Extensions of best approximation and coincidence theorems

Let X be a Hausdorff compact space, E a topological vector space on which E * separates points, F : X → 2 E 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 x 0 ∈ X such that g x 0 ∈ F x 0 or (2) there exist an ( x 0 , z 0 ) on the graph of F and a continuous seminorm p on E such that 0 < p ( g x 0 − z 0 ) ≤ p ( y − z 0 ) 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.