Common fixed point theorems under an implicit contractive condition on metric spaces endowed with an arbitrary binary relation and an application

0.17 MB·

by  Md Ahmadullah

Common fixed point theorems under an implicit contractive condition on metric
spaces endowed with an arbitrary binary relation and an application
MdAhmadullaha,∗,MohammadImdadb,MohammadArifc
aDepartmentofMathematics,AligarhMuslimUniversity,Aligarh,-202002,U.P.,India.
bDepartmentofMathematics,AligarhMuslimUniversity,Aligarh,-202002,U.P.,India.
cDepartmentofMathematics,AligarhMuslimUniversity,Aligarh,-202002,U.P.,India.
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2 Abstract

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Theaimofthispaperisto establishsomemetricalcoincidenceandcommonfixedpointtheoremswithanarbitrary
a
J relationunderanimplicitcontractiveconditionwhichisgeneralenoughtocoveramultitudeofwellknowncontraction

0 conditionsinonegobesidesyieldingseveralnewones. Wealsoprovideanexampletodemonstratethegeneralityof
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ourresultsoverseveralwellknowncorrespondingresultsoftheexistingliterature. Finally,weutilizeourresultsto

] proveanexistencetheoremforensuringthesolutionofanintegralequation.
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Keywords: Fixedpoint,completemetricspaces,binaryrelations,implicitrelation,contractionmappings
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2010MSC: 47H10,54H25
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[ 1. Introduction

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The origin of metric fixed point theory is solely attributed to classical Banach contraction principle which was
v
4 originatedinthePh.D.thesisofBanachin1920. Thisworkwaslaterpublishedintheformofaresearcharticle[11]
5
in 1922 which has already earned around 2000 google citations. The strength of Banach contractionprinciple lies
1
3 in its applicationswhichfall within the severaldomainsuch as: FunctionalAnalysis, GeneralTopology,Algerbaic
0 Topology,DifferentialEquation,LinearAlgebra,EngineeringMathematics,DiscreteMathematics,Economicsetc. In
.
1 thelongcourseoflastseveraldecade,thisnaturalprinciplehasbeengeneralizedandimprovedbyseveralresearchers
0
inthedifferentdirectionsnamely:
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1
: • byweakeningtheinvolvedmetricalnotions,
v
i
X • byenlargingtheclassofunderlyingspaces,
r
a • byreplacingcontractionconditionwithrelativelyweakercontractivecondition,
andsuchpracticeisstillinbusiness.
Popa [30] initiated the idea of an implicit relation which is designed to cover several well known contraction
conditionsoftheexistingliteratureinonegobesidesadmittingseveralnewones. Indeed,thestrengthofanimplicit
relation lies in their unifyingpower besides being general enoughto yield new contractionconditions. For further
detailsonimplicitrelation,onecanconsult[1,8,9,12,13,19–21,30,31]andreferencescitedtherein.
∗Correspondingauthor
Emailaddresses:[email protected](MdAhmadullah),[email protected](MohammadImdad),
[email protected](MohammadArif)
Preprintsubmittedto... January13,2017
The initiationoforder-theoreticmetric fixedpointtheorycanbe attributedto Turinici[36]. Often itis believed
thatsuchresultswereinitiatedintheinterestingarticleofRanandReurings[32]butthisisnotareality. Indeedthe
resultsandapplicationpresentedin Ran andReuringsare morenaturalandinspiringascomparedto otherrelevant
result of this kind. Thereafter, this natural result due to Ran and Reurings was notably generalized by Nieto and
Rodr´ıguez-Lo´pez [28, 29] which also remain the core results in this direction. In the recent year, various type of
relation-theoreticfixed andcommonfixed pointresults were proved. For the work of thiskind onecan be referred
[1–10,14,15,24,28–30,33,36]andreferencescitedtherein.
Recently, Ahmadullah et al. [1] established unified metrical fixed point theorems via an implicit contractive
conditionemployingrelation-theoreticnotions,whichgeneralizeseveralwellknownresultsoftheexistingliterature.
