Improved generalized quasilinearization (GQL) method. (English) Zbl 0830.65068

The authors consider the initial value problem (1) \(x'= f(t, x)\), \(x(0)= x_0\), \(t\in \langle 0, T\rangle\). The main result is formulated in Theorem 2.3. Using the idea of lower and upper solutions for (1) it is proved that under some conditions there exist monotone sequences \(\{a_n(t)\}\), \(\{b_n(t)\}\) which converge uniformly to the unique solution of (1) and the convergence is quadratic. The functions \(a_n(t)\), \(b_n(t)\) are the solutions of some nonlinear initial value problems which are derived from (1).
Next, an improvement of this technique is given in such a way that the functions \(\alpha_n(t)\), \(\beta_n(t)\) which again converge uniformly and quadratically to the unique solution of (1) are the solutions of some linear initial value problems.


65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
Full Text: DOI


[4] McRae, F. A., Generalized quasilinearization of stochastic initial value problems, Stoch. Analysis Applic., 13 (1995) · Zbl 0828.60047
[5] Bellman, R., (Methods of Nonlinear Analysis, Vol. II (1973), Academic Press: Academic Press Prague)
[6] Belman, R.; Kalaba, R., Quasilinearization and Nonlinear Boundary Value Problems (1965), Elsevier: Elsevier New York · Zbl 0139.10702
[7] Ladde, G. S.; Lakshimikantham, V.; Vatsala, A. S., Monotone Iterative Techniques for Nonlinear Differential Equations (1985), Pitman: Pitman New York · Zbl 0658.35003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.