Weighted $L^1$-Remainder Theorems for Resolvents of Volterra Equations

Abstract

Conditions are found that guarantee that the resolvent of a linear Volterra integral or integrodifferential equation may be written as a finite sum of products of polynomials and exponentials, plus a remainder term which belongs to a weighted $L^1$-space. The kernel has the form $a(t) = c + b(t)$; here c is a constant and $b(t)$ belongs to the same weighted $L^1$-space. It is assumed that $b(t)$ satisfies a combination of moment and monotonicity hypotheses that is determined by the maximum of the orders of the zeros on $\operatorname{Re} z = 0$ of certain Laplace transform equations. The results extend to weighted $L^1$-spaces some recent $L^1$-remainder theorems due to K. B. Hannsgen (Indiana Univ. Math. J., 29 (1980), pp. 103–120). The results for resolvents are deduced from more general results for linear Volterra-Stieltjes equations. The proofs employ extensions of Banach algebra techniques used by the authors in an earlier related paper, where the hypotheses involve only moment conditions.