Degenerate Abstract Volterra Equations in Locally Convex Spaces

Abstract

In the paper under review, we analyze various types of degenerate abstract Volterra integro-differential equations in sequentially complete locally convex spaces. From the theory of non-degenerate equations, it is well known that the class of (a; k)-regularized C-resolvent families provides an ecient tool for dealing with abstract Volterra integro-differential equations of scalar type. Following the approach of T.-J. Xiao and J. Liang [41]-[43], we introduce the class of degenerate exponentially equicontinuous (a, k)-regularized C-resolvent families and discuss its basic structural properties. In the final section of paper, we will look at generation of degenerate fractional resolvent operator families associated with abstract differential operators.