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Pseudo-almost-automorphic solutions to some neutral partial functional differential equations in Banach spaces. (English) Zbl 1165.34418

Summary: We study the existence and uniqueness of pseudo-almost-automorphic solutions for the following neutral partial functional differential equation
\[ \frac{d}{dt}\,[u(t)-f(t,u(t-r))]=A[u(t)-f(t,u(t-r))]+g(t,u(t-r))\quad \text{for }r\in\mathbb R,\tag{1.1} \]
where \(A\) is a linear operator on a Banach space \(X\) satisfying the following well-known Hille-Yosida condition.
Recall that the new concept of pseudo-almost-automorphy generalizes the one of the pseudo-almost-periodicity and it has been recently introduced in the literature. Here we assume that the undelayed part is not necessarily densely defined and satisfies the well-known Hille-Yosida condition, the delayed parts are assumed to be pseudo-almost-automorphic with respect to the first argument and Lipschitz continuous with respect to the second argument.

MSC:

34K30 Functional-differential equations in abstract spaces
34K40 Neutral functional-differential equations
34K14 Almost and pseudo-almost periodic solutions to functional-differential equations
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
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References:

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