Existence, upper and lower solutions and quasilinearization for singular differential equations

In this paper, we discuss existence theorems in the presence of upper and lower solutions as well as the method of quasilinearization (QSL) for general non-linear second-order singular ordinary differential equations. We show the existence of solutions under the assumption of weak continuity of the non-linear part. If the non-linear part is monotone decreasing, a solution may be obtained by the QSL method as the strong limit of a quadratically convergent sequence of approximate solutions. Under stronger assumptions on the linear and the non-linear parts, a solution is quadratically bracketed between two monotone sequences of approximate solutions of certain related linear equations.