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Generalized quasilinearization method
for a second order ordinary differential
equation with Dirichlet boundary conditions

Author: Juan J. Nieto
Journal: Proc. Amer. Math. Soc. 125 (1997), 2599-2604
MSC (1991): Primary 34A45, 34B15
MathSciNet review: 1402880
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Abstract | References | Similar Articles | Additional Information

Abstract: We study the existence and approximation of solutions for a nonlinear second order ordinary differential equation with Dirichlet boundary value conditions. We present a generalized quasilinearization technique to obtain a sequence of approximate solutions converging quadratically to a solution.

References [Enhancements On Off] (What's this?)

  • 1. R. Bellman, Methods of Nonlinear Analysis, Vol. II. Academic Press, New York, 1973.
  • 2. Richard E. Bellman and Robert E. Kalaba, Quasilinearization and nonlinear boundary-value problems, Modern Analytic and Computional Methods in Science and Mathematics, Vol. 3, American Elsevier Publishing Co., Inc., New York, 1965. MR 0178571
  • 3. G. S. Ladde, V. Lakshmikantham, and A. S. Vatsala, Monotone iterative techniques for nonlinear differential equations, Monographs, Advanced Texts and Surveys in Pure and Applied Mathematics, 27, Pitman (Advanced Publishing Program), Boston, MA; distributed by John Wiley & Sons, Inc., New York, 1985. MR 855240
  • 4. V. Lakshmikantham, An extension of the method of quasilinearization, J. Optim. Theory Appl. 82 (1994), no. 2, 315–321. MR 1286689, 10.1007/BF02191856
  • 5. V. Lakshmikantham and S. Malek, Generalized quasilinearization, Nonlinear World 1 (1994), no. 1, 59–63. MR 1282859
  • 6. V. Lakshmikantham and J. J. Nieto, Generalized quasilinearization for nonlinear first order ordinary differential equations, Nonlinear Times Digest 2 (1995), no. 1, 1–9. MR 1333329
  • 7. V. Lakshmikantham, N. Shahzad and J. J. Nieto, Methods of Generalized Quasilinearization for Periodic Boundary Value Problems, Nonlinear Anal. 27 (1996), 143-151. CMP 96:12
  • 8. Juan J. Nieto and Alberto Cabada, A generalized upper and lower solutions method for nonlinear second order ordinary differential equations, J. Appl. Math. Stochastic Anal. 5 (1992), no. 2, 157–165. MR 1214298, 10.1155/S1048953392000133
  • 9. N. Shahzad and S. Sivasundaram, Further generalization of quasilinearization method for boundary value problems, Nonlinear Times Digest 2 (1995), no. 1, 59–67. MR 1333334
  • 10. N. Shahzad and A. S. Vatsala, Improved generalized quasilinearization method for second order boundary value problem, Dynam. Systems Appl. 4 (1995), no. 1, 79–85. MR 1312481

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Additional Information

Juan J. Nieto
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, Spain

Keywords: Dirichlet problem, generalized quasilinearization, quadratic convergence
Received by editor(s): March 13, 1996
Additional Notes: The author’s research was partially supported by D.G.I.C.Y.T. (Spain), project PB94-0610, and by EC Network, CHRX-CT94-0555
Communicated by: Hal L. Smith
Article copyright: © Copyright 1997 American Mathematical Society