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Almost automorphic and pseudo-almost automorphic mild solutions to an abstract differential equation in Banach spaces. (English) Zbl 1189.34116
Consider the existence and uniqueness of almost automorphic and pseudo almost automorphic mild solutions to the differential equation
$\frac{d u(t)}{dt}=Au(t)+\frac{d}{dt}\;F_1(t, u(h_1(t)))+F_2(t,u(h_2(t)))$ where $$A$$ is a linear operator on a Banach space that generates an exponentially stable $$C_0$$-semigroup $$(T(t))_{t\geq 0}$$.
The proofs are achieved by means of the contraction mapping principle. This problem goes back to a result by the reviewer [Semigroup Forum 69, No. 1, 80–86 (2004; Zbl 1077.47058)].
The authors use the following statement in the proofs of their main results: If $$u:\mathbb R\to X$$ is almost automorphic and $$h:\mathbb R\to\mathbb R$$ is a continuous function, then $$u(h(t))$$ is almost automorphic. This remained to be proved. An application to a semilinear partial differential equation with Dirichlet conditions is presented.

##### MSC:
 34G20 Nonlinear differential equations in abstract spaces 43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions 47N20 Applications of operator theory to differential and integral equations
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