Uniform convergence and sequence of maps on a compact metric space with some chaotic properties

Abstract

Recently, C. Tain and G. Chen introduced a new concept of sequence of time invariant function. In this paper we try to investigate the chaotic behavior of the uniform limit function f : X → X of a sequence of continuous topologically transitive (in strongly successive way) functions f n : X → X, where X is a compact interval. Surprisingly, we find that the uniform limit function is chaotic in the sense of Devaney. Lastly, we give an example to show that the denseness property of Devaney's definition is lost on the limit function.Key words: uniform convergence, chaos in the sense of Devaney, topological transitivity in strongly successive way

Stability results for fixed point sets of α*- Ψ contractive multivalued mappings

“…More recent references are [4][5][6][7]. It may be mentioned that there are other interesting studies related to the limits of sequence of mappings, as, for instance, the preservance of chaotic properties in the limit under uniform convergence has been discussed in [2]. In this paper we consider α * − ψ contractive multivalued mappings which are defined by Asl et al [1] as a multivalued extensions of a generalized contraction known as α − ψ contraction [8] .…”

Stability of Fixed Point Sets of a Class of Multivalued Nonlinear Contractions

“…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][4][5] the chaotic behavior under uniform convergence is investigated. Particularly, stability of fixed point sets has been considered in [6][7][8][9][10].…”

Fixed points of multivalued $$\alpha $$ α -admissible mappings and stability of fixed point sets in metric spaces

“…It has been studied in various contexts of discrete and continuous dynamical systems [31,35]. The study of the relationship between the convergence of a sequence of mappings and their fixed points, known as the stability of fixed points has also been widely studied in various settings [6,7,15,25,27,28,30]. The fixed point sets of sequence of mappings are said to be stable if they converge to the set of fixed points of the limit mapping in the Hausdorff metric.…”