Periodic Point, Endpoint, and Convergence Theorems for Dissipative Set-Valued Dynamic Systems with Generalized Pseudodis
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Research Article Periodic Point, Endpoint, and Convergence Theorems for Dissipative Set-Valued Dynamic Systems with Generalized Pseudodistances in Cone Uniform and Uniform Spaces Kazimierz Włodarczyk and Robert Plebaniak Department of Nonlinear Analysis, Faculty of Mathematics and Computer Science, University of Ł´od´z, Banacha 22, 90-238 Ł´od´z, Poland Correspondence should be addressed to Kazimierz Włodarczyk, [email protected] Received 29 September 2009; Accepted 17 November 2009 Academic Editor: Mohamed A. Khamsi Copyright q 2010 K. Włodarczyk and R. Plebaniak. 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. In cone uniform and uniform spaces, we introduce the three kinds of dissipative set-valued dynamic systems with generalized pseudodistances and not necessarily lower semicontinuous entropies, we study the convergence of dynamic processes and generalized sequences of iterations of these dissipative dynamic systems, and we establish conditions guaranteeing the existence of periodic points and endpoints of these dissipative dynamic systems and the convergence to these periodic points and endpoints of dynamic processes and generalized sequences of iterations of these dissipative dynamic systems. The paper includes examples.
1. Introduction A set-valued dynamic system is defined as a pair X, T , where X is a certain space and T is a set-valued map T : X → 2X ; in particular, a set-valued dynamic system includes the usual dynamic system where T is a single-valued map. Here 2X denotes the family of all nonempty subsets of a space X. Let X, T be a dynamic system. By FixT , PerT , and EndT we denote the sets of all fixed points, periodic points, and endpoints of T , respectively, that is, FixT {w ∈ X : w ∈ T w}, PerT {w ∈ X : w ∈ T q w for some q ∈ N} and EndT {w ∈ X : {w} T w}. For each x ∈ X, a sequence wm : m ∈ {0} ∪ N such that ∀m∈{0}∪N {wm1 ∈ T wm },
w0 x,
1.1
2
Fixed Point Theory and Applications
is called a dynamic process or a trajectory starting at w0 x of the system X, T for details see Aubin and Siegel 1, Aubin and Ekeland 2, and Aubin and Frankowska 3. For each x ∈ X, a sequence wm : m ∈ {0} ∪ N such that ∀m∈{0}∪N wm1 ∈ T m1 x ,
w0 x,
1.2
T m T ◦ T ◦ · · · ◦ T m-times, m ∈ N, is called a generalized sequence of iterations starting at w0 x of the system X, T for details see Yuan 4, page 557, Tarafdar and Vyborny 5 and Tarafdar and Yuan 6. Each dynamic process starting from w0 is a generalized sequence of iterations starting from w0 , but the converse may not be true; the set T m w0 is, in general, bigger than T wm−1 . If X, T is single valued, then, for each x ∈ X, a sequence wm : m ∈ {0} ∪ N such that ∀m∈{0}∪N wm1 T m1 x ,
w0 x,
1.3
is called a Picard iteration starting at w0 x of the system X, T .
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