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On \(S\)-asymptotically \(\omega \)-periodic functions on Banach spaces and applications. (English) Zbl 1146.43004

A continuous function \(f: [0,+\infty]\to X\) (Banach space) for which there exists \(\omega> 0\) such that
\[ \lim_{t\to+\infty} (f(t+ \omega)- f(t))= 0 \]
is called \(S\)-asymptotically \(\omega\)-periodic. The authors show that the abstract Cauchy problem:
\[ \begin{aligned} u'(t)&= Au(t)+ G(t, u(t)),\quad t\geq 0,\\ u(0)&= x_0\in X, \end{aligned} \]
where \(A: D(A)\subseteq X\to X\) is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators \((T\cos)_{t\geq 0}\) on \(X\) and \(G: [0,+\infty)\times X\to X\) is a continuous function, under some supplementary conditions concerning \(G\), has a unique \(S\)-asymptotically \(\omega\)-periodic mild solution.

MSC:

43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
42A75 Classical almost periodic functions, mean periodic functions
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