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Almost automorphic solutions of semilinear evolution equations. (English) Zbl 1073.34073

In this nicely written paper, the authors give sufficient conditions for the existence of an almost automorphic mild solution of the semilinear differential equation

${x}^{\text{'}}\left(t\right)=Ax\left(t\right)+F\left(t,x\left(t\right)\right),\phantom{\rule{1.em}{0ex}}t\in ℝ,\phantom{\rule{2.em}{0ex}}\left(1\right)$

in a Banach space $X$, where $A:D\left(A\right)\subset X\to X$ is the infinitesimal generator of an exponentially stable ${C}_{0}$-semigroup and $F:ℝ×X\to X$ is jointly continuous. The main result improves a recent theorem proved by G. M. N’Guérékata [Semigroup Forum 69, 80–86 (2004; Zbl 1077.47058)].

Throughout the paper, $AA\left(X\right)$ denotes the Banach space of all almost automorphic functions $f:ℝ\to X$, endowed with the sup-norm and $Y$ is a Banach space algebraically contained in $X$ with compact injection. The authors suppose that $F\left(t,x\right)=P\left(t\right)Q\left(x\right)$, for all $t\in ℝ$ and $x\in X$, where $P$ is continuous with $P\left(t\right)\in AA\left(B\left(X,Y\right)\right)$ for every $t\in ℝ$ and ${sup}_{t\in ℝ}||P\left(t\right)||<\infty$ and $Q:BC\left(ℝ,X\right)\to BC\left(ℝ,X\right)$ is a continuous mapping such that there is $M\in C\left({ℝ}_{+},{ℝ}_{+}\right)$ with ${lim}_{r\to \infty }\left(M\left(r\right)/r\right)=0$ with the property that ${||Q\varphi ||}_{\infty }\le {M\left(||\varphi ||}_{\infty }\right)$ for every $\varphi \in BC\left(ℝ,X\right)$. Under these hypotheses, the main result states that equation (1) has a mild solution in $AA\left(X\right)$.

The presentation is very interesting, the central idea of the proof is based on Schauder’s fixed-point theorem. By an illustrative example, the authors show that generally, the almost automorphic mild solution of the above equation is not uniquely determined.

##### MSC:
 34G20 Nonlinear ODE in abstract spaces 47D06 One-parameter semigroups and linear evolution equations