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Zbl 1182.54024
Włodarczyk, Kazimierz; Plebaniak, Robert; Banach, Artur
Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces. (English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 9, A, 3332-3341 (2009); erratum ibid. 71, No. 7-8, A, 3585-3586 (2009; doi:10.1016/j.na.2008.11.020). ISSN 0362-546X

Authors' abstract: Given a uniform space $X$ and nonempty subsets $A$ and $B$ of $X$, we introduce the concepts of some families $\cal V$ 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 $\cal V$ 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.
[Nawab Hussain (Jeddah)]

MSC 2000:: *54C60 Set-valued maps
47H09 Mappings defined by "shrinking" properties
54E15 Uniform structures and generalizations
46A03 General theory of locally convex spaces
54E50 Complete metric spaces

Keywords: (non-)cyclic set-valued dynamic systems; relatively quasi-asymptotic contraction; best proximity point; family of generalised pseudodistances; uniform space; locally convex space; metric space; closed map; upper semicontinuous map; generalised sequence of iterations; dynamic process

Cited in: Zbl 1171.54311

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