Abstract
Best proximity point theorem furnishes sufficient conditions for the existence and computation of an approximate solution ω that is optimal in the sense that the error σ ( ω , J ω ) assumes the global minimum value σ ( θ , ϑ ) . The aim of this paper is to define the notion of Suzuki α - Θ -proximal multivalued contraction and prove the existence of best proximity points ω satisfying σ ( ω , J ω ) = σ ( θ , ϑ ) , where J is assumed to be continuous or the space M is regular. We derive some best proximity results on a metric space with graphs and ordered metric spaces as consequences. We also provide a non trivial example to support our main results. As applications of our main results, we discuss some variational inequality problems and dynamical programming problems.
Highlights
Introduction and PreliminariesIn 1969, Fan [1] initiated and obtained a classical best approximation result, that is, if θ is a nonempty compact convex subset of a Hausdorff locally convex topological vector space θ andJ : θ → θ is a continuous mapping, there exists ω in θ such that σ(ω, J ω ) = σ(J ω, θ )
If C be a compact subset of a metric space (M, σ ) and ω ∈ M, there exists v ∈ C
We prove that ω ∗ is a best proximity point of J
Summary
In 1969, Fan [1] initiated and obtained a classical best approximation result, that is, if θ is a nonempty compact convex subset of a Hausdorff locally convex topological vector space θ and. J : θ → θ is a continuous mapping, there exists ω in θ such that σ(ω, J ω ) = σ(J ω, θ ). In 2010, Basha [2] introduced the notion of best proximity point of a non-self mapping. He gave a generalization of the Banach fixed point theorems by a best proximity theorem. Let θ and θ be nonempty subsets of a metric space (M, σ ). A point ω is called a best proximity point of mapping. Sankar Raj [3] and
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