On the Solvability of a Class of Second Kind Integral Equations on Unbounded Domains

We consider integral equations of the form c x. s f x. q H k x, y.z y.c y.dy in operator form c sf qK c., where V is some subset V z of Rn nG1.. The functions k, z, and f are assumed known, with zgL V.and ` f gY, the space of bounded continuous functions on V. The function c gY is to be determined. The class of domains V and kernels k considered includes the case VsRn and k x, y.sk xyy. with k gL1 Rn., in which case, if z is the characteristic function of some set G, the integral equation is one of Wiener]Hopf type. The main theorems, proved using arguments derived from collectively com- pact operator theory, are conditions on a set W;L V. which ensure that if ` IyKzis injective for all zgW then IyKzis also surjective and, moreover, the inverse operators IyKz.y1on Y are bounded uniformly in z. These general theorems are used to recover classical results on Wiener]Hopf integral operators of H. Widom Inst. Hautes E´tudes Sci. Publ. Math. 44, 1975, 191]240.and I. B. Simonenko Math. USSR-Sb. 3, 1967, 279]293., and generalisations of these results, and are applied to analyse the Lippmann]Schwinger integral equation