Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces |
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Authors: | Kazimierz W?odarczyk Robert Plebaniak |
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Institution: | Department of Nonlinear Analysis, Faculty of Mathematics and Computer Science, University of ?ód?, Banacha 22, 90-238 ?ód?, Poland |
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Abstract: | In cone uniform spaces X, using the concept of the D-family of cone pseudodistances, the distance between two not necessarily convex or compact sets A and B in X is defined, the concepts of cyclic and noncyclic set-valued dynamic systems of D-relatively quasi-asymptotic contractions T:A∪B→2A∪B are introduced and the best approximation and best proximity point theorems for such contractions are proved. Also conditions are given which guarantee that for each starting point each generalized sequence of iterations of these contractions (in particular, each dynamic process) converges and the limit is a best proximity point. Moreover, D-families are constructed, characterized and compared. The results are new for set-valued and single-valued dynamic systems in cone uniform, cone locally convex and cone metric spaces. Various examples illustrating ideas, methods, definitions and results are constructed. |
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Keywords: | 54C60 47H10 54E15 46A40 46A03 54E50 |
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