Discrete Fixed Point Analysis and Its Applications

We have carried out a thorough study on the existence of discrete fixed points and applications. As a result, we establish several general existence theorems of fixed and zero points on a broad class of subsets of the integer lattice Z and also on arbitrary discrete sets. Most results have their well-known continuous counterparts. One of the major conditions is that of locally gross direction preservation imposed on the mappings under consideration. Furthermore, we establish several theorems for the existence of a unique discrete zero point. In addition, three applications are discussed. We first study a pure exchange competitive economy with indivisibilities and show the existence of a discrete Walrasian equilibrium. Second, we show the existence of a pure strategy Nash equilibrium in games where the set of each player’s strategies is a discrete set and the mixed strategies are not played. Third, we discuss a general Samuelson’s spatial price equilibrium model in a discrete setting.