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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 $T:A\cup B\to {2}^{A\cup B}$ with respect to $𝒱$ and best proximity points for $T$ in $A\cup B$, and describe the methods to establish the conditions guaranteeing the existence of best proximity points for $T$ when $T$ is cyclic (i.e., $T:A\to {2}^{B}$ and $T:B\to {2}^{A}$) or when $T$ is noncyclic (i.e., $T:A\to {2}^{A}$ and $T:B\to {2}^{B}$). 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 $A\cup B$. These best proximity points for $T$ are determined by unique endpoints in $A\cup B$ for a map ${T}^{\left[2\right]}$ 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.

##### MSC:
 54C60 Set-valued maps (general topology) 47H09 Mappings defined by “shrinking” properties 54E15 Uniform structures and generalizations 46A03 General theory of locally convex spaces 54E50 Complete metric spaces