Vol. 20, No. 1, 1967

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ISSN: 0030-8730
On the equation φ(x) = xx+1K(ξ)f[φ(ξ)]

Morton Lincoln Slater

Vol. 20 (1967), No. 1, 155–166
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) = up,p 1.) Theorem: The equation ()φ(x) = xx+1K(ξ)f[φ(ξ)] has a solution φ(x) satisfying 0 < φ(x) 1 for x (−∞,) if and only if eαx[1 K(x)]dx < where α is the largest real root of α = f(1)(1 eα). Furthermore, if φ is any such solution of (), then the limits φ(±∞) exist and satisfy

                 ∫
φ(+-∞)-−-φ(− ∞-)   ∞
2       =  − ∞[φ(x)− K (x )f[φ(x)]]dx.

Mathematical Subject Classification
Primary: 45.30
Milestones
Received: 7 June 1965
Published: 1 January 1967
Authors
Morton Lincoln Slater