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Zbl 1021.47027
Sadiq Basha, S.; Veeramani, P.; Pai, D.V.
Best proximity pair theorems. (English)
[J] Indian J. Pure Appl. Math. 32, No.8, 1237-1246 (2001). ISSN 0019-5588; ISSN 0975-7465/e

Let $X$ and $Y$ be any two topological spaces. A multifunction $T:X\to 2^Y$ is said to be\par (i) upper semi-continuous if $T^{-1}(B)= \{x\in X:(Tx)\cap B\ne\emptyset\}$ is closed in $X$ whenever $B$ is a closed subset of $Y$;\par (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;\par (iii) Kakutani factorizable if $T$ can be expressed as a composition of finitely many Kakutani multifunctions.\par 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 $\{x_\alpha\}$ in $A$ satisfying $d_p (x_\alpha,y)\to d_p(y,A)\equiv\inf\{p(y-a):a\in A\}$, there is a subset of $\{ x_\alpha\}$ converging to an element of $A$.\par 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 $$d_p(g x_0,Tx_0)= d_p(A,B)\equiv \inf\{p(a-b): a\in A,b\in B\},$$ 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.
[T.D.Narang (Amritsar)]

MSC 2000:: *47H04 Set-valued operators

Keywords: approximately $p$-compact set; upper semi-continuous multifunction; Kakutani factorizable multifunction; Kakutani multifunction; best proximity pair theorems

Cited in: Zbl 1021.47028

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