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On the Existence of a Solution of a Boundary Value Problem on a Graph for a Nonlinear Equation of the Fourth Order

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Abstract

A fourth-order nonlinear differential equation on a network that is a model of a system of Euler–Bernoulli rods is considered. Based on the monotone iteration method, the existence of a solution of a boundary value problem on a graph for this equation is established using the positiveness of the Green’s function and the maximum principle for the corresponding linear differential equation. An example is given to illustrate the results.

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REFERENCES

  1. Borovskikh, A.V., Mustafokulov, R., Lazarev, K.P., and Pokornyi, Yu.V., On one class of fourth-order differential equations on a spatial network, Dokl. Ross. Akad. Nauk, 1995, vol. 345, no. 6, pp. 730–732.

    MATH  Google Scholar 

  2. Borovskikh, A.V. and Lazarev, K.P., Fourth-order differential equations on geometric graphs, J. Math. Sci., 2004, vol. 119, no. 6, pp. 719–738.

    Article  MathSciNet  MATH  Google Scholar 

  3. Dekoninck, B. and Nicase, S., The eigenvalue problem for network of beams in generalized functions, Linear Algebra Appl., 2000, vol. 314, no. 1–3, pp. 165–189.

    Article  MathSciNet  Google Scholar 

  4. Pokornyi, Yu.V. and Mustafokulov, R., On the positivity of the Green’s function of linear boundary value problems for fourth-order equations on a graph, Russ. Math., 1999, vol. 43, no. 2, pp. 71–78.

    MathSciNet  MATH  Google Scholar 

  5. Ammari, K., Bchatnia, A., and Mehenaoui, N., Exponential stability for the nonlinear Schrödinger equation on a star-shaped network, Z. Angew. Math. Phys., 2021, vol. 72, pp. 1–19.

    Article  MATH  Google Scholar 

  6. Cerpa, E., Crepeau, E., and Moreno, C., On the boundary controllability of the Korteweg–de Vries equation on a star-shaped network, IMA J. Math. Control Inf., 2020, vol. 37, no. 1, pp. 226–240.

    MathSciNet  MATH  Google Scholar 

  7. Grigor’yan, A., Lin, Y., and Yang, Y., Existence of positive solutions to some nonlinear equations on locally finite graphs, Sci. China Math., 2017, vol. 60, pp. 1311–1324.

    Article  MathSciNet  MATH  Google Scholar 

  8. Han, Zh-J. and Zuazua, E., Decay rates for elastic-thermoelastic star-shaped networks, Networks Heterogeneous Media, 2017, vol. 12, no. 3, pp. 461–488.

    Article  MathSciNet  MATH  Google Scholar 

  9. Bondarenko, N.P., A partial inverse Sturm–Liouville problem on an arbitrary graph, Math. Meth. Appl. Sci., 2021, vol. 44, no. 8, pp. 6896–6910.

    Article  MathSciNet  MATH  Google Scholar 

  10. Yurko, V.A., Inverse spectral problems for differential operators on spatial networks, Russ. Math. Surv., 2016, vol. 71, no. 3, pp. 539–584.

    Article  MathSciNet  MATH  Google Scholar 

  11. Pokornyi, Yu.V., Penkin, O.M., Pryadiev, V.L., Borovskikh, A.V., Lazarev, K.P., and Shabrov, S.A., Differentsial’nye uravneniya na geometricheskikh grafakh (Differential Equations on Geometric Graphs), Moscow: Fizmatlit, 2007.

    Google Scholar 

  12. Kulaev, R.Ch., The Green function of the boundary-value problem on a star-shaped graph, Russ. Math., 2013, vol. 57, no. 2, pp. 48–57.

    Article  MathSciNet  MATH  Google Scholar 

  13. Kulaev, R.Ch., Disconjugacy of fourth-order equations on graphs, Sb. Math., 2015, vol. 206, no. 12, pp. 1731–1770.

    Article  MathSciNet  MATH  Google Scholar 

  14. Kulaev, R.Ch., Oscillation of the Green function of a multipoint boundary value problem for a fourth-order equation, Differ. Equations, 2015, vol. 51, no. 4, pp. 449–463.

    Article  MathSciNet  MATH  Google Scholar 

  15. Kulaev, R.Ch., The qualitative theory of fourth-order differential equations on a graph, Mediterr. J. Math., 2022, vol. 19, p. 73.

    Article  MathSciNet  MATH  Google Scholar 

  16. Huang, A., Lin, Y., and Yau, H., Existence of solutions to mean field equations on graphs, Commun. Math. Phys., 2019, vol. 377, pp. 613–621.

