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

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 →2M 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 →2M 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. © 2006 Elsevier Inc. All rights reserved