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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.

##### MSC:
 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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