An Approach to Homotopy and Degree Theory

Abstract

Spawned by Scarf's pioneering work on the calculation of fixed points, an entire new field in mathematical programming has emerged. A wide array of problems that can be posed as fixed point problems, such as problems involving equilibria, games, systems of equations, global optimization, and structural mechanics, have come into the purview of these new constructive techniques. Underlying these techniques are certain mathematical concepts from differential topology, more specifically degree theory (Brouwer) and homotopies. Usually, the mathematical machinery required for these concepts necessitates a keen familiarity with piecewise linear and/or differential topology. In this paper, the development is made by following the paths arising from an ordinary differential equation. The differential equation results from Cramer's rule for linear equations. The applicability and ease of our approach are shown by obtaining simple proofs of theorems in differential topology.