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Bounded mild solutions for semilinear integro differential equations in Banach spaces. (English) Zbl 1209.45007

The authors study the structure of several classes of spaces of vector-valued functions (;X); here X denotes a Banach space. The integro-differential equation

u ' (t)=Au(t)+ - t a(t-s)Au(s)ds+f(t,u(t))(1)

is considered, where A is a closed linear operator defined in X and aL loc 1 ( + ) is a scalar-valued kernel. Using a unified approach for various spaces (;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.

45N05Abstract integral equations, integral equations in abstract spaces
43A60Almost periodic functions on groups, etc.; almost automorphic functions
45J05Integro-ordinary differential equations
45G05Singular nonlinear integral equations
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