Boundary Value Problems on Infinite Intervals: A Topological Approach

Abstract

The aim of this chapter is twofold. First we wish to survey most of the fixed point theorems available in the literature for compact operators defined on Fréchet spaces. In particular we present the three “most applicable” results from the literature in Section 9.2. The first result is the well-known Schauder-Tychonoff theorem, the second, a Furi-Pera type result and the third, a fixed point result based on a diagonalization argument. Applications of these fixed point theorems to differential and difference equations can be found in a recent book of Agarwal and O’Regan [17]. Our second aim is to survey the results in the literature concerning time scale problems on infinite intervals. Only a handful of results are known, and the theory we present in Section 9.3 is based on the diagonalization approach in Section 9.2; this approach seems to give the most general and natural results. In Section 9.4 we consider linear systems on infinite intervals.