International Journal of Mathematics and Mathematical Sciences
Volume 20 (1997), Issue 4, Pages 689-698
doi:10.1155/S016117129700094X

Extensions of best approximation and coincidence theorems

Abstract

Let X be a Hausdorff compact space, E a topological vector space on which E* separates points, F:X→2E an upper semicontinuous multifunction with compact acyclic values, and g:X→E a continuous function such that g(X) is convex and g−1(y) is acyclic for each y∈g(X). Then either (1) there exists an x0∈X such that gx0∈Fx0 or (2) there exist an (x0,z0) on the graph of F and a continuous seminorm p on E such that 0<p(gx0−z0)≤p(y−z0)         for all         y∈g(X). A generalization of this result and its application to coincidence theorems are obtained. Our aim in this paper is to unify and improve almost fifty known theorems of others.