Weighted Asymptotic Behavior of Solutions to a Sobolev-Type Differential Equation with Stepanov Coefficients in Banach Spaces

Abstract

In this paper, we investigate weighted asymptotic behavior of solutions to the Sobolev-type differential equation

\begin{equation*}

\frac{d}{dt}[u(t)+f(t,u(t))]=A(t)u(t)+g(t,u(t)), \ \ t\in \mathbb{R},

\end{equation*}%

where $A(t) : D \subset \mathbb{X} \rightarrow \mathbb{X}$ for $t \in \mathbb{R}$ is a family of densely defined closed linear operator on a domain $D$, independent of $t$, and $f\ : \mathbb{R} \times \mathbb{X} \rightarrow \mathbb{X}$ is a weighted pseudo almost automorphic function and $g\ : \mathbb{R} \times \mathbb{X} \rightarrow \mathbb{X}$ is an $S^{p}$-weighted pseudo almost automorphic function and satisfying suitable conditions. Some sufficient conditions are established by properties of $S^{p}$-weighted pseudo almost automorphic functions combined with theories of asymptotically stable of operators.