Positive solutions of singular nonlinear boundary value problems

For a class of problems of the form y″=g(t,y)f(y′), 0<t<1, y′(0)=m⩽0, y(1)=0, we prove existence of a unique positive solution with y″(t)<0, give necessary and sufficient conditions for y′(1)>−∞, and find asymptotic formulae for y(t) and y′(t) as t→1−. We particularly include cases where the nonlinearity causes singularities at the endpoints of (0,1).