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

A. Cabada and J. J. Nieto
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A. Cabada: Universidade de Santiago de Compostela
J. J. Nieto: Universidade de Santiago de Compostela

Journal of Optimization Theory and Applications, 2001, vol. 108, issue 1, No 4, 97-107

Abstract: 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 (search for similar items in EconPapers)
Date: 2001
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DOI: 10.1023/A:1026413921997

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