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Perturbed functional and neutral functional evolution equations with infinite delay in Fréchet spaces. (English) Zbl 1190.34098

The authors consider evolution equations of the form:
\[ \begin{aligned} &\frac {d}{dt}[y(t) - h(t, y_t )] = A(t)y(t) + f (t, y_t ) + g(t, y_t ),\quad t \in J : = [0,\infty)\\ & y_0 = \phi \in B,\end{aligned} \]
where \(f,g,h\) and \(\phi\) are given functions, \(B\) is the phase space and \(\{A(t)\}\) is a family of linear closed operators from a real Banach space \((E, | . | )\) into \(E\) that generates an evolution system of operators.
Sufficient conditions for the existence of mild solutions on the positive half-line are obtained. As main tools the nonlinear alternative proved by Avramescu and the semigroup theory are used. Two examples which illustrate the results are given.

MSC:

34K30 Functional-differential equations in abstract spaces
34K40 Neutral functional-differential equations
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