On the solvability of second kind integral equations on the real line

This paper considers general second kind integral equations of the form phi(s) - integral(R)k(s,t)phi(t) dt = psi(s) (in operator form phi-K(k)phi = psi), where the functions k and psi are ass umed known. with psi is an element of Y,the space of bounded continuous fun ctions on R, and k such that the mapping s --> k(s,.), from R to L-1(R), is bounded and continuous. The function phi is an element of Y is the solutio n to be determined. Conditions on a set W subset of BC(R L-1(R)) are obtain ed such that a generalised Fredholm alternative holds: If Mi satisfies thes e conditions and I - K-k is injective for all k is an element of W then I - K-k is also surjective for all k is an element of W and, moreover, the inv erse operators (I - K-k)(-1) on Y are uniformly bounded for k is an element of W. The approximation of the kernel in the integral equation by a sequen ce (k(n)) converging in a weak sense to k is also considered and results on stability and convergence are obtained. These general theorems are used to establish results for two special classes of kernels: k(s, t) = kappa(s - t)z(t) and k(s, t) = kappa(s - t)lambda(s - t, t), where kappa is an elemen t of L-1(R), z is an element of L-infinity(R), and lambda is an element of BC((R\{0}) x R). Kernels of both classes arise in problems of time harmonic wave scattering by unbounded surfaces. The general integral equation resul ts are here applied to prove the existence of a solution for a boundary int egral equation formulation of scattering by an infinite rough surface and t o consider the stability and convergence of approximation of the rough surf ace problem by a sequence of diffraction grating problems of increasingly l arge period. (C) 2000 Academic Press.