Volterra and Functional Differential Equations

WEIGHTED L¹-SPACES AND RESOLVENTS OF VOLTERRA EQUATION

G. Samuel Jordan

Department of Mathematics Univerysity of Tennessee Knoxville, Tennessee

DOI: 10.1201/9781003420026-18

1. INTRODUCTION

The resolvent kernels r₁ and r₂ associated with the scalar Volterra integral and integrodifferential equation

$$\text{x}\left(\text{t}\right)+{{\displaystyle \int }}_0^{\text{t}}x\left(\text{t}-\text{s}\right)\text{a}\left(\text{s}\right)\text{ds}=\text{f}\left(\text{t}\right),\text{t}\in {ℝ}^{+}\equiv \left[\right[0,\infty )$$ (1.1)

$$\begin{array}{ll}\hfill & \hfill \\ \hfill & {\text{x}}^{\prime }\left(\text{t}\right)+{{\displaystyle \int }}_0^{\text{t}}\text{x}\left(\text{t}-\text{s}\right)\text{dμ}\left(\text{s}\right)=\text{f}\left(\text{t}\right),\text{t}\in R^{+},\text{x}\left(0\right)={\text{x}}_{0,}\hfill \end{array}$$ (1.2)

are defined by

$$r_1\left(t\right)+{{\displaystyle \int }}_0^t{r}_1\left(t-s\right)a\left(s\right)ds=a\left(t\right),t\in {ℝ}^{+},$$ (1.3)

$$\begin{array}{ll}\hfill & \hfill \\ \hfill & {\text{r}}_2^1\left(\text{t}\right)+{{\displaystyle \int }}_0^{\text{t}}{\text{r}}_2\left(\text{t}-\text{s}\right)\text{dμ}\left(\text{s}\right)=0,\text{t}=\hfill \end{array}$$