    Article  MathSciNet  MATH  Google Scholar 

  17. Ge, H., Kazdan–Warner equation on graph in the negative case, J. Math. Anal. Appl., 2017, vol. 453, no. 2, pp. 1022–1027.

    Article  MathSciNet  MATH  Google Scholar 

  18. Lin, Y. and Wu, Y., Blow-up problems for nonlinear parabolic equations on locally finite graphs, Acta Math. Scientia, 2018, vol. 38, no. 3, pp. 843–856.

    Article  MathSciNet  MATH  Google Scholar 

  19. Mehandiratta, V., Mehra, M., and Leugering, G., Existence and uniqueness results for a nonlinear Caputo fractional boundary value problem on a star graph, J. Math. Anal. Appl., 2019, vol. 477, no. 2, pp. 1243–1264.

    Article  MathSciNet  MATH  Google Scholar 

  20. Harjani, J. and Sadarangani, K., Existence and uniqueness of positive solutions for a nonlinear fourth-order boundary value problem, Positivity, 2010, vol. 14, pp. 849–858.

    Article  MathSciNet  MATH  Google Scholar 

  21. Ma, R., Zhang, J., and Fu, Sh., The method of lower and upper solutions for fourth-order two-point boundary value problems, J. Math. Anal. Appl., 1997, vol. 215, no. 1, pp. 415–422.

    MathSciNet  MATH  Google Scholar 

  22. Song, W. and Gao, W., A fourth-order boundary value problem with one-sided Nagumo condition, Boundary Value Probl., 2011, p. 569191.

  23. Graef, J.R., Qian, Ch., and Yang, B., A three point boundary value problem for nonlinear fourth order differential equations, J. Math. Anal. Appl., 2003, vol. 187, no. 1, pp. 217–233.

    Article  MathSciNet  MATH  Google Scholar 

  24. Wei, Z. and Pang, C., Positive solutions and multiplicity of fourth-order \(m \)-point boundary value problems with two parameters, Nonlinear Anal., 2007, vol. 67, no. 5, pp. 1586–1598.

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhang, Q., Chen, S., and Lú, J., Upper and lower solution method for fourth-order four-point boundary value problems, J. Comput. Appl. Math., 2006, vol. 196, no. 2, pp. 387–393.

  26. Mustafokulov, R., Positive solutions of nonlinear boundary value problems for a fourth-order equation on a graph, Dokl. Nats. Akad. Nauk. Tajik., 1999, vol. 42, no. 3, pp. 40–46.

  27. Kulaev, R.Ch., On the disconjugacy property of an equation on a graph, Sib. Math. J., 2016, vol. 57, no. 1, pp. 64–73.

    Article  MathSciNet  MATH  Google Scholar 

  28. Kulaev, R.Ch., Criterion for the positiveness of the Green function of a many-point boundary value problem for a fourth-order equation, Differ. Equations, 2015, vol. 51, no. 2, pp. 163–176.

    Article  MathSciNet  MATH  Google Scholar 

  29. Kulaev, R.Ch. and Urtaeva, A.A., Sturm separation theorems for a fourth-order equation on a graph, Math. Notes, 2022, vol. 112, no. 6, pp. 977–981.

    Article  MathSciNet  MATH  Google Scholar 

  30. Kulaev, R.Ch. and Urtaeva, A.A., Spectral properties of a fourth-order differential operator on a network, Math. Meth. Appl. Sci., 2023, pp. 1–21.

  31. Li, Y. and Gao, Y., Existence and uniqueness results for the bending elastic beam equations, Appl. Math. Lett., 2019, vol. 95, pp. 72–77.

    Article  MathSciNet  MATH  Google Scholar 

  32. Xu, G.Q. and Mastorakis, N.E., Differential Equations on Metric Graph, WSEAS Press, 2010.

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Funding

This work was supported by the Ministry of Science and Higher Education of the Russian Federation, agreement no. 075-02-2023-939.

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Authors and Affiliations

  1. Southern Mathematical Institute, Vladikavkaz Scientific Center, Russian Academy of Sciences, Vladikavkaz, 362025, Russia

    R. Ch. Kulaev

  2. Khetagurov North Ossetian State University, Vladikavkaz, 362025, Russia

    A. A. Urtaeva

Authors
  1. R. Ch. Kulaev
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  2. A. A. Urtaeva
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Correspondence to R. Ch. Kulaev or A. A. Urtaeva.

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Translated by V. Potapchouck

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Kulaev, R.C., Urtaeva, A.A. On the Existence of a Solution of a Boundary Value Problem on a Graph for a Nonlinear Equation of the Fourth Order. Diff Equat 59, 1175–1184 (2023). https://doi.org/10.1134/S0012266123090033

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  • DOI: https://doi.org/10.1134/S0012266123090033

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