Solvability and spectral properties of integral equations on the real line: I. Weighted spaces of continuous functions

AbstractWe consider in this paper the solvability of linear integral equations on the real line, in operator form (λ−K)φ=ψ, where λ∈C and K is an integral operator. We impose conditions on the kernel, k, of K which ensure that K is bounded as an operator on X:=BC(R). Let Xa denote the weighted space Xa:={χ∈X:χ(s)=O(|s|−a) as |s|→∞}. Our first result is that if, additionally, |k(s,t)|⩽κ(s−t), with κ∈L1(R) and κ(s)=O(|s|−b) as |s|→∞, for some b>1, then the spectrum of K is the same on Xa as on X, for 0<a⩽b. Using this result we then establish conditions on families of operators, {Kk:k∈W}, which ensure that, if λ≠0 and λφ=Kkφ has only the trivial solution in X, for all k∈W, then, for 0⩽a⩽b, (λ−K)φ=ψ has exactly one solution φ∈Xa for every k∈W and ψ∈Xa. These conditions ensure further that (λ−K)−1:Xa→Xa is bounded uniformly in k∈W, for 0⩽a⩽b. As a particular application we consider the case when the kernel takes the form k(s,t)=κ(s−t)z(t), with κ∈L1(R), z∈L∞(R), and κ(s)=O(|s|−b) as |s|→∞, for some b>1. As an example where kernels of this latter form occur we discuss a boundary integral equation formulation of an impedance boundary value problem for the Helmholtz equation in a half-plane