On a class of retarded integrodifferential equations

The following class of retarded integro-differential equations in a Banach space $$\dot{x}\left(t\right)=Ax\left(t\right)+{\int }_0^{t}b(t-\tau )L{x}_{\tau }d\tau +K{x}_t;t\ge 0,$$ are taken into consideration in this study. The delay term $L{x}_{\tau }$ of this equation is inserted into the integral as a convolution product with a scalar kernel. We prove the well-posedness of the problem under investigation using the Miyadera-Voigt perturbation and the theory of semigroups. We also explore the spectral analysis of an associated abstract Cauchy problem.