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Zbl 1209.45007
Lizama, Carlos; N'Guérékata, Gaston M.
Bounded mild solutions for semilinear integro differential equations in Banach spaces. (English)
[J] Integral Equations Oper. Theory 68, No. 2, 207-227 (2010). ISSN 0378-620X; ISSN 1420-8989/e

The authors study the structure of several classes of spaces of vector-valued functions $\mathcal{M}(\mathbb{R};X)$; here $X$ denotes a Banach space. The integro-differential equation $$ u'(t)=Au(t)+\int_{-\infty}^t a(t-s)Au(s)ds+f(t,u(t)) \tag1$$ is considered, where $A$ is a closed linear operator defined in $X$ and $a\in L^1_{\text{loc}}(\mathbb{R}_+)$ is a scalar-valued kernel. Using a unified approach for various spaces $\mathcal{M}(\mathbb{R};X)$, the authors establish conditions on $A$ and $f$ ensuring that the solution $u$ of (1) exists and has the same asymptotic behaviour as $f$. In particular, almost automorphic, pseudo-almost automorphic, asymptotically periodic and almost periodic classes of functions are investigated. Moreover, asymptotically compact almost automorphic functions and pseudo compact almost automorphic functions are introduced in the paper.
[Mirosława Zima (Rzeszów)]

MSC 2000:: *45N05 Integral equations in abstract spaces
43A60 Almost periodic functions on groups, etc.
45J05 Integro-ordinary differential equations
45G05 Singular nonlinear integral equations

Keywords: linear and semilinear integro-differential equations; regularized operator families; bounded mild solutions; Banach space; asymptotic behaviour; almost automorphic; pseudo-almost automorphic; asymptotically periodic; almost periodic

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