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On the Spectrum of a Quasi-Differential Boundary Value Problem of the Second-Order

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Abstract

This paper studies the structure of the spectrum of a second-order quasi-differential boundary value problem \(\left( {{}_{P}^{2}x} \right)(t)\) = \( - \lambda \left( {{}_{P}^{0}x} \right)(t)\) \((t \in [a,b],\lambda \in \mathbb{R})\) (the coefficients in the equation are real-valued functions) with given homogeneous boundary conditions at the ends of the interval, \({}_{P}^{0}x(a)\) = \({}_{P}^{0}x(b)\) = 0. First, an auxiliary Cauchy problem with a real parameter \(\beta \) in the coefficient \({{p}_{{20}}}(t)\) of the equation, namely, \({{p}_{{22}}}(t)\left( {{{p}_{{11}}}(t){v}{\kern 1pt} '(t)} \right)'\) + \(({{p}_{{20}}}(t) + \beta ){v}(t)\) = 0, \({v}(a) = 0\), \({{p}_{{11}}}(a){v}{\kern 1pt} '(a)\) = 1, is considered. A fundamental theorem on either the continuity or discreteness of the real spectrum of the original boundary value problem is formulated in terms of the solution of this problem. Examples illustrating the cases of both continuous and discrete spectra of the original boundary value problem are given.

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REFERENCES

  1. W. N. Everitt and L. Marcus, Boundary Value Problems and Symplectic Algebra for Ordinary Differential and Quasi-Differential Operators, Mathematical Surveys and Monographs, Vol. 61 (American Mathematical Society, 1999).

  2. J. Eckhardt, F. Gestezy, R. Nichols, and G. Teschl, “Weyl–Titchmarsh theory for Sturm–Liuville operators with distributional potentials,” Opuscula Math. 33, 467–563 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  3. W. N. Everitt and D. Race, “The regular representation of singular second order differential expressions using quasi-derivatives,” Proc. London Math. Soc. s3-65, 383–404 (1992). https://doi.org/10.1112/plms/s3-65.2.383

  4. X. Lv, J. Ao, and A. Zettl, “Dependence of eigenvalues of fourth-order differential equations with discontinuous boundary conditions on the problem,” J. Math. Anal. Appl. 456, 671–685 (2017). https://doi.org/10.1016/j.jmaa.2017.07.021

    Article  MathSciNet  MATH  Google Scholar 

  5. Bao. Qinglan, Sun. Jiong, Hao. Xiaoling, and A. Zettl, “Characterization of self-adjoint domains for regular even order C-symmetric differential operators,” Electron. J. Qualitative Theory Differ. Equations 62, 1–17 (2019).

  6. A. Zettl, Sturm–Liouville Theory (American Mathematical Society, 2005).

    MATH  Google Scholar 

  7. A. Zettl, Recent Developments in Sturm–Liouville Theory, De Gruyter Studies in Mathematics, Vol. 76 (De Gruyter, Berlin, 2021). https://doi.org/10.1515/9783110719000

  8. J. Qin, K. Li, Zh. Zheng, and J. Cai, “Dependence of eigenvalues of discontinuous fourth-order differential operators with eigenparameter dependent boundary conditions,” J. Nonlinear Math. Phys. 29, 776–793 (2022). https://doi.org/10.1007/s44198-022-00060-x

    Article  MathSciNet  MATH  Google Scholar 

  9. K. A. Mirzoev, “Sturm–Liouville operators,” Trans. Moscow Math. Soc. 2014, 281–289 (2014). https://doi.org/10.1090/S0077-1554-2014-00234-X

    Article  MathSciNet  MATH  Google Scholar 

  10. V. E. Vladykina and A. A. Shkalikov, “Asymptotics of the solutions of the Sturm–Liouville equation with singular coefficients,” Math. Notes 98, 891–899 (2015). https://doi.org/10.1134/S0001434615110218

