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Stepanov-like pseudo almost periodic mild solutions to nonautonomous neutral partial evolution equations. (English) Zbl 1236.42006
Authors’ abstract: We obtain new existence and uniqueness theorems of pseudo almost periodic mild solutions to nonautonomous neutral partial evolution equations \[ \begin{aligned} {d\over dt}[u(t)+ f(t, u(t))]= A(t)[u(t)- f(t,u(t))]+ g(t,u(t)),\quad & t\in\mathbb{R},\\ {d\over dt} [u(t)+ f(t,Bu(t))]= A(t)[u(t)+ f(t,Bu(t))]+ g(t,Cu(t)),\quad & t\in\mathbb{R},\end{aligned} \] assuming that \(A(t)\) satisfy “Acquistapace-Terreni” conditions, the evolution family generated by \(A(t)\) has exponential dichotomy, \(R(\lambda_0, A(\cdot))\) is almost periodic, \(B\), \(C\) are densely defined closed linear operators, \(f\), \(g\) are Lipschitz with respect to the second argument uniformyl in the first argument, \(f\) is pseudo almost periodic in the first argument, \(g\) is Stepanov-like pseudo almost periodic in the first argument for \(p> 1\) and jointly continuous. To illustrate our abstract results, two examples are given.

MSC:
42A75 Classical almost periodic functions, mean periodic functions
47D06 One-parameter semigroups and linear evolution equations
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