Volterra and Functional Differential Equations

INTEGRABLE RESOLVENT OPERATORS FOR AN ABSTRACT VOLTERRA EQUATION IN HILBERT SPACE

R. W. Carr

Department of Mathematics

St. Cloud State University

St. Cloud, Minnesota

DOI: 10.1201/9781003420026-15

We study the asymptotic behavior of the solution to the initial value problem

$$\underset{\sim }{\text{y}}\text{'(t) + }{\displaystyle {\int }_0^{\text{t}}[\text{d + a(t - s)}}]\underset{\sim }{\text{L}}\underset{\sim }{\dot{\text{y}}}(\text{s})\text{ds = }\underset{\sim }{\text{f}}(\text{t}),\text{ t > 0}$$ (1)

$$\underset{\sim }{\text{y}}\text{'(t) = }{\underset{\sim }{\text{y}}}_{\text{0}}\text{ ε H},$$

(‘= d/dt) where $\underset{\sim }{\text{L}}$ is a positive self-adjoint linear operator defined on a dense subspace D of the Hilbert space H. The kernel d + a(t) satisfies

$$\{\begin{array}{l}{\text{a ε L}}_{\text{l}0c}^1\left({ℝ}^{+},{\overline{ℝ}}^{+}\right),\left({ℝ}^{+}=(0,\infty ),{\overline{ℝ}}^{+}=[0,\infty )\right);\\ \text{ a is nonincreasing and convex with a }(\infty )=0<\text{a}\left(0^{+}\right)\end{array}$$