On the Solvability of Second Kind Integral Equations on the Real Line

This paper considers general second kind integral equations of the form f s. yH k s, t.f t. dtsc s. R in operator form f yKkf sc., where the functions k and c are assumed known, with c gY, the space of bounded continuous functions on R, and k such that the mapping sªk s, ?., from R to L1 R., is bounded and continuous. The function f gY is the solution to be determined. Conditions on a set W; BC R, L1 R..are obtained such that a generalised Fredholm alternative holds: If W satisfies these conditions and IyKkis injective for all kgW then IyKkis also surjective for all kgW and, moreover, the inverse operators IyKk.y1 on Y are uniformly bounded for kgW. The approximation of the kernel in the integral equation by a sequence kn.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.sk syt.z t. and k s, t.sk syt.l syt, t., where k gL1 R., zgL` R., and l gBC R_ 04.= 1 E-mail: Simon.Chandler-Wilde@brunel.ac.uk. 2 E-mail: B.Zhang@coventry.ac.uk. 28 0022-247Xr00 $35.00 Copyright Q 2000 by Academic Press All rights of reproduction in any form reserved. SECOND KIND INTEGRAL EQUATIONS 29 R.. Kernels of both classes arise in problems of time harmonic wave scattering by unbounded surfaces. The general integral equation results are here applied to prove the existence of a solution for a boundary integral equation formulation of scattering by an infinite rough surface and to consider the stability and convergence of approximation of the rough surface problem by a sequence of diffraction grating problems of increasingly large period.