sdot sdot = 119860119898

= 119860 1198611

= 1198612

= sdot sdot sdot =

119861119899

= 119861 1198781

= 1198782

= sdot sdot sdot = 119878119901

= 119878 and 1198791

= 1198792

= sdot sdot sdot = 119879119902

= 119879

in Theorem 16 one gets the following corollary

Corollary 18 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861 119878 119879 119884 rarr 119883 Suppose that

(1) the pairs (119860119898 119878119901) and (119861119899 119879119902) share the (119862119871119877(119878

119901119879

119902)

)

property where119898 119899 119901 119902 are fixed positive integers(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860

119898

119909119861

119899

119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905) 119889119905) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(49)

10 Abstract and Applied Analysis

where 119898(119909 119910) = max119872119896119860

119898119861

119899119878

119901119879

119902(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Moreover if 119884 = 119883 then 119860 119861 119878 and 119879 have a uniquecommon fixed point provided 119860119878 = 119878119860 and 119861119879 = 119879119861

Remark 19 The above Corollary 18 is a slight but partial gen-eralization of Theorem 9 as the commutativity requirements(ie 119860119878 = 119878119860 and 119861119879 = 119879119861) in this corollary are relativelystronger as compared to weak compatibility in Theorem 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The first author gratefully acknowledges the support fromthe Deanship of Scientific Research (DSR) at King AbdulazizUniversity (KAU) during this research

References

[1] T L Hicks and B E Rhoades ldquoFixed point theory in symmetricspaces with applications to probabilistic spacesrdquo NonlinearAnalysis ATheory andMethods vol 36 no 3 pp 331ndash344 1999

[2] R P Pant ldquoNoncompatible mappings and common fixedpointsrdquo Soochow Journal of Mathematics vol 26 no 1 pp 29ndash35 2000

[3] G Jungck ldquoCompatible mappings and common fixed pointsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 9 no 4 pp 771ndash779 1986

[4] S Sessa ldquoOn a weak commutativity condition of mappings infixed point considerationsrdquo Institut Mathematique vol 32(46)pp 149ndash153 1982

[5] G Jungck and B E Rhoades ldquoFixed points for set valued func-tions without continuityrdquo Indian Journal of Pure and AppliedMathematics vol 29 no 3 pp 227ndash238 1998

[6] S L Singh and A Tomar ldquoWeaker forms of commuting mapsand existence of fixed pointsrdquo Journal of the Korea Society ofMathematical Education B vol 10 no 3 pp 145ndash161 2003

[7] P P Murthy ldquoImportant tools and possible applications ofmetric fixed point theoryrdquo Nonlinear Analysis Theory Methodsamp Applications vol 47 no 53479 3490 pages 2001

[8] M A Al-Thagafi andN Shahzad ldquoGeneralized 119868-nonexpansiveselfmaps and invariant approximationsrdquo Acta MathematicaSinica vol 24 no 5 pp 867ndash876 2008

[9] M Abbas and B E Rhoades ldquoCommon fixed point theoremsfor occasionally weakly compatible mappings satisfying a gen-eralized contractive conditionrdquoMathematical Communicationsvol 13 no 2 pp 295ndash301 2008

[10] A Aliouche and V Popa ldquoCommon fixed point theorems foroccasionally weakly compatible mappings via implicit rela-tionsrdquo Filomat vol 22 no 2 pp 99ndash107 2008

[11] A Bhatt H Chandra and D R Sahu ldquoCommon fixed pointtheorems for occasionally weakly compatible mappings underrelaxed conditionsrdquo Nonlinear Analysis A Theory and Methodsvol 73 no 1 pp 176ndash182 2010

[12] M Imdad S Chauhan Z Kadelburg andCVetro ldquoFixed pointtheorems for non-self mappings in symmetric spaces under 120593-weak contractive conditions and an application to functionalequations in dynamic programmingrdquo Applied Mathematics andComputation vol 227 pp 469ndash479 2014

[13] D ETHoric Z Kadelburg and S Radenovic ldquoA note on occasion-ally weakly compatible mappings and common fixed pointsrdquoFixed Point Theory vol 13 no 2 pp 475ndash479 2012

[14] M Aamri and D El Moutawakil ldquoSome new common fixedpoint theorems under strict contractive conditionsrdquo Journal ofMathematical Analysis and Applications vol 270 no 1 pp 181ndash188 2002

[15] K P R Sastry and I S R Krishna Murthy ldquoCommon fixedpoints of two partially commuting tangential selfmaps on amet-ric spacerdquo Journal of Mathematical Analysis and Applicationsvol 250 no 2 pp 731ndash734 2000

[16] Y Liu J Wu and Z Li ldquoCommon fixed points of single-valuedand multivalued mapsrdquo International Journal of Mathematicsand Mathematical Sciences no 19 pp 3045ndash3055 2005

[17] M Imdad S Chauhan and Z Kadelburg ldquoFixed point theoremfor mappings with common limit range property satisfyinggeneralized (120595 120593)-weak contractive conditionsrdquo MathematicalSciences vol 7 article 16 2013

[18] A H SolimanM Imdad andMHasan ldquoProving unified com-mon fixed point theorems via common property (EA) in sym-metric spacesrdquo Korean Mathematical Society vol 25 no 4 pp629ndash645 2010

[19] R P Pant ldquoCommon fixed points of Lipschitz type mappingpairsrdquo Journal of Mathematical Analysis and Applications vol240 no 1 pp 280ndash283 1999

[20] W Sintunavarat andPKumam ldquoCommonfixed point theoremsfor a pair of weakly compatible mappings in fuzzy metricspacesrdquo Journal of Applied Mathematics vol 2011 Article ID637958 14 pages 2011

[21] M Imdad B D Pant and S Chauhan ldquoFixed point theoremsinMenger spaces using the (119862119871119877

119878119879

) property and applicationsrdquoJournal of Nonlinear Analysis and Optimization Theory andApplications vol 3 no 2 pp 225ndash237 2012

[22] E Karapınar D K Patel M Imdad and D Gopal ldquoSomenonunique common fixed point theorems in symmetric spacesthrough 119862119871119877

(119878119879)

propertyrdquo International Journal of Mathemat-ics and Mathematical Sciences Article ID 753965 8 pages 2013

[23] D Gopal R P Pant and A S Ranadive ldquoCommon fixed pointof absorbingmapsrdquo BulletinMarathwadaMathematical Societyvol 9 no 1 pp 43ndash48 2008

[24] Ya I Alber and S Guerre-Delabriere ldquoPrinciple of weaklycontractive maps in Hilbert spacesrdquo in New Results in OperatorTheory and its Applications vol 98 ofOperatorTheory Advancesand Applications pp 7ndash22 1997

[25] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis vol 47 pp 2683ndash2693 2001

[26] A Branciari ldquoA fixed point theorem for mappings satisfyinga general contractive condition of integral typerdquo InternationalJournal of Mathematics and Mathematical Sciences vol 29 no9 pp 531ndash536 2002

[27] A Djoudi and A Aliouche ldquoCommon fixed point theoremsof Gregus type for weakly compatible mappings satisfyingcontractive conditions of integral typerdquo Journal ofMathematicalAnalysis and Applications vol 329 no 1 pp 31ndash45 2007

[28] Z Liu X Li S M Kang and S Y Cho ldquoFixed point theoremsfor mappings satisfying contractive conditions of integral type

Abstract and Applied Analysis 11

and applicationsrdquo Fixed Point Theory and Applications vol 64article 18 2011

[29] B Samet and C Vetro ldquoAn integral version of Cirics fixed pointtheoremrdquo Mediterranean Journal of Mathematics vol 9 no 1pp 225ndash238 2012

[30] W Sintunavarat and P Kumam ldquoGregus-type common fixedpoint theorems for tangential multivalued mappings of integraltype inmetric spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2011 Article ID 923458 12 pages2011

[31] W Sintunavarat and P Kumam ldquoGregus type fixed points for atangential multi-valued mappings satisfying contractive condi-tions of integral typerdquo Journal of Inequalities and Applicationsvol 2011 article 3 2011

[32] T Suzuki ldquoMeir-Keeler contractions of integral type are stillMeir-Keeler contractionsrdquo International Journal ofMathematicsand Mathematical Sciences vol 2007 Article ID 39281 6 pages2007

[33] C Vetro S Chauhan E Karapnar and W Shatanawi ldquoFixedpoints of weakly compatible mappings satisfying generalized120593-weak contractionsrdquo Bulletin of the Malaysian MathematicalSciences Society In press

[34] Q Zhang and Y Song ldquoFixed point theory for generalized 120593-weak contractionsrdquo Applied Mathematics Letters vol 22 no 1pp 75ndash78 2009

[35] D ETHoric ldquoCommon fixed point for generalized (120595 120593)-weakcontractionsrdquo Applied Mathematics Letters vol 22 no 12 pp1896ndash1900 2009

[36] P N Dutta and B S Choudhury ldquoA generalisation of con-traction principle in metric spacesrdquo Fixed Point Theory andApplications Article ID 406368 8 pages 2008

[37] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[38] S Radenovic Z Kadelburg D Jandrlic and A Jandrlic ldquoSomeresults on weakly contractive mapsrdquo Iranian MathematicalSociety Bulletin vol 38 no 3 pp 625ndash645 2012

[39] W A Wilson ldquoOn semi-metric spacesrdquo American Journal ofMathematics vol 53 no 2 pp 361ndash373 1931

[40] A Aliouche ldquoA common fixed point theorem for weakly com-patible mappings in symmetric spaces satisfying a contractivecondition of integral typerdquo Journal ofMathematical Analysis andApplications vol 322 no 2 pp 796ndash802 2006

[41] F Galvin and S D Shore ldquoCompleteness in semimetric spacesrdquoPacific Journal of Mathematics vol 113 no 1 pp 67ndash75 1984

[42] S-H Cho G-Y Lee and J-S Bae ldquoOn coincidence and fixed-point theorems in symmetric spacesrdquo Fixed Point Theory andApplications Article ID 562130 9 pages 2008

[43] M Aamri and D El Moutawakil ldquoCommon fixed points undercontractive conditions in symmetric spacesrdquo Applied Math-ematics E-Notes vol 3 pp 156ndash162 2003

[44] DGopalMHasan andM Imdad ldquoAbsorbing pairs facilitatingcommon fixed point theorems for Lipschitzian type mappingsin symmetric spacesrdquo Korean Mathematical Society Communi-cations vol 27 no 2 pp 385ndash397 2012

[45] D Gopal M Imdad and C Vetro ldquoCommon fixed pointtheorems for mappings satisfying common property (EA) insymmetric spacesrdquo Filomat vol 25 no 2 pp 59ndash78 2011

