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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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