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Let Pαβλ be the singular
boundary value problem on 0 ≦ t < ∞ consisting of the nonlinear ordinary
differential equation y′′′ + y′′ + λ(1 − y′2) = 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 Paβ0 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,λ has a solution if α > Aλ but no solution if
α < Aλ.
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