Ouraimofthispaperistoproverelation-theoreticcoincidenceandcommonfixedpointresultsunderanimplicit
contractivecondition.Themainresultsofthispaperarebasedonthefollowingmotivationsandobservations:
(i) toextendtheresultsofAhmadullahetal. [1](especiallyTheorems1and2)toapairofself-mappings,
(ii) theconditionR-completenessontheinvolvedspace X intheearliermentionedtheorems(duetoAhmadullah
et al. [1]) are replaced by relatively weaker condition of R-completeness of any subspace Y ⊆ X, wherein
T(X)⊆Y ⊆ X,
(iii) widening the class of continuous implicit relations by replacing it with the class of lower semi-continuous
implicitrelations,whichalsocovercertainnonlinearcontractionsaswell,
(iv) examplesareutilizedtohighlightthegenueinessofournewlyprovedresults,and
(v) asanapplicationofourmainresult,theexistenceofthesolutionofanintegralequationisproved.
2. Preliminaries
Thissectiondealswithsomebasicrelevantdefinitions,lemmasandpropositions.
2.1. ImplicitRelation
Inordertodescribeourimplicitrelation,letΦbethesetofallnon-negativerealvaluedfunctionsφ : R+ → R+
satisfyingthefollowingconditions:
(i) φisincreasingandφ(0)=0,
∞
(ii) φn(t)<∞, fort>0,whereφnisnth-iterate.
Xn=1
LetGbethecollectionofalllowersemi-continuousrealvaluedfunctionsG :R6 →Rwhichsatisfythefollowing
+
conditions:
(G ) G is decreasingin the fifth and sixth variables; andG(r,s,s,r,r+ s,0) ≤ 0 forall r,s ≥ 0 implies thatthere
1
existsφ∈Φsuchthatr≤φ(s);
(G ) G(r,0,r,0,0,r)>0, forallr>0.
2
Let F be collection of all lower semi-continuous real valued functions which is relativity smaller than G. Let
G :R6 →Rwhichsatisfy(G )and(G )alongwiththefollowingadditionalcondition:
+ 1 2
2
(G ) G(r,r,0,0,r,r)>0,forallr>0.
3
Example2.1. ThefunctionG :R6 →Rdefinedby
+
r +r
r −ϕ r 5 6 , if r +r ,0;
G(r1,r2,r3,r4,r5,r6)=r11−r2(cid:16),2r3+r4(cid:17) if r3+3 r4 =4 0,
whereϕ:R+ →R+isuppersemi-continuousmapping,satisfiestheproperties(G1)and(G2)withφ=ϕbutdoesnot
satisfytheproperty(G ).
3
Example2.2. TheimplicitrelationsG :R6 →Rdefinedbelowsatisfytheforegoingrequirements(see[1,8,12,19,
+
20,31]):
I.G(r ,r ,r ,r ,r ,r )=r −kr , wherek∈[0,1);
1 2 3 4 5 6 1 2
II.G(r1,r2,r3,r4,r5,r6)=r1−ϕ r2 , whereϕ:R+ →R+ is anuppersemi-continuous mappingsuch that
ϕ(t)<t, ∀t>0; (cid:0) (cid:1)
III.G(r ,r ,r ,r ,r ,r )=r −k(r +r ), wherek∈[0,1/2);
1 2 3 4 5 6 1 3 4
IV.G(r ,r ,r ,r ,r ,r )=r −k(r +r ), wherek∈[0,1/2);
1 2 3 4 5 6 1 5 6
V.G(r ,r ,r ,r ,r ,r )=r −a r −a (r +r )−a (r +r ), wherea ,a ,a ∈[0,1)anda +2a +2a <1;
1 2 3 4 5 6 1 1 2 2 3 4 3 5 6 1 2 3 1 2 3
VI.G(r ,r ,r ,r ,r ,r )=r −kr −Lmin{r ,r ,r ,r }, wherek∈[0,1)andL≥0;
1 2 3 4 5 6 1 2 3 4 5 6
