Kernel-resolvent relations for an integral equation

We consider a scalar integral equation where |G(t,z)| ≤ ϕ(t)|z|, C is convex, and . Related to this is the linear equation and the resolvent equation . 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 as t→∞ and is in L²[0, ∞] implies that as t→∞ and is in L²[0, ∞]. Such results are fundamental in proving that the solution z satisfies z(t) →a(t) as t→∞ and that .

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