Abstract degenerate Volterra inclusions in locally convex spaces

Abstract

In this paper, we analyze the abstract degenerate Volterra integro-differential equations in sequentially complete locally convex spaces by using multivalued linear operators and vector-valued Laplace transform. We follow the method which is based on the use of (a, k)-regularized C-resolvent families generated by multivalued linear operators and which suggests a very general way of approaching abstract Volterra equations. Among many other themes, we consider the Hille-Yosida type theorems for \((a, k)\)-regularized C-resolvent families, differential and analytical properties of \((a, k)\)-regularized $C$-resolvent families, the generalized variation of parameters formula, and subordination principles. We also introduce and analyze the class of \((a, k)\)-regularized \((C_1,C_2)\)-existence and uniqueness families. The main purpose of third section, which can be viewed of some independent interest, is to introduce a relatively simple and new theoretical concept useful in the analysis of operational properties of Laplace transform of non-continuous functions with values in sequentially complete locally convex spaces. This concept coincides with the classical concept of vector-valued Laplace transform in the case that \(X\) is a Banach space. For more information see https://ejde.math.txstate.edu/Volumes/2023/63/abstr.html