A best proximity point theorem for weakly contractive non-self-mappings

Let us consider a map T : A → B , where A and B are two nonempty subsets of a metric space X . The aim of this article is to provide sufficient conditions for the existence of a unique point x ∗ in A , called the best proximity point, which satisfies d ( x ∗ , T x ∗ ) = dist ( A , B ) : = inf { d (...

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Bibliographic Details
Published in:Nonlinear analysis 2011-10, Vol.74 (14), p.4804-4808
Main Author: Sankar Raj, V.
Format: Article
Language:English
Subjects:
[formula omitted]-contraction
[formula omitted]-property
Best proximity points
Exact sciences and technology
Fixed points
Global analysis, analysis on manifolds
Mathematical analysis
Mathematics
Metric space
Nonlinearity
Proximity
Sciences and techniques of general use
Theorems
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
Weakly contractive non-self-maps
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Summary:Let us consider a map T : A → B , where A and B are two nonempty subsets of a metric space X . The aim of this article is to provide sufficient conditions for the existence of a unique point x ∗ in A , called the best proximity point, which satisfies d ( x ∗ , T x ∗ ) = dist ( A , B ) : = inf { d ( a , b ) : a ∈ A , b ∈ B } . Our result generalizes a result due to Rhoades [B.E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Analysis TMA, 47(2001), 2683–2693] and hence it provides an extension of Banach’s contraction principle to the case of non-self-mappings.
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2011.04.052