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Fixed point of set-valued graph contractive mappings

Abstract

Let (X,d) be a metric space and let F, H be two set-valued mappings on X. We obtained sufficient conditions for the existence of a common fixed point of the mappings F, H in the metric space X endowed with a graph G such that the set of vertices of G, V(G)=X and the set of edges of G, E(G)X×X.

MSC:47H10, 47H04, 47H07, 54C60, 54H25.

1 Introduction and preliminaries

Edelstein [1] generalized classical Banach’s contraction mapping principle and Nadler [2] proved Banach’s fixed point theorem for set-valued mappings. Recently several extensions of Nadler’s theorem in different directions were obtained; see [315]. Beg and Azam [5] extended Edelstein’s theorem by considering a pair of set-valued mappings with a general 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 graph G. Our results improve/generalize [1, 2, 16] and several other known results in the literature.

Let (X,d) be a complete metric space and let CB(X) be a class of all nonempty closed and bounded subsets of X. For A,BCB(X), let

D(A,B):=max { sup b B d ( b , A ) , sup a A d ( a , B ) } ,

where

d(a,B):= inf b B d(a,b).

Mapping D is said to be a Hausdorff metric induced by d.

Definition 1.1 Let F:XX be a set-valued mapping, i.e., XxFx is a subset of X. A point xX is said to be a fixed point of the set-valued mapping F if xFx.

Definition 1.2 A metric space (X,d) is called a ε-chainable metric space for some ε>0 if given x,yX, there is nN and a sequence ( x i ) i = 0 n such that

x 0 =x, x n =yandd( x i 1 , x i )<εfor i=1,,n.

Let FixF:={xX:xFx} denote the set of fixed points of the mapping F.

Definition 1.3 Let (X,d) be a metric space, ε>0, 0κ<1 and x,yX. A mapping f:XX is called (ε,κ) uniformly locally contractive if 0<d(x,y)<εd(fx,fy)<κd(x,y).

The following significant generalization of Banach’s contraction principle [[17], Theorem 2.1 ] was obtained by Edelstein [1].

Theorem 1.4 [1]

Let (X,d) be a ε-chainable complete metric space. If f:XX is a (ε,κ) uniformly locally contractive mapping, then f has a unique fixed point.

Afterwards, in 1969, Nadler [2] proved a set-valued extension of Banach’s theorem and obtained the following result.

Theorem 1.5 [2]

Let (X,d) be a complete metric space and F:XCB(X). If there exists κ(0,1) such that

D(Fx,Fy)κd(x,y) for all x,yX,

then F has a fixed point in X.

Nadler [2] also extended Edelstein’s theorem for set-valued mappings.

Theorem 1.6 [2]

Let (X,d) be a ε-chainable complete metric space for some ε>0 and let F:XC(X) be a set-valued mapping such that Fx is a nonempty compact subset of X. If F satisfies the following condition:

x,yXand0<d(x,y)<εD(Fx,Fy)<κd(x,y),

then F has a fixed point.

Consider a directed graph G such that the set of its vertices coincides with X (i.e., V(G):=X) and the set of its edges E(G):={(x,y):(x,y)X×X,xy}. We assume that G has no parallel edges and weighted graph by assigning to each edge the distance between the vertices; for details about definitions in graph theory, see [18].

We can identify G as (V(G),E(G)). G 1 denotes the conversion of a graph G, the graph obtained from G by reversing the direction of its edges. G ˜ denotes the undirected graph obtained from G by ignoring the direction of edges of G. We consider G ˜ as a directed graph for which the set if its edges is symmetric, thus we have

E( G ˜ ):=E(G)E ( G 1 ) .

Definition 1.7 A subgraph of a graph G is a graph H such that V(H)V(G) and E(H)E(G) and for any edge (x,y)E(H), x,yV(H).

Definition 1.8 Let x and y be vertices in a graph G. A path in G from x to y of length n (nN{0}) is a sequence ( x i ) i = 0 n of n+1 vertices such that x 0 =x, x n =y and ( x i 1 , x i )E(G) for i=1,2,,n.

Definition 1.9 The number of edges in G constituting the path is called the length of the path.

Definition 1.10 A graph G is connected if there is a path between any two vertices of G.

If a graph G is not connected, then it is called disconnected. Moreover, G is weakly connected if G ˜ is connected.

Assume that G is such that E(G) is symmetric, and x is a vertex in G, then the subgraph G x consisting of all edges and vertices, which are contained in some path in G beginning at x, is called the component of G containing x. In this case the equivalence class [ x ] G defined on V(G) by the rule R (uRv if there is a path from u to v) is such that V( G x )= [ x ] G .

