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.
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Sadiq Basha, S. Common best proximity points: global minimization of multi-objective functions. J Glob Optim 54, 367–373 (2012). https://doi.org/10.1007/s10898-011-9760-8
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DOI: https://doi.org/10.1007/s10898-011-9760-8