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Stepanov-like pseudo-almost periodic mild solutions to perturbed nonautonomous evolution equations with infinite delay. (English) Zbl 1173.42308
Summary: We prove the invariance of Stepanov-like pseudo-almost periodic functions under bounded linear operators. Furthermore, we obtain existence and uniqueness theorems of pseudo-almost periodic mild solutions to evolution equations \(u{^{\prime}}(t)=A(t)u(t)+h(t)\) and \(u^\prime (t)= A(t)u(t)+f(t,Bu(t))+\int_{-\infty}^t C(t,s)u(s)\text ds+F(t)\) on \(\mathbb R\), assuming that \(A(t)\) satisfy “Acquistapace-Terreni” conditions, that the evolution family generated by \(A(t)\) has exponential dichotomy, that \(R(\lambda _{0},A(\cdot))\) is almost periodic, that \(B,C(t,s)_{t\geq s}\) are bounded linear operators, that \(f\) is Lipschitz with respect to the second argument uniformly in the first argument and that \(h, f, F\) are Stepanov-like pseudo-almost periodic for \(p>1\) and continuous. To illustrate our abstract result, a concrete example is given.

MSC:
42A75 Classical almost periodic functions, mean periodic functions
42A85 Convolution, factorization for one variable harmonic analysis
44A35 Convolution as an integral transform
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