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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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##### References:
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