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On Solutions of a Boundary Value Problem for a Second-Order Differential Equation with a Parameter and Discontinuous Right-Hand Side

  • ORDINARY DIFFERENTIAL EQUATIONS
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Abstract

A boundary value problem for a second-order ordinary differential equation with a parameter and discontinuous right-hand side is considered. Theorems on the number of solutions to the problem are established. The resulting solutions are illustrated by plots. The process of numerically solving the problem is described.

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REFERENCES

  1. D. K. Potapov, “Sturm–Liouville’s problem with discontinuous nonlinearity,” Differ. Equations 50 (9), 1272–1274 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  2. A. M. Kamachkin, D. K. Potapov, and V. V. Yevstafyeva, “Solution to second-order differential equations with discontinuous right-hand side,” Electron. J. Differ. Equations, No. 221, 1–6 (2014).

  3. J. Llibre and M. A. Teixeira, “Periodic solutions of discontinuous second order differential systems,” J. Singularities 10, 183–190 (2014).

    MathSciNet  MATH  Google Scholar 

  4. D. K. Potapov, “Existence of solutions, estimates for the differential operator, and a “separating” set in a boundary value problem for a second-order differential equation with a discontinuous nonlinearity,” Differ. Equations 51 (7), 967–972 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  5. A. M. Samoilenko and I. L. Nizhnik, “Differential equations with bistable nonlinearity,” Ukr. Math. J. 67 (4), 584–624 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  6. G. Bonanno, G. D’Agui, and P. Winkert, “Sturm–Liouville equations involving discontinuous nonlinearities,” Minimax Theory Appl. 1 (1), 125–143 (2016).

    MathSciNet  MATH  Google Scholar 

  7. A. M. Kamachkin, D. K. Potapov, and V. V. Yevstafyeva, “Non-existence of periodic solutions to non-autonomous second-order differential equation with discontinuous nonlinearity,” Electron. J. Differ. Equations, No. 4, 1–8 (2016).

  8. A. M. Kamachkin, D. K. Potapov, and V. V. Yevstafyeva, “Existence of solutions for second-order differential equations with discontinuous right-hand side,” Electron. J. Differ. Equations, No. 124, 1–9 (2016).

  9. S. Bensid and J. I. Diaz, “Stability results for discontinuous nonlinear elliptic and parabolic problems with a S‑shaped bifurcation branch of stationary solutions,” Discrete Contin. Dyn. Syst. Ser. B 22 (5), 1757–1778 (2017).

    MathSciNet  MATH  Google Scholar 

  10. C. E. L. da Silva, P. R. da Silva, and A. Jacquemard, “Sliding solutions of second-order differential equations with discontinuous right-hand side,” Math. Methods Appl. Sci. 40 (14), 5295–5306 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  11. V. N. Pavlenko and E. Yu. Postnikova, “Sturm–Liouville problem for an equation with a discontinuous nonlinearity,” Chelyab. Fiz.-Mat. Zh. 4 (2), 142–154 (2019).

    MathSciNet  MATH  Google Scholar 

  12. C. E. L. da Silva, A. Jacquemard, and M. A. Teixeira, “Periodic solutions of a class of non-autonomous discontinuous second-order differential equations,” J. Dyn. Control Syst. 26 (1), 17–44 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  13. N. N. Kalitkin, Numerical Methods (Nauka, Moscow, 1978) [in Russian].

    Google Scholar 

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Funding

This research was funded by the Russian Science Foundation, project no. 23-21-00069, https://rscf.ru/en/project/23-21-00069/.

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Authors and Affiliations

  1. St. Petersburg State University, 199034, St. Petersburg, Russia

    O. V. Baskov & D. K. Potapov

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  1. O. V. Baskov
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  2. D. K. Potapov
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Correspondence to D. K. Potapov.

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The authors declare that they have no conflicts of interest.

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Translated by I. Ruzanova

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Baskov, O.V., Potapov, D.K. On Solutions of a Boundary Value Problem for a Second-Order Differential Equation with a Parameter and Discontinuous Right-Hand Side. Comput. Math. and Math. Phys. 63, 1424–1436 (2023). https://doi.org/10.1134/S096554252308002X

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  • DOI: https://doi.org/10.1134/S096554252308002X

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