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MONOTONE ITERATION SCHEME FOR A FORCED DUFFING EQUATION WITH NONLOCAL THREE-POINT CONDITIONS

  • Alsaedi, Ahmed (Department of Mathematics Faculty of Science, King Abdul Aziz University)
  • Published : 2007.01.31

Abstract

In this paper, we apply the generalized quasilinearization technique to a forced Duffing equation with three-point mixed nonlinear nonlocal boundary conditions and obtain sequences of upper and lower solutions converging monotonically and quadratically to the unique solution of the problem.

Keywords

References

  1. B. Ahmad, A quasilinearization method for a class of integro-differential equations with mixed nonlinearities, Nonlinear Analysis: Real World Appl. 7 (2006), 997-1004 https://doi.org/10.1016/j.nonrwa.2005.09.003
  2. B. Ahmad, Monotone iteration scheme for general three-point nonlinear boundary value problems, New Zealand J. Math. (to appear)
  3. B. Ahmad, A. Al-Saedi, and S. Sivasundaram, Approximation of the solution of non-linear second order integro-differential equations, Dynamic Systems Appl. 14 (2005), 253-263
  4. B. Ahmad, J. J. Nieto, and N. Shahzad, The Bellman-Kalaba-Lakshamikantham quasi-linearization method for Neumann problems, J. Math. Anal. Appl. 257 (2001), 356-363 https://doi.org/10.1006/jmaa.2000.7352
  5. C. Bai and J. Fang, Existence of multiple positive solutions for nonlinear m-point bound-ary value problems, J. Math. Anal. Appl. 281 (2003), 76-85 https://doi.org/10.1016/S0022-247X(02)00509-7
  6. R. Bellman, Methods of Nonlinear Analysis, Vol. 2, Academic Press, New York, 1973
  7. R. Bellman and R. Kalaba, Quasilinearization and Nonlinear Boundary Value Problems, Amer. Elsevier, New York, 1965
  8. A. Buica, Quasilinearization for the forced Duffing equation, Studia Uni. BabenC S-Bolyia Math. 47 (2000), 21-29
  9. A. Cabada and J. J. Nieto, Quasilinearization and rate of convergence for higher order nonlinear periodic boundary value problems, J. Optim. Theory Appl. 108 (2001), 97-107 https://doi.org/10.1023/A:1026413921997
  10. W. Coppel, Disconjugacy, Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, NewYork/Berlin, 1971
  11. P. W. Eloe and B. Ahmad, Positive solutions of a nonlinear nth order boundary value problem with nonlocal conditions, Appl. Math. Lett. 18 (2005), no. 5, 521-527 https://doi.org/10.1016/j.aml.2004.05.009
  12. P. Eloe and Y. Gao, The method of quasilinearization and a three-point boundary value problem, J. Korean Math. Soc. 39 (2002), no. 2, 319-330 https://doi.org/10.4134/JKMS.2002.39.2.319
  13. C. P. Gupta, A second order m-point boundary value problem at resonance, Nonlinear Anal. 24 (1995), 1483-1489 https://doi.org/10.1016/0362-546X(94)00204-U
  14. C. P. Gupta and S. Trofimchuck, A priori estimates for the existence of a solution for a multi-point boundary value problem, J. Inequal. Appl. 5 (2000), 351-365 https://doi.org/10.1155/S1025583400000187
  15. L. Jackson, Boundary value problems for ordinary differential equations, Studies in ordinary differential equations, pp. 93-127. Stud. in Math., Vol. 14, Math. Assoc. of America, Washington, D.C., 1977
  16. I. T. Kiguradze and A. G. Lomtatidze, On certain boundary value problems for second-order linear ordinary differential equations with singularities, J. Math. Anal. Appl. 101 (1984), 325-347 https://doi.org/10.1016/0022-247X(84)90107-0
  17. V. Lakshmikantham, An extension of the method of quasilinearization, J. Optim. Theory Appl. 82 (1994), 315-321 https://doi.org/10.1007/BF02191856
  18. V. Lakshmikantham, Further improvement of generalized quasilinearization, Nonlinear Anal. 27 (1996), 223-227 https://doi.org/10.1016/0362-546X(94)00281-L
  19. V. Lakshmikantham and A. S.Vatsala, Generalized Quasilinearization for Nonlinear Problems, Kluwer Academic Publishers, Dordrecht, 1998
  20. E. S. Lee, Quasilinearization and Invariant Embedding, Academic Press, New York, 1968
  21. R. Ma, Existence theorems for a second order three-point boundary value problem, J. Math. Anal. Appl. 212 (1997), 430-442 https://doi.org/10.1006/jmaa.1997.5515
  22. R. Ma, Existence and uniqueness of solutions to first-order three-point boundary value problems, Appl. Math. Lett. 15 (2002), 211-216 https://doi.org/10.1016/S0893-9659(01)00120-3
  23. V. B. Mandelzweig and F. Tabakin, Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs, Computer Physics Comm. 141 (2001), 268-281 https://doi.org/10.1016/S0010-4655(01)00415-5
  24. J. J. Nieto and A. Torres, A nonlinear biomathematical model for the study of intracra-nial aneurysms, J. Neurological Science 177 (2000), 18-23 https://doi.org/10.1016/S0022-510X(00)00315-4
  25. S. Nikolov, S. Stoytchev, A. Torres, and J. J. Nieto, Biomathematical modeling and analysis of blood flow in an intracranial aneurysms, Neurological Research 25 (2003), 497-504 https://doi.org/10.1179/016164103101201724
  26. I. Yermachenko and F. Sadyrbaev, Quasilinearization and multiple solutions of the Emden-Fowler type equation, Math. Model. Anal. 10 (2005), no. 1, 41-50

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  2. On nonlocal three-point boundary value problems of Duffing equation with mixed nonlinear forcing terms vol.2011, pp.1, 2011, https://doi.org/10.1186/1687-2770-2011-47