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Periodic point, endpoint, and convergence theorems for dissipative set-valued dynamic systems with generalized pseudodistances in cone uniform and uniform spaces. (English) Zbl 1193.37101
A set-valued dynamical system is defined as a pair $(X,T)$, where $X$ is a certain space and $T$ is a set-valued map $T: X\rightarrow 2^X $. The authors introduce in cone uniform and uniform spaces three kinds of dissipative set-valued dynamical systems with generalized pseudodistances and with not necessarily lower semicontinuous entropies, and the methods which are useful for establishing general conditions guaranteeing the existence of periodic points and endpoints of the set-valued dynamical systems and conditions that for each starting point the dynamical process or generalized sequences of iterations converge and the limit is a periodic point or endpoint, are presented. The definitions and results of the paper are more general and different from those given in the literature and are new even for single-valued and set-valued dynamical systems in metric spaces. The paper is a continuation of two papers of the authors. Examples are given.

37L05General theory, nonlinear semigroups, evolution equations
37L99Infinite-dimensional dissipative dynamical systems
Full Text: DOI EuDML