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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.

##### MSC:
 90C26 Nonconvex programming, global optimization 90C30 Nonlinear programming 41A65 Abstract approximation theory 46B20 Geometry and structure of normed linear spaces 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces 54H25 Fixed-point and coincidence theorems in topological spaces
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