VII.G(r ,r ,r ,r ,r ,r )=r −kmax r ,r ,r ,r5+r6 −Lmin{r ,r ,r ,r },wherek ∈[0,1)andL≥0;
1 2 3 4 5 6 1 2 3 4 2 3 4 5 6
(cid:8) (cid:9)
VIII.G(r ,r ,r ,r ,r ,r )=r −kmax{r ,r ,r ,r ,r }, wherek∈[0,1/2);
1 2 3 4 5 6 1 2 3 4 5 6
IX.G(r ,r ,r ,r ,r ,r )=r −(a r +a r +a r +a r +a r ),wherea’s>0(fori=1,2,3,4,5); and
1 2 3 4 5 6 1 1 2 2 3 3 4 4 5 5 6 i
sumofthemisstrictlylessthan1;
r r
X.G(r ,r ,r ,r ,r ,r )=r −kmax r ,r ,r , 5, 6 ,wherek∈[0,1);
1 2 3 4 5 6 1 2 3 4
2 2
n o
XI.G(r ,r ,r ,r ,r ,r )=r −k max{r ,r ,r }−(1−k)(ar +br ), wherek ∈[0,1)and0≤a,b<1/2;
1 2 3 4 5 6 1 2 3 4 5 6
XII.G(r ,r ,r ,r ,r ,r )=r2−r a r +a r +a r −a r r ,wherea >0;a ,a ,a ≥0;
1 2 3 4 5 6 1 1 1 2 2 3 3 4 4 5 6 1 2 3 4
a +a +a <1anda +a <1; (cid:0) (cid:1)
1 2 3 1 4
r +r
r −kr 5 6, if r +r ,0;
XIII.G(r1,r2,r3,r4,r5,r6)=r11, 2r1+r2 if r11+r22=0, wherek∈[0,1);
XIV.G(r ,r ,r ,r ,r ,r )=r2−a max{r2,r2,r2}−a max{r r ,r r }−a r r ,wherea’s≥0(fori=1,2,3);
1 2 3 4 5 6 1 1 2 3 4 2 3 5 4 6 3 5 6 i
a +2a <1anda +a <1;
1 2 1 3
XV.G(r ,r ,r ,r ,r ,r )=r3−k r3+r3+r3+r3+r3 ,wherek∈[0,1/11);
1 2 3 4 5 6 1 2 3 4 5 6
(cid:0) (cid:1)
r r r r
r −a 2 4 −a 3 6 , if r +r ,0;
XVI.G(r1,r2,r3,r4,r5,r6)=r11, 1r2+r4 2r5+r6+1 if r22+r44 =0,
 3
wherea ,a >0anda <2.
1 2 1
2.2. Relevantrelation-theoreticnotions
Withaviewtohaveapossiblyself-containedpresentation,werecallsomebasicdefinitions,lemmasandproposi-
tionsneededinoursubsequentdiscussion.
Definition2.1. [22,23]LetT andgbetwoself-mappingsdefinedonanon-emptysetX. Then
(i) apointx∈X issaidtobeacoincidencepointofT andgifTx=gx,
(ii) apointx∈X issaidtobeapointofcoincidenceofT andgifthereexistssomex∈Xsuchthatx=Tx=gx,
(iii) acoincidencepointx∈ XofT andg,issaidtobeacommonfixedpointif x=Tx=gx,
(iv) T andgarecalledcommutingifT(gx)=g(Tx),∀x∈ X.
Definition2.2. [23,34,35]LetT andgbetwoself-mappingsdefinedonametricspace(X,d).Then
(i) T andgaresaidtobeweaklycommutingifforallx∈X, d(T(gx),g(Tx))≤d(Tx,gx),
(ii) T andg aresaidtobecompatibleiflim d(T(gx ),g(Tx )) = 0whenever{x } ⊂ X isa sequencesuchthat
n→∞ n n n
lim gx =lim Tx ,
n→∞ n n→∞ n
d d
(iii) T issaidtobeag-continuousatx∈XifTx −→Txwhenevergx −→gx,forallsequence{x }⊂ X.Moreover,
n n n
T issaidtobeag-continuousifitiscontinuousateverypointofX.
Definition2.3. [26] A subsetR of X ×X is called a binaryrelationon X. We say that“x relatesy underR” if and
onlyif(x,y)∈R.
Throughoutthispaper,Rstandsfora‘non-emptybinaryrelation’(i.e.,R , ∅)insteadof‘binaryrelation’while
N denotesthesetofwholenumbersi.e.,N =N∪{0}.
0 0
Definition2.4. [27]AbinaryrelationRdefinedonanon-emptysetXiscalledcompleteifeverypairofelementsof
Xarecomparableunderthatrelationi.e.,forallx,yinX,either(x,y)∈Ror(y,x)∈Rwhichisdenotedby[x,y]∈R.
