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

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  • A. Cabada

    (Universidade de Santiago de Compostela)

  • J. J. Nieto

    (Universidade de Santiago de Compostela)

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.

Suggested Citation

  • A. Cabada & J. J. Nieto, 2001. "Quasilinearization and Rate of Convergence for Higher-Order Nonlinear Periodic Boundary-Value Problems," Journal of Optimization Theory and Applications, Springer, vol. 108(1), pages 97-107, January.
    • Handle: RePEc:spr:joptap:v:108:y:2001:i:1:d:10.1023_a:1026413921997
    DOI: 10.1023/A:1026413921997
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