Existence and uniqueness of solutions of the fractional integro-differential equations in vector-valued function space

The aim of this work is to study the existence and uniqueness of solutions of the fractional integro-differential equations $\frac{d}{dt}[x(t) - L(x_{t})]= A[x(t)- L(x_{t})]+G(x_{t})+ \frac{1}{\Gamma (\alpha )} \int _{- \infty }^{t} (t-s)^{\alpha - 1} ( \int _{- \infty }^{s}a(s-\xi )x(\xi ) d \xi )ds+f(t)$, ($\alpha > 0$) 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.