On the equation φ(x) = ∫ _x^x+1K(ξ)f[φ(ξ)]dξ

Morton Lincoln Slater

Vol. 20 (1967), No. 1, 155–166

DOI: 10.2140/pjm.1967.20.155

Abstract

Suppose K(x) measurable and 0 < K(x) ≦ 1 for x ∈ (−∞,∞). Suppose f(u) convex for u ∈ [0,1],f(0) = 0,f(u) > 0 for u ∈ (0,1), and f(u) = 1 − f′(1)(1 − u) + O(1 − u)^1+δ as u → 1 for some δ > 0. (Example: f(u) = u^p,p ≧ 1.) Theorem: The equation (∗)φ(x) = ∫ _x^x+1K(ξ)f[φ(ξ)]dξ has a solution φ(x) satisfying 0 < φ(x) ≦ 1 for x ∈ (−∞,∞) if and only if ∫ ^∞e^αx[1 − K(x)]dx < ∞ where α is the largest real root of α = f′(1)(1 − e^−α). Furthermore, if φ is any such solution of (∗), then the limits φ(±∞) exist and satisfy

Mathematical Subject Classification

Primary: 45.30

Milestones

Received: 7 June 1965

Published: 1 January 1967