Date of Award

1995

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

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 $\alpha\in(0,{\pi\over 2}\rbrack.$.

Pages

81

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