Simplicial Methods, Fixed-Point Theory, and Mathematical Economics

In the following two chapters, we present two ways of introducing fixed-point theory which have been developed intensely during the last years and provide an effective method for solving nonlinear equations on computers:(i)Simplicial methods (Chapter 77).(ii)Homotopy methods (Chapter 78). In (i) we use triangulations and suitable labelings. The starting point is the lemma of Sperner of Section 77.1. In (ii), curves, i.e., one-dimensional manifolds are being followed. Thereby differential topological methods, especially Sard’s theorem are employed. The disadvantage of (i) over (ii), in view of computer usage, is that with an increasing number of variables in the nonlinear equations, the necessary storage place increases very rapidly.