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Stepanov-like pseudo-almost periodicity and its applications to some nonautonomous differential equations. (English) Zbl 1169.34330
Let $$X$$ be a Banach space, and $$p\geq 1.$$ The author proves the existence and uniqueness of a pseudo-almost periodic mild solution to the nonautonomous differential equation
$u'(t)=A(t)u(t)+f(t),\quad t\in\mathbb R,$
where $$A(t):D(A(t))\subset X\rightarrow X$$ is a family of densely defined closed linear operators with a common domain $$D=D(A(t)),$$ independent of $$t,$$ whose associated evolution family of operators is asymptotically stable, and $$f:R\rightarrow X$$ is a continuous and Stepanov-like pseudo-almost periodic (or $$S^{p}$$-pseudo-almost periodic) function. This notion of $$S^{p}$$-pseudo-almost periodicity was introduced by the author [Commun. Math. Anal. 3, No. 1, 9–18 (2007; Zbl 1286.44007)], and generalizes the classical notion of pseudo-almost periodicity due to C. Y. Zhang [J. Math. Anal. Appl. 181, No. 1, 62–76 (1994; Zbl 0796.34029) and 192, No. 2, 543–561 (1995; Zbl 0826.34040)].

##### MSC:
 34G20 Nonlinear differential equations in abstract spaces 34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations 42A75 Classical almost periodic functions, mean periodic functions
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##### References:
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