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

Abstract

This paper considers general second kind integral equations of the formφs−∫Rks,tφtdt=ψs(in operator form φ−Kkφ=ψ), where the functions k and ψ are assumed known, with ψ∈Y, the space of bounded continuous functions on R, and k such that the mapping s→k(s,·), from R to L1(R), is bounded and continuous. The function φ∈Y 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 I−Kk is injective for all k∈W then I−Kk is also surjective for all k∈W and, moreover, the inverse operators (I−Kk)−1 on Y are uniformly bounded for k∈W. 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)=κ(s−t)z(t) and k(s,t)=κ(s−t)λ(s−t,t), where κ∈L1(R), z∈L∞(R), and λ∈BC((R\\{0})×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.