Electronic Journal of Differential Equations,
Vol. 2014 (2014), No. 91, pp. 1-16.
Title: Weighted asymptotic behavior of solutions to semilinear
integro-differential equations in Banach spaces
Authors: Yan-Tao Bian (Lanzhou Jiaotong Univ., Lanzhou, China)
Yong-Kui Chang (Lanzhou Jiaotong Univ., Lanzhou, China)
Juan J. Nieto (Univ. de Santiago de Compostela, Spain)
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 02, 2014.
Math Subject Classifications: 4K14, 60H10, 35B15, 34F05.
Key Words: Integro-differential equations; uniformly exponentially stable;
Stepanov-like weighted pseudo almost automorphy.