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Almost automorphic solutions for abstract functional differential equations. (English) Zbl 1046.34088
Let $$X$$ be a Banach space and let $$B= B((-\infty,0], X)$$ be the so-called fading memory space consisting of functions $$\varphi: (-\infty, 0]\to X$$ which satisfy some conditions. The authors consider the abstract functional-differential equation (1) $$du(t)/dt= Au(t)+ F(t)u_t+ f(t)$$, where $$A$$ is the infinitesimal generator of a strongly continuous semigroup of linear operators on $$X$$, $$F(t)$$ is a bounded linear operator from $$B$$ into $$X$$ which is periodic in $$t$$. Moreover, $$f\in BC(\mathbb{R}, X)$$. A result on the existence of almost automorphic solutions of equation (1) is established. It is assumed that the function $$f$$ is almost automorphic.

MSC:
 34K30 Functional-differential equations in abstract spaces
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References:
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