The authors apply the method of quasilinearization to the differential equation
with the linear boundary conditions
or with the nonlinear boundary conditions
Here, and are supposed to be continuous functions. The authors assume that there are lower and upper solutions to problem (1), (2) or (1), (3) and that are continuous and , on . In the case of problem (1), (3), they additionaly assume that are continuous and , on . Then they prove the existence of a monotone sequence of lower solutions and of a monotone sequence of upper solutions to problem (1), (2) or (1), (3). Both sequences converge to the unique solution to the problem under consideration.