Open Access

Fixed point of set-valued graph contractive mappings

Journal of Inequalities and Applications20132013:252

https://doi.org/10.1186/1029-242X-2013-252

Received: 26 November 2012

Accepted: 30 April 2013

Published: 17 May 2013

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.

Keywords

fixed pointdirected graphmetric spaceset-valued mapping

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 C B ( X ) be a class of all nonempty closed and bounded subsets of X. For A , B C B ( 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 : X X be a set-valued mapping, i.e., X x F x is a subset of X. A point x X is said to be a fixed point of the set-valued mapping F if x F x .

Definition 1.2 A metric space ( X , d ) is called a ε-chainable metric space for some ε > 0 if given x , y X , there is n N and a sequence ( x i ) i = 0 n such that
x 0 = x , x n = y and d ( x i 1 , x i ) < ε for  i = 1 , , n .

Let Fix F : = { x X : x F x } denote the set of fixed points of the mapping F.

Definition 1.3 Let ( X , d ) be a metric space, ε > 0 , 0 κ < 1 and x , y X . A mapping f : X X is called ( ε , κ ) uniformly locally contractive if 0 < d ( x , y ) < ε d ( f x , f y ) < κ d ( 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 : X X 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 : X C B ( X ) . If there exists κ ( 0 , 1 ) such that
D ( F x , F y ) κ d ( x , y ) for all x , y X ,

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 : 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 0 < d ( x , y ) < ε D ( F x , F y ) < κ d ( 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 , x y } . 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 , y V ( 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 ( n N { 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 ( u R v 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 n N , then ( x n , x ) E ( G ) .

Definition 1.11 Let ( X , d ) be a metric space and F , H : X C B ( X ) . The mappings F, H are said to be graph contractive if there exists κ ( 0 , 1 ) such that
( x y ) , ( x , y ) E ( G ) D ( F x , H y ) < κ d ( x , y ) ,
and if u F x and v H y 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.

    x x (reflexivity);

     
  2. 2.

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

     
  3. 3.

    if x y and y z , then x z (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 , y X . Elements x and y are said to be comparable elements of X if either x y or y x .

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 , y X , 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 , B C B ( X ) with D ( A , B ) < ϵ , then for each a A there exists an element b B such that d ( a , b ) < ϵ .

Lemma 1.14 Let { A n } be a sequence in C B ( X ) and lim n D ( A n , A ) = 0 for A C B ( X ) . If x n A n and lim n d ( x n , x ) = 0 , 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 and G is a directed graph such that E ( G ) is symmetric.

Theorem 2.1 Let F , H : X C B ( X ) be graph contractive mappings and let the triple ( X , d , G ) have the property A. Set X F : = { x X : ( x , u ) E ( G ) for some u F x } . 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 F E ( 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 ) < κ d ( 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 ) < κ d ( 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 ) < κ d ( 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 x F x H x .

For n even: By property A, we have ( x n , x ) E ( G ) . Therefore, by using graph contractivity, we have
D ( F x n , H x ) < κ d ( x n , x ) .

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

For n odd: As ( x , x n ) E ( G ) ,
D ( F x , H x n ) < κ d ( x , x n ) .

Now, by following the same arguments as above, x F x .

Next as ( x n , x n + 1 ) E ( G ) , also ( x n , x ) E ( G ) for n N . 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.

    F E ( G ) implies that all x X are such that there exists some u F x 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 : = { x X : ( x , u ) E ( G )  for some  u H x } in conditions 1-3 of Theorem 2.1, then the conclusion remains true. That is, if X F X H , then we have Fix F Fix H , which follows easily from 1-3. Similarly, in condition 4, we can replace F E ( G ) by H E ( 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 : X C B ( 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 : X C B ( X ) be such that there exists κ ( 0 , 1 ) with
0 < d ( x , y ) < ε D ( F x , H x ) < κ d ( 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 ( F x , H y ) < κ d ( x , y ) < κ ε < ε

and by using Lemma 1.13, for each u F x , we have the existence of v H y 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 n N , 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 : X C B ( X ) be a graph contractive mapping and let the triple ( X , d , G ) have the property A. Set X F : = { x X : ( x , u ) E ( G ) for some u F x } . 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 F E ( G ) , then F has a fixed point.

     
  5. 5.

    If X F , then Fix F .

     

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].

     

Declarations

Authors’ Affiliations

(1)
Centre for Mathematics and Statistical Sciences, Lahore School of Economics
(2)
Department of Mathematics, University of Engineering and Technology

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© Beg and Butt; licensee Springer 2013

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