Abstract
In the present paper, we deal with mild solutions for the semilinear evolution equation \begin{displaymath} \frac{d}{dt}x(t)=Ax(t)+f(t,x(t)),\qquad t\in \R, \end{displaymath} under the sectoriality of $A$, a linear operator with not necessarily dense domain, in a Banach space $X$ and $\sigma(A)\cap i\R=\emptyset$. We prove the existence and uniqueness of an almost automorphic solution in an intermediate space $X_{\alpha}$, when the function $f\colon \ \R\times X_{\alpha}\longrightarrow X$ is almost automorphic. An example illustrating the obtained result is given.
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Boulite, S., Maniar, L. & N'Guerekata, G. Almost Automorphic Solutions for Hyperbolic Semilinear Evolution Equations. Semigroup Forum 71, 231–240 (2005). https://doi.org/10.1007/s00233-005-0524-y
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DOI: https://doi.org/10.1007/s00233-005-0524-y