Equilibrium Heat Distribution

Abstract

The rest of Part I will be devoted to demonstrating the usefulness of two of the tools we have developed: a fixed point existence theorem (the Leray-Schauder alternative, a corollary of the Schauder theorem) and a compactness property of C ^k-spaces that is a consequence of the Ascoli-Arzela theorem. We used information from the Ascoli-Arzela and Schauder theories in Chapter 1, to prove the Cauchy-Peano theorem by topological methods. We will see a more recent application of these theories in the next chapter, in which we demonstrate the existence of solutions to a different type of problem in the theory of ordinary differential equations.