Existence and uniqueness of endpoints of closed set-valued asymptotic contractions in metric spaces

Abstract

Investigations of problems of set-valued asymptotic fixed point theory of set-valued dynamic systems are initiated. The concepts of the set-valued asymptotic contractions of set-valued dynamic systems T : M → 2 M in metric space M are introduced and conditions guaranteeing the existence and uniqueness of endpoints v ∈ M of set-valued dynamic systems of set-valued asymptotic contractions T : M → 2 M and convergence to v of all generalized sequences of iterations of T are established. Examples and remarks show a fundamental difference between our results and the well-known ones.

A stationary point theorem characterizing metric completeness

“…Thus y ∈ Fx and diam Fy < ε, which proves (2). So F is a fixed point free set-valued mapping satisfying (1) and (2), which contradicts (iii).…”

“…Very recently, a problem of the existence and uniqueness of stationary points has been studied extensively (see, e.g., [1][2][3][4][5][6][7][8]). In this paper we give a stationary point theorem for some set-valued mappings on a metric space.…”

Endpoints of Multivalued Contraction Operators

“…The results related to existence of endpoints or strict fixed-points were first given by Rus [9] in 2003. Thereafter, a number of authors established interesting results concerning existence and uniqueness of endpoints for multivalued contractions in different settings; see, for example, [10][11][12][13][14][15]. The main purpose of this paper is to establish some existence and uniqueness results for endpoints using different multivalued contractions.…”

Note on asymptotic contractions

“…Chen [4], P. Gerhardy [5], [6], T. Suzuki [9], H. K. Xu [12], M. Arav, F. E. C. Santos, S. Reich, A. Zaslavski [2] and K. W lodarczyk, D. Klim, R. Plebaniak [10], [11]. The papers [10] and [11] presents some ideas for application of the theory of asymptotic contractions in the analysis of set-valued dynamic systems.…”