[46] M Imdad and J Ali ldquoCommon fixed point theorems in sym-metric spaces employing a new implicit function and common

property (EA)rdquo Bulletin of the Belgian Mathematical Societyvol 16 no 3 pp 421ndash433 2009

[47] M Imdad J Ali and L Khan ldquoCoincidence and fixed pointsin symmetric spaces under strict contractionsrdquo Journal ofMathematical Analysis and Applications vol 320 no 1 pp 352ndash360 2006

[48] D Turkoglu and I Altun ldquoA common fixed point theorem forweakly compatible mappings in symmetric spaces satisfying animplicit relationrdquo Sociedad Matematica Mexicana vol 13 no 1pp 195ndash205 2007

[49] M Imdad A Sharma Chauhan and S ldquoUnifying a multitudeof metrical fixed point theorems in symmetric spacesrdquo FilomatIn press

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

where

119898(119908 119911) = max 119889 (119878119908 119879119911) 119889 (119878119908 119860119908) 119889 (119879119911 119861119911)

119889 (119878119908 119861119911) 119889 (119879119911 119860119908)

= max 119889 (119911 119861119911) 119889 (119911 119911) 119889 (119861119911 119861119911)

119889 (119911 119861119911) 119889 (119861119911 119911)

= 119889 (119911 119861119911)

(38)

Hence inequality (37) implies

120595(int119889(119911119861119911)

0

120601 (119905) 119889119905)

le 120595(int119889(119911119861119911)

0

120601 (119905) 119889119905) minus 120593(int119889(119911119861119911)

0

120601 (119905) 119889119905)

(39)

so that 120593(int119889(119911119861119911)0

120601(119905)119889119905) = 0 that is 119889(119911 119861119911) = 0 Therefore119861119911 = 119911 = 119879119911 which shows that 119911 is a common fixed pointof the pair (119861 119879) Hence 119911 is a common fixed point of thepairs (119860 119878) and (119861 119879) Uniqueness of common fixed point isan easy consequence of the inequality (13)This concludes theproof

Now we furnish an illustrative example which demon-strates the validity of the hypotheses and degree of generalityof Theorem 9

Example 10 Consider 119883 = 119884 = [2 11) equipped with thesymmetric 119889(119909 119910) = (119909 minus 119910)

2 for all 119909 119910 isin 119883 which alsosatisfies (1119862) and (119867119864) Define the mappings 119860 119861 119878 and 119879

by

119860119909 = 2 if 119909 isin 2 cup (5 11)

5 if 119909 isin (2 5]

119861119909 = 2 if 119909 isin 2 cup (5 11)

4 if 119909 isin (2 5]

119878119909 =

2 if 119909 = 2

6 if 119909 isin (2 5] 3119909 + 1

8 if 119909 isin (5 11)

119879119909 =

2 if 119909 = 2

8 if 119909 isin (2 5]

119909 minus 3 if 119909 isin (5 11)

(40)

Then 119860(119883) = 2 5 119861(119883) = 2 4 119879(119883) = [2 8] and 119878(119883) =[2 174) cup 6 Now consider the sequences 119909

119899

= 5 + 1119899

and 119910119899

= 2 Then

lim119899rarrinfin

119860119909119899

= lim119899rarrinfin

119878119909119899

= lim119899rarrinfin

119861119910119899

= lim119899rarrinfin

119879119910119899

= 2 isin 119878 (119883) cap 119879 (119883)

(41)

that is both the pairs (119860 119878) and (119861 119879) satisfy the (CLR119878119879

)

property

Let Lebesgue-integrable 120601 [0 +infin) rarr [0 +infin) definedby 120601(119905) = 119890

119905 Take 120595 isin Ψ and 120593 isin Φ given by 120595(119905) = 2119905 and120593(119905) = (27)119905 In order to check the contractive condition (13)consider the following nine cases

(i) 119909 = 119910 = 2

(ii) 119909 = 2 119910 isin (2 5]

(iii) 119909 = 2 119910 isin (5 11)

(iv) 119909 isin (2 5] 119910 = 2

(v) 119909 119910 isin (2 5]

(vi) 119909 isin (2 5] 119910 isin (5 11)

(vii) 119909 isin (5 11) 119910 = 2

(viii) 119909 isin (5 11) 119910 isin (2 5]

(ix) 119909 119910 isin (5 11)

In the cases (i) (iii) (vii) and (ix) we get that119889(119860119909 119861119910) =0 and (13) is trivially satisfied

In the cases (ii) and (viii) we have 119889(119860119909 119861119910) = 4 and119898(119909 119910) = 36 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int4

0

119890119905

119889119905)

= 120595 (1198904

minus 1)

= 2 (1198904

minus 1)

le12

7(11989036

minus 1)

= 2 (11989036

minus 1) minus2

7(11989036

minus 1)

= 120595(int36

0

119890119905

119889119905) minus 120593(int36

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(42)

Therefore we get the fact that (13) holds

8 Abstract and Applied Analysis

In the case (iv) we get 119889(119860119909 119861119910) = 9 and 119898(119909 119910) = 16Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int9

0

119890119905

119889119905)

= 120595 (1198909

minus 1)

= 2 (1198909

minus 1)

le12

7(11989016

minus 1)

= 2 (11989016

minus 1) minus2

7(11989016

minus 1)

= 120595(int16

0

119890119905

119889119905) minus 120593(int16

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(43)

This implies that (13) holdsIn the case (vi) we have 119889(119860119909 119861119910) = 9 and 119889(119878119909 119861119910) =

16 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int9

0

119890119905

119889119905)

= 120595 (1198909

minus 1)

= 2 (1198909

minus 1)

le12

7(11989016

minus 1)

=12

7(119890119889(119878119909119861119910)

minus 1)

le12

7(119890119898(119909119910)

minus 1)

= 2 (119890119898(119909119910)

minus 1) minus2

7(119890119898(119909119910)

minus 1)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(44)

This shows that (13) holds

Finally in the case (v) we obtain 119889(119860119909 119861119910) = 1 and119898(119909 119910) = 16 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int1

0

119890119905

119889119905)

= 120595 (119890 minus 1)

= 2 (119890 minus 1)

le12

7(11989016

minus 1)

= 2 (11989016

minus 1) minus2

7(11989016

minus 1)

= 120595(int16

0

119890119905

119889119905) minus 120593(int16

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(45)

that is the contractive condition (13) holdsHence all the conditions ofTheorem 9 are satisfied and 2

is a unique common fixed point of the pairs (119860 119878) and (119861 119879)which also remains a point of coincidence as well Here onemay notice that all the involved mappings are discontinuousat their unique common fixed point 2

Corollary 11 Let (119883 119889) be a symmetric space wherein 119889

satisfies the condition (119862119862)whereas119884 is an arbitrary nonemptyset with 119860 119861 119878 119879 119884 rarr 119883 satisfying all the hypothesesof Lemma 8 Then (119860 119878) and (119861 119879) have a coincidence pointeach Moreover if 119884 = 119883 then 119860 119861 119878 and 119879 have a uniquecommon fixed point provided both the pairs (119860 119878) and (119861 119879)are weakly compatible

Proof Owing to Lemma 8 it follows that the pairs (119860 119878) and(119861 119879) enjoy the (CLR

119878119879

) property Hence all the conditionsof Theorem 9 are satisfied and 119860 119861 119878 and 119879 have a uniquecommon fixed point provided both the pairs (119860 119878) and (119861 119879)are weakly compatible

Remark 12 The conclusions of Lemma 8 Theorem 9and Corollary 11 remain true if we choose 119898(119909 119910) =

max1198724119860119861119878119879

(119909 119910) or119898(119909 119910) = max1198723119860119861119878119879

(119909 119910)

Our next result shows the importance of common limitrange property over common property (EA)

Theorem 13 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861 119878 119879 119884 rarr 119883 which satisfy theinequalities (13) and (15) of Lemma 8 Suppose that

(1) the pairs (119860 119878) and (119861 119879) satisfy the common property(EA)

(2) 119878(119884) and 119879(119884) are closed subsets of119883

Abstract and Applied Analysis 9

Then (119860 119878) and (119861 119879) have a coincidence point eachMoreover if 119884 = 119883 then119860 119861 119878 and 119879 have a unique commonfixed point provided both the pairs (119860 119878) and (119861 119879) are weaklycompatible

Proof Since the pairs (119860 119878) and (119861 119879) satisfy the commonproperty (EA) there exist two sequences 119909

119899

and 119910119899

in 119884such that

lim119899rarrinfin

119889 (119860119909119899

119911) = lim119899rarrinfin

119889 (119878119909119899

119911) = lim119899rarrinfin

119889 (119861119910119899

119911)

= lim119899rarrinfin

119889 (119879119910119899

119911) = 0(46)

for some 119911 isin 119883 Since 119878(119884) and 119879(119884) are closed subsets of119883therefore 119911 isin 119878(119884)cap119879(119884) Since 119911 isin 119878(119884) there exists a point119908 isin 119884 such that 119878119908 = 119911 Also since 119911 isin 119879(119884) there exists apoint V isin 119883 such that 119879V = 119911 The rest of the proof runs onthe lines of the proof of Theorem 9

Remark 14 The conclusions of Theorem 13 remain true ifcondition (2) of Theorem 13 is replaced by one of thefollowing conditions

(2)1015840

119860(119884) sub 119879(119884) and 119861(119884) sub 119878(119884) where 119860(119884) and 119861(119884)denote the closure of ranges of themappings119860 and 119861

(2)10158401015840

119860(119884) and 119861(119884) are closed subsets of 119883 provided119860(119884) sub 119879(119884) and 119861(119884) sub 119878(119884)

By setting119860119861 119878 and119879 suitably we can deduce corollariesinvolving two and three self-mappings As a sample we candeduce the following corollary involving two self-mappings

Corollary 15 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119878 119884 rarr 119883 Suppose that

(1) the pair (119860 119878) satisfies the (119862119871119877119878

) property(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860119909119860119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905) 119889119905) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(47)

where 119898(119909 119910) = max119872119896119860119878

(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Then (119860 119878) has a coincidence point Moreover if 119884 = 119883then119860 and 119878 have a unique common fixed point in119883 providedthe pair (119860 119878) is weakly compatible

As an application of Theorem 9 we have the followingresult involving four finite families of self-mappings

Theorem 16 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860

119894

119898

119894=1

119861119895

119899

119895=1

119878119903

119901

119903=1

119879119897

119902

119897=1

119884 rarr

119883 satisfying the inequalities (13) and (15) of Lemma 8 where119860 = 119860

1

1198602

sdot sdot sdot 119860119898

119861 = 1198611

1198612

sdot sdot sdot 119861119899

119878 = 1198781

1198782

sdot sdot sdot 119878119901

and119879 = 119879

1

1198792

sdot sdot sdot 119879119902

Suppose that the pairs (119860 119878) and (119861 119879) satisfythe (119862119871119877

