Monotone approximations of unstable solutions

Assume that for a nonlinear two-point problem we have a subsolution lying above a supersolution. It is well-known that in general one cannot conclude the existence of a solution in between. However, if one restricts the growth of the nonlinearity, then under some restrictions on the super- and subsolution, one gets both existence of a solution and two sequences of monotone approximations. In addition, we can assert some qualitative property of the solution.