Some considerations to the fundamental theory of infinite delay equations

Abstract

In recent years we see an increasing interest in infinite delay equations. The main reason is that equations of this type become more and more important for different applications. Regardless of the specific problem one has to deal with it is in most cases necessary to establish some fundamental theory as, for instance, existence, uniqueness of solutions and continuous dependence on initial data. Also in the last few years we find an increasing number of papers where the general theory of linear and nonlinear semigroups or evolution operators is applied to functional differential equations. Of fundamental importance for all approaches is the right choice of the state space which in most cases is a Banach space of functions or of equivalence classes of functions. For equations with bounded delays this in general is not a difficult problem. But for infinite delay equations the choice of an appropriate state space is no more trivial. It was natural to investigate which properties of the state space are sufficient in order to establish the fundamental theory for infinite delay equations. Up to now the most thorough discussion of this problem is contained in [12]. The set of axioms given there seems to be in more or less final form. Other systems of axioms are just slight modifications of those given in [12] (cf., for instance, [18, 211) or consider more special cases (as in [6, 17). Section 1 of this paper can be considered as a discussion of the axioms given in [12] which results in a somewhat streamlined version of these axioms. It should be mentioned here that state spaces for equations with bounded retardation are included as special case, of course. In Section 2 we prove existence, uniqueness and continuous dependence of solutions concentrating on the nonstandard situation where the right-hand side of the equation is not defined for elements in the state space but only for representatives of elements in the state space. Such situations occur for instance if difference-differential equations are considered in a space like UP