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Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces. (English) Zbl 1182.54024
Authors’ abstract: Given a uniform space X and nonempty subsets A and B of X, we introduce the concepts of some families 𝒱 of generalized pseudodistances on X, of set-valued dynamic systems of relatively quasi-asymptotic contractions T:AB2 AB with respect to 𝒱 and best proximity points for T in AB, and describe the methods to establish the conditions guaranteeing the existence of best proximity points for T when T is cyclic (i.e., T:A2 B and T:B2 A ) or when T is noncyclic (i.e., T:A2 A and T:B2 B ). Moreover, we establish conditions guaranteeing that for each starting point each generalized sequence of iterations (in particular, each dynamic process) converges and the limit is a best proximity point for T in AB. These best proximity points for T are determined by unique endpoints in AB for a map T [2] when T is cyclic and for a map T when T is noncyclic. The results and the methods are new for set-valued and single-valued dynamic systems in uniform, locally convex, metric and Banach spaces. Various examples illustrating the ideas of our definitions and results, and fundamental differences between our results and the well-known ones are given.

54C60Set-valued maps (general topology)
47H09Mappings defined by “shrinking” properties
54E15Uniform structures and generalizations
46A03General theory of locally convex spaces
54E50Complete metric spaces