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Best proximity point theorems on partially ordered sets. (English) Zbl 1267.90104
Summary: The main purpose of this article is to address a problem that amalgamates approximation and optimization in the setting of a partially ordered set that is endowed with a metric. Indeed, if $A$ and $B$ are non-void subsets of a partially ordered set that is equipped with a metric, and $S$ is a non-self mapping from $A$ to $B$, this paper scrutinizes the existence of an optimal approximate solution, called a best proximity point of the mapping $S$, to the operator equation $Sx~=~x$ where $S$ is a continuous, proximally monotone, ordered proximal contraction. Further, this paper manifests an iterative algorithm for discovering such an optimal approximate solution. As a special case of the result obtained in this article, an interesting fixed point theorem on partially ordered sets is deduced.

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
Full Text: DOI
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