119878119879

) property Then (119860 119878) and (119861 119879) have a point ofcoincidence each

Moreover if 119884 = 119883 then 119860119894

119898

119894=1

119861119895

119899

119895=1

119878119903

119901

119903=1

and119879119897

119902

119897=1

have a unique common fixed point provided the families(119860119894

119878119903

) and (119861119895

119879119897

) commute pairwise where 119894 isin

1 2 119898 119903 isin 1 2 119901 119895 isin 1 2 119899 and 119897 isin

1 2 119902

Now we indicate that Theorem 16 can be utilized toderive common fixed point theorems for any finite number ofmappings As a sample we can derive a common fixed pointtheorem for six mappings by setting two families of twomembers while setting the rest two of single members

Corollary 17 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861119867 119877 119878 119879 119884 rarr 119883 Suppose that

(1) the pairs (119860 119878119877) and (119861 119879119867) share the (119862119871119877(119878119877)(119879119867)

)

property(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905)) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(48)

where 119898(119909 119910) = max119872119896119860119861119867119877119878119879

(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Then (119860 119878119877) and (119861 119879119867) have a coincidence point eachMoreover if 119884 = 119883 then 119860 119861 119867 119877 119878 and 119879 have a uniquecommon fixed point provided 119860119878 = 119878119860 119860119877 = 119877119860 119878119877 = 119877119878119861119879 = 119879119861 119861119867 = 119867119861 and 119879119867 = 119867119879

By setting 1198601

= 1198602

= sdot sdot sdot = 119860119898

= 119860 1198611

= 1198612

= sdot sdot sdot =

119861119899

= 119861 1198781

= 1198782

= sdot sdot sdot = 119878119901

= 119878 and 1198791

= 1198792

= sdot sdot sdot = 119879119902

= 119879

in Theorem 16 one gets the following corollary

Corollary 18 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861 119878 119879 119884 rarr 119883 Suppose that

(1) the pairs (119860119898 119878119901) and (119861119899 119879119902) share the (119862119871119877(119878

119901119879

119902)

)

property where119898 119899 119901 119902 are fixed positive integers(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860

119898

119909119861

119899

119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905) 119889119905) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(49)

10 Abstract and Applied Analysis

where 119898(119909 119910) = max119872119896119860

119898119861

119899119878

119901119879

119902(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Moreover if 119884 = 119883 then 119860 119861 119878 and 119879 have a uniquecommon fixed point provided 119860119878 = 119878119860 and 119861119879 = 119879119861

Remark 19 The above Corollary 18 is a slight but partial gen-eralization of Theorem 9 as the commutativity requirements(ie 119860119878 = 119878119860 and 119861119879 = 119879119861) in this corollary are relativelystronger as compared to weak compatibility in Theorem 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The first author gratefully acknowledges the support fromthe Deanship of Scientific Research (DSR) at King AbdulazizUniversity (KAU) during this research

References

[1] T L Hicks and B E Rhoades ldquoFixed point theory in symmetricspaces with applications to probabilistic spacesrdquo NonlinearAnalysis ATheory andMethods vol 36 no 3 pp 331ndash344 1999

[2] R P Pant ldquoNoncompatible mappings and common fixedpointsrdquo Soochow Journal of Mathematics vol 26 no 1 pp 29ndash35 2000

[3] G Jungck ldquoCompatible mappings and common fixed pointsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 9 no 4 pp 771ndash779 1986

[4] S Sessa ldquoOn a weak commutativity condition of mappings infixed point considerationsrdquo Institut Mathematique vol 32(46)pp 149ndash153 1982

[5] G Jungck and B E Rhoades ldquoFixed points for set valued func-tions without continuityrdquo Indian Journal of Pure and AppliedMathematics vol 29 no 3 pp 227ndash238 1998

[6] S L Singh and A Tomar ldquoWeaker forms of commuting mapsand existence of fixed pointsrdquo Journal of the Korea Society ofMathematical Education B vol 10 no 3 pp 145ndash161 2003

[7] P P Murthy ldquoImportant tools and possible applications ofmetric fixed point theoryrdquo Nonlinear Analysis Theory Methodsamp Applications vol 47 no 53479 3490 pages 2001

[8] M A Al-Thagafi andN Shahzad ldquoGeneralized 119868-nonexpansiveselfmaps and invariant approximationsrdquo Acta MathematicaSinica vol 24 no 5 pp 867ndash876 2008

[9] M Abbas and B E Rhoades ldquoCommon fixed point theoremsfor occasionally weakly compatible mappings satisfying a gen-eralized contractive conditionrdquoMathematical Communicationsvol 13 no 2 pp 295ndash301 2008

[10] A Aliouche and V Popa ldquoCommon fixed point theorems foroccasionally weakly compatible mappings via implicit rela-tionsrdquo Filomat vol 22 no 2 pp 99ndash107 2008

[11] A Bhatt H Chandra and D R Sahu ldquoCommon fixed pointtheorems for occasionally weakly compatible mappings underrelaxed conditionsrdquo Nonlinear Analysis A Theory and Methodsvol 73 no 1 pp 176ndash182 2010

[12] M Imdad S Chauhan Z Kadelburg andCVetro ldquoFixed pointtheorems for non-self mappings in symmetric spaces under 120593-weak contractive conditions and an application to functionalequations in dynamic programmingrdquo Applied Mathematics andComputation vol 227 pp 469ndash479 2014

[13] D ETHoric Z Kadelburg and S Radenovic ldquoA note on occasion-ally weakly compatible mappings and common fixed pointsrdquoFixed Point Theory vol 13 no 2 pp 475ndash479 2012

[14] M Aamri and D El Moutawakil ldquoSome new common fixedpoint theorems under strict contractive conditionsrdquo Journal ofMathematical Analysis and Applications vol 270 no 1 pp 181ndash188 2002

[15] K P R Sastry and I S R Krishna Murthy ldquoCommon fixedpoints of two partially commuting tangential selfmaps on amet-ric spacerdquo Journal of Mathematical Analysis and Applicationsvol 250 no 2 pp 731ndash734 2000

[16] Y Liu J Wu and Z Li ldquoCommon fixed points of single-valuedand multivalued mapsrdquo International Journal of Mathematicsand Mathematical Sciences no 19 pp 3045ndash3055 2005

[17] M Imdad S Chauhan and Z Kadelburg ldquoFixed point theoremfor mappings with common limit range property satisfyinggeneralized (120595 120593)-weak contractive conditionsrdquo MathematicalSciences vol 7 article 16 2013

[18] A H SolimanM Imdad andMHasan ldquoProving unified com-mon fixed point theorems via common property (EA) in sym-metric spacesrdquo Korean Mathematical Society vol 25 no 4 pp629ndash645 2010

[19] R P Pant ldquoCommon fixed points of Lipschitz type mappingpairsrdquo Journal of Mathematical Analysis and Applications vol240 no 1 pp 280ndash283 1999

[20] W Sintunavarat andPKumam ldquoCommonfixed point theoremsfor a pair of weakly compatible mappings in fuzzy metricspacesrdquo Journal of Applied Mathematics vol 2011 Article ID637958 14 pages 2011

[21] M Imdad B D Pant and S Chauhan ldquoFixed point theoremsinMenger spaces using the (119862119871119877

119878119879

) property and applicationsrdquoJournal of Nonlinear Analysis and Optimization Theory andApplications vol 3 no 2 pp 225ndash237 2012

[22] E Karapınar D K Patel M Imdad and D Gopal ldquoSomenonunique common fixed point theorems in symmetric spacesthrough 119862119871119877

(119878119879)

propertyrdquo International Journal of Mathemat-ics and Mathematical Sciences Article ID 753965 8 pages 2013

[23] D Gopal R P Pant and A S Ranadive ldquoCommon fixed pointof absorbingmapsrdquo BulletinMarathwadaMathematical Societyvol 9 no 1 pp 43ndash48 2008

[24] Ya I Alber and S Guerre-Delabriere ldquoPrinciple of weaklycontractive maps in Hilbert spacesrdquo in New Results in OperatorTheory and its Applications vol 98 ofOperatorTheory Advancesand Applications pp 7ndash22 1997

[25] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis vol 47 pp 2683ndash2693 2001

[26] A Branciari ldquoA fixed point theorem for mappings satisfyinga general contractive condition of integral typerdquo InternationalJournal of Mathematics and Mathematical Sciences vol 29 no9 pp 531ndash536 2002

[27] A Djoudi and A Aliouche ldquoCommon fixed point theoremsof Gregus type for weakly compatible mappings satisfyingcontractive conditions of integral typerdquo Journal ofMathematicalAnalysis and Applications vol 329 no 1 pp 31ndash45 2007

[28] Z Liu X Li S M Kang and S Y Cho ldquoFixed point theoremsfor mappings satisfying contractive conditions of integral type

Abstract and Applied Analysis 11

and applicationsrdquo Fixed Point Theory and Applications vol 64article 18 2011

[29] B Samet and C Vetro ldquoAn integral version of Cirics fixed pointtheoremrdquo Mediterranean Journal of Mathematics vol 9 no 1pp 225ndash238 2012

[30] W Sintunavarat and P Kumam ldquoGregus-type common fixedpoint theorems for tangential multivalued mappings of integraltype inmetric spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2011 Article ID 923458 12 pages2011

[31] W Sintunavarat and P Kumam ldquoGregus type fixed points for atangential multi-valued mappings satisfying contractive condi-tions of integral typerdquo Journal of Inequalities and Applicationsvol 2011 article 3 2011

[32] T Suzuki ldquoMeir-Keeler contractions of integral type are stillMeir-Keeler contractionsrdquo International Journal ofMathematicsand Mathematical Sciences vol 2007 Article ID 39281 6 pages2007

[33] C Vetro S Chauhan E Karapnar and W Shatanawi ldquoFixedpoints of weakly compatible mappings satisfying generalized120593-weak contractionsrdquo Bulletin of the Malaysian MathematicalSciences Society In press

[34] Q Zhang and Y Song ldquoFixed point theory for generalized 120593-weak contractionsrdquo Applied Mathematics Letters vol 22 no 1pp 75ndash78 2009

[35] D ETHoric ldquoCommon fixed point for generalized (120595 120593)-weakcontractionsrdquo Applied Mathematics Letters vol 22 no 12 pp1896ndash1900 2009

[36] P N Dutta and B S Choudhury ldquoA generalisation of con-traction principle in metric spacesrdquo Fixed Point Theory andApplications Article ID 406368 8 pages 2008

