The method of quasilinearization for the periodic boundary value problem for systems of impulsive differential equations

The method of quasilinearization of Bellman and Kalaba [2] has been extended, refined, and generalized when the forcing function is the sum of a convex and concave function using coupled lower and upper solutions. This method is now known as the method of generalized quasilinearization. It has all the advantages of the quasilinearization method such that the iterates are solutions of linear systems and the sequences simultaneously converge to the unique solution of the nonlinear problem. See [1–4, 6–8] for details. In this paper, we extend themethod of generalized quasilinearization to system of nonlinear impulsive differential equations with periodic boundary conditions. For this purpose, we develop a linear comparison theorem for system of impulsive differential equations with periodic boundary conditions. We develop two iterates which are solutions of linear impulsive system with periodic boundary conditions which converge monotonically and quadratically to the unique solution of the nonlinear problem. Results related to different types of coupled lower and upper solutions are developed.We note that the results of [1] are a special case of our results where the forcing function is made to be convex and in addition they obtain semiquadratic convergence only. The results of [3] can be obtained as the scalar case of our result.