Ciric-type δ-contractions in metric spaces endowed with a graph
© Chifu and Petruşel; licensee Springer. 2014
Received: 15 October 2013
Accepted: 13 January 2014
Published: 17 February 2014
The purpose of this paper is to present some fixed-point and strict fixed-point results in a metric space endowed with a graph, using a contractive condition of Ćirić type with respect to the functional δ. The data dependence of the fixed-point set, the well-posedness of the fixed-point problem, and the limit shadowing property are also studied.
A recent research direction in fixed-point theory is the study of the fixed-point problem for single-valued and multivalued operators in the context of a metric space endowed with a graph. This approach was recently considered by Jachymski in , Gwóźdź-Lukawska and Jachymski in , and then it was developed in many other papers ([3–5], etc.).
On the other hand, fixed points and strict fixed points (also called end-points) are important elements in mathematical economics and game theory. It represents optimal preferences in some Arrow-Debreu type models or Nash type equilibrium points for some abstract noncooperative games, see, for example,  and . From this perspective, it is important to give fixed and strict fixed-point theorems for multivalued operators.
We shall begin by presenting some notions and notations that will be used throughout the paper.
Let be a metric space and Δ be the diagonal of . Let G be a directed graph such that the set of its vertices coincides with X and , being the set of the edges of the graph. Assuming that G has no parallel edges, we will suppose that G can be identified with the pair .
If x and y are vertices of G, then a path in G from x to y of length is a finite sequence of vertices such that , and , for .
Let us denote by the undirected graph obtained from G by ignoring the direction of edges. Notice that a graph G is connected if there is a path between any two vertices and it is weakly connected if is connected.
Denote by the diameter of the set A.
Let be a multivalued operator and the graphic of T. is called fixed point for T if and only if , and it is called strict fixed point if and only if .
The set is called the fixed-point set of T, while is called the strict fixed-point set of T. Notice that .
We will write if and only if for every with we have .
For the particular case of a single-valued operator the above notations should be considered accordingly. In particular, the condition means that the operator t is edge preserving (in the sense of the Jachymski’s definition of a Banach contraction), i.e. for each with we have (see ). We will also denote by the orbit of order n of the operator t corresponding to .
In this paper we prove some fixed-point and strict fixed-point theorems for single-valued and multivalued operators satisfying a contractive condition of Ćirić type with respect to the functional δ. Our results also generalize and extend some fixed-point theorems in partially ordered complete metric spaces given in Harjani, Sadarangani , Nieto, Rodríguez-López  and , Nieto et al. , O’Regan, Petruşel , Petruşel, Rus  and Ran, Reurings . For other general results concerning Ćirić type fixed-point theorems see Rus .
In the main section of the paper we give results concerning the existence and uniqueness of the fixed point and of the strict fixed point of a Ćirić type (single-valued and multivalued) contraction. Then the well-posedness of the fixed-point problem, the data dependence of the fixed-point set, and the limit shadowing property are also studied. Our results complement and extend some recent theorems given in  for multivalued Reich-type operators.
2 Main results
In this section we present the main results of the paper concerning the fixed-point problem and, respectively, the strict fixed-point problem for a single-valued, respectively, of a multivalued Ćirić type contraction.
If is a sequence in X such that , as , then , as .
Definition 2.2 Let be a metric space and be a multivalued operator. By definition T has the limit shadowing property if for any sequence from X such that , as , there exists , a sequence of successive approximation of T, such that , as .
In order to prove the limit shadowing property we shall need Cauchy’s lemma.
Lemma 2.1 (Cauchy’s lemma)
The first main result of this paper is the following result for the case of single-valued operators. The proof of this result is inspired by the proof of Ćirić’s fixed-point theorem in  and the approach introduced for metric spaces endowed with a graph by Jachymski in .
- (i)there exists such that
for all ;
there exists such that ;
if and , then .
- (b)(uniqueness) If, in addition, the following implication holds
Thus, (2.4) shows that the sequence is Cauchy and, as a consequence of the completeness of the space, it converges to an element .
We will show now that .
Letting we get .
(b) Suppose that there exist with .
which is a contradiction. □
Remark 2.1 Notice that, from the proof of the above theorem, it follows that the sequence converges to in .
then we get again .
Based on the above theorem, we can prove now our second main result.
- (i)there exists such that
for all ;
there exists such that, for all , we have ;
if and , then .
- (b)(uniqueness) If, in addition, the following implication holds:
(well-posedness of the fixed-point problem) If T has closed graphic and for any sequence , with , as , we have , then the fixed-point problem is well-posed for T with respect to H;
- (d)(limit shadowing property of T) If and is a sequence in X such that the following implication holds:
then T has the limit shadowing property.
Thus, the operator t satisfies all the hypotheses of Theorem 2.1 and, as a consequence, it has a fixed point . Then . If we suppose now that there exists , then, since , from the condition (i) (with ), we get , which implies (since ) that . Thus . This is a contradiction with , proving that .
(b) Suppose that there exist .
which implies that . Hence .
(c) Let , , be a sequence with the property that , as . It is obvious that .
and, as a consequence, the fixed-point problem is well-posed for T with respect to H.
(d) Let be a sequence in X such that , as and let be a sequence of successive approximation of T starting from , constructed as in the proof of Theorem 2.1. We shall prove that , as .
If we consider and and using the fact that , we obtain, via Cauchy’s lemma, , as . Thus , as , and hence, the operator T has the limit shadowing property. □
Proof Since every selection t of T has closed graphic, the conclusion follows by Theorem 2.1, via Remark 2.2. □
If is a sequence in X such that , as , then there exists a subsequence of such that , as .
Remark 2.4 If in Theorem 2.1 instead of (c) we suppose the following assumption, (c′):
(c′) If every selection t of T has closed graphic and, in addition, we suppose that for any sequence with , as , there exists a subsequence such that and , then the fixed-point problem is well-posed in the generalized sense for T with respect to H.
In what follows we shall present some examples of operators satisfying the hypotheses of our main results.
Let and .
Then all the hypotheses of Theorem 2.1 are satisfied, and, if we take , then converges to .
Notice that and all the hypotheses in Theorem 2.2 are satisfied (the condition (i) is verified for ).
Remark 2.5 It is also important to notice that, if we suppose that there exists , then, since , from the condition (i) in the above theorem (with ), we get , which implies (since ). This is a contradiction with , showing that we cannot get fixed points which are not strict fixed points in the presence of the condition (i) of the above theorem. It is an open problem to prove a similar theorem to the above one for a more general class of multivalued operators T.
The next result presents the data dependence of the fixed-point set of a multivalued operator which satisfies a contractive condition of Ćirić type.
- (i)for , there exist such that
for all ;
for each and each we have , for ;
if and , then ;
there exists such that for all .
Under these conditions we have:
- (b)If, in addition, for , the following implication holds:
then , for each ;
Proof Conclusions (a) and (b) are immediate if we apply Theorem 2.2.
A second relation of this type will be obtained by interchanging the role of and . Hence, the conclusion follows by the properties of the functional H. □
The work of the second author is supported by financial support of a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0094.
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