Fixed point of set valued graph contractive mappings

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R E S E A R C H

Open Access

Fixed point of set-valued graph contractive

mappings

Ismat Beg

1*

_{and Asma Rashid Butt}

2

*_{Correspondence:}

begismat@yahoo.com

1_{Centre for Mathematics and}

Statistical Sciences, Lahore School of Economics, Lahore, Pakistan Full list of author information is available at the end of the article

Abstract

Let (X,d) be a metric space and letF,Hbe two set-valued mappings onX. We obtained suﬃcient conditions for the existence of a common ﬁxed point of the mappingsF,Hin the metric spaceXendowed with a graphGsuch that the set of vertices ofG,V(G) =Xand the set of edges ofG,E(G)⊆X×X.

MSC: Primary 47H10; secondary 47H04; 47H07; 54C60; 54H25

Keywords: ﬁxed point; directed graph; metric space; set-valued mapping

1 Introduction and preliminaries

Edelstein [] generalized classical Banach’s contraction mapping principle and Nadler [] proved Banach’s fixed point theorem for set-valued mappings. Recently several extensions of Nadler’s theorem in different directions were obtained; see [–]. Beg and Azam [] extended Edelstein’s theorem by considering a pair of set-valued mappings with a gen-eral contractive condition. The aim of this paper is to study the existence of common fixed points for set-valued graph contractive mappings in metric spaces endowed with a graphG. Our results improve/generalize [, , ] and several other known results in the literature.

Let (X,d) be a complete metric space and letCB(X) be a class of all nonempty closed and bounded subsets ofX. ForA,B∈CB(X), let

D(A,B) :=max

sup

b∈B

d(b,A),sup

a∈A

d(a,B)

,

where

d(a,B) :=inf

b∈Bd(a,b).

MappingDis said to be aHausdorﬀ metricinduced byd.

Deﬁnition . LetF:X→Xbe a set-valued mapping,i.e.,Xx→Fxis a subset ofX. A pointx∈Xis said to be aﬁxed pointof the set-valued mappingFifx∈Fx.

Deﬁnition . A metric space (X,d) is called aε-chainable metric space for someε>  if givenx,y∈X, there isn∈Nand a sequence (xi)ni=such that

x=x, xn=y and d(xi–,xi) <ε fori= , . . . ,n.

LetFixF:={x∈X:x∈Fx}denote the set of ﬁxed points of the mappingF.

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Deﬁnition . Let (X,d) be a metric space,ε> , ≤κ<  andx,y∈X. A mappingf :

X→Xis called (ε,κ) uniformly locally contractive if  <d(x,y) <ε⇒d(fx,fy) <κd(x,y).

The following signiﬁcant generalization of Banach’s contraction principle [, Theo-rem . ] was obtained by Edelstein [].

Theorem .[] Let(X,d)be aε-chainable complete metric space.If f :X→X is a(ε,κ)

uniformly locally contractive mapping,then f has a unique ﬁxed point.

Afterwards, in , Nadler [] proved a set-valued extension of Banach’s theorem and obtained the following result.

Theorem .[] Let(X,d)be a complete metric space and F:X→CB(X).If there exists

κ∈(, )such that

D(Fx,Fy)≤κd(x,y) for all x,y∈X,

then F has a ﬁxed point in X.

Nadler [] also extended Edelstein’s theorem for set-valued mappings.

Theorem .[] Let(X,d)be aε-chainable complete metric space for someε> and let F:X→C(X)be a set-valued mapping such that Fx is a nonempty compact subset of X.If F satisﬁes the following condition:

x,y∈X and  <d(x,y) <ε ⇒ D(Fx,Fy) <κd(x,y),

then F has a ﬁxed point.

Consider a directed graphGsuch that the set of its vertices coincides withX(i.e.,V(G) :=

X) and the set of its edgesE(G) :={(x,y) : (x,y)∈X×X,x=y}. We assume thatGhas no parallel edges and weighted graph by assigning to each edge the distance between the vertices; for details about deﬁnitions in graph theory, see [].

We can identifyGas (V(G),E(G)).G–_{denotes the conversion of a graph}_G_{, the graph}

obtained fromGby reversing the direction of its edges.Gdenotes the undirected graph obtained fromGby ignoring the direction of edges ofG. We considerGas a directed graph for which the set if its edges is symmetric, thus we have

E(G) :=E(G)∪EG–.

Deﬁnition . Asubgraphof a graphGis a graphHsuch thatV(H)⊆V(G) andE(H)⊆

E(G) and for any edge (x,y)∈E(H),x,y∈V(H).

