Abstract weighted pseudo almost automorphic functions, convolution invariance and neutral integral equations with applications

This paper deals with a systematic study of the convolution operator defined on the weighted pseudo almost automorphic functions space ($,\mathit{PAA} (\mathbb {X}, \rho )$). The purpose of this, is to ensure the existence and uniqueness of solutions in $,\mathit{PAA} (\mathbb {X},\rho )$ for general abstract neutral integral equations of convolution type. Upon making different assumptions on the kernel $k$ of the convolution operator and the weight $\rho $ we obtain results about the convolution invariance of the operator on $,\mathit{PAA} (\mathbb {X}, \rho )$. Essentially the assumptions are of two type, one is a new condition on $\rho $ valid for every kernel $k$ and the other is a ($k$, $\rho $)-type condition, in which the kernel $k$ helps the weight $\rho $. These conditions are not known in the literature. Explicit examples show the utility of these two different assumptions. Taking advantage of the rich properties of the convolution we have obtained new results about composition which permits study the existence and uniqueness of mild solutions in $,\mathit{PAA} (\mathbb {X}, \rho )$ for general abstract neutral integral equations. The results obtained are directly applied to integro differential equations, partial differential equations, logistic equations and differential equations of first and fractional order, among other. Several examples and concrete classes of differential equations illustrate our results.