Existence results, regularity and compactness properties, in the \(\alpha \)-norm, for semilinear partial functional integrodifferential equations with nonlinear Kernel and delay argument

In this paper, we consider a finite delay integro-differential equation with a nonlinear kernel in a general Banach space. The nonlinear part is assumed to be continuous with respect to a fractional power of the linear part in the second variable. We prove, using the semigroup theory, the local existence, continuous dependence of the initial data, the phenomena of blowing up, regularity, and compactness properties of the so-called mild solution. An application is provided to illustrate our results.