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Fourier mutlipliers and integro-differential equations in Banach spaces. (English) Zbl 1053.45008
Periodic solutions of the linear integro-differential equation \[ u'(t)=Au(t)+\int_{-\infty}^ta(t-s)Au(s)ds+f(t) \] with a closed operator \(A\) are studied by Fourier series techniques in Lebesgue-Bochner spaces and Besov spaces (and as a special case in Hölder spaces). Under some hypotheses on the Laplace transform \(\tilde a\) of \(a\), the existence and uniqueness of solutions (for each \(f\)) is equivalent to the fact that the particular resolvent sequence \((I-(1+\tilde a(ik))A/ik)^{-1}\) is a Fourier multiplier in the considered space. Moreover, under some additional assumptions, this is the case if and only if this sequence is bounded.

45N05 Abstract integral equations, integral equations in abstract spaces
42A45 Multipliers in one variable harmonic analysis
44A10 Laplace transform
45J05 Integro-ordinary differential equations
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