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

By Gaston M. N'Guérékata

ISBN-10: 1441933735

ISBN-13: 9781441933737

ISBN-10: 147574482X

ISBN-13: 9781475744828

Almost Automorphic and virtually Periodic capabilities in summary Spaces introduces and develops the speculation of virtually automorphic vector-valued capabilities in Bochner's experience and the examine of virtually periodic services in a in the community convex house in a homogenous and unified demeanour. It additionally applies the consequences acquired to check virtually automorphic options of summary differential equations, increasing the middle issues with a plethora of groundbreaking new effects and functions. For the sake of readability, and to spare the reader pointless technical hurdles, the ideas are studied utilizing classical tools of practical analysis.

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Example text

F : lR x X --+ X be almost automorphic in t for each x E X and assume that f satisfies a Lipschitz condition in x uniformly in t E JR. Let 1fJ : lR --+ X be almost automorphic. f(t, lfJ(t)) is almost automorphic. Proof: Let (s~) be a sequence of real numbers. T EX, = ¢(t), for each t E IR, iii) limn_, 00 g(t- Sn,x) = f(t,x), for each t E lR and x EX, and Almost Automorphic Functions iv) limn-+oo ¢(t- sn) = rp(t), for each t E JR. Let us consider the function G: lR---+ X defined by G(t) = g(t, ¢(t)), F(t + sn) = G(t), for each t E lR and limn-+oo G(t- sn) = F(t), for each t E JR.

Proof: Let (s~) be a sequence of real numbers. ; then we have lim x* f(t + sn) = x*g(t) n--too 26 Gaston M. N'Guerekata and lim x* g(l- sn) = x* f(l) n-+oo for every x* E X*, the dual space of X. Since the range Rt off is relatively compact in X, we can deduce that lim f(l + sn) = g(l) n-+oo in the strong sense. Observe that the range R9 of g is also relatively compact in X. Indeed, for every l E JR, g(l) is the strong limit of the sequence (J(f + sn)), which is contained in Rt, the closure of Rt; whence g(f) E Rt, a compact set in X.

N'Guerekata so that f(a + s) Finally let us put s Vs 2: 0. , =t 2: 0. , Vt 2: a. The proof is complete. D Definition 2. 5 w+(xo) = {y EX (3 0 y} is called the w-limit set of T(t)x 0 . ~ tn---+ oo such that limn-HJO T(tn)x 0 = wj(x0 ) = {y EX /30 ~ tn---+ oo such that J1~f(tn) = y} is called thew-limit set of f(t), the principal term of T(t)x 0 . +} is the trajectory of T(t)x 0 . We have the following properties. r 0 ) i= ¢. Proof: We let tn = n, n = 1, 2, · · ·. Since f(t) is almost automorphic, there exists a subsequence (tnk) C (tn) with tnk = nk such that lim f(tnJ = g(O).

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


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