Simplicial fixed point algorithms and applications

Fixed point theory is an important branch of modern mathematics and has always been a major theoretical tool in fields such as differential equations, topology, function analysis, optimal control, economics, and game theory. Its applicability has been increased enormously by the development of simplicial fixed point methods starting in the late 1960s. These methods have become powerful tools for finding solutions to general systems of nonlinear equations, price equilibria, Nash equilibria, core elements, and so on. In this monograph, new simplicial fixed point methods are developed to solve various new fixed point problems to which the existing methods do not apply. The convergence and performance of the methods are thoroughly studied. The theoretical problems considered range from the stability and refinement of stationary points, through the existence of a continuum of fixed points, the existence of equilibria in competitive economies with several indivisible commodities and money, and the existence of equilibria in competitive economies with production, to the feasibility of integer programs and generalizations of the classical Sperner lemma and KKM lemma.