Existence results on nonautonomous partial functional differential equations with state-dependent infinite delay

Abstract

The aim of this work is to establish the existence of mild solutions for some nondensely nonau-tonomous partial functional differential equations with state-dependent infinite delay in Banachspace. We assume that, the linear part is not necessarily densely defined and generates an evolution family under the hyperbolique conditions. We use the classic Shauder Fixed Point Theorem, the Nonlinear Alternative Leray-Schauder Fixed Point Theorem and the theory of evolution family, we show the existence of mild solutions. Secondly, we obtain the existence of mild solution in a maximal interval using Banach’s Fixed Point Theorem which may blow up at the finite time, weshow that this solution depends continuously on the initial data under the global Lipschitz condition on the second argument of F and we get the existence of global mild solution. We proposesome model arising in dynamic population for the application of our results.