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

Abstract

This paper considers general second kind integral equations of the form φ(s) - ∫R k(s,t)φ(t) dt = ψ(s) (in operator form φ script Kk φ = 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 - script Kk is injective for all k ∈ W then I - script Kk is also surjective for all k ∈ W and, moreover, the inverse operators (I - script 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}) X 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. © 2000 Academic Press.