## LSU Historical Dissertations and Theses

#### Title

Abstract Volterra Equations.

1995

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

Frank Neubrander

#### Abstract

This dissertation is devoted to the study of the abstract Volterra equation $$v(t) = A\int\sbsp{0}{t}\ v(t - s)d\mu(s) + f(t)\qquad{\rm for}\ t\ge0,\eqno&(\rm VE)$$. where A is a closed linear operator in a complex Banach space $X,\ \mu$ is a complex valued function of local bounded variation, and $f:\lbrack0,\infty)\to X$ is continuous and Laplace transformable. Laplace transform methods are used to characterize the existence and uniqueness of exponentially bounded solutions v for a given forcing term f, an operator A, and a given kernel $\mu$. We extend the methods of a solution family (or a resolvent) for (VE) by studying integrated and analytic integrated solution operator families. These notions are employed to characterize those pairs $(A,\mu)$ for which (VE) has unique solutions for all sufficiently regular forcing terms f. Besides existence, uniqueness and wellposedness results for (VE), new results include Trotter-Kato type theorems for integrated solution operator families and a characterization of those pairs $(A,\mu$) for which the integrated solution operator families are analytic in an open sector $\{\lambda\in\doubc\ \vert\ \vert$arg $\lambda\vert<\alpha\}$ for some $\alpha\in(0,{\pi\over 2}\rbrack.$.

81

COinS