Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 63, pp. 1-55.

Title: Abstract degenerate Volterra inclusions in locally convex spaces
       
Author: Marko Kostic (Univ. of Novi Sad, Serbia)
         
Abstract: 
In this article, 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.


Submitted July 10, 2022. Published September 25, 2023.
Math Subject Classifications: 34G25, 45D05, 47D06, 46G12, 47D60, 47D62.
Key Words: Abstract degenerate Volterra inclusion; locally convex space;
 Abstract degenerate fractional differential equation;
 (a,k)-regularized C-resolvent family; multivalued linear operator.