Abstract degenerate Volterra integro-differential equations: inverse generator problem

TL;DR

This paper studies the inverse generator problem for abstract degenerate Volterra integro-differential equations in locally convex spaces, introducing new classes of regularized resolvent families and applying results to fractional differential equations.

Contribution

It introduces novel classes of degenerate regularized resolvent families and applies these theoretical results to fractional differential equations with Caputo derivatives.

Findings

Established existence and uniqueness of solutions for the inverse generator problem.

Developed new theoretical framework for degenerate Volterra integro-differential equations.

Provided applications to fractional differential equations with Caputo derivatives.

Abstract

In this paper, we investigate the inverse generator problem for abstract degenerate Volterra integro-differential equations in locally convex spaces. The classes of degenerate $(a,k)$-regularized $C$-resolvent families and degenerate mild $(a,k)$-regularized $(C_{1},C_{2})$-existence and uniqueness families play an important role in our analysis. We provide several illustrative applications of obtained theoretical results, primarily to abstract degenerate fractional differential equations with Caputo derivatives.