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Almost automorphic solutions for hyperbolic semilinear evolution equations. (English) Zbl 1086.34052
Summary: Here, we deal with mild solutions for the semilinear evolution equation $$\tfrac{d}{dt}x(t) = Ax(t) + f(t,x(t)),\qquad t\in \Bbb R,$$ under the sectoriality of $A$, a linear operator with not necessarily dense domain, in a Banach space $X$ and $\sigma(A)\cap i\Bbb R = \emptyset$. We discuss the existence and uniqueness of an almost automorphic solution in an intermediate space $X_{\alpha}$, when the function $f \colon \Bbb R \times X_{\alpha} \longrightarrow X$ is almost automorphic. An example illustrating the obtained result is given (a partial differential equation).

34G20Nonlinear ODE in abstract spaces
47A55Perturbation theory of linear operators
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