The Fold Complementarity Problem and the Order Complementarity Problem

We consider the Fold Complementarity Problem, which is one of the recent subjects in complementarity theory. It is a mathematical model used in economics in the study of distributive problems (cf. [25], [5]). A particular case is the k-Fold Complementarity Problem, studied using a variant of the notion of Z-function by A. Villar [25]. In this way, Villar obtained the solution of this problem as a solution of a minimization problem. We will study the Fold Complementarity Problem by a topological method. We will show that this method is also applicable to systems of Fold Complementarity Problems. We work towards two aims. First, we show that the Fold Complementarity Problem is exactly equivalent to an Order Complementarity Problem. This is important, since in this way the Fold Complementarity Problem is transformed into a nonlinear equation, or a fixed point problem, and hence we can use the theory of Order Complementarity Problems [9]–[15]. Second, we show that the Order Complementarity Problem associated with the Fold Complementarity Problem is naturally prepared to apply the topological index defined by Opŏıtsev [20] for continuous admissible mappings, defined on solid convex cones. We remark that to define this topological index it is not necessary to introduce a complicated