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Composition of pseudo almost-periodic functions and Cauchy problems with operator of nondense domain. (English) Zbl 0941.34059
The authors deal with the problem of existence of so-called pseudo-periodic solutions to an equation of the form \(x'(t) =Ax(t) +f(t,x(t)),\) for \(t\) in the real line, where \(A\) is an unbounded Hille-Yosida linear operator with negative type and non-dense domain in a Banach space \(X\) and \(f\) is a continuous function. A function \(x\) defined on the real line and taking values in \(X\) is called pseudo almost-periodic if it can be written in the form \(g+\phi\) where \(g\) is almost-periodic and \(\phi\) is continuous, bounded and it satisfies the relation \(lim_{_{r\to +\infty}}\int_{-r}^{r}||\phi(t)||dt=0.\)
The authors after presenting some composition results for pseudo almost-periodic functions make use of the Banach fixed point theorem to prove that the equation given admits one and only one bounded pseudo almost-periodic solution (defined on the whole real line).

MSC:
34G20 Nonlinear differential equations in abstract spaces
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
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