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On Solutions of Nonlinear Boundary-Value Problems Whose Components Vanish at Certain Points

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Ukrainian Mathematical Journal Aims and scope

We show how an appropriate parametrization technique and successive approximations can help to investigate nonlinear boundary-value problems for systems of differential equations under the condition that the components of solutions vanish at certain unknown points. The technique can be applied to nonlinearities involving the signs of the absolute value and positive or negative parts of functions under boundary conditions of various types.

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References

  1. A. Capietto, J. Mawhin, and F. Zanolin, “On the existence of two solutions with a prescribed number of zeros for a superlinear two-point boundary-value problem,” Topol. Methods Nonlin. Anal., 6, No. 1, 175–188 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Mawhin, “Topological degree methods in nonlinear boundary-value problems,” CBMS Reg. Conf. Ser. Math., 40 (1979).

  3. K. McLeod, W. C. Troy, and F. B. Weissler, “Radial solutions of δu + f (u) = 0 with prescribed numbers of zeros,” J. Differ. Equ., 83, No. 2, 368–378 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Naito and Y. Naito, “Solutions with prescribed numbers of zeros for nonlinear second-order differential equations,” Funkcial. Ekvac., 37, No. 3, 505–520 (1994).

    MathSciNet  MATH  Google Scholar 

  5. Y. Naito, “Bounded solutions with prescribed numbers of zeros for the Emden–Fowler differential equation,” Hiroshima Math. J., 24, No. 1, 177–220 (1994).

    MathSciNet  MATH  Google Scholar 

  6. A. Rontó and M. Rontó, “Successive approximation techniques in nonlinear boundary-value problems for ordinary differential equations,” in: Handbook of Differential Equations: Ordinary Differential Equations, Elsevier/North-Holland, Amsterdam, Vol. 4 (2008), pp. 41–592.

  7. A. Rontó and M. Rontó, “Existence results for three-point boundary-value problems for systems of linear functional differential equations,” Carpathian J. Math., 28, No. 1, 163–182 (2012).

    MathSciNet  MATH  Google Scholar 

  8. A. Rontó, M. Rontó, G. Holubová, and P. Nečesal, “Numerical-analytic technique for investigation of solutions of some nonlinear equations with Dirichlet conditions,” Bound. Value Probl., 58, No. 20 (2011).

  9. A. Rontó, M. Rontó, and N. Shchobak, “On the parametrization of three-point nonlinear boundary value problems,” Nonlin. Oscillat., 7, No. 3, 384–402 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Rontó, M. Rontó, and N. Shchobak, “On finding solutions of two-point boundary value problems for a class of non-linear functional differential systems,” Electron. J. Qual. Theory Different. Equat., 13, 1–17 (2012).

    MATH  Google Scholar 

  11. A. Rontó, M. Rontó, and N. Shchobak, “On boundary-value problems with prescribed number of zeros of solutions,” Miskolc Math. Notes, 18, No. 1, 431–452 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Rontó, M. Rontó, and N. Shchobak, “Constructive analysis of periodic solutions with interval halving,” Bound. Value Probl., 57, No. 34 (2013).

  13. A. Rontó, M. Rontó, and N. Shchobak, “Notes on interval halving procedure for periodic and two-point problems,” Bound. Value Probl., 164, No. 20 (2014).

  14. A. Rontó, M. Rontó, and J. Varha, “A new approach to non-local boundary value problems for ordinary differential systems,” Appl. Math. Comput., 250, 689–700 (2015).

    MathSciNet  MATH  Google Scholar 

  15. A. Rontó, M. Rontó, and J. Varha, “On non-linear boundary-value problems and parametrization at multiple nodes,” Electron. J. Qual. Theory Differ. Equ., 18, Paper 80 (2016).

  16. M. Rontó and J. Varha, “Constructive existence analysis of solutions of non-linear integral boundary value problems,” Miskolc Math. Notes, 15, No. 2, 725–742 (2014).

    MathSciNet  MATH  Google Scholar 

  17. M. Rontó and J. Varha, “Successive approximations and interval halving for integral boundary-value problems,” Miskolc Math. Notes, 16, No. 2, 1129–1152 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Rontó and N. Shchobak, “On parametrized problems with non-linear boundary conditions,” Electron. J. Qual. Theory Differ. Equ., 7, No. 20, 1–25 (2004).

    MathSciNet  MATH  Google Scholar 

  19. M. Rontó, J. Varha, and K. Marynets, “Further results on the investigation of solutions of integral boundary-value problems,” Tatra Mt. Math. Publ., 63, 247–267 (2015).

    MathSciNet  MATH  Google Scholar 

  20. A. M. Samoilenko, “A numerical-analytic method for the investigation of periodic systems of ordinary differential equations. I,” Ukr. Mat. Zh., 17, No. 4, 82–93 (1965).

    Article  MathSciNet  Google Scholar 

  21. A. M. Samoilenko, “A numerical-analytic method for the investigation of periodic systems of ordinary differential equations. II,” Ukr. Mat. Zh., 18, No. 2, 50–59 (1966).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Brno University of Technology, Brno, Czech Republic

    B. Půža

  2. Institute of Mathematics, Czech Academy of Sciences, Brno, Czech Republic

    A. Rontó

  3. Institute of Mathematics, University of Miskolc, Miskolc, Hungary

    M. Rontó

  4. Brno University of Technology, Brno, Czech Republic

    N. Shchobak

Authors
  1. B. Půža
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  2. A. Rontó
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  3. M. Rontó
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  4. N. Shchobak
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, No. 1, pp. 94–114, January, 2018.

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Půža, B., Rontó, A., Rontó, M. et al. On Solutions of Nonlinear Boundary-Value Problems Whose Components Vanish at Certain Points. Ukr Math J 70, 101–123 (2018). https://doi.org/10.1007/s11253-018-1490-3

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  • DOI: https://doi.org/10.1007/s11253-018-1490-3

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