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Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces. (English) Zbl 1185.54020
Authors’ abstract: In cone uniform spaces $$X$$, using the concept of the $$\mathcal{D}$$-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 $$\mathcal{D}$$-relatively quasi-asymptotic contractions $$T:A\cup B\to 2^{A\cup B}$$ 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, $$\mathcal{D}$$-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.

##### MSC:
 54C60 Set-valued maps in general topology 47H10 Fixed-point theorems 54E15 Uniform structures and generalizations 46A40 Ordered topological linear spaces, vector lattices 46A03 General theory of locally convex spaces 54E50 Complete metric spaces
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