Almost automorphic mild solutions to some semilinear abstract differential equations with deviated argument in Fréchet spaces

C. Gal, Morgan State University, Baltimore, MD, U.S.A.

E. J. Qualitative Theory of Diff. Equ., No. 16. (2006), pp. 1-8.

Abstract: In this paper we consider the semilinear differential equation with deviated argument in a Fr\'echet space $x^{\prime}(t) = A x(t) + f(t, x(t), x[\alpha(x(t),t)]),$ $t \in {\mathbb{R}}$, where $A$ is the infinitesimal (bounded) generator of a $C_{0}$-semigroup satisfying some conditions of exponential stability. Under suitable conditions on the functions $f$ and $\alpha$ we prove the existence and uniqueness of an almost automorphic mild solution to the equation.

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