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

Ž . Ž . We consider integral equations of the form c x s f x q Ž . Ž . Ž . Ž . H k x, y z y c y dy in operator form c s f q K c , where V is some subset V z n Ž . Ž . of R n G 1 . The functions k, z, and f are assumed known, with z g L V and ` f g Y, the space of bounded continuous functions on V. The function c g Y is to be determined. The class of domains V and kernels k considered includes the case n Ž . Ž . Ž n. V s R and k x, y s k x y y with k g L R , in which case, if z is the 1 characteristic function of some set G, the integral equation is one of Wiener]Hopf type. The main theorems, proved using arguments derived from collectively comŽ . pact operator theory, are conditions on a set W ; L V which ensure that if ` I y K is injective for all z g W then I y K is also surjective and, moreover, the z z Ž .y1 inverse operators I y K on Y are bounded uniformly in z. These general z theorems are used to recover classical results on Wiener]Hopf integral operators ́ Ž . of H. Widom Inst. Hautes Etudes Sci. Publ. Math. 44, 1975, 191]240 and I. B. Ž . Simonenko Math. USSR-Sb. 3, 1967, 279]293 , and generalisations of these results, and are applied to analyse the Lippmann]Schwinger integral equation.