Abstract
We consider a problem of stability of fixed point sets for a sequence of multivalued mappings defined on a metric space converging to a limit function where the convergence is with respect to the Pompeiu-Hausdorff distance. The members of the sequence are assumed to be multivalued almost contractions. We show that the fixed point sets of this sequence of mappings are stable.
1. Introduction and Preliminaries
There are various notions of stability, both in discrete and continuous dynamical systems [1, 2]. It is a concept in dynamical system related to limiting behaviors. In this paper we consider a problem of stability related to a sequence of multivalued mappings on metric spaces. The limiting behaviors of sequences of mappings have been considered in a number of papers in recent time as, for instances, in [3–5] the chaotic behavior under uniform convergence is investigated. Particularly, stability of fixed point sets has been considered in [6–10].
Here our interest is in the stability of fixed point sets for a convergent sequence of multivalued mappings, that is, how they are related, in the limit, to the fixed point set of the function to which the sequence converges. We say that the fixed point sets are stable when they converge in the Pompeiu-Hausdorff distance to the set of fixed points of the limiting function. More often than not, in the above mentioned problem of stability, sequences of multivalued mappings are considered. One of the reasons behind this is that multivalued mappings often have more fixed points than their single valued counterparts. For instance, in the theorem of Nadler Jr. [3], which is the multivalued generalization of the Banach contraction principle and, incidentally, which is also the first work appearing on multivalued contractive fixed point studies, the fixed point is not unique in contrast to the case of single valued Banach’s contractions in complete metric spaces. In those situations the fixed point set becomes larger and, hence, more interesting for the study of stability.
Almost contractions are generalizations of Banach’s contraction. It was introduced by Berinde [11, 12].
In this paper we study the stability problem of fixed point sets for a uniformly convergent sequence of set valued almost contractions.
The following are some concepts, definitions, and results based on which we develop our main results.
Let be a metric space and let be the class of nonempty closed and bounded subsets of . For , functions and are defined as follows:
If , then we write and . Also in addition, if , then and . Obviously, . For all , the definition of yields the following [13, 14] clearly:
Let be a metric space and let be the family of all nonempty closed and bounded subsets of . The Pompeiu-Hausdorff distance [15] is defined on bywhere and
We recall that a sequence of functions , , converges uniformly on to a function if for every there is an integer such that implies for all .
Let be a metric space and let be the family of all subsets of . Let be a mapping; a fixed point of is such that .
We denote as a set of all fixed points of
Our purpose is to establish a stability result for the fixed point sets for a sequence of generalized multivalued almost contractions. The fixed point set for such a contraction is nonempty by Theorem 2 which is a consequence of a recent result of Choudhury and Metiya [16]. Nontrivial examples of the contractions of the above type are provided in [16].
Definition 1. Let be a complete metric space. Let be a multivalued mapping. is called generalized multivalued almost contraction iffor all , where and is a nondecreasing and continuous function with for each
The above mappings actually belong weakly to Picard operators which have been considered in [17–21], for instance.
The following theorem is a consequence of a result of Choudhury and Metiya [16].
Theorem 2. Let be a complete metric space. Let be a generalized multivalued almost contraction; that is, satisfies Then has a fixed point.
Nadler Jr. [3] established the following lemma.
Lemma 3 (see [3]). Let be a metric space and . Let . Then, for each , there exists such that .
In fact, Choudhury and Metiya [16] proved the result noted in Theorem 2 in a partially ordered metric space with some additional order conditions. Here we work with a metric space having no ordering defined on it.
2. Main Result
Theorem 4. Let be a complete metric space and let be two multivalued generalized almost contractions with the same and , where satisfies the following additional condition: with as . Then
, where
Proof. By Theorem 2, and are nonempty.
Let be any number. By Lemma 3, for , there exists such thatWe construct a sequence as, for all , . ThenBy (4), It then follows that, for all ,Then by repeated application we haveTherefore, This implies that is a Cauchy sequence.
Since is complete, as . ConsiderTaking limit in the above inequality and from the properties of , we havewhich implies Since is closed, so
Now using triangular inequality we have Thus, given arbitrary , we can find for which Reversing the roles of and we also conclude that, for each , there exist and such that
From the above and from the definition of the Pompeiu-Hausdorff distance, we conclude thatThis completes the proof.
Lemma 5. If is a sequence of multivalued generalized almost contractions which is uniformly convergent to , then is a multivalued generalized almost contraction.
Proof. Since each is multivalued generalized almost contraction, thereforeNow, taking limit we get hence the result.
Theorem 6. If is a sequence of multivalued generalized almost contractions which is uniformly convergent to , then the fixed point sets of are stable; that is,
Proof. converges to uniformly on .
Therefore, by Lemma 5, is a multivalued generalized almost contraction.
Let
Since uniformly with respect to the Pompeiu-Hausdorff distance, therefore .
Then, from Theorem 4, we obtainThis proves the theorem.
Example 7. Let , and is usual metric on . Let be defined as follows: Let be defined by Let . Then is multivalued generalized almost contraction.
Also we observe that as , where .
Now for all , and , for all .
Therefore we have and . One has . ConsiderLet ; then
It is also observed that and
Hence we conclude that as with respect to Pompeiu-Hausdorff distance. Then all of the conditions of Theorems 4 and 6 are satisfied which verify the above example.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.