Stepanov-like pseudo-almost periodicity and its applications to some nonautonomous differential equations
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 \textit{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)].
scientific article; zbMATH DE number 5382857