[37] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[38] S Radenovic Z Kadelburg D Jandrlic and A Jandrlic ldquoSomeresults on weakly contractive mapsrdquo Iranian MathematicalSociety Bulletin vol 38 no 3 pp 625ndash645 2012

[39] W A Wilson ldquoOn semi-metric spacesrdquo American Journal ofMathematics vol 53 no 2 pp 361ndash373 1931

[40] A Aliouche ldquoA common fixed point theorem for weakly com-patible mappings in symmetric spaces satisfying a contractivecondition of integral typerdquo Journal ofMathematical Analysis andApplications vol 322 no 2 pp 796ndash802 2006

[41] F Galvin and S D Shore ldquoCompleteness in semimetric spacesrdquoPacific Journal of Mathematics vol 113 no 1 pp 67ndash75 1984

[42] S-H Cho G-Y Lee and J-S Bae ldquoOn coincidence and fixed-point theorems in symmetric spacesrdquo Fixed Point Theory andApplications Article ID 562130 9 pages 2008

[43] M Aamri and D El Moutawakil ldquoCommon fixed points undercontractive conditions in symmetric spacesrdquo Applied Math-ematics E-Notes vol 3 pp 156ndash162 2003

[44] DGopalMHasan andM Imdad ldquoAbsorbing pairs facilitatingcommon fixed point theorems for Lipschitzian type mappingsin symmetric spacesrdquo Korean Mathematical Society Communi-cations vol 27 no 2 pp 385ndash397 2012

[45] D Gopal M Imdad and C Vetro ldquoCommon fixed pointtheorems for mappings satisfying common property (EA) insymmetric spacesrdquo Filomat vol 25 no 2 pp 59ndash78 2011

[46] M Imdad and J Ali ldquoCommon fixed point theorems in sym-metric spaces employing a new implicit function and common

property (EA)rdquo Bulletin of the Belgian Mathematical Societyvol 16 no 3 pp 421ndash433 2009

[47] M Imdad J Ali and L Khan ldquoCoincidence and fixed pointsin symmetric spaces under strict contractionsrdquo Journal ofMathematical Analysis and Applications vol 320 no 1 pp 352ndash360 2006

[48] D Turkoglu and I Altun ldquoA common fixed point theorem forweakly compatible mappings in symmetric spaces satisfying animplicit relationrdquo Sociedad Matematica Mexicana vol 13 no 1pp 195ndash205 2007

[49] M Imdad A Sharma Chauhan and S ldquoUnifying a multitudeof metrical fixed point theorems in symmetric spacesrdquo FilomatIn press

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

In the case (iv) we get 119889(119860119909 119861119910) = 9 and 119898(119909 119910) = 16Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int9

0

119890119905

119889119905)

= 120595 (1198909

minus 1)

= 2 (1198909

minus 1)

le12

7(11989016

minus 1)

= 2 (11989016

minus 1) minus2

7(11989016

minus 1)

= 120595(int16

0

119890119905

119889119905) minus 120593(int16

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(43)

This implies that (13) holdsIn the case (vi) we have 119889(119860119909 119861119910) = 9 and 119889(119878119909 119861119910) =

16 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int9

0

119890119905

119889119905)

= 120595 (1198909

minus 1)

= 2 (1198909

minus 1)

le12

7(11989016

minus 1)

=12

7(119890119889(119878119909119861119910)

minus 1)

le12

7(119890119898(119909119910)

minus 1)

= 2 (119890119898(119909119910)

minus 1) minus2

7(119890119898(119909119910)

minus 1)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(44)

This shows that (13) holds

Finally in the case (v) we obtain 119889(119860119909 119861119910) = 1 and119898(119909 119910) = 16 Now we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905) = 120595(int1

0

119890119905

119889119905)

= 120595 (119890 minus 1)

= 2 (119890 minus 1)

le12

7(11989016

minus 1)

= 2 (11989016

minus 1) minus2

7(11989016

minus 1)

= 120595(int16

0

119890119905

119889119905) minus 120593(int16

0

119890119905

119889119905)

= 120595(int119898(119909119910)

0

120601 (119905) 119889119905)

minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(45)

that is the contractive condition (13) holdsHence all the conditions ofTheorem 9 are satisfied and 2

is a unique common fixed point of the pairs (119860 119878) and (119861 119879)which also remains a point of coincidence as well Here onemay notice that all the involved mappings are discontinuousat their unique common fixed point 2

Corollary 11 Let (119883 119889) be a symmetric space wherein 119889

satisfies the condition (119862119862)whereas119884 is an arbitrary nonemptyset with 119860 119861 119878 119879 119884 rarr 119883 satisfying all the hypothesesof Lemma 8 Then (119860 119878) and (119861 119879) have a coincidence pointeach Moreover if 119884 = 119883 then 119860 119861 119878 and 119879 have a uniquecommon fixed point provided both the pairs (119860 119878) and (119861 119879)are weakly compatible

Proof Owing to Lemma 8 it follows that the pairs (119860 119878) and(119861 119879) enjoy the (CLR

119878119879

) property Hence all the conditionsof Theorem 9 are satisfied and 119860 119861 119878 and 119879 have a uniquecommon fixed point provided both the pairs (119860 119878) and (119861 119879)are weakly compatible

Remark 12 The conclusions of Lemma 8 Theorem 9and Corollary 11 remain true if we choose 119898(119909 119910) =

max1198724119860119861119878119879

(119909 119910) or119898(119909 119910) = max1198723119860119861119878119879

(119909 119910)

Our next result shows the importance of common limitrange property over common property (EA)

Theorem 13 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861 119878 119879 119884 rarr 119883 which satisfy theinequalities (13) and (15) of Lemma 8 Suppose that

(1) the pairs (119860 119878) and (119861 119879) satisfy the common property(EA)

(2) 119878(119884) and 119879(119884) are closed subsets of119883

Abstract and Applied Analysis 9

Then (119860 119878) and (119861 119879) have a coincidence point eachMoreover if 119884 = 119883 then119860 119861 119878 and 119879 have a unique commonfixed point provided both the pairs (119860 119878) and (119861 119879) are weaklycompatible

Proof Since the pairs (119860 119878) and (119861 119879) satisfy the commonproperty (EA) there exist two sequences 119909

119899

and 119910119899

in 119884such that

lim119899rarrinfin

119889 (119860119909119899

119911) = lim119899rarrinfin

119889 (119878119909119899

119911) = lim119899rarrinfin

119889 (119861119910119899

119911)

= lim119899rarrinfin

119889 (119879119910119899

119911) = 0(46)

for some 119911 isin 119883 Since 119878(119884) and 119879(119884) are closed subsets of119883therefore 119911 isin 119878(119884)cap119879(119884) Since 119911 isin 119878(119884) there exists a point119908 isin 119884 such that 119878119908 = 119911 Also since 119911 isin 119879(119884) there exists apoint V isin 119883 such that 119879V = 119911 The rest of the proof runs onthe lines of the proof of Theorem 9

Remark 14 The conclusions of Theorem 13 remain true ifcondition (2) of Theorem 13 is replaced by one of thefollowing conditions

(2)1015840

119860(119884) sub 119879(119884) and 119861(119884) sub 119878(119884) where 119860(119884) and 119861(119884)denote the closure of ranges of themappings119860 and 119861

(2)10158401015840

119860(119884) and 119861(119884) are closed subsets of 119883 provided119860(119884) sub 119879(119884) and 119861(119884) sub 119878(119884)

By setting119860119861 119878 and119879 suitably we can deduce corollariesinvolving two and three self-mappings As a sample we candeduce the following corollary involving two self-mappings

Corollary 15 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119878 119884 rarr 119883 Suppose that

(1) the pair (119860 119878) satisfies the (119862119871119877119878

) property(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860119909119860119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905) 119889119905) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(47)

where 119898(119909 119910) = max119872119896119860119878

(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Then (119860 119878) has a coincidence point Moreover if 119884 = 119883then119860 and 119878 have a unique common fixed point in119883 providedthe pair (119860 119878) is weakly compatible

As an application of Theorem 9 we have the followingresult involving four finite families of self-mappings

Theorem 16 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860

119894

119898

119894=1

119861119895

119899

119895=1

119878119903

119901

119903=1

119879119897

119902

119897=1

119884 rarr

119883 satisfying the inequalities (13) and (15) of Lemma 8 where119860 = 119860

1

1198602

sdot sdot sdot 119860119898

119861 = 1198611

1198612

sdot sdot sdot 119861119899

119878 = 1198781

1198782

sdot sdot sdot 119878119901

and119879 = 119879

1

1198792

sdot sdot sdot 119879119902

Suppose that the pairs (119860 119878) and (119861 119879) satisfythe (119862119871119877

119878119879

) property Then (119860 119878) and (119861 119879) have a point ofcoincidence each

Moreover if 119884 = 119883 then 119860119894

119898

119894=1

119861119895

119899

119895=1

119878119903

119901

119903=1

and119879119897

119902

119897=1

have a unique common fixed point provided the families(119860119894

119878119903

) and (119861119895

119879119897

) commute pairwise where 119894 isin

1 2 119898 119903 isin 1 2 119901 119895 isin 1 2 119899 and 119897 isin

1 2 119902

Now we indicate that Theorem 16 can be utilized toderive common fixed point theorems for any finite number ofmappings As a sample we can derive a common fixed pointtheorem for six mappings by setting two families of twomembers while setting the rest two of single members

Corollary 17 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861119867 119877 119878 119879 119884 rarr 119883 Suppose that

(1) the pairs (119860 119878119877) and (119861 119879119867) share the (119862119871119877(119878119877)(119879119867)

)

property(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905)) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(48)

where 119898(119909 119910) = max119872119896119860119861119867119877119878119879

(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Then (119860 119878119877) and (119861 119879119867) have a coincidence point eachMoreover if 119884 = 119883 then 119860 119861 119867 119877 119878 and 119879 have a uniquecommon fixed point provided 119860119878 = 119878119860 119860119877 = 119877119860 119878119877 = 119877119878119861119879 = 119879119861 119861119867 = 119867119861 and 119879119867 = 119867119879

By setting 1198601

= 1198602

= sdot sdot sdot = 119860119898

= 119860 1198611

= 1198612

= sdot sdot sdot =

119861119899

= 119861 1198781

= 1198782

= sdot sdot sdot = 119878119901

= 119878 and 1198791

= 1198792

= sdot sdot sdot = 119879119902

= 119879

in Theorem 16 one gets the following corollary

Corollary 18 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861 119878 119879 119884 rarr 119883 Suppose that

(1) the pairs (119860119898 119878119901) and (119861119899 119879119902) share the (119862119871119877(119878

