A Digital Version of the Kakutani Fixed Point Theorem for Convex-valued Multifunctions

Abstract

In this paper, the concepts of power graphs and power complexes are introduced. The multifunctions for graphs are denned and will be classified. The concept of simplicial mappings for complexes then is extended to multifunctions. A notion of weak convexity is defined in the intersection graphs of (3− 1)-adjacent n-dimensional real digital pictures based on the usual Euclidean convex closure operator. It is shown that any (3 − 1)-adjacent n-dimensional digital picture has the simplicial weak convex almost fixed point property, which may be considered as a digital version of the Kakutani fixed point theorem for convex-valued multifunctions.