    Article  MathSciNet  MATH  Google Scholar 

  11. N. N. Konechnaya and K. A. Mirzoev, “Asympototics of solutions to linear differential equations of odd order,” Moscow Univ. Math. Bull. 75, 22–26 (2020). https://doi.org/10.3103/S0027132220010039

    Article  MathSciNet  MATH  Google Scholar 

  12. A. M. Savchuk and A. A. Shkalikov, “Asymptotic analysis of solutions of ordinary differential equations with distribution coefficient,” Sb. Math. 211, 1623–1659 (2020). https://doi.org/10.1070/SM9340

    Article  MathSciNet  MATH  Google Scholar 

  13. A. M. Savchuk and I. V. Sadovnichaya, “On the existence of an operator group generated by the one-dimensional Dirac system,” Trans. Moscow Math. Soc. 2019, 235–250 (2019). https://doi.org/10.1090/mosc/297

    Article  MathSciNet  MATH  Google Scholar 

  14. A. M. Savchuk and I. V. Sadovnichaya, “Spectral analysis of one-dimensional Dirac system with summable potential and Sturm–Liouville operator with distribution coefficients,” Sovrem. Mat. Fundam. Napravleniya 66, 373–530 (2020). https://doi.org/10.22363/2413-3639-2020-66-3-373-530

    Article  MathSciNet  Google Scholar 

  15. A. M. Savchuk and I. V. Sadovnichaya, “Equiconvergence of spectral decompositions for Sturm–Liouville operators with a distributional potential in scales of spaces,” Dokl. Math. 103, 47–49 (2021). https://doi.org/10.1134/S1064562421010117

    Article  MathSciNet  MATH  Google Scholar 

  16. V. Ya. Derr, “Nonoscillation of solutions to linear quasi-differential equation,” Izv. Inst. Mat. Inf. Udmurtsk. Gos. Univ., No. 1, 3–105 (1999).

  17. V. Ya. Derr, “On an adequate description of adjoint operator,” Vestn. Udmurtsk. Univ. Mat. Mekh. Komp’yuternye Nauki, No. 3, 43–63 (2011).

    Google Scholar 

  18. D. Yu. Shin, “On the solutions to linear nth-order quasi-differential equation,” Mat. Sb. 7, 479–532 (1940).

    Google Scholar 

  19. D. Yu. Shin, “Quasi-differential operators in Hilbert space,” Mat. Sb. 13, 39–70 (1943).

    MathSciNet  MATH  Google Scholar 

  20. M. A. Naimark, Linear Differential Operators (Nauka, Moscow, 1969).

    MATH  Google Scholar 

  21. Z. Nehari, “Disconjugate linear differential operators,” Trans. Am. Math. Soc. 129, 500–516 (1969). https://doi.org/10.1090/S0002-9947-1967-0219781-0

    Article  MathSciNet  MATH  Google Scholar 

  22. U. Elias, “The extremal solutions of the equation Ly + p(x)y = 0, 11,” J. Math. Anal. Appl. 55, 253–265 (1976). https://doi.org/10.1016/0022-247X(76)90156-6

    Article  MathSciNet  MATH  Google Scholar 

  23. W. N. Everitt, “Fourth order singular differential equation,” Math. Ann. 149, 320–340 (1963). https://doi.org/10.1007/BF01471126

    Article  MathSciNet  MATH  Google Scholar 

  24. T. Kusano and M. Naito, “Oscillation criteria of general linear ordinary differential equations,” Pac. J. Math. 92, 345–355 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  25. W. F. Trench, “Canonical forms and principal systems for general disconjugate equations,” Trans. Am. Math. Soc. 189, 319–327 (1974). https://doi.org/10.1090/S0002-9947-1974-0330632-X

    Article  MathSciNet  MATH  Google Scholar 

  26. W. F. Trench, “Asymptotic theory perturbed general disconjugate equations,” Hiroshima Math. J. 12, 43–58 (1982). https://doi.org/10.32917/hmj/1206133874

    Article  MathSciNet  MATH  Google Scholar 

  27. W. N. Everitt and J. D. Key, “On some properties of matrices associated with linear ordinary quasi-differential equations,” Proc. R. Soc. Edinburg 96, 211–219 (1984). https://doi.org/10.1017/S030821050002535X