Proposition2.1. [4]LetRbeabinaryrelationdefinedonanon-emptysetX. Then(x,y)∈Rs ⇐⇒[x,y]∈R.
Definition2.5. [4] LetT bea self-mappingdefinedonanon-emptyset X. Thena binaryrelationR on X is called
T-closedif(Tx,Ty)∈Rwhenever(x,y)∈R, forallx,y∈ X.
Definition2.6. [5]LetT andgbetwoself-mappingsdefinedonanon-emptysetX. ThenabinaryrelationRonXis
called(T,g)-closedif(Tx,Ty)∈Rwhenever(gx,gy)∈R, forallx,y∈ X.
Noticethatonsettingg= I,theidentitymappingonX,Definition2.6reducestoDefinition2.5.
Definition2.7. [4]LetRbeabinaryrelationdefinedonanon-emptysetX. Thenasequence{x } ⊂ X issaidtobe
n
R-preservingif(xn,xn+1)∈R, ∀n∈N0.
Definition2.8. [5]Let(X,d)beametricspaceequippedwithabinaryrelationR. Then(X,d)issaidtobeR-complete
ifeveryR-preservingCauchysequenceinXconvergestoapointinX.
4
Remark2.1. [5]EverycompletemetricspaceisR-complete,whereRdenotesabinaryrelation. Particularly,ifRis
universalrelation,thennotionsofcompletenessandR-completenesscoincide.
Definition2.9. [5]Let(X,d)beametricspaceequippedwithabinaryrelationR. Thenaself-mappingT onXissaid
d d
tobeR-continuousat xifTx −→ Txwhenever x −→ x,foranyR-preservingsequence{x } ⊂ X. Moreover,T is
n n n
saidtobeR-continuousifitisR-continuousateverypointofX.
Definition2.10. [5]Let(X,d)beametricspaceequippedwithabinaryrelationRandgaself-mappingonX. Thena
d
self-mappingT onX issaidtobe(g,R)-continuousatxifTx −→Tx,foranyR-preservingsequence{x }⊂ Xwith
n n
d
gx −→gx. Moreover,T iscalled(g,R)-continuousifitis(g,R)-continuousateverypointofX.
n
Noticethatonsettingg= I,theidentitymappingonX,Definition2.10reducestoDefinition2.9.
Remark2.2. Every continuous mapping is R-continuous, where R denotes a binary relation. Particularly, if R is
universalrelation,thennotionsofR-continuityandcontinuitycoincide.
Definition2.11. [4]Let(X,d)beametricspace. ThenabinaryrelationRonX issaidtobed-self-closedifforany
R-preservingsequence{x }withx −→d x,thereisasubsequence{x }of {x }suchthat[x ,x]∈R, forallk ∈N .
n n nk n nk 0
Definition2.12. [33]Let(X,d)beametricspaceequippedwithabinaryrelationR. ThenasubsetDofX issaidto
beR-directedifforeverypairofpointsx,yinD,thereiszinX suchthat(x,z)∈Rand(y,z)∈R.
Definition2.13. [33]Let(X,d)beametricspaceequippedwithabinaryrelationRandgaself-mappingonX.Then
asubsetDofXissaidtobe(g,R)-directedifforeverypairofpointsx,yinD,thereiszinXsuchthat(x,gz)∈Rand
(y,gz)∈R.
Definition2.14. [25] Let(X,d)bea metricspaceequippedwitha binaryrelationR andT,gtwo self-mappingson
X. ThenT andgaresaidtobeR-compatibleif lim d(g(Tx ),T(gx ))=0,whenever lim g(x )= lim T(x ),forany
n n n n
n→∞ n→∞ n→∞
sequence{x }⊂ Xsuchthatthesequences{Tx }and{gx }areR-preserving.
n n n
Definition2.15. [17]LetRbeabinaryrelationdefinedonanon-emptyset X and x,yapairofpointsinX. Ifthere
isafinitesequence{w0,w1,w2,...,wl}⊂ Xsuchthatw0 = x,wl =yand(wi,wi+1)∈Rforeachi∈{0,1,2,···,l−1},
thenthisfinitesequenceiscalledapathoflengthl(wherel∈N)joiningxtoyinR.
