Abstract
We describe a method for solution of a linear integral equation of the form φ (s) =f (s)+(1/π) ℱ∞4{[B (s)−B (t)]/(s−t)} q (t) φ (t) dt, where q (t) = (1−4/t)1/2. The specified functions f (s) and B (s) have the asymptotic behaviors f (s) ∼f0 (lns)−α, α≳1, B (s) ∼b (lns)−1, s→∞; in addition, B is subject to smoothness conditions. The equation is analyzed on a Banach space S of continuous functions φ (s) which have asymptotes of the form φ0(lns)−α. It is found that the integral operator K is bounded but not compact on the space S, so that the equation is not of Fredholm type on S. We separate K into a noncompact part K1 and a compact part K2, and construct explicitly the inverse of 1−K1 by solving an associated differential equation. We then convert the original equation φ=f+Kφ into an equivalent Fredholm equation φ= (1−K1)−1(f+K2φ).
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