On the solvability of a class of second kind integral equations on unbounded domains

[摘要] We consider integral equations of the form psi(x) = phi(x) + integral(Omega)k(x, y)z(y)psi(y) dy (in operator form psi = phi + K-z psi), where Omega is some subset of R-n (n greater than or equal to 1). The functions k, z, and phi are assumed known, with z is an element of L-infinity(Omega) and phi is an element of Y, the space of bounded continuous functions on <(Omega)over bar>. The function psi is an element of Y is to be determined. The class of domains n and kernels k considered includes the case Omega = R-n and k(x,y) = k(x - y) with k is an element of L-1(R-n), 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 subset of L-infinity(Omega) which ensure that if I - K-z is injective for all z is an element of W then I - K-z is also surjective and, moreover, the inverse operators (I - K-z)(-1) on 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 Etudes 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. (C) 1997 Academic Press.

[发布日期] 1997-10-15