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Almost automorphic solutions of semilinear evolution equations. (English) Zbl 1073.34073
Proc. Am. Math. Soc. 133, No. 8, 2401-2408 (2005); corrigendum ibid. 140, No. 3, 1111-1112 (2012).
In this nicely written paper, the authors give sufficient conditions for the existence of an almost automorphic mild solution of the semilinear differential equation \[ x^\prime(t) = Ax(t) +F(t, x(t)), \quad t \in {\mathbb R}, \tag{1} \] in a Banach space \(X\), where \(A : D(A) \subset X \to X\) is the infinitesimal generator of an exponentially stable \(C_0\)-semigroup and \(F: {\mathbb R} \times X \to X\) 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, \(AA(X)\) denotes the Banach space of all almost automorphic functions \(f : {\mathbb R} \to X\), endowed with the sup-norm and \(Y\) is a Banach space algebraically contained in \(X\) with compact injection. The authors suppose that \(F(t, x) = P(t)Q(x)\), for all \(t \in {\mathbb R}\) and \(x \in X\), where \(P\) is continuous with \(P(t) \in AA(B(X, Y))\) for every \(t \in {\mathbb R}\) and \(\sup_{t \in {\mathbb R}} | | P(t)| | < \infty\) and \(Q: BC({\mathbb R}, X) \to BC({\mathbb R}, X)\) is a continuous mapping such that there is \(M\in C({\mathbb R}_+, {\mathbb R}_+)\) with \(\lim_{r \to \infty} (M(r)/r ) = 0\) with the property that \(| | Q\varphi| | _\infty \leq {M}(| | \varphi| | _\infty)\) for every \(\varphi \in BC({\mathbb R}, X)\). Under these hypotheses, the main result states that equation (1) has a mild solution in \(AA(X)\).
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 differential equations in abstract spaces
47D06 One-parameter semigroups and linear evolution equations
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