Dynamical systems with memory on history-spaces with monotonic seminorms

Abstract

During the last 20 years the study of solutions of retarded functional-differential equations has played an important part in theoretical physics, chemistry, and biology. For the case in which delay is bounded by the number r 3 0 and the equation describes the behavior in time of n real variables, Hale was the first who systematically studied the solution operator as a transformation in C([-Y, 01, W) [7]. If the delay is infinite, a seminormed vector space X of functions on the half axis (-co, 0] may take the place of C([-Y, 01, W”). But in this case, provided that the right-hand side of the equation is continuous and bounded on bounded subsets of X, it is already difficult to characterize the seminorms such that existence, continuability, and continuous dependence of solutions and the relative compactness of bounded orbits hold true. The first systematic study in this direction is due to Coleman and Mizel [14, and references therein]. As the class of spaces X underlying the work of Coleman and Mizel does not contain C([-Y, 01, UP), Hale [5, 141 suggested the definition of another class and he has shown that in his spaces bounded orbits of retarded functional-differential equations are relatively compact. In recent years the techniques for the theoretical treatement of functional-differential equations with infinite delay have been further developed; cf. [6, 10, 12-14, and references]. The present essay studies dynamical systems with memory, in which the seminorm of X is monotonic with respect to the natural partial ordering of W-valued functions on (--co, 01. Some problems already treated in [3, 5, 6, 141, e.g., the problem when bounded orbits are relatively compact or when the solution operator is condensing with respect to a measure of noncompactness (cf. [ll]), are raised at once. It will be shown that the componentwise monotony of the seminorm of ,ri is a powerful hypothesis that enables us to dispense with various other hypotheses which in absence of the monotony would be required; cf. [5, hypothesis (h3), (h4)]. In the current literature this fact has not yet been pointed out; the present paper will remove this deficiency.