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Quasilinearization and Rate of Convergence for Higher-Order Nonlinear Periodic Boundary-Value Problems

Authors:

Institution:

(1) Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, Spain

Abstract:

We study the convergence of a sequence of approximate solutions for thefollowing higher-order nonlinear periodic boundary–value problem:

$\begin{gathered} u^{(n)} (t) = f(t,u(t)),{\text{ }}t \in I = 0,T], \hfill \\ u^{(i)} (0) - u^{(i)} (T) = c_i ,{\text{ }}i = 0,...,n - 1. \hfill \\ \end{gathered}$

Here, $f \in C(I \times \mathbb{R},\mathbb{R})$ is such that,for some k

1, $k \geqslant 1,\partial ^i f/\partial u^i$ exists and isa continuous function for i=0, 1, . . . , k. We prove thatit is possible to construct two sequences of approximate solutionsconverging to the extremal solution with rate of convergence of order k.

Keywords:

quasilinearization rapid convergence upper and lower solutions higher-order periodic problems

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