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Zbl 0941.34059
Amir, B.; Maniar, L.
Composition of pseudo almost-periodic functions and Cauchy problems with operator of nondense domain. (English)
[J] Ann. Math. Blaise Pascal 6, No.1, 1-11 (1999). ISSN 1259-1734

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 $lim_{_{r\to +\infty}}\int_{-r}^{r}||\phi(t)||dt=0.$ \par 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).
[George Karakostas (Ioannina)]

MSC 2000:: *34G20 Nonlinear ODE in abstract spaces
34C27 Almost periodic solutions of ODE

Keywords: differential equations in abstract spaces; pseudo almost-periodic solutions; unbounded Hille-Yosida linear operator

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