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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 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:AB2 AB 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.
MSC:
54C60Set-valued maps (general topology)
47H10Fixed point theorems for nonlinear operators on topological linear spaces
54E15Uniform structures and generalizations
46A40Ordered topological linear spaces, vector lattices
46A03General theory of locally convex spaces
54E50Complete metric spaces