Common best proximity points: global minimization of multi-objective functions

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Abstract

Given non-empty subsets A and B of a metric space, let \$\${S{:}A{\longrightarrow} B}\$\$ and \$\${T {:}A{\longrightarrow} B}\$\$ be non-self mappings. Due to the fact that S and T are non-self mappings, the equations Sx=x and Tx=x are likely to have no common solution, known as a common fixed point of the mappings S and T. Consequently, when there is no common solution, it is speculated to determine an element x that is in close proximity to Sx and Tx in the sense that d(x, Sx) and d(x, Tx) are minimum. As a matter of fact, common best proximity point theorems inspect the existence of such optimal approximate solutions, called common best proximity points, to the equations Sx=x and Tx=x in the case that there is no common solution. It is highlighted that the real valued functions \$\${x{\longrightarrow}d(x, Sx)}\$\$ and \$\${x{\longrightarrow}d(x, Tx)}\$\$ assess the degree of the error involved for any common approximate solution of the equations Sx=x and Tx=x. Considering the fact that, given any element x in A, the distance between x and Sx, and the distance between x and Tx are at least d(A, B), a common best proximity point theorem affirms global minimum of both functions \$\${x{\longrightarrow}d(x, Sx)}\$\$ and \$\${x{\longrightarrow}d(x, Tx)}\$\$ by imposing a common approximate solution of the equations Sx=x and Tx=x to satisfy the constraint that d(x, Sx)=d(x, Tx)=d(A, B). The purpose of this article is to derive a common best proximity point theorem for proximally commuting non-self mappings, thereby producing common optimal approximate solutions of certain simultaneous fixed point equations in the event there is no common solution. Copyright Springer Science+Business Media, LLC. 2012

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Bibliographic Info

Article provided by Springer in its journal Journal of Global Optimization.

Volume (Year): 54 (2012)
Issue (Month): 2 (October)
Pages: 367-373

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Handle: RePEc:spr:jglopt:v:54:y:2012:i:2:p:367-373

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Keywords: Optimal approximate solution; Common best proximity point; Common fixed point; Proximally commuting mappings;

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