# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] S. Bochner, Continuous mappings of almost automorphic and almost periodic functions, Proc. Nat. Acad. Sci. U.S.A. 52 (1964), 907 – 910. · Zbl 0134.30102 [2] Toka Diagana, Gaston Nguerekata, and Nguyen Van Minh, Almost automorphic solutions of evolution equations, Proc. Amer. Math. Soc. 132 (2004), no. 11, 3289 – 3298. · Zbl 1053.34050 [3] Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. · Zbl 0084.10402 [4] Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. · Zbl 0592.47034 [5] Gaston M. N’Guérékata, Existence and uniqueness of almost automorphic mild solutions to some semilinear abstract differential equations, Semigroup Forum 69 (2004), no. 1, 80 – 86. · Zbl 1077.47058 · doi:10.1007/s00233-003-0021-0 · doi.org [6] Gaston M. N’Guerekata, Almost automorphic and almost periodic functions in abstract spaces, Kluwer Academic/Plenum Publishers, New York, 2001. · Zbl 0974.34058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.