ABSTRACT
In this chapter, we prove some existence, uniqueness, stability and even uniform asymptotic stability results regarding C0-solutions for a large class of nonlinear delay evolution equations with nonlocal initial data. For the sake of simplicity here, we confine ourselves to the study of the autonomous case which, under fairly reasonable hypotheses, can be handled with the results from Section 1.12. Moreover, from now on, we will focus our attention mainly on bounded C0-solutions. The bias of this preference is explained by the fact that many phenomena exhibit a long-time bounded behavior. Accordingly, if the corresponding mathematical models do not have bounded solutions, they are usually considered inappropriate and rejected. This is the case of the wellknown demographic model proposed by Malthus – see Vrabie [254, Section 1.4.6, pp. 3-34] – which was severely criticized even by his contemporaries.
