Kernel-resolvent relations for and integral equation

Abstract

We consider a scalar integral equation $z(t)=a(t)-\int^t_0 C(t,s)[z(s)+G(s,z(s))]ds$ where $|G(t,z)|\leq \phi(t)|z|$, $C$ is convex, and $a\in (L^{\infty}\cap L^2)[0,\infty)$.  Related to this is the linear equation $x(t)=a(t)-\int^t_0 C(t,s)x(s)ds$ and the resolvent equation $R(t,s)=C(t,s)-\int^t_s C(t,u)R(u,s)du$.  A Liapunov functional is constructed which gives qualitative results about all three equations.  We have two goals.  First, we are interested in conditions under which properties of $C$ are transferred into properties of the resolvent $R$ which is used in the variation-of-parameters formula.  We establish conditions on $C$ and functions $b$ so that $\int^t_0 C(t,s)b(s)ds \to 0$ as $t \to \infty$ and is in $L^2[0,\infty)$ implies that $\int^t_0 R(t,s)b(s)ds \to 0$ as $t \to \infty$ and is in $L^2[0,\infty)$.  Such results are fundamental in proving that the solution $z$ satisfies $z(t) \to a(t)$ as $t \to \infty$ and that $\int^{\infty}_0 (z(t)-a(t))^2dt <\infty$.