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Best proximity pair theorems. (English) Zbl 1021.47027

Let $X$ and $Y$ be any two topological spaces. A multifunction $T:X\to {2}^{Y}$ is said to be

(i) upper semi-continuous if ${T}^{-1}\left(B\right)=\left\{x\in X:\left(Tx\right)\cap B\ne \varnothing \right\}$ is closed in $X$ whenever $B$ is a closed subset of $Y$;

(ii) Kakutani multifunction if (a) $T$ is upper semi-continuous, (b) either $Tx$ is a singleton for each $x\in X$ or $Tx$ is a non-empty compact convex subset of $Y$, assuming $Y$ to be a non-empty convex set in a Hausdorff topological vector space;

(iii) Kakutani factorizable if $T$ can be expressed as a composition of finitely many Kakutani multifunctions.

Let $E$ be a Hausdorff locally convex topological vector space with a continuous seminorm $p$. A non-empty subset $A$ of $E$ is said to be approximately $p$-compact if for each $y\in E$ and each net $\left\{{x}_{\alpha }\right\}$ in $A$ satisfying ${d}_{p}\left({x}_{\alpha },y\right)\to {d}_{p}\left(y,A\right)\equiv inf\left\{p\left(y-a\right):a\in A\right\}$, there is a subset of $\left\{{x}_{\alpha }\right\}$ converging to an element of $A$.

In the present paper, the authors prove best proximity pair theorems which furnish sufficient conditions ensuring the existence of an element ${x}_{0}\in A$ such that

${d}_{p}\left(g{x}_{0},T{x}_{0}\right)={d}_{p}\left(A,B\right)\equiv inf\left\{p\left(a-b\right):a\in A,b\in B\right\},$

when $A$ is a non-empty approximately $p$-compact convex subset, $B$ a non-empty closed convex subset of $E$, $T:A\to {2}^{B}$ is a Kakutani factorizable multifunction and $g:A\to A$ is a single-valued function.

##### MSC:
 47H04 Set-valued operators