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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:
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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