Recent zbMATH articles in MSC 45M15https://zbmath.org/atom/cc/45M152021-07-26T21:45:41.944397ZWerkzeugExistence and uniqueness of solutions of the fractional integro-differential equations in vector-valued function space.https://zbmath.org/1463.450522021-07-26T21:45:41.944397Z"Rachid, Bahloul"https://zbmath.org/authors/?q=ai:rachid.bahloulSummary: The aim of this work is to study the existence and uniqueness of solutions of the fractional integro-differential equations
\[
\frac{d}{dt}[x(t)-L(x_t)]=A[x(t)-L(x_t)]+G(x_t)+\frac{1}{\Gamma(\alpha)}\int_{-\infty}^t(t-s)^{\alpha-1}\left(\int_{-\infty}^sa(s-\xi)x(\xi)d\xi\right)ds+f(t),\quad (\alpha>0)
\]
with the periodic condition \(x(0)=x(2\pi)\), where \(a\in L^1(\mathbb{R}_{+})\). Our approach is based on the R-boundedness of linear operators \(L^p\)-multipliers and UMD-spaces.