Extended method of quasilinearization for systems of nonlinear differential equations

Abstract

The extended quasilinearization method of Lakshmikantham et al. (J. Optim. Theory Appl. 87 (1995), 379-401) for the first order initial value problems is applied to the nonlinear systems. It is shown that there exist monotone sequences which converge uniformly to the unique solution of the system and the convergence is quadratic. Futhermore, a variety of results are obtained by splitting the functions involved into the difference of two convex or two concave functions, each of which is interesting by itself, with the same conclusion. Moreover, new results are extracted, as a byproduct, from the present results which offer simultaneous bounds for the cases where there is no splitting involved

Keywords:

• Quasilinearization
• systems of equations
• existence
• quadratic convergence
• upper and lower solutions