Property A: For any sequence ( x n ) n N in X, if x n x and ( x n , x n + 1 )E(G) for nN, then ( x n ,x)E(G).

Definition 1.11 Let (X,d) be a metric space and F,H:XCB(X). The mappings F, H are said to be graph contractive if there exists κ(0,1) such that

(xy),(x,y)E(G)D(Fx,Hy)<κd(x,y),

and if uFx and vHy are such that

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

then (u,v)E(G).

Definition 1.12 A partial order is a binary relation over a set X which satisfies the following conditions:

  1. 1.

    xx (reflexivity);

  2. 2.

    if xy and yx, then x=y (antisymmetry);

  3. 3.

    if xy and yz, then xz (transitivity);

for all x, y and z in X.

A set with a partial order is called a partially ordered set.

Let (X,) be a partially ordered set and x,yX. Elements x and y are said to be comparable elements of X if either xy or yx.

Let be a partial order in X. Define the graph G:= G 1 by

E( G 1 ):= { ( x , y ) X × X : x y , x y } ,

and G:= G 2 by

E( G 2 ):= { ( x , y ) X × X : x y y x , x y } .

The class of G 1 -contractive mappings was considered in [19] and that of G 2 -contractive mappings in [20].

The weak connectivity of G 1 or G 2 means, given x,yX, there is a sequence ( x i ) i = 0 n such that x 0 =x, x n =y and for all i=1,,n, x i 1 and x i are comparable.

We shall make use of the following lemmas due to Nadler [2], Assad and Kirk [21] in the proof of our results in next section.

Lemma 1.13 If A,BCB(X) with D(A,B)<ϵ, then for each aA there exists an element bB such that d(a,b)<ϵ.

Lemma 1.14 Let { A n } be a sequence in CB(X) and lim n D( A n ,A)=0 for ACB(X). If x n A n and lim n d( x n ,x)=0, then xA.

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 and G is a directed graph such that E(G) is symmetric.

Theorem 2.1 Let F,H:XCB(X) be graph contractive mappings and let the triple (X,d,G) have the property A. Set X F :={xX:(x,u)E(G) for some uFx}. Then the following statements hold.

  1. 1.

    For any x X F , F, H | [ x ] G have a common fixed point.

  2. 2.

    If X F and G is weakly connected, then F, H have a common fixed point in X.

  3. 3.

    If X :={ [ x ] G :x X F }, then F, H | X have a common fixed point.

  4. 4.

    If FE(G), then F, H have a common fixed point.

Proof 1. Let x 0 X F , then there exists x 1 F x 0 such that ( x 0 , x 1 )E(G). Since F, H are graph contractive mappings, we have

D(F x 0 ,H x 1 )<κd( x 0 , x 1 ).

Using Lemma 1.13, we have the existence of x 2 H x 1 such that

d( x 1 , x 2 )<κd( x 0 , x 1 ).
(1)

Again, because F, H are graph contractive ( x 1 , x 2 )E(G), also ( x 2 , x 1 )E(G), since E(G) is symmetric, we have

D(F x 2 ,H x 1 )<κd( x 1 , x 2 )< κ 2 d( x 0 , x 1 ),

and Lemma 1.13 gives the existence of x 3 F x 2 such that

d( x 2 , x 3 )< κ 2 d( x 0 , x 1 ).
(2)

Continuing in this way, we have x 2 n + 1 F x 2 n and x 2 n + 2 H x 2 n + 1 , n=0,1,2, . Also, ( x n , x n + 1 )E(G) such that

d( x n , x n + 1 )< κ n d( x 0 , x 1 ).
(3)

Next we show that ( x n ) is a Cauchy sequence in X. Let m>n. Then

d ( x n , x m ) d ( x n , x n + 1 ) + d ( x n + 1 , x n + 2 ) + d ( x n + 2 , x n + 3 ) + + d ( x m 1 , x m ) < [ κ n + κ n + 1 + κ n + 2 + + κ m 1 ] d ( x 0 , x 1 ) = κ n [ 1 + κ + κ 2 + + κ m n 1 ] d ( x 0 , x 1 ) = κ n [ 1 κ m n 1 κ ] d ( x 0 , x 1 )

because κ(0,1), 1 κ m n <1.

Therefore d( x n , x m )0 as n implies that ( x n ) is a Cauchy sequence and hence converges to some point (say) x in the complete metric space X.

Now we have to show that xFxHx.

