Differential equations in abstract spaces

Abstract

In this chapter, we are concerned with the initial value problem:

$$\left\{ {\begin{array}{*{20}{c}} {y'\left( t \right) = f\left( {t,y\left( t \right)} \right){\text{ }}t \in \left[ {0,T} \right]} \\ {y\left( 0 \right) = a \in E;} \end{array}} \right.$$

(1.1)

where E is a real Banach space and f : [0, T] × E → E has a decomposition f = g + h with g and h Carathéodory functions satisfying respectively, a compactness and Lipschitz assumptions. Our results rely on Krasnoselskii fixed point theorem for contraction plus compact mappings and don’t use homotopy arguments. It is worth remarking here that the periodic problem could also be discussed in this setting (we leave this as an exercise). Our main existence principle is obtained in section 16.2. This result will be used in section 16.3 to obtain more applicable existence results. More precisely, in section 16.3, we give existence theorems of Wintner type. Also in section 16.3, existence theorems are obtained under an assumption which is equivalent to an assumption of existence of upper and lower solutions to (1.1) in the scalar case. No growth condition is assumed.