A discrete fixed point theorem utilizing the direction preserving condition

This paper aims to characterize the direction preserving condition that guarantees the existence of a fixed point of discrete mappings defined on an integer rectangle X into itself. We deal with a discrete fixed point theorem based on Brouwer's fixed point theorem, which depends on the simplicial decomposition of the convex hull of X. We first review an arbitrary simplicial decomposition in ℝ² and the Preudenthal decomposition in ℝⁿ. Next we characterize the direction preserving condition for an arbitrary consistent simplicial decomposition in ℝⁿ, which implies a sufficient condition for the strategic game to have a pure-strategy equilibrium.