## 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.

### MSC:

 65L05 Numerical methods for initial value problems involving ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems
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### References:

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