A class of singular nonlinear boundary value problems

We study nonlinear boundary value problems of the form y′ = g(t, y) f(y′), y′(0) = m ≤ 0, y(1) = 0, where g > 0 on [0, 1) × (0, ∞), f < 0 on (m0, m1), m0 < m < m1, and f(mo) = 0. We specifically allow g to be singular in t at t = 1 and f to be singular in y at y = 0. We provide conditions under which the boundary value problem has a unique positive solution, which then satisfies mo < y′(t) ≤ m. We also obtain information about the qualitative behavior of this solution as t → 1−. For example, we distinguish cases for which limt → 1− y′(t) either equals mo or is greater than mo, and obtain asymptotic formulae for y(t) and y′(t) as t → 1−.