Quasilinearization and boundary value problems at resonance

A quasilinearization algorithm is developed for boundary value problems at resonance. To do so, a standard monotonicity condition is assumed to obtain the uniqueness of solutions for the boundary value problem at resonance. Then the method of upper and lower solutions and the shift method are applied to obtain the existence of solutions. A quasilinearization algorithm is developed and sequences of approxi- mate solutions are constructed, which converge monotonically and quadratically to the unique solution of the boundary value problem at resonance. Two examples are provided in which explicit upper and lower solutions are exhibited.