Quasilinearization method for nonlinear differential equations via coupled lower and upper solutions with initial time difference

In this thesis, the quasilinearization method (QLM) is used to obtain approximate solutions of the given nonlinear initial value problem (IVP) through the usage of coupled lower and upper solutions with initial time difference. For this purpose, it is tested under which conditions the method we use can give results. First of all, the comparison results related to the coupled lower and upper solutions with initial time difference for the given nonlinear IVP are discussed. Then, by weakening the convexity assumption in the QLM, monotonic function sequences converging to the unique solution of the problem and bounding the solution from below and above have been established. It is also shown that these sequences converge uniformly and quadratically to the unique solution of the given nonlinear problem via coupled lower and upper solutions with the initial time difference.