Differential equations in abstract spaces

  • Donal O’Regan
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 398)

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.

Keywords

Banach Space Existence Result Fixed Point Theorem Real Hilbert Space Real Banach Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Copyright information

© Springer Science+Business Media Dordrecht 1997

Authors and Affiliations

  • Donal O’Regan
    • 1
  1. 1.Department of MathematicsUniversity College GalwayGalwayIreland

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