119901119879

119902)

)

property where119898 119899 119901 119902 are fixed positive integers(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860

119898

119909119861

119899

119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905) 119889119905) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(49)

10 Abstract and Applied Analysis

where 119898(119909 119910) = max119872119896119860

119898119861

119899119878

119901119879

119902(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Moreover if 119884 = 119883 then 119860 119861 119878 and 119879 have a uniquecommon fixed point provided 119860119878 = 119878119860 and 119861119879 = 119879119861

Remark 19 The above Corollary 18 is a slight but partial gen-eralization of Theorem 9 as the commutativity requirements(ie 119860119878 = 119878119860 and 119861119879 = 119879119861) in this corollary are relativelystronger as compared to weak compatibility in Theorem 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The first author gratefully acknowledges the support fromthe Deanship of Scientific Research (DSR) at King AbdulazizUniversity (KAU) during this research

References

[1] T L Hicks and B E Rhoades ldquoFixed point theory in symmetricspaces with applications to probabilistic spacesrdquo NonlinearAnalysis ATheory andMethods vol 36 no 3 pp 331ndash344 1999

[2] R P Pant ldquoNoncompatible mappings and common fixedpointsrdquo Soochow Journal of Mathematics vol 26 no 1 pp 29ndash35 2000

[3] G Jungck ldquoCompatible mappings and common fixed pointsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 9 no 4 pp 771ndash779 1986

[4] S Sessa ldquoOn a weak commutativity condition of mappings infixed point considerationsrdquo Institut Mathematique vol 32(46)pp 149ndash153 1982

[5] G Jungck and B E Rhoades ldquoFixed points for set valued func-tions without continuityrdquo Indian Journal of Pure and AppliedMathematics vol 29 no 3 pp 227ndash238 1998

[6] S L Singh and A Tomar ldquoWeaker forms of commuting mapsand existence of fixed pointsrdquo Journal of the Korea Society ofMathematical Education B vol 10 no 3 pp 145ndash161 2003

[7] P P Murthy ldquoImportant tools and possible applications ofmetric fixed point theoryrdquo Nonlinear Analysis Theory Methodsamp Applications vol 47 no 53479 3490 pages 2001

[8] M A Al-Thagafi andN Shahzad ldquoGeneralized 119868-nonexpansiveselfmaps and invariant approximationsrdquo Acta MathematicaSinica vol 24 no 5 pp 867ndash876 2008

[9] M Abbas and B E Rhoades ldquoCommon fixed point theoremsfor occasionally weakly compatible mappings satisfying a gen-eralized contractive conditionrdquoMathematical Communicationsvol 13 no 2 pp 295ndash301 2008

[10] A Aliouche and V Popa ldquoCommon fixed point theorems foroccasionally weakly compatible mappings via implicit rela-tionsrdquo Filomat vol 22 no 2 pp 99ndash107 2008

[11] A Bhatt H Chandra and D R Sahu ldquoCommon fixed pointtheorems for occasionally weakly compatible mappings underrelaxed conditionsrdquo Nonlinear Analysis A Theory and Methodsvol 73 no 1 pp 176ndash182 2010

[12] M Imdad S Chauhan Z Kadelburg andCVetro ldquoFixed pointtheorems for non-self mappings in symmetric spaces under 120593-weak contractive conditions and an application to functionalequations in dynamic programmingrdquo Applied Mathematics andComputation vol 227 pp 469ndash479 2014

[13] D ETHoric Z Kadelburg and S Radenovic ldquoA note on occasion-ally weakly compatible mappings and common fixed pointsrdquoFixed Point Theory vol 13 no 2 pp 475ndash479 2012

[14] M Aamri and D El Moutawakil ldquoSome new common fixedpoint theorems under strict contractive conditionsrdquo Journal ofMathematical Analysis and Applications vol 270 no 1 pp 181ndash188 2002

[15] K P R Sastry and I S R Krishna Murthy ldquoCommon fixedpoints of two partially commuting tangential selfmaps on amet-ric spacerdquo Journal of Mathematical Analysis and Applicationsvol 250 no 2 pp 731ndash734 2000

[16] Y Liu J Wu and Z Li ldquoCommon fixed points of single-valuedand multivalued mapsrdquo International Journal of Mathematicsand Mathematical Sciences no 19 pp 3045ndash3055 2005

[17] M Imdad S Chauhan and Z Kadelburg ldquoFixed point theoremfor mappings with common limit range property satisfyinggeneralized (120595 120593)-weak contractive conditionsrdquo MathematicalSciences vol 7 article 16 2013

[18] A H SolimanM Imdad andMHasan ldquoProving unified com-mon fixed point theorems via common property (EA) in sym-metric spacesrdquo Korean Mathematical Society vol 25 no 4 pp629ndash645 2010

[19] R P Pant ldquoCommon fixed points of Lipschitz type mappingpairsrdquo Journal of Mathematical Analysis and Applications vol240 no 1 pp 280ndash283 1999

[20] W Sintunavarat andPKumam ldquoCommonfixed point theoremsfor a pair of weakly compatible mappings in fuzzy metricspacesrdquo Journal of Applied Mathematics vol 2011 Article ID637958 14 pages 2011

[21] M Imdad B D Pant and S Chauhan ldquoFixed point theoremsinMenger spaces using the (119862119871119877

119878119879

) property and applicationsrdquoJournal of Nonlinear Analysis and Optimization Theory andApplications vol 3 no 2 pp 225ndash237 2012

[22] E Karapınar D K Patel M Imdad and D Gopal ldquoSomenonunique common fixed point theorems in symmetric spacesthrough 119862119871119877

(119878119879)

propertyrdquo International Journal of Mathemat-ics and Mathematical Sciences Article ID 753965 8 pages 2013

[23] D Gopal R P Pant and A S Ranadive ldquoCommon fixed pointof absorbingmapsrdquo BulletinMarathwadaMathematical Societyvol 9 no 1 pp 43ndash48 2008

[24] Ya I Alber and S Guerre-Delabriere ldquoPrinciple of weaklycontractive maps in Hilbert spacesrdquo in New Results in OperatorTheory and its Applications vol 98 ofOperatorTheory Advancesand Applications pp 7ndash22 1997

[25] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis vol 47 pp 2683ndash2693 2001

[26] A Branciari ldquoA fixed point theorem for mappings satisfyinga general contractive condition of integral typerdquo InternationalJournal of Mathematics and Mathematical Sciences vol 29 no9 pp 531ndash536 2002

[27] A Djoudi and A Aliouche ldquoCommon fixed point theoremsof Gregus type for weakly compatible mappings satisfyingcontractive conditions of integral typerdquo Journal ofMathematicalAnalysis and Applications vol 329 no 1 pp 31ndash45 2007

[28] Z Liu X Li S M Kang and S Y Cho ldquoFixed point theoremsfor mappings satisfying contractive conditions of integral type

Abstract and Applied Analysis 11

and applicationsrdquo Fixed Point Theory and Applications vol 64article 18 2011

[29] B Samet and C Vetro ldquoAn integral version of Cirics fixed pointtheoremrdquo Mediterranean Journal of Mathematics vol 9 no 1pp 225ndash238 2012

[30] W Sintunavarat and P Kumam ldquoGregus-type common fixedpoint theorems for tangential multivalued mappings of integraltype inmetric spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2011 Article ID 923458 12 pages2011

[31] W Sintunavarat and P Kumam ldquoGregus type fixed points for atangential multi-valued mappings satisfying contractive condi-tions of integral typerdquo Journal of Inequalities and Applicationsvol 2011 article 3 2011

[32] T Suzuki ldquoMeir-Keeler contractions of integral type are stillMeir-Keeler contractionsrdquo International Journal ofMathematicsand Mathematical Sciences vol 2007 Article ID 39281 6 pages2007

[33] C Vetro S Chauhan E Karapnar and W Shatanawi ldquoFixedpoints of weakly compatible mappings satisfying generalized120593-weak contractionsrdquo Bulletin of the Malaysian MathematicalSciences Society In press

[34] Q Zhang and Y Song ldquoFixed point theory for generalized 120593-weak contractionsrdquo Applied Mathematics Letters vol 22 no 1pp 75ndash78 2009

[35] D ETHoric ldquoCommon fixed point for generalized (120595 120593)-weakcontractionsrdquo Applied Mathematics Letters vol 22 no 12 pp1896ndash1900 2009

[36] P N Dutta and B S Choudhury ldquoA generalisation of con-traction principle in metric spacesrdquo Fixed Point Theory andApplications Article ID 406368 8 pages 2008

[37] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[38] S Radenovic Z Kadelburg D Jandrlic and A Jandrlic ldquoSomeresults on weakly contractive mapsrdquo Iranian MathematicalSociety Bulletin vol 38 no 3 pp 625ndash645 2012

[39] W A Wilson ldquoOn semi-metric spacesrdquo American Journal ofMathematics vol 53 no 2 pp 361ndash373 1931

[40] A Aliouche ldquoA common fixed point theorem for weakly com-patible mappings in symmetric spaces satisfying a contractivecondition of integral typerdquo Journal ofMathematical Analysis andApplications vol 322 no 2 pp 796ndash802 2006

[41] F Galvin and S D Shore ldquoCompleteness in semimetric spacesrdquoPacific Journal of Mathematics vol 113 no 1 pp 67ndash75 1984

[42] S-H Cho G-Y Lee and J-S Bae ldquoOn coincidence and fixed-point theorems in symmetric spacesrdquo Fixed Point Theory andApplications Article ID 562130 9 pages 2008

[43] M Aamri and D El Moutawakil ldquoCommon fixed points undercontractive conditions in symmetric spacesrdquo Applied Math-ematics E-Notes vol 3 pp 156ndash162 2003

[44] DGopalMHasan andM Imdad ldquoAbsorbing pairs facilitatingcommon fixed point theorems for Lipschitzian type mappingsin symmetric spacesrdquo Korean Mathematical Society Communi-cations vol 27 no 2 pp 385ndash397 2012

[45] D Gopal M Imdad and C Vetro ldquoCommon fixed pointtheorems for mappings satisfying common property (EA) insymmetric spacesrdquo Filomat vol 25 no 2 pp 59ndash78 2011

[46] M Imdad and J Ali ldquoCommon fixed point theorems in sym-metric spaces employing a new implicit function and common

property (EA)rdquo Bulletin of the Belgian Mathematical Societyvol 16 no 3 pp 421ndash433 2009

[47] M Imdad J Ali and L Khan ldquoCoincidence and fixed pointsin symmetric spaces under strict contractionsrdquo Journal ofMathematical Analysis and Applications vol 320 no 1 pp 352ndash360 2006

