Differential Equations on Closed Subsets of a Banach Space

Abstract

In this paper the problem of existence of solutions to the initial value problem u\t) = A(t, u(t)), u(a) = z, is considered where A : [a, b) x D - E is continuous, O is a closed subset of a Banach space E, and z e D. With a dissipative type condition on A, we establish sufficient conditions for this initial value problem to have a solution.Using these results, we are able to characterize all continuous functions which are generators of nonlinear semigroups on D. I. Introduction.Suppose that F is a Banach space over the real or complex field and that |-| denotes the norm on E. Let [a,b) be a number interval, D a closed (or locally closed) subset of F, and A a continuous function from [a, b)x D into F.In this paper we study the existence and uniqueness of solutions to the initial value problem (IVP) u'=A(t,u), u(a) = z, where z is in D. By a solution to (IVP) on an interval [a, c) c [a, b), we mean a continuously differentiable function u from [a, c) into D such that u(a) = z and u'(t) = A(t, u(t)) for all t in [a, c).Throughout this paper we assume that E has the norm topology.The main technique employed here is the construction of approximate solutions to (IVP) by using conditions on A similar to those of H. Brzis in [2].We then place dissipative type conditions on A which ensure the convergence of our approximate solutions and the uniqueness of solutions to (IVP).Using these results we are able to give a characterization of continuous generators of semigroups of nonlinear transformations on D. For other recent papers which deal with similar problems, see F. Browder [4], M. Crandall [5], [6], P. Hartman [8], J. Herod [9], R. Martin [13], [14], R. Redheffer [15], and G. Webb [16], [17].Acknowledgement.The original version of this paper contained results similar to those presented here, but