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Best proximity points: Global optimal approximate solutions. (English) Zbl 1208.90128
Summary: Let $A$ and $B$ be non-empty subsets of a metric space. As a non-self mapping ${T : A \longrightarrow B}$ does not necessarily have a fixed point, it is of considerable interest to find an element $x$ in $A$ that is as close to $Tx$ in $B$ as possible. In other words, if the fixed point equation $Tx = x$ has no exact solution, then it is contemplated to find an approximate solution $x$ in $A$ such that the error $d(x, Tx)$ is minimum, where $d$ is the distance function. Indeed, best proximity point theorems investigate the existence of such optimal approximate solutions, called best proximity points, to the fixed point equation $Tx = x$ when there is no exact solution. As the distance between any element $x$ in $A$ and its image $Tx$ in $B$ is at least the distance between the sets $A$ and $B$, a best proximity pair theorem achieves global minimum of $d(x, Tx)$ by stipulating an approximate solution $x$ of the fixed point equation $Tx = x$ to satisfy the condition that $d(x, Tx) = d(A, B)$. The purpose of this article is to establish best proximity point theorems for contractive non-self mappings, yielding global optimal approximate solutions of certain fixed point equations. Besides establishing the existence of best proximity points, iterative algorithms are also furnished to determine such optimal approximate solutions.

90C20Quadratic programming
Full Text: DOI
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