P.J.J. Herings

A General Existence Theorem of Zero Points

AbstractLet X be a nonempty, compact, convex set in

Modelling producer decisions in a spatial continuum

\mathbb{R}^n

The Average Tree permission value for games with a permission tree

and let φ be an upper semicontinuous mapping from X to the collection of nonempty, compact, convex subsets of

Equilibria with coordination failures

\mathbb{R}^n

A general existence theorem of zero points

. It is well known that such a mapping has a stationary point on X; i.e., there exists a point X such that its image under φ has a nonempty intersection with the normal cone of X at the point. In the case where, for every point in X, it holds that the intersection of the image under φ with the normal cone of X at the point is either empty or contains the origin 0n, then φ must have a zero point on X; i.e., there exists a point in X such that 0n lies in the image of the point. Another well-known condition for the existence of a zero point follows from the Ky Fan coincidence theorem, which says that, if for every point the intersection of the image with the tangent cone of X at the point is nonempty, 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 previous two results are obtained as special cases. We discuss also what kind of solutions may exist when no further conditions are stated on the mapping φ. Finally, we show how our results can be used to establish several new intersection results on a compact, convex set.