A General Existence Theorem of Zero Points

Abstract

Let X be a non-empty, compact, convex set in R and o an upper semi-continuous mapping from X to the collection of non-empty, compact, convex subsets of R;. It is well known that such a mapping has a stationary point on X, i.e. there exists a point in X satisfying that its image under o has a non-empty intersection with the normal cone of X at the point. In case for every point in X it holds that the intersection of the image under o with the normal cone of X at the point is either empty or contains the origin O; then o must have a zero point on X, i.e. there exists a point in X satisfying that O lies in the image of the point. Another well-known condition for the existence of a zero point follows from Ky Fan's coincidence theorem, which says that if for every point the intersection of the image with the tangent cone of X at the point is non-empty, the mapping must have a zero point. In this paper we extend all these existence results by giving a general zero point existence theorem, of which the two results are obtained as special cases. We also discuss what kind of solutions may exist when no further conditions are stated on the mapping o. Finally, we show how our results can be used to establish several new intersection results on a compact, convex set.