Deﬁnition . Letxandybe vertices in a graphG. ApathinGfromxtoyof lengthn

(n∈N∪ {}) is a sequence (xi)ni=ofn+  vertices such thatx=x,xn=yand (xi–,xi)∈

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Deﬁnition . The number of edges inGconstituting the path is called thelength of the path.

Deﬁnition . A graphGisconnectedif there is a path between any two vertices ofG.

If a graphGis not connected, then it is calleddisconnected. Moreover,Gis weakly con-nected ifGis connected.

Assume thatGis such thatE(G) is symmetric, andxis a vertex inG, then the subgraph

Gxconsisting of all edges and vertices, which are contained in some path inGbeginning at

x, is called the component ofGcontainingx. In this case the equivalence class [x]Gdeﬁned onV(G) by the ruleR(uRvif there is a path fromutov) is such thatV(Gx) = [x]G.

PropertyA: For any sequence (xn)n∈NinX, ifxn→xand (xn,xn+)∈E(G) forn∈N, then

(xn,x)∈E(G).

Deﬁnition . Let (X,d) be a metric space andF,H:X→CB(X). The mappingsF,H

are said to be graph contractive if there existsκ∈(, ) such that

(x=y), (x,y)∈E(G) ⇒ D(Fx,Hy) <κd(x,y),

and ifu∈Fxandv∈Hyare such that

d(u,v) <d(x,y),

then (u,v)∈E(G).

Deﬁnition . Apartial order is a binary relation over a setXwhich satisﬁes the following conditions:

. xx(reﬂexivity);

. ifxyandyx, thenx=y(antisymmetry); . ifxyandyz, thenxz(transitivity);

for allx,yandzinX.

A set with a partial orderis called apartially ordered set.

Let (X,) be a partially ordered set andx,y∈X. Elementsxandyare said to be compa-rable elementsofXif eitherxyoryx.

Letbe a partial order inX. Deﬁne the graphG:=Gby

E(G) :=(x,y)∈X×X:xy,x=y, andG:=Gby

E(G) :=(x,y)∈X×X:xy∨yx,x=y.

The class ofG-contractive mappings was considered in [] and that ofG-contractive

mappings in [].

The weak connectivity ofGorGmeans, givenx,y∈X, there is a sequence (xi)ni=such

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We shall make use of the following lemmas due to Nadler [], Assad and Kirk [] in the proof of our results in next section.

Lemma . If A,B∈CB(X)with D(A,B) <,then for each a∈A there exists an element b∈B such that d(a,b) <.

Lemma . Let{An}be a sequence in CB(X)andlimn→∞D(An,A) = for A∈CB(X).If

xn∈Anandlimn→∞d(xn,x) = ,then x∈A.

2 Common ﬁxed point

We begin with the following theorem that gives the existence of a common ﬁxed point (not necessarily unique) in metric spaces endowed with a graph for the set-valued mappings. Further, we assume that (X,d) is a complete metric space andGis a directed graph such thatE(G) is symmetric.

Theorem . Let F,H:X→CB(X) be graph contractive mappings and let the triple

(X,d,G)have the propertyA.Set XF :={x∈X: (x,u)∈E(G)for some u∈Fx}.Then the

following statements hold.

. For anyx∈XF,F,H|[x]G have a common ﬁxed point.

. IfXF=∅andGis weakly connected,thenF,Hhave a common ﬁxed point inX.

. IfX:= {[x]G:x∈XF},thenF,H|X have a common ﬁxed point.

. IfF⊆E(G),thenF,Hhave a common ﬁxed point.

Proof . Letx∈XF, then there existsx∈Fxsuch that (x,x)∈E(G). SinceF,H are graph contractive mappings, we have

D(Fx,Hx) <κd(x,x).

Using Lemma ., we have the existence ofx∈Hxsuch that

d(x,x) <κd(x,x). ()

Again, becauseF,Hare graph contractive (x,x)∈E(G), also (x,x)∈E(G), sinceE(G) is symmetric, we have

D(Fx,Hx) <κd(x,x) <κd(x,x),

and Lemma . gives the existence ofx∈Fxsuch that

d(x,x) <κd(x,x). ()

Continuing in this way, we havexn+ ∈Fxn andxn+ ∈Hxn+, n= , , , . . . . Also,

(xn,xn+)∈E(G) such that

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Next we show that (xn) is a Cauchy sequence inX. Letm>n. Then

d(xn,xm)≤d(xn,xn+) +d(xn+,xn+) +d(xn+,xn+) +· · ·+d(xm–,xm)

<κn+κn++κn++· · ·+κm–d(x,x) =κn +κ+κ+· · ·+κm–n–d(x,x)

=κn

 –κm–n  –κ

d(x,x)

becauseκ∈(, ),  –κm–n_{< .}

Therefored(xn,xm)→ asn→ ∞implies that (xn) is a Cauchy sequence and hence converges to some point (say)xin the complete metric spaceX.