[48] D Turkoglu and I Altun ldquoA common fixed point theorem forweakly compatible mappings in symmetric spaces satisfying animplicit relationrdquo Sociedad Matematica Mexicana vol 13 no 1pp 195ndash205 2007

[49] M Imdad A Sharma Chauhan and S ldquoUnifying a multitudeof metrical fixed point theorems in symmetric spacesrdquo FilomatIn press

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

Then (119860 119878) and (119861 119879) have a coincidence point eachMoreover if 119884 = 119883 then119860 119861 119878 and 119879 have a unique commonfixed point provided both the pairs (119860 119878) and (119861 119879) are weaklycompatible

Proof Since the pairs (119860 119878) and (119861 119879) satisfy the commonproperty (EA) there exist two sequences 119909

119899

and 119910119899

in 119884such that

lim119899rarrinfin

119889 (119860119909119899

119911) = lim119899rarrinfin

119889 (119878119909119899

119911) = lim119899rarrinfin

119889 (119861119910119899

119911)

= lim119899rarrinfin

119889 (119879119910119899

119911) = 0(46)

for some 119911 isin 119883 Since 119878(119884) and 119879(119884) are closed subsets of119883therefore 119911 isin 119878(119884)cap119879(119884) Since 119911 isin 119878(119884) there exists a point119908 isin 119884 such that 119878119908 = 119911 Also since 119911 isin 119879(119884) there exists apoint V isin 119883 such that 119879V = 119911 The rest of the proof runs onthe lines of the proof of Theorem 9

Remark 14 The conclusions of Theorem 13 remain true ifcondition (2) of Theorem 13 is replaced by one of thefollowing conditions

(2)1015840

119860(119884) sub 119879(119884) and 119861(119884) sub 119878(119884) where 119860(119884) and 119861(119884)denote the closure of ranges of themappings119860 and 119861

(2)10158401015840

119860(119884) and 119861(119884) are closed subsets of 119883 provided119860(119884) sub 119879(119884) and 119861(119884) sub 119878(119884)

By setting119860119861 119878 and119879 suitably we can deduce corollariesinvolving two and three self-mappings As a sample we candeduce the following corollary involving two self-mappings

Corollary 15 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119878 119884 rarr 119883 Suppose that

(1) the pair (119860 119878) satisfies the (119862119871119877119878

) property(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860119909119860119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905) 119889119905) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(47)

where 119898(119909 119910) = max119872119896119860119878

(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Then (119860 119878) has a coincidence point Moreover if 119884 = 119883then119860 and 119878 have a unique common fixed point in119883 providedthe pair (119860 119878) is weakly compatible

As an application of Theorem 9 we have the followingresult involving four finite families of self-mappings

Theorem 16 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860

119894

119898

119894=1

119861119895

119899

119895=1

119878119903

119901

119903=1

119879119897

119902

119897=1

119884 rarr

119883 satisfying the inequalities (13) and (15) of Lemma 8 where119860 = 119860

1

1198602

sdot sdot sdot 119860119898

119861 = 1198611

1198612

sdot sdot sdot 119861119899

119878 = 1198781

1198782

sdot sdot sdot 119878119901

and119879 = 119879

1

1198792

sdot sdot sdot 119879119902

Suppose that the pairs (119860 119878) and (119861 119879) satisfythe (119862119871119877

119878119879

) property Then (119860 119878) and (119861 119879) have a point ofcoincidence each

Moreover if 119884 = 119883 then 119860119894

119898

119894=1

119861119895

119899

119895=1

119878119903

119901

119903=1

and119879119897

119902

119897=1

have a unique common fixed point provided the families(119860119894

119878119903

) and (119861119895

119879119897

) commute pairwise where 119894 isin

1 2 119898 119903 isin 1 2 119901 119895 isin 1 2 119899 and 119897 isin

1 2 119902

Now we indicate that Theorem 16 can be utilized toderive common fixed point theorems for any finite number ofmappings As a sample we can derive a common fixed pointtheorem for six mappings by setting two families of twomembers while setting the rest two of single members

Corollary 17 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861119867 119877 119878 119879 119884 rarr 119883 Suppose that

(1) the pairs (119860 119878119877) and (119861 119879119867) share the (119862119871119877(119878119877)(119879119867)

)

property(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860119909119861119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905)) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(48)

where 119898(119909 119910) = max119872119896119860119861119867119877119878119879

(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Then (119860 119878119877) and (119861 119879119867) have a coincidence point eachMoreover if 119884 = 119883 then 119860 119861 119867 119877 119878 and 119879 have a uniquecommon fixed point provided 119860119878 = 119878119860 119860119877 = 119877119860 119878119877 = 119877119878119861119879 = 119879119861 119861119867 = 119867119861 and 119879119867 = 119867119879

By setting 1198601

= 1198602

= sdot sdot sdot = 119860119898

= 119860 1198611

= 1198612

= sdot sdot sdot =

119861119899

= 119861 1198781

= 1198782

= sdot sdot sdot = 119878119901

= 119878 and 1198791

= 1198792

= sdot sdot sdot = 119879119902

= 119879

in Theorem 16 one gets the following corollary

Corollary 18 Let (119883 119889) be a symmetric space wherein 119889

satisfies the conditions (1119862) and (119867119864)whereas119884 is an arbitrarynonempty set with 119860 119861 119878 119879 119884 rarr 119883 Suppose that

(1) the pairs (119860119898 119878119901) and (119861119899 119879119902) share the (119862119871119877(119878

119901119879

119902)

)

property where119898 119899 119901 119902 are fixed positive integers(2) there exist 120593 isin Φ and 120595 isin Ψ such that for all 119909 119910 isin 119884

we have

120595(int119889(119860

119898

119909119861

119899

119910)

0

120601 (119905) 119889119905)

le 120595(int119898(119909119910)

0

120601 (119905) 119889119905) minus 120593(int119898(119909119910)

0

120601 (119905) 119889119905)

(49)

10 Abstract and Applied Analysis

where 119898(119909 119910) = max119872119896119860

119898119861

119899119878

119901119879

119902(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Moreover if 119884 = 119883 then 119860 119861 119878 and 119879 have a uniquecommon fixed point provided 119860119878 = 119878119860 and 119861119879 = 119879119861

Remark 19 The above Corollary 18 is a slight but partial gen-eralization of Theorem 9 as the commutativity requirements(ie 119860119878 = 119878119860 and 119861119879 = 119879119861) in this corollary are relativelystronger as compared to weak compatibility in Theorem 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The first author gratefully acknowledges the support fromthe Deanship of Scientific Research (DSR) at King AbdulazizUniversity (KAU) during this research

References

[1] T L Hicks and B E Rhoades ldquoFixed point theory in symmetricspaces with applications to probabilistic spacesrdquo NonlinearAnalysis ATheory andMethods vol 36 no 3 pp 331ndash344 1999

[2] R P Pant ldquoNoncompatible mappings and common fixedpointsrdquo Soochow Journal of Mathematics vol 26 no 1 pp 29ndash35 2000

[3] G Jungck ldquoCompatible mappings and common fixed pointsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 9 no 4 pp 771ndash779 1986

[4] S Sessa ldquoOn a weak commutativity condition of mappings infixed point considerationsrdquo Institut Mathematique vol 32(46)pp 149ndash153 1982

[5] G Jungck and B E Rhoades ldquoFixed points for set valued func-tions without continuityrdquo Indian Journal of Pure and AppliedMathematics vol 29 no 3 pp 227ndash238 1998

[6] S L Singh and A Tomar ldquoWeaker forms of commuting mapsand existence of fixed pointsrdquo Journal of the Korea Society ofMathematical Education B vol 10 no 3 pp 145ndash161 2003

[7] P P Murthy ldquoImportant tools and possible applications ofmetric fixed point theoryrdquo Nonlinear Analysis Theory Methodsamp Applications vol 47 no 53479 3490 pages 2001

[8] M A Al-Thagafi andN Shahzad ldquoGeneralized 119868-nonexpansiveselfmaps and invariant approximationsrdquo Acta MathematicaSinica vol 24 no 5 pp 867ndash876 2008

[9] M Abbas and B E Rhoades ldquoCommon fixed point theoremsfor occasionally weakly compatible mappings satisfying a gen-eralized contractive conditionrdquoMathematical Communicationsvol 13 no 2 pp 295ndash301 2008

[10] A Aliouche and V Popa ldquoCommon fixed point theorems foroccasionally weakly compatible mappings via implicit rela-tionsrdquo Filomat vol 22 no 2 pp 99ndash107 2008

[11] A Bhatt H Chandra and D R Sahu ldquoCommon fixed pointtheorems for occasionally weakly compatible mappings underrelaxed conditionsrdquo Nonlinear Analysis A Theory and Methodsvol 73 no 1 pp 176ndash182 2010

[12] M Imdad S Chauhan Z Kadelburg andCVetro ldquoFixed pointtheorems for non-self mappings in symmetric spaces under 120593-weak contractive conditions and an application to functionalequations in dynamic programmingrdquo Applied Mathematics andComputation vol 227 pp 469ndash479 2014

[13] D ETHoric Z Kadelburg and S Radenovic ldquoA note on occasion-ally weakly compatible mappings and common fixed pointsrdquoFixed Point Theory vol 13 no 2 pp 475ndash479 2012

[14] M Aamri and D El Moutawakil ldquoSome new common fixedpoint theorems under strict contractive conditionsrdquo Journal ofMathematical Analysis and Applications vol 270 no 1 pp 181ndash188 2002

[15] K P R Sastry and I S R Krishna Murthy ldquoCommon fixedpoints of two partially commuting tangential selfmaps on amet-ric spacerdquo Journal of Mathematical Analysis and Applicationsvol 250 no 2 pp 731ndash734 2000

[16] Y Liu J Wu and Z Li ldquoCommon fixed points of single-valuedand multivalued mapsrdquo International Journal of Mathematicsand Mathematical Sciences no 19 pp 3045ndash3055 2005

[17] M Imdad S Chauhan and Z Kadelburg ldquoFixed point theoremfor mappings with common limit range property satisfyinggeneralized (120595 120593)-weak contractive conditionsrdquo MathematicalSciences vol 7 article 16 2013

[18] A H SolimanM Imdad andMHasan ldquoProving unified com-mon fixed point theorems via common property (EA) in sym-metric spacesrdquo Korean Mathematical Society vol 25 no 4 pp629ndash645 2010

[19] R P Pant ldquoCommon fixed points of Lipschitz type mappingpairsrdquo Journal of Mathematical Analysis and Applications vol240 no 1 pp 280ndash283 1999

