On a class of retarded integrodifferential equations

TL;DR

This paper investigates a class of retarded integro-differential equations in Banach spaces, establishing well-posedness and spectral properties using semigroup theory and Miyadera-Voigt perturbation techniques.

Contribution

It introduces a novel analysis of retarded integro-differential equations with convolution delay terms, proving well-posedness and spectral characteristics.

Findings

Proved well-posedness of the equations using semigroup theory.

Analyzed the spectral properties of the associated operator.

Applied Miyadera-Voigt perturbation to handle delay terms.

Abstract

The following class of retarded integro-differential equations in a Banach space \[ \dot{x}\left(t\right)=Ax\left(t\right)+\int_{0}^{t}b\left(t-\tau\right)Lx_{\tau}d\tau+Kx_{t};\,\,t\geq0, \] are taken into consideration in this study. The delay term $Lx_{\tau}$ 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.