Lyapunov functions and autonomous differential equations in a Banach space

Abstract

Let R denote the space of real numbers, let E be a Banach space over the real or complex field, and let [. [ denote the norm on E. In this paper we study the existence and behavior of solutions to the autonomous differential equationwhere A is a continuous function from E into E. In particular, sufficient conditions are established to ensure that (1) has a unique critical point which is globally asymptotically stable and, with additional conditions on A, an iterative method is developed which converges to this critical point. The main purpose of this paper is to point out that many of the techniques used in the theory of monotone operators can be applied in a more general situation (see, for example, Hartman [8]). Instead of using the norm on E to study the solutions to (1), we assume the existence of a function V from E× E into [0, or) which has the following basic properties: 66MATHEMATICAL SYSTEMS THEORY t VOI. 7, NO. i .

On the rest points of a nonlinear nonexpansive semigroup

“…M. Crandall and A. Pazy [2] give an affirmative answer, when X is a Hubert space. However, the converse is not true in general (see R. Martin [4]). …”