[20] W Sintunavarat andPKumam ldquoCommonfixed point theoremsfor a pair of weakly compatible mappings in fuzzy metricspacesrdquo Journal of Applied Mathematics vol 2011 Article ID637958 14 pages 2011

[21] M Imdad B D Pant and S Chauhan ldquoFixed point theoremsinMenger spaces using the (119862119871119877

119878119879

) property and applicationsrdquoJournal of Nonlinear Analysis and Optimization Theory andApplications vol 3 no 2 pp 225ndash237 2012

[22] E Karapınar D K Patel M Imdad and D Gopal ldquoSomenonunique common fixed point theorems in symmetric spacesthrough 119862119871119877

(119878119879)

propertyrdquo International Journal of Mathemat-ics and Mathematical Sciences Article ID 753965 8 pages 2013

[23] D Gopal R P Pant and A S Ranadive ldquoCommon fixed pointof absorbingmapsrdquo BulletinMarathwadaMathematical Societyvol 9 no 1 pp 43ndash48 2008

[24] Ya I Alber and S Guerre-Delabriere ldquoPrinciple of weaklycontractive maps in Hilbert spacesrdquo in New Results in OperatorTheory and its Applications vol 98 ofOperatorTheory Advancesand Applications pp 7ndash22 1997

[25] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis vol 47 pp 2683ndash2693 2001

[26] A Branciari ldquoA fixed point theorem for mappings satisfyinga general contractive condition of integral typerdquo InternationalJournal of Mathematics and Mathematical Sciences vol 29 no9 pp 531ndash536 2002

[27] A Djoudi and A Aliouche ldquoCommon fixed point theoremsof Gregus type for weakly compatible mappings satisfyingcontractive conditions of integral typerdquo Journal ofMathematicalAnalysis and Applications vol 329 no 1 pp 31ndash45 2007

[28] Z Liu X Li S M Kang and S Y Cho ldquoFixed point theoremsfor mappings satisfying contractive conditions of integral type

Abstract and Applied Analysis 11

and applicationsrdquo Fixed Point Theory and Applications vol 64article 18 2011

[29] B Samet and C Vetro ldquoAn integral version of Cirics fixed pointtheoremrdquo Mediterranean Journal of Mathematics vol 9 no 1pp 225ndash238 2012

[30] W Sintunavarat and P Kumam ldquoGregus-type common fixedpoint theorems for tangential multivalued mappings of integraltype inmetric spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2011 Article ID 923458 12 pages2011

[31] W Sintunavarat and P Kumam ldquoGregus type fixed points for atangential multi-valued mappings satisfying contractive condi-tions of integral typerdquo Journal of Inequalities and Applicationsvol 2011 article 3 2011

[32] T Suzuki ldquoMeir-Keeler contractions of integral type are stillMeir-Keeler contractionsrdquo International Journal ofMathematicsand Mathematical Sciences vol 2007 Article ID 39281 6 pages2007

[33] C Vetro S Chauhan E Karapnar and W Shatanawi ldquoFixedpoints of weakly compatible mappings satisfying generalized120593-weak contractionsrdquo Bulletin of the Malaysian MathematicalSciences Society In press

[34] Q Zhang and Y Song ldquoFixed point theory for generalized 120593-weak contractionsrdquo Applied Mathematics Letters vol 22 no 1pp 75ndash78 2009

[35] D ETHoric ldquoCommon fixed point for generalized (120595 120593)-weakcontractionsrdquo Applied Mathematics Letters vol 22 no 12 pp1896ndash1900 2009

[36] P N Dutta and B S Choudhury ldquoA generalisation of con-traction principle in metric spacesrdquo Fixed Point Theory andApplications Article ID 406368 8 pages 2008

[37] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[38] S Radenovic Z Kadelburg D Jandrlic and A Jandrlic ldquoSomeresults on weakly contractive mapsrdquo Iranian MathematicalSociety Bulletin vol 38 no 3 pp 625ndash645 2012

[39] W A Wilson ldquoOn semi-metric spacesrdquo American Journal ofMathematics vol 53 no 2 pp 361ndash373 1931

[40] A Aliouche ldquoA common fixed point theorem for weakly com-patible mappings in symmetric spaces satisfying a contractivecondition of integral typerdquo Journal ofMathematical Analysis andApplications vol 322 no 2 pp 796ndash802 2006

[41] F Galvin and S D Shore ldquoCompleteness in semimetric spacesrdquoPacific Journal of Mathematics vol 113 no 1 pp 67ndash75 1984

[42] S-H Cho G-Y Lee and J-S Bae ldquoOn coincidence and fixed-point theorems in symmetric spacesrdquo Fixed Point Theory andApplications Article ID 562130 9 pages 2008

[43] M Aamri and D El Moutawakil ldquoCommon fixed points undercontractive conditions in symmetric spacesrdquo Applied Math-ematics E-Notes vol 3 pp 156ndash162 2003

[44] DGopalMHasan andM Imdad ldquoAbsorbing pairs facilitatingcommon fixed point theorems for Lipschitzian type mappingsin symmetric spacesrdquo Korean Mathematical Society Communi-cations vol 27 no 2 pp 385ndash397 2012

[45] D Gopal M Imdad and C Vetro ldquoCommon fixed pointtheorems for mappings satisfying common property (EA) insymmetric spacesrdquo Filomat vol 25 no 2 pp 59ndash78 2011

[46] M Imdad and J Ali ldquoCommon fixed point theorems in sym-metric spaces employing a new implicit function and common

property (EA)rdquo Bulletin of the Belgian Mathematical Societyvol 16 no 3 pp 421ndash433 2009

[47] M Imdad J Ali and L Khan ldquoCoincidence and fixed pointsin symmetric spaces under strict contractionsrdquo Journal ofMathematical Analysis and Applications vol 320 no 1 pp 352ndash360 2006

[48] D Turkoglu and I Altun ldquoA common fixed point theorem forweakly compatible mappings in symmetric spaces satisfying animplicit relationrdquo Sociedad Matematica Mexicana vol 13 no 1pp 195ndash205 2007

[49] M Imdad A Sharma Chauhan and S ldquoUnifying a multitudeof metrical fixed point theorems in symmetric spacesrdquo FilomatIn press

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

10 Abstract and Applied Analysis

where 119898(119909 119910) = max119872119896119860

119898119861

119899119878

119901119879

119902(119909 119910) 119896 isin 3 4 5 and 120601

[0 +infin) rarr [0 +infin) is a Lebesgue-integrable mapping whichis summable and nonnegative such that (15) holds

Moreover if 119884 = 119883 then 119860 119861 119878 and 119879 have a uniquecommon fixed point provided 119860119878 = 119878119860 and 119861119879 = 119879119861

Remark 19 The above Corollary 18 is a slight but partial gen-eralization of Theorem 9 as the commutativity requirements(ie 119860119878 = 119878119860 and 119861119879 = 119879119861) in this corollary are relativelystronger as compared to weak compatibility in Theorem 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The first author gratefully acknowledges the support fromthe Deanship of Scientific Research (DSR) at King AbdulazizUniversity (KAU) during this research

References

[1] T L Hicks and B E Rhoades ldquoFixed point theory in symmetricspaces with applications to probabilistic spacesrdquo NonlinearAnalysis ATheory andMethods vol 36 no 3 pp 331ndash344 1999

[2] R P Pant ldquoNoncompatible mappings and common fixedpointsrdquo Soochow Journal of Mathematics vol 26 no 1 pp 29ndash35 2000

[3] G Jungck ldquoCompatible mappings and common fixed pointsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 9 no 4 pp 771ndash779 1986

[4] S Sessa ldquoOn a weak commutativity condition of mappings infixed point considerationsrdquo Institut Mathematique vol 32(46)pp 149ndash153 1982

[5] G Jungck and B E Rhoades ldquoFixed points for set valued func-tions without continuityrdquo Indian Journal of Pure and AppliedMathematics vol 29 no 3 pp 227ndash238 1998

[6] S L Singh and A Tomar ldquoWeaker forms of commuting mapsand existence of fixed pointsrdquo Journal of the Korea Society ofMathematical Education B vol 10 no 3 pp 145ndash161 2003

[7] P P Murthy ldquoImportant tools and possible applications ofmetric fixed point theoryrdquo Nonlinear Analysis Theory Methodsamp Applications vol 47 no 53479 3490 pages 2001

[8] M A Al-Thagafi andN Shahzad ldquoGeneralized 119868-nonexpansiveselfmaps and invariant approximationsrdquo Acta MathematicaSinica vol 24 no 5 pp 867ndash876 2008

[9] M Abbas and B E Rhoades ldquoCommon fixed point theoremsfor occasionally weakly compatible mappings satisfying a gen-eralized contractive conditionrdquoMathematical Communicationsvol 13 no 2 pp 295ndash301 2008

[10] A Aliouche and V Popa ldquoCommon fixed point theorems foroccasionally weakly compatible mappings via implicit rela-tionsrdquo Filomat vol 22 no 2 pp 99ndash107 2008

[11] A Bhatt H Chandra and D R Sahu ldquoCommon fixed pointtheorems for occasionally weakly compatible mappings underrelaxed conditionsrdquo Nonlinear Analysis A Theory and Methodsvol 73 no 1 pp 176ndash182 2010

[12] M Imdad S Chauhan Z Kadelburg andCVetro ldquoFixed pointtheorems for non-self mappings in symmetric spaces under 120593-weak contractive conditions and an application to functionalequations in dynamic programmingrdquo Applied Mathematics andComputation vol 227 pp 469ndash479 2014

[13] D ETHoric Z Kadelburg and S Radenovic ldquoA note on occasion-ally weakly compatible mappings and common fixed pointsrdquoFixed Point Theory vol 13 no 2 pp 475ndash479 2012

[14] M Aamri and D El Moutawakil ldquoSome new common fixedpoint theorems under strict contractive conditionsrdquo Journal ofMathematical Analysis and Applications vol 270 no 1 pp 181ndash188 2002

[15] K P R Sastry and I S R Krishna Murthy ldquoCommon fixedpoints of two partially commuting tangential selfmaps on amet-ric spacerdquo Journal of Mathematical Analysis and Applicationsvol 250 no 2 pp 731ndash734 2000

[16] Y Liu J Wu and Z Li ldquoCommon fixed points of single-valuedand multivalued mapsrdquo International Journal of Mathematicsand Mathematical Sciences no 19 pp 3045ndash3055 2005

[17] M Imdad S Chauhan and Z Kadelburg ldquoFixed point theoremfor mappings with common limit range property satisfyinggeneralized (120595 120593)-weak contractive conditionsrdquo MathematicalSciences vol 7 article 16 2013

