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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.

90C26Nonconvex programming, global optimization
90C30Nonlinear programming
41A65Abstract approximation theory
46B20Geometry and structure of normed linear spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
54H25Fixed-point and coincidence theorems in topological spaces
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