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

Abstract

We 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 X a denote the weighted space X a:={χ∈X: χ(s)=O(|s| −a) as | s|→∞}. Our first result is that if, additionally, | k( s, t)|⩽ κ( s− t), with κ∈L 1( R) and κ( s)= O(| s| − b ) as | s|→∞, for some b>1, then the spectrum of K is the same on X a as on X, for 0< a⩽ b. Using this result we then establish conditions on families of operators, {K k: k∈W} , which ensure that, if λ≠0 and λφ= K k φ 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∈ X a for every k∈ W and ψ∈ X a . These conditions ensure further that (λ−K) −1 :X a→X a 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∈L 1( 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.