Weighted asymptotic behavior of solutions to semilinear integro-differential equations in Banach spaces

Abstract

In this article, we study weighted asymptotic behavior of solutions to the semilinear integro-differential equation $$ u'(t)=Au(t)+\alpha\int_{-\infty}^{t}e^{-\beta(t-s)}Au(s)ds+f(t,u(t)), \quad t\in \mathbb{R}, $$ where $\alpha, \beta \in \mathbb{R}$, with $\beta > 0, \alpha \neq 0$ and $\alpha+\beta >0$, A is the generator of an immediately norm continuous $C_0$-Semigroup defined on a Banach space $\mathbb{X}$, and $f:\mathbb{R}\times \mathbb{X} \to \mathbb{X}$ is an $S^p$-weighted pseudo almost automorphic function satisfying suitable conditions. Some sufficient conditions are established by using properties of $S^p$-weighted pseudo almost automorphic functions combined with theories of uniformly exponentially stable and strongly continuous family of operators.

Keywords

• Integro-differential equations
• uniformly exponentially stable
• Stepanov-like weighted pseudo almost automorphy