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Nonlinear Oscillations

, Volume 14, Issue 3, pp 379–413 | Cite as

On the parametrization of boundary-value problems with two-point nonlinear boundary conditions

  • M. I. Ronto
  • K. V. Marynets’
Article

We obtain some results for the solutions of nonlinear boundary-value problems of a certain type with two-point nonlinear boundary conditions and reveal the efficiency of the procedure of reduction of the analyzed problem to a parametrized boundary-value problem with linear boundary conditions containing certain artificially introduced parameters. To study the transformed two-point problem, we propose a method based on the approximations of a special type constructed in the analytic form. It is shown that these approximations uniformly converge to a parametrized limit function and establish the relationship between this function and the exact solution. The proposed procedure leads to a certain system of algebraic equations whose solutions give numerical values of the parameters corresponding to the solution of the given two-point nonlinear boundary-value problem.

Keywords

Limit Function Determine Equation Nonlinear Boundary Condition Linear Boundary Condition Parametrized Boundary Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    A. M. Samoilenko and Le Lyiong Tai, “Method for studying boundary value problems with nonlinear boundary conditions,” Ukr. Mat. Zh., 42, No. 7, 951–957 (1990); English translation: Ukr. Math. J., 42, No. 7, 844–850 (1990).CrossRefGoogle Scholar
  2. 2.
    A. M. Samoilenko and N. I. Ronto, Numerical-Analytic Methods in the Theory of Boundary-Value Problems for Ordinary Differential Equations [in Russian], Naukova Dumka, Kiev (1992).Google Scholar
  3. 3.
    M. Ronto and A. M. Samoilenko, Numerical-Analytic Methods in the Theory of Boundary-Value Problems, World Scientific, River Edge, NJ (2000).MATHCrossRefGoogle Scholar
  4. 4.
    A. Ronto and M. Ronto, “On the investigation of some boundary-value problems with nonlinear boundary conditions,” Miskolc: Math. Notes, 1, No. 1, 43–55 (2000).MathSciNetMATHGoogle Scholar
  5. 5.
    A. Ronto and M. Ronto, “A note on the numerical-analytic method for nonlinear two-point boundary-value problems,” Nonlin. Oscillations, 4, No. 1, 112–128 (2001).MathSciNetMATHGoogle Scholar
  6. 6.
    M. Ronto, “On nonlinear boundary-value problems containing parameters,” Arch. Math. (Brno), 36, 585–593 (2000).MathSciNetMATHGoogle Scholar
  7. 7.
    M. Ronto and N. Shchobak, “On the numerical-analytic investigation of parametrized problems with nonlinear boundary conditions,” Nelin. Kolyvannya, 6, No. 4, 482–510 (2003); English translation: Nonlinear Oscillations, 6, No. 4, 469–496 (2003).MathSciNetCrossRefGoogle Scholar
  8. 8.
    M. I. Ronto, N. M. Shchobak, and K. V. Marynets’, “On the parametrization of boundary-value problems of the Cauchy–Nicoletti type,” Nauk. Visn. Uzhhorod. Nats. Univ., Ser. Mat. Inform., No. 16, 163–173 (2008).Google Scholar
  9. 9.
    K. V. Marynets’, “On a nonlinear three-point problem of the Cauchy–Nicoletti type,” Nauk. Visn. Uzhhorod. Nats. Univ., Ser. Mat. Inform., No. 19, 53–63 (2009).Google Scholar
  10. 10.
    K. V. Marynets’, “Investigation of the solutions of three-point problems of the Cauchy–Nicoletti type and their reduction to two-point problems,” Nauk. Visn. Kyiv. Nats. Univ., Ser. Fiz.-Mat. Nauk., No. 3, 85–90 (2009).Google Scholar
  11. 11.
    M. Ronto and J. Meszaros, “Some remarks concerning the convergence of the numerical-analytic method of successive approximations,” Ukr. Mat. Zh., 48, No. 1, 90–95 (1996); English translation: Ukr. Math. J., 48, No. 1, 101–107 (1996).MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    A. Ronto and M. Ronto, “On a Cauchy–Nicoletti-type three-point boundary value problem for linear differential equations with argument deviations,” Miskolc: Math. Notes, 10, No. 2, 173–205 (2009).MathSciNetMATHGoogle Scholar
  13. 13.
    M. Farkas, Periodic Motions, Applied Mathematical Sciences, Springer, London (1994).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  1. 1.Institute of MathematicsMiskolc UniversityMiskolcHungary
  2. 2.Uzhhorod National UniversityUzhhorodUkraine

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