[18] A H SolimanM Imdad andMHasan ldquoProving unified com-mon fixed point theorems via common property (EA) in sym-metric spacesrdquo Korean Mathematical Society vol 25 no 4 pp629ndash645 2010

[19] R P Pant ldquoCommon fixed points of Lipschitz type mappingpairsrdquo Journal of Mathematical Analysis and Applications vol240 no 1 pp 280ndash283 1999

[20] W Sintunavarat andPKumam ldquoCommonfixed point theoremsfor a pair of weakly compatible mappings in fuzzy metricspacesrdquo Journal of Applied Mathematics vol 2011 Article ID637958 14 pages 2011

[21] M Imdad B D Pant and S Chauhan ldquoFixed point theoremsinMenger spaces using the (119862119871119877

119878119879

) property and applicationsrdquoJournal of Nonlinear Analysis and Optimization Theory andApplications vol 3 no 2 pp 225ndash237 2012

[22] E Karapınar D K Patel M Imdad and D Gopal ldquoSomenonunique common fixed point theorems in symmetric spacesthrough 119862119871119877

(119878119879)

propertyrdquo International Journal of Mathemat-ics and Mathematical Sciences Article ID 753965 8 pages 2013

[23] D Gopal R P Pant and A S Ranadive ldquoCommon fixed pointof absorbingmapsrdquo BulletinMarathwadaMathematical Societyvol 9 no 1 pp 43ndash48 2008

[24] Ya I Alber and S Guerre-Delabriere ldquoPrinciple of weaklycontractive maps in Hilbert spacesrdquo in New Results in OperatorTheory and its Applications vol 98 ofOperatorTheory Advancesand Applications pp 7ndash22 1997

[25] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis vol 47 pp 2683ndash2693 2001

[26] A Branciari ldquoA fixed point theorem for mappings satisfyinga general contractive condition of integral typerdquo InternationalJournal of Mathematics and Mathematical Sciences vol 29 no9 pp 531ndash536 2002

[27] A Djoudi and A Aliouche ldquoCommon fixed point theoremsof Gregus type for weakly compatible mappings satisfyingcontractive conditions of integral typerdquo Journal ofMathematicalAnalysis and Applications vol 329 no 1 pp 31ndash45 2007

[28] Z Liu X Li S M Kang and S Y Cho ldquoFixed point theoremsfor mappings satisfying contractive conditions of integral type

Abstract and Applied Analysis 11

and applicationsrdquo Fixed Point Theory and Applications vol 64article 18 2011

[29] B Samet and C Vetro ldquoAn integral version of Cirics fixed pointtheoremrdquo Mediterranean Journal of Mathematics vol 9 no 1pp 225ndash238 2012

[30] W Sintunavarat and P Kumam ldquoGregus-type common fixedpoint theorems for tangential multivalued mappings of integraltype inmetric spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2011 Article ID 923458 12 pages2011

[31] W Sintunavarat and P Kumam ldquoGregus type fixed points for atangential multi-valued mappings satisfying contractive condi-tions of integral typerdquo Journal of Inequalities and Applicationsvol 2011 article 3 2011

[32] T Suzuki ldquoMeir-Keeler contractions of integral type are stillMeir-Keeler contractionsrdquo International Journal ofMathematicsand Mathematical Sciences vol 2007 Article ID 39281 6 pages2007

[33] C Vetro S Chauhan E Karapnar and W Shatanawi ldquoFixedpoints of weakly compatible mappings satisfying generalized120593-weak contractionsrdquo Bulletin of the Malaysian MathematicalSciences Society In press

[34] Q Zhang and Y Song ldquoFixed point theory for generalized 120593-weak contractionsrdquo Applied Mathematics Letters vol 22 no 1pp 75ndash78 2009

[35] D ETHoric ldquoCommon fixed point for generalized (120595 120593)-weakcontractionsrdquo Applied Mathematics Letters vol 22 no 12 pp1896ndash1900 2009

[36] P N Dutta and B S Choudhury ldquoA generalisation of con-traction principle in metric spacesrdquo Fixed Point Theory andApplications Article ID 406368 8 pages 2008

[37] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[38] S Radenovic Z Kadelburg D Jandrlic and A Jandrlic ldquoSomeresults on weakly contractive mapsrdquo Iranian MathematicalSociety Bulletin vol 38 no 3 pp 625ndash645 2012

[39] W A Wilson ldquoOn semi-metric spacesrdquo American Journal ofMathematics vol 53 no 2 pp 361ndash373 1931

[40] A Aliouche ldquoA common fixed point theorem for weakly com-patible mappings in symmetric spaces satisfying a contractivecondition of integral typerdquo Journal ofMathematical Analysis andApplications vol 322 no 2 pp 796ndash802 2006

[41] F Galvin and S D Shore ldquoCompleteness in semimetric spacesrdquoPacific Journal of Mathematics vol 113 no 1 pp 67ndash75 1984

[42] S-H Cho G-Y Lee and J-S Bae ldquoOn coincidence and fixed-point theorems in symmetric spacesrdquo Fixed Point Theory andApplications Article ID 562130 9 pages 2008

[43] M Aamri and D El Moutawakil ldquoCommon fixed points undercontractive conditions in symmetric spacesrdquo Applied Math-ematics E-Notes vol 3 pp 156ndash162 2003

[44] DGopalMHasan andM Imdad ldquoAbsorbing pairs facilitatingcommon fixed point theorems for Lipschitzian type mappingsin symmetric spacesrdquo Korean Mathematical Society Communi-cations vol 27 no 2 pp 385ndash397 2012

[45] D Gopal M Imdad and C Vetro ldquoCommon fixed pointtheorems for mappings satisfying common property (EA) insymmetric spacesrdquo Filomat vol 25 no 2 pp 59ndash78 2011

[46] M Imdad and J Ali ldquoCommon fixed point theorems in sym-metric spaces employing a new implicit function and common

property (EA)rdquo Bulletin of the Belgian Mathematical Societyvol 16 no 3 pp 421ndash433 2009

[47] M Imdad J Ali and L Khan ldquoCoincidence and fixed pointsin symmetric spaces under strict contractionsrdquo Journal ofMathematical Analysis and Applications vol 320 no 1 pp 352ndash360 2006

[48] D Turkoglu and I Altun ldquoA common fixed point theorem forweakly compatible mappings in symmetric spaces satisfying animplicit relationrdquo Sociedad Matematica Mexicana vol 13 no 1pp 195ndash205 2007

[49] M Imdad A Sharma Chauhan and S ldquoUnifying a multitudeof metrical fixed point theorems in symmetric spacesrdquo FilomatIn press

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 11

and applicationsrdquo Fixed Point Theory and Applications vol 64article 18 2011

[29] B Samet and C Vetro ldquoAn integral version of Cirics fixed pointtheoremrdquo Mediterranean Journal of Mathematics vol 9 no 1pp 225ndash238 2012

[30] W Sintunavarat and P Kumam ldquoGregus-type common fixedpoint theorems for tangential multivalued mappings of integraltype inmetric spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2011 Article ID 923458 12 pages2011

[31] W Sintunavarat and P Kumam ldquoGregus type fixed points for atangential multi-valued mappings satisfying contractive condi-tions of integral typerdquo Journal of Inequalities and Applicationsvol 2011 article 3 2011

[32] T Suzuki ldquoMeir-Keeler contractions of integral type are stillMeir-Keeler contractionsrdquo International Journal ofMathematicsand Mathematical Sciences vol 2007 Article ID 39281 6 pages2007

[33] C Vetro S Chauhan E Karapnar and W Shatanawi ldquoFixedpoints of weakly compatible mappings satisfying generalized120593-weak contractionsrdquo Bulletin of the Malaysian MathematicalSciences Society In press

[34] Q Zhang and Y Song ldquoFixed point theory for generalized 120593-weak contractionsrdquo Applied Mathematics Letters vol 22 no 1pp 75ndash78 2009

[35] D ETHoric ldquoCommon fixed point for generalized (120595 120593)-weakcontractionsrdquo Applied Mathematics Letters vol 22 no 12 pp1896ndash1900 2009

[36] P N Dutta and B S Choudhury ldquoA generalisation of con-traction principle in metric spacesrdquo Fixed Point Theory andApplications Article ID 406368 8 pages 2008

[37] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[38] S Radenovic Z Kadelburg D Jandrlic and A Jandrlic ldquoSomeresults on weakly contractive mapsrdquo Iranian MathematicalSociety Bulletin vol 38 no 3 pp 625ndash645 2012

[39] W A Wilson ldquoOn semi-metric spacesrdquo American Journal ofMathematics vol 53 no 2 pp 361ndash373 1931

[40] A Aliouche ldquoA common fixed point theorem for weakly com-patible mappings in symmetric spaces satisfying a contractivecondition of integral typerdquo Journal ofMathematical Analysis andApplications vol 322 no 2 pp 796ndash802 2006

[41] F Galvin and S D Shore ldquoCompleteness in semimetric spacesrdquoPacific Journal of Mathematics vol 113 no 1 pp 67ndash75 1984

[42] S-H Cho G-Y Lee and J-S Bae ldquoOn coincidence and fixed-point theorems in symmetric spacesrdquo Fixed Point Theory andApplications Article ID 562130 9 pages 2008

[43] M Aamri and D El Moutawakil ldquoCommon fixed points undercontractive conditions in symmetric spacesrdquo Applied Math-ematics E-Notes vol 3 pp 156ndash162 2003

[44] DGopalMHasan andM Imdad ldquoAbsorbing pairs facilitatingcommon fixed point theorems for Lipschitzian type mappingsin symmetric spacesrdquo Korean Mathematical Society Communi-cations vol 27 no 2 pp 385ndash397 2012

[45] D Gopal M Imdad and C Vetro ldquoCommon fixed pointtheorems for mappings satisfying common property (EA) insymmetric spacesrdquo Filomat vol 25 no 2 pp 59ndash78 2011

[46] M Imdad and J Ali ldquoCommon fixed point theorems in sym-metric spaces employing a new implicit function and common

property (EA)rdquo Bulletin of the Belgian Mathematical Societyvol 16 no 3 pp 421ndash433 2009

[47] M Imdad J Ali and L Khan ldquoCoincidence and fixed pointsin symmetric spaces under strict contractionsrdquo Journal ofMathematical Analysis and Applications vol 320 no 1 pp 352ndash360 2006

[48] D Turkoglu and I Altun ldquoA common fixed point theorem forweakly compatible mappings in symmetric spaces satisfying animplicit relationrdquo Sociedad Matematica Mexicana vol 13 no 1pp 195ndash205 2007

[49] M Imdad A Sharma Chauhan and S ldquoUnifying a multitudeof metrical fixed point theorems in symmetric spacesrdquo FilomatIn press

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