Uniqueness of positive solutions of some semilinear Sturm-Liouville problems on the half line

This note gives a simple proof of uniqueness for positive solutions of certain non-linear boundary value problems on R⁺ which are typified by the equation —u”(x) = -u(x) + r(x)u(x)^1+σwith boundary conditions u'(0) = u(+∞) - 0. In the autonomous case (r≡l), this is easy to see, by quadrature. The proof here supposes r to be non-increasing on R⁺.