Forourfutureuse,wealsointroducethefollowingdefinition:
Definition2.16. LetRbeabinaryrelationdefinedonanon-emptysetX andgaself-mappingonX. Ifforapairof
points x,y in X, there is a finite sequence {w0,w1,w2,...,wl} ⊂ X such that gw0 = x,gwl = y and (gwi,gwi+1) ∈ R
foreachi ∈ {0,1,2,···,l−1},thenthefinitesequence{w ,w ,w ,...,w}iscalledag-pathoflengthl(wherel ∈N)
0 1 2 l
joiningxtoyinR.
Noticethat,apathoflengthlinvolves(l+1)elementsofXandneednotbedistinctingeneral.Observethatwith
g= I (theidentitymappingonX),Definition2.16reducestoDefinition2.15.
Lemma 2.1. [18] Let g be a self-mapping defined on a non-empty set X. Then there exists a subset Z ⊆ X with
g(Z)=g(X)andg:Z →Xisone-one.
Givenanon-emptyset X,abinaryrelationRonX,self-mappingsT,gonX andaR-directedsubsetDof X,we
usethefollowingnotations:
5
• C(T,g):thecollectionofallcoincidencepointsofTandg;
• X(T,g,R):thesetofallpointsinw∈ Xsuchthat(gw,Tw)∈R;
• ∆(D,g,R):=∪ z∈ X :(x,gz)∈Rand(y,gz)∈R ;
x,y∈D
(cid:8) (cid:9)
• Υ (x,y,R): thecollectionofallg-pathsjoiningxtoyinRwherex,y∈ X;
g
• Υ (x,y,T,R): the collectionof all g-paths{w ,w ,w ,...,w} joining x to y in R suchthat [gw,Tw] ∈ R for
g 0 1 2 l i i
eachi∈{1,2,3,···,l−1}.
Noticethat,withg= I,identitymappingonX,thefamilyΥ (x,y,T,R)coincideswithΥ(x,y,T,R).
g
3. Mainresults
Now,weareequippedtoproveourmainresultasunder:
Theorem3.1. Let(X,d)bea metric spaceequippedwith abinaryrelationR andY anR-completesubspaceof X.
LetT andgbetwoself-mappingsonX. Assumethatthefollowingconditionshold:
(a) ∃ x ∈X suchthat(gx ,Tx )∈R,
0 0 0
(b) T(X)⊆Y∩g(X),
(c) Ris(T,g)-closed,
(d) thereexistsanimplicitrelationG ∈Gsuchthat forallx,y∈ X with(gx,gy)∈R
(cid:0) (cid:1)
G d(Tx,Ty),d(gx,gy),d(gx,Tx),d(gy,Ty),d(gx,Ty),d(gy,Tx) ≤0,
(cid:0) (cid:1)
(e) (e ) Y ⊆g(X)
1
(e ) eitherT is(g,R)-continuousorT andgarecontinuousorR| isd-self-closed,
2 Y
or,alternatively
(e′) (e′) T andgareR-compatible,
1
(e′) T andgareR-continuous.
2
ThenT andghaveacoincidencepoint.
Proof. Suppose x ∈ X such that(gx ,Tx ) ∈ R hypothesis(a) . In view of (b), T(X) ⊆ Y and T(X) ⊆ g(X),we
0 0 0
choosex ∈ Xsothatgx =Tx . Next,choosex ∈(cid:0) Xsuchthatgx(cid:1) =Tx . Continuinginthisway,weget
1 1 0 2 2 1
gxn+1 =Txn, ∀n∈N0. (1)
Usingthehypothesis(c),wehave
(Tx ,T2x ),(T2x ,T3x ),··· ,(Tnx ,Tn+1x ),···∈R.
0 0 0 0 0 0
Noticethat,
(gxn,gxn+1)∈R, ∀n∈N0, (2)
6
sothatthesequence{gx }isR-preserving.Onusingthecondition(d),wehave(foralln∈N )
n 0
G d(Txn,Txn+1),d(gxn,gxn+1),d(gxn,Txn),d(gxn+1,Txn+1),d(gxn,Txn+1),d(gxn+1,Txn) ≤0,
(cid:0) (cid:1)
or,
G d(gxn+1,gxn+2),d(gxn,gxn+1),d(gxn,gxn+1),d(gxn+1,gxn+2),d(gxn,gxn+2),d(gxn+1,gxn+1) ≤0.
