Ž. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 214, 482�502 1997 ARTICLE NO. AY975585 On the Solvability of a Class of Second Kind Integral Equations on Unbounded Domains

We consider integral equations of the form �Ž. x � �Ž. x � H kŽ x, y. zŽ y. �Ž y. dyŽ in operator form � � � � K �. � z, where � is some subset n of R Ž n � 1.. The functions k, z, and � are assumed known, with z � L Ž �. � and � � Y, the space of bounded continuous functions on �. The function � � Y is to be determined. The class of domains � and kernels k considered includes the case n Ž. Ž. Ž n ��R and k x, y � � x�y with � � L R. 1, 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 compact operator theory, are conditions on a set W � L Ž. � � which ensure that if I � K z is injective for all z � W then I � K z is also surjective and, moreover, the Ž. �1 inverse operators I � K z on Y are bounded uniformly in z. These general theorems are used to recover classical results on Wiener�Hopf integral operator