Vol. 36, No. 1, 1971

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On third order, nonlinear, singular boundary value problems

Philip Hartman

Vol. 36 (1971), No. 1, 165–180
Abstract

Let Pαβλ be the singular boundary value problem on 0 t < consisting of the nonlinear ordinary differential equation y′′′ + y′′ + λ(1 y2) = 0, the boundary conditions y(0) = α,y(0) = β and y() = 1, and the condition β < y(t) < 1 for t > 0. The problems Pαβλ arise in boundary layer theory and questions of existence and uniqueness have been settled for parameters on the range: (λ,α) arbitrary, 0 β < 1. If λ = 0, a “discontinuity” occurs in the existence theory at β = 0 in the following sense: if 0 < β < 1, then P0 has a solution for all α, but if β = 0, then there is a number A0 with the property that Pα00 has a solution if and only if α A0. In this paper, it is shown that, if λ > 0, a similar “discontinuity” occurs at β = 1; namely, if λ > 0 and 1 < β < 1, then Pαβλ has a solution for arbitrary α, while if β = 1, then there exists a number Aλ such that Pα,1 has a solution if α > Aλ but no solution if α < Aλ.

Mathematical Subject Classification
Primary: 34.36
Milestones
Received: 9 March 1970
Published: 1 January 1971
Authors
Philip Hartman