Kernel-resolvent relations for an integral equation

Rok, strany: 2011, 25 - 40

We consider a scalar integral equation $ z(t)=a(t)-\int^t₀ C(t,s)[z(s)+G(s,z(s))]ds $ where $|G(t,z)|≤ φ(t)|z|$, $C$ is convex, and $a\in (L^∞\cap L²)[0,∞)$. Related to this is the linear equation $ x(t)=a(t)-\int^t₀ 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 $\int^t₀ C(t,s)b(s)ds \to 0$ as $t \to ∞$ and is in $L²[0,∞)$ implies that $\int^t₀ R(t,s)b(s)ds \to 0$ as $t \to ∞$ and is in $L²[0,∞)$. Such results are fundamental in proving that the solution $z$ satisfies $z(t) \to a(t)$ as $t \to ∞$ and that $\int^∞₀ (z(t)-a(t))²dt <∞$. \par This is in final form and no other version will be submitted.

ISO 690:

Burton, T. 2011. Kernel-resolvent relations for an integral equation. In Tatra Mountains Mathematical Publications, vol. 48, no.1, pp. 25-40. 1210-3195.

APA:

Burton, T. (2011). Kernel-resolvent relations for an integral equation. Tatra Mountains Mathematical Publications, 48(1), 25-40. 1210-3195.