A Liapunov functional for a linear integral equation

Abstract

In this note we consider a scalar integral equation $x(t)= a(t)-\int^t_0 C(t,s)x(s)ds$, together with its resolvent equation, $R(t,s)= C(t,s)-\int^t_s C(t,u) R(u,s)du$, where $C$ is convex. Using a Liapunov functional we show that for fixed $s$ then $|R(t,s) - C(t,s)| \to 0$ as $t \to \infty$ and $\int^{\infty}_s (R(t,s)-C(t,s))^2 dt < \infty$. We then show that the variation of parameters formula $x(t)=a(t)-\int^t_0 R(t,s) a(s)ds$ can be replaced by $X(t)=a(t)-\int^t_0 C(t,s)a(s)ds$ when $a \in L^1[0,\infty)$ and that $|X(t) - x(t)|\to 0$ as $t \to \infty$ and $\int^{\infty}_0 (x(t)-X(t))^2 dt < \infty$. A mild nonlinear extension is given.