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International Journal of Mathematics and Mathematical Sciences
Volume 20 (1997), Issue 4, Pages 689-698

Extensions of best approximation and coincidence theorems

Department of Mathematics, Seoul National University, Seoul 151–742, Korea

Received 15 December 1995

Copyright © 1997 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let X be a Hausdorff compact space, E a topological vector space on which E* separates points, F:X2E an upper semicontinuous multifunction with compact acyclic values, and g:XE a continuous function such that g(X) is convex and g1(y) is acyclic for each yg(X). Then either (1) there exists an x0X such that gx0Fx0 or (2) there exist an (x0,z0) on the graph of F and a continuous seminorm p on E such that 0<p(gx0z0)p(yz0)         for all         yg(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.