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Best proximity point theorems for reckoning optimal approximate solutions. (English) Zbl 1277.90097

Summary: Given a non-self mapping from \(A\) to \(B\), where \(A\) and \(B\) are subsets of a metric space, in order to compute an optimal approximate solution of the equation \(Sx = x\), a bestproximity point theorem probes into the global minimization of the error function \(x\to d(x, Sx)\) corresponding to approximate solutions of the equation \(Sx = x\). This paper presents a best proximity point theorem for generalized contractions, thereby furnishing optimal approximate solutions, called best proximity points, to some non-linear equations. Also, an iterative algorithm is presented to compute such optimal approximate solutions.

MSC:

90C26 Nonconvex programming, global optimization
90C30 Nonlinear programming
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46B20 Geometry and structure of normed linear spaces
47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
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