Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces

Given a uniform space XX and nonempty subsets AA and BB of XX, we introduce the concepts of some families VV of generalized pseudodistances on XX, of set-valued dynamic systems of relatively quasi-asymptotic contractions T:A∪B→2A∪BT:A∪B→2A∪B with respect to VV and best proximity points for TT in A∪BA∪B, and we describe the methods which we use to establish the conditions guaranteeing the existence of best proximity points for TT when TT is cyclic (i.e. T:A→2BT:A→2B and T:B→2AT:B→2A) or when TT is noncyclic (i.e. T:A→2AT:A→2A and T:B→2BT:B→2B). Moreover, we establish conditions guaranteeing 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 for TT in A∪BA∪B. These best proximity points for TT are determined by unique endpoints in A∪BA∪B for a map T[2]T[2] when TT is cyclic and for a map TT when TT 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.