Retracts , fixed point index and differential equations ∗ Rafael Ortega

Many problems in the theory of differential equations were initially treated with analytic techniques and later evolved towards more topological approaches. Perhaps the most paradigmatic case is found in the study of nonlinear boundary value problems. The classical proofs based on successive approximations or in the implicit function theorem were soon replaced by the use of fixed points and degree theory. The modern point of view is already found in the famous paper by Leray and Schauder [15]. The same process has been experienced by other branches of differential equations. The next pages are an attempt to illustrate this evolution in two concrete problems. First we will discuss the existence of asymptotic solutions. These are non-trivial solutions tending to the origin as time increases to infinity and they appear in systems of differential equations having the trivial solution. Asymptotic solutions have been studied since Poincaré’s times. The classical method for proving their existence consists in the reduction of the problem to an integral equation. Once this equation has been found one uses the method of successive approximations or the contraction principle. This analytical method leads to the Principle of Linearization and to the Stable Manifold