By V. Lakshmikantham, A.S. Vatsala

ISBN-10: 1441947949

ISBN-13: 9781441947949

ISBN-10: 1475728743

ISBN-13: 9781475728743

The booklet presents a scientific improvement of generalized quasilinearization indicating the notions and technical problems which are encountered within the unified procedure. It complements significantly the usefulness of the tactic of quasilinearization which has proved to be very powerful in different components of research and in purposes. additional it comprises the well known monotone iterative method as a different case. *Audience:* Researchers, business and engineering scientists.

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**Extra resources for Generalized Quasilinearization for Nonlinear Problems**

**Example text**

We develop two results in this set up. The first result provides nonlinear iterates due to the Lipschitzian term and gives quadratic convergence. In the second result, we have totally linear iterates and obtain semi-quadratic convergence. 4 will be special cases of this result. Finally we study the situation when f(t, x) can be split as a convex function and a continuous function. In this case we get weak quadratic convergence only. For convenience of notation, we consider the IVP, x' = N(t, x), x(o) = Xo on J = [0, T].

40) is true. 1) on J. Next we shall prove that the convergence is quadratic. For this purpose, consider Note that Pn+l (0) = 0 and qn+l (0) = O. We have I Pn+l I I an+l f(t, x) - [f(t, an) + {Fx(t, an) -

The above result generalizes the method of quasilinearization under less restrictive assumptions. The question is whether it is possible to avoid the multistage process and develop a simple algorithmic approach which directly offers the construction of monotone sequences that are the solutions of linear differential equations. The answer is affirmative, and the next result yields a technique which does this precisely. 2 Assume that (AI) ao, 130 E el[J, R] such that a~::; I(t,ao), f3b 2: I(t, (30) and ao(t)::; f3o(t), t E J; (A2) I,

0 on n where n = [(t,x) : ao(t) ::; x::; f3o(t), t E J].

### Generalized Quasilinearization for Nonlinear Problems by V. Lakshmikantham, A.S. Vatsala

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