(cid:0) (cid:1)
Puttingr =d(gxn+1,gxn+2)ands=d(gxn,gxn+1)intheaboveinequality,wehave
G r,s,s,r,d(gxn,gxn+2),0 ≤0.
(cid:0) (cid:1)
OnusingtriangularinequalityanddecreasingpropertyofGinthefifthvariable,wehave
G r,s,s,r,r+s,0 ≤0,
(cid:0) (cid:1)
implyingthereby(owingto(G ))theexistenceofsomeφ∈Φsuchthatr≤φ(s), i.e.,
1
d(gxn+1,gxn+2)≤φ d(gxn,gxn+1) ,
(cid:0) (cid:1)
whichinductivelygivesarise
d(gxn+1,gxn+2)≤φn+1 d(gx0,gx1) , ∀n∈N0. (3)
(cid:0) (cid:1)
Using(3)andtriangularinequality,foralln,m∈N withm>n,wehave
0
d(gxn,gxm) ≤ d(gxn,gxn+1)+d(gxn+1,gxn+2)+···+d(gxm−1,gxm)
m−1
≤ φj d(gx ,gx )
0 1
Xj=n (cid:0) (cid:1)
≤ φj d(gx ,gx )
0 1
X
j≥n (cid:0) (cid:1)
→ 0asn→∞.
Therefore, {gx } is a Cauchy sequence in Y (in view (1) and T(X) ⊆ Y). Hence, {gx } is an R-preserving Cauchy
n n
d
sequenceinY. SinceY isR-complete,∃y∈Y suchthatgx −→y.AsY ⊆g(X)thereexistssomew∈ Xsuchthat
n
lim gx =y=gw. (4)
n
n→∞
Inviewofthehypothesis(e ),firstlyweassumethatT is(g,R)-continuous.Onusing(2)and(4),weget
2
lim gxn+1 = lim Txn =Tw.
n→∞ n→∞
Bytheuniquenessoflimit,wehaveTw=gw,sothatwisacoincidencepointofT andg.
Next,supposethatT andgarecontinuous. FromLemma2.1,thereexistsasubsetZ ⊆ X suchthatg(Z) = g(X)
andg:Z →X isone-one.Now,defineh:g(Z)→g(X)by
h(gz)=Tz, ∀gz∈g(Z)wherez∈Z. (5)
Sincegisone-oneandT(X)⊆g(Z),hiswelldefined.AsT andgarecontinuous,soish.Onusingthefactg(Z)=g(X)
andtheconditions(b)and(e ),wehaveT(X)⊆g(Z)∩Y andY ⊆g(X)whichensuresthatavailabilityofasequence
1
{x }⊂Zsatisfying(1). Takew∈Z. Onusing(4),(5)andthecontinuityofh,weget
n
Tw=h(gw)=h(lim gx )= lim h(gx )= lim Tx =gw,
n n n
n→∞ n→∞ n→∞
7
sothatwisacoincidencepointT andg.
d
Finally, assume thatR| is d-self-closed. Since {gx }is an R-preservingin Y andgx −→ gw, thereis a subse-
Y n n
quence{gx }of {gx }with[gx ,gw] ∈ R| ⊆ R, ∀k ∈ N .Noticethat,∀k ∈ N , [gx ,gw] ∈ Rimpliesthateither
nk n nk Y 0 0 nk
(gx ,gw)∈Ror,(gw,gx )∈R. Applyingthecondition(d)to(gx ,gw)∈R, ∀k ∈N ,wehave
nk nk nk 0
G d(Tx ,Tw),d(gx ,gw),d(gx ,Tx ),d(gw,Tw),d(gx ,Tw),d(gw,Tx ) ≤0,
nk nk nk nk nk nk
(cid:0) (cid:1)
or,
G d(gxnk+1,Tw),d(gxnk,gw),d(gxnk,gxnk+1),d(gw,Tw),d(gxnk,Tw),d(gw,gxnk+1) ≤0.
