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By Gaston M. N'Guérékata

This monograph offers fresh contributions to the subjects of just about periodicity and nearly automorphy. numerous new tools, together with the equipment of invariant subspaces and uniform spectrum, in addition to a variety of classical tools, equivalent to mounted aspect theorems, are used to acquire virtually periodic and nearly automorphic suggestions to a couple linear and non-linear evolution equations and dynamical platforms. nearly periodicity and virtually automorphy also are intensively built at the extra common buildings known as fuzzy-number style areas. they've got additional power purposes to the research of differential equations, which version the real-world difficulties ruled through imprecision because of uncertainty or vagueness, instead of randomness. In end, the writer exhibits a number of open difficulties and instructions for destiny study. This monograph is a brilliant resource of knowledge and notion for researchers and graduate scholars from many mathematical fields.

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There exists a subsequence (sn) such that the sequence (f{t+Sn,x)) is uniformly convergent int eR andx G E. 8. 69. Let f : R x E —^ E be almost periodic in t for each t G E, and assume that f satisfies a Lipschitz condition in x uniformly int E:R, that is p{f{t^x)J{t,y)) for all t E R and x, (j> :R-^

Such that (fn) converges uniformly to f on R, then f is also almost periodic. The following Criterion due to Bochner is a key result. 61. ^ o c h n e r ' s Criterion). Let E be a Frechet space, that is a Hausdorff locally convex space whose topology is induced by a complete and invariant metric. Then f € C(R, E) is almost periodic if and only if for every sequence of real numbers {s'^, there exists a subsequence (sn) such that {f(t + Sn)) converges uniformly m t € M. Now we denote AP(E) the set of all almost periodic functions R -> £*, where £* is a Frechet space.

1) We will present various conditions for ensuring almost automorphy of the classical and/or mild solutions. We start with the simplest case ^ = A G C. 1. Let X be a uniformly convex (complex) Banach space. Suppose f G AA{X). 1) is in AA{X). 1) admits solutions of the form e^<*-'")/(r)dr, if R e A > 0 and X2it) = f e^^*-^^f(r)dr, if ReA < 0. J —OO Let us prove that they are almost automorphic. We start with Let s = t — r; then we can write x^(t) = - I e^'f(t-s)ds. J —CX) Let (5^) be an arbitrary sequence of real numbers.

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