Institution: | a Department of Mathematics, The University of Electro-Communications, Chofu, Tokyo 182-8585, Japan b Department of Applied Mathematics, Okayama University of Science, Ridai, Okayama, Okaya 700, Japan |
Abstract: | We first introduce the notion of positive linear Volterra integral equations. Then, we offer a criterion for positive equations in terms of the resolvent. In particular, equations with nonnegative kernels are positive. Next, we obtain a variant of the Paley-Wiener theorem for equations of this class and its extension to perturbed equations. Furthermore, we get a Perron-Frobenius type theorem for linear Volterra integral equations with nonnegative kernels. Finally, we give a criterion for positivity of the initial function semigroup of linear Volterra integral equations and provide a necessary and sufficient condition for exponential stability of the semigroups. |