ACTA issues

Best proximity pair theorems

P. S. Srinivasan

Acta Sci. Math. (Szeged) 67:1-2(2001), 421-429

Abstract. Let $A$ be a non-empty approximately compact convex subset, $B$ be a non-empty closed convex subset and $C$ be a non-empty convex subset of a normed linear space $E$. Given a multifunction $T_1\colon A \longrightarrow2^C$ with open fibres, a Kakutani factorizable multifunction $T_2\colon C\longrightarrow2^B$ and a single valued function $g\colon A \longrightarrow A$, best proximity pair theorems furnishing the sufficient conditions for the existence of an element $x_\circ\in A$ such that $$ d(gx_{\circ }, T_2 T_1 x_{\circ }) = d(A, B) $$ are explored. As a consequence, a generalization of Ko and Tan's coincidence theorem is obtained.

AMS Subject Classification (1991): 47H10, 54H25

Keyword(s): Best proximity pairs, Kakutani factorizable multifunctions, Multifunctions with open fibres, Proper map, Quasi affine map

Received March 9, 2000. (Registered under 2793/2009.)