  28. W. N. Everitt, “A note of linear ordinary quasi-differential equations,” Proc. R. Soc. Edinburg 101, 1–14 (1985). https://doi.org/10.1017/S0308210500026111

  29. L. S. Nicolson, “Disconjugate systems of linear differential equation,” J. Differ. Equations 7, 570–583 (1970). https://doi.org/10.1016/0022-0396(70)90102-6

    Article  MathSciNet  MATH  Google Scholar 

  30. V. Ya. Derr, “On the transformation of some multipoint boundary value problems into the de la Vallée–Poussin problem and of the conditions of solvability,” Differ. Uravn. 23, 598–608 (1987).

    Google Scholar 

  31. V. Ya. Derr, “On the generalized de la Vallée–Poussin problem,” Differ. Uravn. 23, 1861–1872 (1987).

    MATH  Google Scholar 

  32. O. Sh. Mukhtarov and M. Kadakal, “Some spectral properties of one Sturm–Liouville type problem with discontinuous weight,” Sib. Math. J. 46, 681–694 (2005). https://doi.org/10.1007/s11202-005-0069-z

    Article  MATH  Google Scholar 

  33. A. A. Vladimirov and I. A. Sheipak, “Asympotics of the eigenvalues of the Sturm–Liouville problem with discrete self-similar weight,” Math. Notes 88, 637–646 (2010). https://doi.org/10.1134/S0001434610110039

    Article  MathSciNet  MATH  Google Scholar 

  34. A. A. Vladimirov and I. A. Sheipak, “Asymptotics of eigenvalues in a problem of high even order with discrete self-similar weight,” St. Petersburg Math. J. 24, 263–273 (2013). https://doi.org/10.1090/S1061-0022-2013-01237-4

    Article  MathSciNet  MATH  Google Scholar 

  35. A. A. Vladimirov and E. S. Karulina, “On an a priori majorant of the least eigenvalues of the Sturm–Liouville problem,” Itogi Nauki Tekh. Ser. Sovrem. Mat. Ee Prilozheniya 193, 25–27 (2021). https://doi.org/10.36535/0233-6723-2021-193-25-27

    Article  Google Scholar 

  36. M. Yu. Vatolkin and V. Ya. Derr, “On the representation of solutions to a quasidifferential equation,” Russ. Math. 39 (10), 25–32 (1995).

    MathSciNet  MATH  Google Scholar 

  37. M. Yu. Vatolkin, “On the structure of spectrum of one second-order boundary value problem,” in Control Theory and Mathematical Modeling: Proc. All-Russ. Conf. with Int. Participation Dedicated to the Memory of Prof. N.V. Azbelev and Prof. E.L. Tonkov, Ed. by A. S. Bannikov, V. A. Zaitsev, N. N. Petrov, and S. N. Popova (Udmurtskii Univ., Izhevsk, 2020), pp. 56–58.

  38. M. Yu. Vatolkin, “Examples on studying the spectrum of one second-order boundary value problem,” in Control Theory and Mathematical Modeling: Proc. All-Russ. Conf. with Int. Participation Dedicated to the Memory of Prof. N.V. Azbelev and Prof. E.L. Tonkov, Ed. by A. S. Bannikov, V. A. Zaitsev, N. N. Petrov, and S. N. Popova (Udmurtskii Univ., Izhevsk, 2020), pp. 58–60.

  39. E. Kamke, Differentialgleichungen: Gewöhnliche Differentialgleichungen (Geest & Portig K.-G., 1969).

    MATH  Google Scholar 

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  1. Izhevsk State Technical University, 426069, Izhevsk, Russia

    M. Yu. Vatolkin

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Translated by E. Chernokozhin

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Vatolkin, M.Y. On the Spectrum of a Quasi-Differential Boundary Value Problem of the Second-Order. Russ Math. 67, 1–19 (2023). https://doi.org/10.3103/S1066369X23010061

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