Almost periodic mild solutions of a class of partial functional differential equations

Abstract

We study the existence of almost periodic mild solutions of a class of partial functional differential equations via semilinear almost periodic abstract functional differential equations of the form $$(*)x^{\prime }=f\left(t,x,x_t\right).$$ To this end, we first associate with every almost periodic semilinear equation $$\left(**\right)x^{\prime }=F\left(t,x\right).$$ a nonlinear semigroup in the space of almost periodic functions. We then give sufficient conditions (in terms of the accretiveness of the generator of this semigroup) for the existence of almost periodic mild solutions of (**) as fixed points of the semigroup. Those results are then carried over to equation (*). The main results are stated under accretiveness conditions of the function $f$ in terms of $x$ and Lipschitz conditions with respect to $x_t$.