(cid:0) (cid:1)
d
Takingliminfask→∞;usinggx −→gw,lowersemi-continuityofGandcontinuityofd,weobtain
nk
G d(gw,Tw),0,0,d(gw,Tw),d(gw,Tw),0 ≤0.
(cid:0) (cid:1)
Hence,owingto(G ),weobtaind(gw,Tw)=0,sothatTw=gw, i.e., wisacoincidencepointofT andg.
1
Similarly,if(gw,x )∈ R, ∀k ∈N ,thenowingto(G ),weobtaind(Tw,gw)=0,sothatTw =gw, i.e., wisa
nk 0 2
coincidencepointofT andg.
Alternatively, suppose that (e′) holds. As {gx } ⊂ T(X) ⊆ Y, (in view (1)) we infer that {gx } is R-preserving
n n
CauchysequenceinY. SinceY isR-complete,thereexistsy∈Y suchthat
lim gx =y and lim Tx =y. (6)
n n
n→∞ n→∞
As{Tx }and{gx }areR-preserving(dueto(1)and(2)),usingthecondition(e′),weobtain
n n 1
lim d(g(Tx ),T(gx ))=0. (7)
n n
n→∞
Using(2),(6)andthecondition(e′),wehave
2
lim g(Tx )=g(limTx )=gy. (8)
n n
n→∞ n→∞
and
lim T(gx )=T(lim gx )=Ty. (9)
n n
n→∞ n→∞
InordertoproveTy=gy,applying(7)-(9)andcontinuityofd,wehave
d(Ty,gy) = d(lim T(gx ), lim g(Tx ))
n n
n→∞ n→∞
= lim d(T(gx ),g(Tx ))
n n
n→∞
= 0,
yieldingtherebyTy=gy.Thisconcludestheproof.
Now,wepresenttheuniquenessofcommonfixedpointresult,whichrunsas:
Theorem3.2. InadditiontothehypothesesofTheorem3.1,supposethatthefollowingconditionshold:
(u ) Υ (α,β,T,R|s )isnon-empty,foreachα,β∈T(X),
1 g g(X)
(u ) T andgarecommuteattheircoincidencepointswhereinGalsoenjoys(G ).
2 3
ThenT andghaveauniquecommonfixedpoint.
8
Proof. Wedividetheproofinthreesteps.
Step 1: Observe that (in view of Theorem 3.1)C(T,g) is non-empty. To substantiate the proof, take two arbitrary
elementsu,vinC(T,g),sothat
Tu=gu= x and Tv=gv=y (10)
Now,wearerequiredtoshowthatx=y.
Inviewofthehypothesis(u ), thereexistsa g-path(say,{w ,w ,w ,...,w}) oflengthlinR|s fromTutoTv,
1 0 1 2 l g(X)
with
gw0 =Tu, gwl =Tv, [gwi,gwi+1]∈R|g(X) ⊆R, foreachi∈{0,1,2,···,l−1} (11)
and
[gw,Tw]∈R| ⊆R, foreachi∈{1,2,··· ,l−1}. (12)
i i g(X)
Definetwoconstantsequences
w0 =uandwl =v.
n n
Thenonusing(10),foralln∈N
0
Tw0 =Tu= x, andTwl =Tv=y
n n
Setting,
wi =w foreachi∈{0,1,2,···,l}, (13)
0 i
weconstructjointsequence{wi},i.e., Twi =gwi correspondingtoeachw. Since[gwi,gwi]∈R inviewof(11)
n n n+1 i 0 1
and(12) ,thenonusing(3)and(T,g)-closednessofR,weget (cid:0)
(cid:1)
lim d(gwi,gwi )=0, foreachi∈{1,2,··· ,l−1}. (14)
n→∞ n n+1
Onusing[gwi,gwi+1]∈R inviewof(11)and(13) and(T,g)-closednessofR,weobtain
0 0
(cid:0) (cid:1)
[Twi,Twi+1]∈R, foreachi∈{0,1,2,···,l−1}andforalln∈N ,
n n 0
or, [gwi,gwi+1]∈R, foreachi∈{0,1,2,···,l−1}andforalln∈N .
