Research Article | Open Access
Stability of Fixed Point Sets of a Class of Multivalued Nonlinear Contractions
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. , 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.
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:
Let be a metric space and let be the family of all nonempty closed and bounded subsets of . The Pompeiu-Hausdorff distance  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 . Nontrivial examples of the contractions of the above type are provided in .
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 following theorem is a consequence of a result of Choudhury and Metiya .
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.  established the following lemma.
Lemma 3 (see ). Let be a metric space and . Let . Then, for each , there exists such that .
In fact, Choudhury and Metiya  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
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.
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.
- C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press, 2nd edition, 1998.
- S. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Westview Press, 2001.
- S. Nadler Jr., “Sequences of contractions and fixed points,” Pacific Journal of Mathematics, vol. 27, no. 3, pp. 579–585, 1968.
- J. T. Markin, “Continuous dependence of fixed point sets,” Proceedings of the American Mathematical Society, vol. 38, no. 3, pp. 545–545, 1973.
- I. Bhaumik and B. S. Choudhury, “Uniform convergence and sequence of maps on a compact metric space with some chaotic properties,” Analysis in Theory and Applications, vol. 26, no. 1, pp. 53–58, 2010.
- J. T. Markin, “A fixed point stability theorem for nonexpansive set valued mappings,” Journal of Mathematical Analysis and Applications, vol. 54, no. 2, pp. 441–443, 1976.
- G. Moţ and A. Petruşel, “Fixed point theory for a new type of contractive multivalued operators,” Nonlinear Analysis, Theory, Methods and Applications, vol. 70, no. 9, pp. 3371–3377, 2009.
- T.-C. Lim, “On fixed point stability for set-valued contractive mappings with applications to generalized differential equations,” Journal of Mathematical Analysis and Applications, vol. 110, no. 2, pp. 436–441, 1985.
- B. S. Choudhury and C. Bandyopadhyay, “A new multivalued contraction and stability of its fixed point sets,” Journal of the Egyptian Mathematical Society, vol. 23, no. 2, pp. 321–325, 2015.
- B. S. Choudhury, N. Metiya, T. Som, and C. Bandyopadhyay, “Multivalued fixed point results and stability of fixed point sets in metric spaces,” Facta Universitatis Series: Mathematics and Informatics, vol. 30, no. 4, pp. 501–512, 2015.
- V. Berinde, “Approximating fixed points of weak contractions using the Picard iteration,” Nonlinear Analysis Forum, vol. 9, no. 1, pp. 43–53, 2004.
- V. Berinde, “General constructive fixed point theorems for Ciric type almost contractions in metric spaces,” Carpathian Journal of Mathematics, vol. 24, no. 2, pp. 10–19, 2008.
- B. Fisher, “Common fixed points of mappings and set-valued mappings,” Rostocker Mathematisches Kolloquium, vol. 18, pp. 69–77, 1981.
- B. Fisher and K. Ise'ki, “Fixed points for set-valued mappings on complete metric spaces,” Mathematica Japonica, vol. 28, pp. 639–646, 1983.
- V. Berinde and M. Pacurar, “The role of the Pompeiu-Hausdorff metric in fixed point theory,” Creative Mathematics and Informatics, vol. 22, no. 2, pp. 143–150, 2013.
- B. S. Choudhury and N. Metiya, “Fixed point theorems for almost contractions in partially ordered metric spaces,” Annali dell'Università di Ferrara, vol. 58, no. 1, pp. 21–36, 2012.
- I. A. Rus, “Picard operators and applications,” Scientiae Mathematicae Japonicae, vol. 58, no. 1, pp. 191–219, 2003.
- I. A. Rus, “Heuristic introduction to weakly Picard operator theory,” Creative Mathematics and Informatics, vol. 23, no. 2, pp. 243–252, 2014.
- I. A. Rus, A. Petrusel, and G. Petrusel, Fixed Point Theory, Cluj University Press, Cluj-Napoca, Romania, 2008.
- I. A. Rus, A. Petruşel, and M. A. Șerban, “Weakly Picard operators: equivalent definitions, applications and open problems,” Fixed Point Theory, vol. 7, no. 1, pp. 3–22, 2006.
- I. A. Rus and M.-A. Şerban, “Basic problems of the metric fixed point theory and the relevance of a metric fixed point theorem,” Carpathian Journal of Mathematics, vol. 29, no. 2, pp. 239–258, 2013.
Copyright © 2015 Binayak S. Choudhury and Chaitali Bandyopadhyay. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.