# Almost Automorphic and Almost Periodic Functions in Abstract by Gaston M. N'Guérékata

By Gaston M. N'Guérékata

*Almost Automorphic and virtually Periodic features in summary Spaces* introduces and develops the idea of virtually automorphic vector-valued services in Bochner's feel and the learn of just about periodic services in a in the community convex house in a homogenous and unified demeanour. It additionally applies the implications bought to check virtually automorphic suggestions of summary differential equations, increasing the center subject matters with a plethora of groundbreaking new effects and purposes. For the sake of readability, and to spare the reader pointless technical hurdles, the strategies are studied utilizing classical equipment of practical analysis.

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B Gaston M. N'Guerekata 56 Putting a =t - sn, we get f(a + sm)- f(a + Sn) ¢ U. (*) Suppose there exists a subsequence (s~) of (sn) such that (J(t + s~)) converges uniformly in t E R. Then for every neighborhood of the origin V, there exists a natural number N = N(V) such that, if n,m > N, (we may take m > n), we have f(t + s~) - f(t + s~) EV for every t E R This contradicts (*) and so establishes the sufficiency of the condition. D The theorem is proved. 9 Let E be a Frechet space and let f, g, fi, h be almost periodic functions lR --t E.

Thus, g and h are called, respectively, the principal and corrective terms of the function f. 2 Every almost automorphic function restricted to JR+ is asymptotically almost automorphic. It suffices to put h(t) = 0. f, >.. an arbitrary scalar, are also asymptotically almost automorphic. 4 The decomposition of an asymptotically almost automorphic function is unique. Proof: Let f : JR+ --+ X be asymptotically almost automorphic with two decompositions: f(t) = g;(t) + h;(t), t E JR+, i = 1, 2 Gaston M.

F. 2 51 G. M. N’Guerekata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces © Springer Science+Business Media New York 2001 52 Gaston M. 1, we observe that for each neighborhood of the origin U, the set of U-translation numbers is relatively dense in JR. ii) It is obvious that every periodic function over lR is almost periodic. We now present some elementary properties of almost periodic functions taking values in locally convex spaces. is any scalar) and jV : lR--+ E defined by jV(t) = f(-t), are also almost periodic.