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

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

By Gaston M. N'Guérékata

Almost Automorphic and virtually Periodic capabilities in summary Spaces introduces and develops the idea of just about automorphic vector-valued services in Bochner's feel and the research of just about periodic services in a in the community convex area in a homogenous and unified demeanour. It additionally applies the implications bought to check nearly automorphic ideas of summary differential equations, increasing the center themes with a plethora of groundbreaking new effects and functions. For the sake of readability, and to spare the reader pointless technical hurdles, the thoughts are studied utilizing classical tools of useful analysis.

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Extra resources for Almost Automorphic and Almost Periodic Functions in Abstract Spaces

Example text

Then f : lR ---+ E is almost periodic if and only if for every sequence of real numbers ( s~), there exists a subsequence ( sn) such that (J (t + sn)) converges uniformly in t E JR. 1. 7 and we now prove that it is sufficient. ], there exists t = ts such that f(t+s)- f(t) ~ U. Let us consider s 1 E lR and an interval (a 1 , bl) with b1 - a 1 > 21 s 1 1which contains no U-translation number of f. Now let s 2 = (a,;b'); then s2 - s1 E ( a 1 , bl) and consequently s2 - s 1 cannot be a U -translation number of f.

Almost Automorphic Functions 45 Put oo lim oo Using the continuity of T(t) we get lim T(s)

U, for every n E R (r~) and E E = a2 E E. Consequently, a 1 - a 2 rJ. U by (*), and using, the Hahn-Bach Theorem (Proposition 1. 4), we can find x* E E* such that x* (a 1 - a2) # 0, hence x* (at) # x*(a 2). 5 Let E be a Frechet space and¢ : lR--+ E be an almost periodic junction. Let (sn) be a sequence of real numbers such that limn~oo ¢(sn + TJk) exists for each k = 1, 2, ... 5: Suppose, by contradiction, that (¢(t + sn)) were not uniformly convergent in t. Then there exists a neighborhood of the origin U such that for every N = 1, 2, ...

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