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Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces. (English) Zbl 1182.54024

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 9, 3332-3341 (2009); erratum ibid. 71, No. 7-8, 3585-3586 (2009; doi:10.1016/j.na.2008.11.020).
Authors’ abstract: Given a uniform space \(X\) and nonempty subsets \(A\) and \(B\) of \(X\), we introduce the concepts of some families \(\mathcal V\) of generalized pseudodistances on \(X\), of set-valued dynamic systems of relatively quasi-asymptotic contractions \(T:A\cup B\rightarrow 2^{A\cup B}\) with respect to \(\mathcal V\) 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., \(T:A\rightarrow 2^B\) and \(T:B\rightarrow 2^A\)) or when \(T\) is noncyclic (i.e., \(T:A\rightarrow 2^A\) and \(T:B\rightarrow 2^B\)). 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 \(T^{[2]}\) 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.

MSC:

54C60 Set-valued maps in general topology
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
54E15 Uniform structures and generalizations
46A03 General theory of locally convex spaces
54E50 Complete metric spaces
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References:

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