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

D(F x n ,Hx)<κd( x n ,x).

Since x n + 1 F x n and x n x, therefore by Lemma 1.14, xHx.

For n odd: As (x, x n )E(G),

D(Fx,H x n )<κd(x, x n ).

Now, by following the same arguments as above, xFx.

Next as ( x n , x n + 1 )E(G), also ( x n ,x)E(G) for nN. We infer that ( x 0 , x 1 ,, x n ,x) is a path in G and so x [ x 0 ] G .

  1. 2.

    Since X F , so there exists x 0 X F , and since G is weakly connected, therefore [ x 0 ] G =X, and by 1, mappings F and H have a common fixed point in X.

  2. 3.

    It follows easily from 1 and 2.

  3. 4.

    FE(G) implies that all xX are such that there exists some uFx with (x,u)E(G) so X F =X and by 2 and 3. F, H have a fixed point. □

Remark 2.2 Replace X F by X H :={xX:(x,u)E(G) for some uHx} in conditions 1-3 of Theorem 2.1, then the conclusion remains true. That is, if X F X H , then we have FixFFixH, which follows easily from 1-3. Similarly, in condition 4, we can replace FE(G) by HE(G).

Corollary 2.3 is a direct consequence of Theorem 2.1(1).

Corollary 2.3 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:XCB(X) such that ( x 0 , x 1 )E(G) for some x 1 F x 0 have a common fixed point.

Corollary 2.4 Let (X,d) be a ε-chainable complete metric space for some ε>0. Let F,H:XCB(X) be such that there exists κ(0,1) with

0<d(x,y)<εD(Fx,Hx)<κd(x,y).

Then F and H have a common fixed point.

Proof Consider the graph G as V(G):=X and

E(G):= { ( x , y ) X × X : 0 < d ( x , y ) < ε } .
(4)

The ε-chainability of (X,d) means G is connected. If (x,y)E(G), then

D(Fx,Hy)<κd(x,y)<κε<ε

and by using Lemma 1.13, for each uFx, we have the existence of vHy such that d(u,v)<ε, which implies (u,v)E(G). Hence F and H are graph contractive mappings. Also, (X,d,G) has property A. Indeed, if x n x and d( x n , x n + 1 )<ε for nN, then d( x n ,x)<ε for sufficiently large n, therefore ( x n ,x)E(G). So, by Theorem 2.1(2), F and H have a common fixed point. □

Theorem 2.5 Let F:XCB(X) be a graph contractive mapping and let the triple (X,d,G) have the property A. Set X F :={xX:(x,u)E(G) for some uFx}. Then the following statements hold.

  1. 1.

    For any x X F , F | [ x ] G has a fixed point.

  2. 2.

    If X F and G is weakly connected, then F has a fixed point in X.

  3. 3.

    If X :={ [ x ] G :x X F }, then F | X has a fixed point.

  4. 4.

    If FE(G), then F has a fixed point.

  5. 5.

    If X F , then FixF.

Proof Statements 1-4 can be proved by taking F=H in Theorem 2.1 and 5 obtained from Remark 2.2.

Note that the assumption that E(G) is symmetric is not needed in our Theorem 2.5. □

Remark 2.6

  1. 1.

    If we assume G is such that E(G):=X×X, then clearly G is connected and our Theorem 2.5(2) improves Nadler’s theorem, and further if F is single-valued, then we improve the Banach contraction theorem.

  2. 2.

    If F is a single-valued mapping, then Theorem 2.5(2, 5) with the graph G 1 improves [[19], Theorem 2.2].

  3. 3.

    If F is a single-valued mapping, then Theorem 2.5(2, 5) with the graph G 2 improves [[20], Theorem 2.1].

  4. 4.

    If F=H is a single-valued mapping, then Theorem 2.1 and Theorem 2.5 partially generalize [[22], Theorem 3.2].

  5. 5.

    If we take F=H as single-valued mappings in Corollary 2.4, then we have [[1], Theorem 5.2].

  6. 6.

    If we take F=H, then Corollary 2.4 becomes Theorem 1.5 due to [2].

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Correspondence to Ismat Beg.

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IB gave the idea. ARB wrote the initial draft. IB and ARB finalized the manuscript. All authors read and approved the final manuscript. Correspondence was mainly done by IB.

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Beg, I., Butt, A.R. Fixed point of set-valued graph contractive mappings. J Inequal Appl 2013, 252 (2013). https://doi.org/10.1186/1029-242X-2013-252

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Keywords

  • fixed point
  • directed graph
  • metric space
  • set-valued mapping