n n 0
Defineti :=d(gwi,gwi+1), foralln∈N andforeachi∈{0,1,2,···,l−1}.Weassertthat, lim ti =0.Supposeon
n n n 0 n
n→∞
contrarythat lim ti =t>0. Since[gwi,gwi+1]∈R,either(gwi,gwi+1)∈Ror,(gwi+1,gwi)∈R. If(gwi,gwi+1)∈R,
n n n n n n n n n
n→∞
thenapplyingthecondition(d),wehave
G d(Twi,Twi+1),d(gwi,gwi+1),d(gwi,Twi),d(gwi+1,Twi+1),d(gwi,Twi+1),d(gwi+1,Twi) ≤0,
n n n n n n n n n n n n
(cid:0) (cid:1)
or, G d(gwi ,gwi+1),d(gwi,gwi+1),d(gwi,gwi ),d(gwi+1,gwi+1),d(gwi,gwi+1),d(gwi+1,gwi ) ≤0.
n+1 n+1 n n n n+1 n n+1 n n+1 n n+1
(cid:0) (cid:1)
Asd(gwi,gwi+1)≤d(gwi,gwi )+d(gwi ,gwi+1)andGisdecreasinginfifthvariable,weget
n n+1 n n+1 n+1 n+1
G d(gwi ,gwi+1),d(gwi,gwi+1),d(gwi,gwi ),d(gwi+1,gwi+1),
n+1 n+1 n n n n+1 n n+1
(cid:0) d(gwi,gwi )+d(gwi ,gwi+1),d(gwi+1,gwi ) ≤0.
n n+1 n+1 n+1 n n+1
(cid:1)
Takingliminfasn→∞andusing lim ti =talongwiththelowersemi-continuityofGand(14),weget
n
n→∞
G t,t,0,0,t,t ≤0,
(cid:0) (cid:1)
9
whichiscontradiction(inviewof(G ))andhence(foreachi∈{0,1,2,···,l−1})
3
lim ti =t=0.
n
n→∞
Similarly,if(gwi+1,gwi)∈R,thenasearlier,weobtain(foreachi∈{0,1,2,···,l−1})
n n
lim ti =t=0.
n
n→∞
Thus,
lim ti := lim d(gwi,gwi+1)=0, foreachi∈{0,1,2,···,l−1}.
n n n
n→∞ n→∞
Using(10), lim ti =0andtriangularinequality,wehave
n
n→∞
l−1
d(x,y)=d(gw0,gwl)≤ d(gwi,gwi+1)
n n n n
Xi=0
l−1
= ti
n
Xi=0
→0 as n→∞,
sothatd(x,y)=0implyingtherebyx=y.Therefore,gx=gy.
Step2:ToprovetheexistenceofcommonfixedpointT andg,letu∈C(T,g), i.e., Tu=gu.SinceT andgcommute
attheircoincidencepoints,wehave
T(gu)=g(Tu)=g(gu). (15)
Putgu=z. Thenfrom(15),Tz=gz.HencezisalsoacoincidencepointofT andg. FromStep1,wehave
z=gu=gz=Tz,
sothatzisacommonfixedpointT andg.
Step3: To provethe uniquenessof commonfixed pointof T and g, letus assume thatw is anothercommonfixed
pointofT andg.Thenw∈C(T,g),byStep1,
w=gw=gz=z.
Thus,T andghaveauniquecommonfixedpoint.Thiscompletestheproof.
IfR| iscompleteorT(X)is(g,R|s )-directed,thenthefollowingcorollaryisworthrecording.
g(X) g(X)
Corollary3.1. TheconclusionsofTheorem3.2remaintrueifthecondition(u )isreplacedbyoneofthefollowing
1
conditionsbesidesretainingtherestofthehypotheses:
(u′) R| iscomplete;
1 g(X)
(u′′) T(X)is(g,R|s )-directedand∆(T(X),g,Rs)⊆X(T,g,Rs).
1 g(X)
Proof. Supposethatthecondition(u′)holds. Takeanarbitrarypairofpointsα, βinT(X). Owingtothehypothesis,
1
T(X) ⊆ g(X), there exist x,y ∈ X such that α = gx, β = gy. As R| is complete, [gx,gy] ∈ R| which shows
g(X) g(X)
that{x,y}is a g-pathoflength1fromαto βinR|s , so thatΥ (α,β,T,R|s ) isnon-empty. Now,on thelinesof
g(X) g g(X)
Theorem3.2,resultfollows.
10