Semigroup Forum Vol. 71 (2005) 201–230 c  2005 Springer DOI: 10.1007/s00233-005-0508-y RESEARCH ARTICLE Almost Automorphic Functions in Fr´ echet Spaces and Applications to Differential Equations Ciprian S. Gal, Sorin G. Gal, and Gaston M. N’Gu´ er´ ekata Communicated by Jerome A. Goldstein Abstract In this paper we first develop a theory of almost automorphic functions with values in Fr´ echet spaces. Then, we consider the semilinear differential equation x ′ (t)= Ax(t)+ f (t, x(t)), t ∈ R in a Fr´ echet space X , where A is the infinitesimal generator of a C 0 -semigroup satisfying some conditions of exponential stability. Under suitable conditions on f , we prove the existence and uniqueness of an almost automorphic mild solution to the equation. Keywords: Almost automorphic, asymptotically almost automorphic, mild so- lutions, semigroups of linear operators, semilinear differential equations, Fr´ echet spaces. 1991 Mathematics Subject Classification: 43A60, 34G10. 1. Introduction Harald Bohr’s interest in which functions could be represented by a Dirichlet series, i.e. of the form ∑ ∞ n=1 a n e -λnz , where a n ,z ∈ C and (λ n ) n∈N is a monotone increasing sequence of real numbers (series which play an important role in complex analysis and analytic number theory), led him to devise a theory of almost periodic real (and complex) functions, founding this theory between the years 1923 and 1926. The theory of almost periodic functions was extended to abstract spaces, see for example the monographs [7], [8], [18], [19] (for Banach space valued functions), [6], [18], [31] (for Fr´ echet space valued functions). Also, in the recent paper [1] (see also Chapter 3 of the book [19]), the theory of real-valued almost periodic functions has been extended to the case of fuzzy-number-valued functions. The concept of almost automorphy is a generalization of almost period- icity. It has been introduced in the literature by S. Bochner in relation to some aspects of differential geometry [2-5]. Important contributions to the theory of almost automorphic functions have been made, for example, with the papers [28], [23]–[26], [15] and the books [27], [18], [19] (concerning almost automorphic functions with values in Banach spaces), and the paper [22] (concerning almost automorphy on groups). Also, the theory of almost automorphic functions with