Keyantuo, Valentin; Lizama, Carlos Fourier mutlipliers and integro-differential equations in Banach spaces. (English) Zbl 1053.45008 J. Lond. Math. Soc., II. Ser. 69, No. 3, 737-750 (2004). 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. Reviewer: Martin Väth (Würzburg) Cited in 1 ReviewCited in 36 Documents 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 Keywords:\(p\)-multiplier; Fourier series; Laplace transform; linear integro-differential equation in Banach spaces; periodic solution; Besov space; Hölder space PDFBibTeX XMLCite \textit{V. Keyantuo} and \textit{C. Lizama}, J. Lond. Math. Soc., II. Ser. 69, No. 3, 737--750 (2004; Zbl 1053.45008) Full Text: DOI