Teoremas de ponto fixo

Abstract

This work is concerned with the Fixed Point theorems, more precisely, we deal with the Banach, Brouwer, Schauder and Schaefer theorems, and their applications. For the proof of Banach?s theorem, the iterative process has been presented providing the successive approximation algorithm for applications. In order to prove the Brouwer Fixed Point theorem, we show that there is no continuous retraction from the unit ball into its boundary. The machinery of the Null- Lagrangian and some Functional Analysis results are used. The Shauder theorem is proven as a generalization of the Brouwer Fixed Point theorem to the infinite-dimensional compact and convex sets. Finite-dimensional projections are used for continuous operators to show the to be possessed with a fixed point. As a corollary, we prove in the sequel the Schaefer Fixed Point theorem which is useful to solve boundary and initial-boundary value problems for non-linear differential equations