Now we have to show thatx∈Fx∩Hx.

For n even: By property A, we have (xn,x)∈E(G). Therefore, by using graph contractivity, we have

D(Fxn,Hx) <κd(xn,x).

Sincexn+∈Fxnandxn→x, therefore by Lemma .,x∈Hx.

For n odd: As (x,xn)∈E(G),

D(Fx,Hxn) <κd(x,xn).

Now, by following the same arguments as above,x∈Fx.

Next as (xn,xn+)∈E(G), also (xn,x)∈E(G) forn∈N. We infer that (x,x, . . . ,xn,x) is a path inGand sox∈[x]G.

. Since XF =∅, so there exists x∈XF, and sinceGis weakly connected, therefore [x]G=X, and by , mappingsFandHhave a common ﬁxed point inX.

. It follows easily from  and .

.F⊆E(G) implies that allx∈Xare such that there exists someu∈Fxwith (x,u)∈E(G)

soXF=Xand by  and .F,Hhave a ﬁxed point.

Remark . ReplaceXFbyXH:={x∈X: (x,u)∈E(G) for someu∈Hx}in conditions - of Theorem ., then the conclusion remains true. That is, ifXF∪XH=∅, then we have

FixF∩FixH=∅, which follows easily from -. Similarly, in condition , we can replace

F⊆E(G) byH⊆E(G).

Corollary . is a direct consequence of Theorem .().

Corollary . Let(X,d)be a complete metric space and let the triple(X,d,G)have the

property A.If G is weakly connected,then graph contractive mappings F,H:X→CB(X)

such that(x,x)∈E(G)for some x∈Fxhave a common ﬁxed point.

Corollary . Let(X,d)be aε-chainable complete metric space for someε> .Let F,H:

X→CB(X)be such that there existsκ∈(, )with

 <d(x,y) <ε ⇒ D(Fx,Hx) <κd(x,y).

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Proof Consider the graphGasV(G) :=Xand

E(G) :=(x,y)∈X×X:  <d(x,y) <ε. ()

Theε-chainability of (X,d) meansGis connected. If (x,y)∈E(G), then

D(Fx,Hy) <κd(x,y) <κε<ε

and by using Lemma ., for each u∈Fx, we have the existence of v∈Hy such that

d(u,v) <ε, which implies (u,v)∈E(G). HenceFandHare graph contractive mappings. Also, (X,d,G) has property A. Indeed, if xn→x and d(xn,xn+) <ε for n∈N, then d(xn,x) <εfor suﬃciently large n, therefore (xn,x)∈E(G). So, by Theorem .(),Fand

Hhave a common ﬁxed point.

Theorem . Let F :X→ CB(X) be a graph contractive mapping and let the triple

(X,d,G)have the propertyA.Set XF :={x∈X: (x,u)∈E(G)for some u∈Fx}.Then the

following statements hold.

. For anyx∈XF,F|[x]Ghas a ﬁxed point.

. IfXF=∅andGis weakly connected,thenFhas a ﬁxed point inX.

. IfX:= {[x]G:x∈XF},thenF|X has a ﬁxed point.

. IfF⊆E(G),thenFhas a ﬁxed point. . IfXF=∅,thenFixF=∅.

Proof Statements - can be proved by takingF=Hin Theorem . and  obtained from Remark ..

Note that the assumption thatE(G) is symmetric is not needed in our Theorem ..

Remark .

. If we assumeGis such thatE(G) :=X×X, then clearlyGis connected and our Theorem .() improves Nadler’s theorem, and further ifFis single-valued, then we improve the Banach contraction theorem.

. IfFis a single-valued mapping, then Theorem .(, ) with the graphGimproves [, Theorem .].

. IfFis a single-valued mapping, then Theorem .(, ) with the graphGimproves [, Theorem .].

. IfF=His a single-valued mapping, then Theorem . and Theorem . partially generalize [, Theorem .].

. If we takeF=Has single-valued mappings in Corollary ., then we have [, Theorem .].

. If we takeF=H, then Corollary . becomes Theorem . due to [].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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Author details

1_{Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore, Pakistan.}2_{Department of}

Mathematics, University of Engineering and Technology, Lahore, Pakistan.

Received: 26 November 2012 Accepted: 30 April 2013 Published: 17 May 2013 References

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doi:10.1186/1029-242X-2013-252