Electron. J. Diff. Equ., Vol. 2014 (2014), No. 91, pp. 1-16.

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

Yan-Tao Bian, Yong-Kui Chang, Juan J. Nieto

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.

Submitted October 21, 2013. Published April 2, 2014.
Math Subject Classifications: 4K14, 60H10, 35B15, 34F05.
Key Words: Integro-differential equations; uniformly exponentially stable; Stepanov-like weighted pseudo almost automorphy.

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Yan-Tao Bian
Department of Mathematics, Lanzhou Jiaotong University
Lanzhou 730070, China
email: 1120689525@qq.com
Yong-Kui Chang
Department of Mathematics, Lanzhou Jiaotong University
Lanzhou 730070, China
email: lzchangyk@163.com
Juan J. Nieto
Departamento de Análisis Matemático, Facultad de Matemáticas
Universidad de Santiago de Compostela
15782, Santiago de Compostela, Spain
email: juanjose.nieto.roig@usc.es

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