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Asymptotical Distributions of Eigenvalues of Periodic and Antiperiodic Boundary Value Problems for Second-Order Differential Equations

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Abstract—We consider asymptotical distributions of characteristic constants in periodic and antiperiodic boundary value problems for a second-order linear equation with periodic coefficients. This allows one to obtain asymptotical properties of stability and instability zones of solutions. We show that if there are no turning points, i.e., if \(r(t) > 0\), then the lengths of instability zones converge to zero as their number increases, while the lengths of stability zones converge to a positive number. If \(r(t) \geqslant 0\) and function \(r(t)\) has zeroes, then the lengths of stability and instability zones have finite nonzero limits as the numbers of the corresponding zones infinitely increase. If function \(r(t)\) is alternating, then the lengths of all stability zones converge to zero and the lengths of all instability zones converge to finite numbers. This yields various stability and instability criteria for solutions of second-order equations with periodic coefficients. The presented results are illustrated by a substantial example. The investigation methods are based on a detailed study of so-called special standard equations and the reduction of original equations to standard equations. Here, asymptotical methods of the theory of singular perturbations and properties of series of special functions are used.

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REFERENCES

  1. Yakubovich, V.A. and Starzhinskii, V.M., Lineinye differentsial’nye uravneniya s periodicheskimi koeffitsientami (Linear Differential Equations with Periodic Coefficients), Moscow: Nauka, 1972.

  2. Lyapunov, A.M., On one linear second-order differential equation, in Sobr. soch. (Collected Works), Moscow, Leningrad: Izd. Akad. Nauk SSSR, 1956, pp. 401–403.

  3. Lyapunov, A.M., On one transcendental equation and second-order linear differential equations with periodic coefficients, in Sobr. soch. (Collected Works), Moscow, Leningrad: Izd. Akad. Nauk SSSR, 1956, pp. 404–406.

  4. Smirnov, V.I., The scientific archive of A.M. Lyapunov on stability problems and the theory of ordinary differential equations, Tr. III Vsesoyuzn. matem. s”ezda (Proc. III All-Union Math. Congress), Moscow: Izd. Akad. Nauk N SSSR, 1956, vol. 1.

  5. Coddington, E. and Levinson, N., Theory of Ordinary Differential Equations, London: McGraw-Hill Book Company, 1955.

    MATH  Google Scholar 

  6. Dorodnitsyn, A.A., Asymptotic laws for the distribution of eigenvalues for certain singular types of second-order differential equations, Usp. Mat. Nauk, 1952, vol. 6, no. 52, pp. 3–96.

    MATH  Google Scholar 

  7. Tikhonov, A.N., Systems of differential equations containing small parameters in derivatives, Mat. Sb., 1952, vol. 31, no. 3, pp. 575–586.

    MathSciNet  Google Scholar 

  8. Butuzov, V.F., Vasil’eva, A.B., and Fedoryuk, M.V., Asymptotic methods in the theory of ordinary differential equations, USSR Comput. Math. Math. Phys., 1970, vol. 8, pp. 1–82.

    MathSciNet  MATH  Google Scholar 

  9. Vasil’eva, A.B. and Butuzov, V.F., Asimptoticheskie razlozheniya reshenii singulyarno vozmushchennykh uravnenii (Asymptotic Expansions of Solutions of Singularly Perturbed Equations), Moscow: Nauka, 1973.

  10. Kreyn, M.G., Fundamental propositions of the theory of λ-stability zones of canonical systems of linear differential equations with periodic coefficients, in Sbornik Pamyati A.A. Andronova (Collection in the Memory of A.A. Andronov), Moscow: Izd. Akad. Nauk SSSR, 1955.

  11. Kashchenko, S.A., Limit values of eigenvalues of the first boundary value problem for a singularly perturbed second-order differential equation with turning points, Vestn. Yarosl. Univ., 1974, vol. 10, pp. 3–39.

    MathSciNet  Google Scholar 

  12. Kashchenko, S.A., Asymptotics of eigenvalues of the first boundary value problem for a singularly perturbed second-order differential equation with turning points, Vestn. Yarosl. Univ., 1974, vol. 10, pp. 40–64.

    MathSciNet  Google Scholar 

  13. Kashchenko, S.A., Asymptotics of eigenvalues of the first boundary-value problem for singularly perturbed second-order differential equation with turning points, Autom. Control Comput. Sci., 2016, vol. 50, no. 7, pp. 636–656.

    Article  Google Scholar 

  14. Kashchenko, S.A., Asymptotics of eigenvalues of periodic and antiperiodic boundary value problems for singularly perturbed second-order differential equations with turning points, Vestn. Yarosl. Univ., 1975, vol. 13, pp. 20–83.

    MathSciNet  Google Scholar 

  15. Kashchenko, S.A., Asymptotic laws for the distribution of eigenvalues of a periodic and antiperiodic boundary value problem for second-order differential equations with turning points, Issled. po ustoych. i teorii koleb. (Studies on Stability and Theory of Oscillations), Yaroslavl: Yarosl. Gos. Univ., 1976, pp. 95–113.

    Google Scholar 

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ACKNOWLEDGMENTS

This work was carried out within the framework of the State Programme of the Ministry of Education and Science of the Russian Federation, project no. 1.12873.2018/12.1.

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Correspondence to S. A. Kashchenko.

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Translated by A. Muravnik

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Kashchenko, S.A. Asymptotical Distributions of Eigenvalues of Periodic and Antiperiodic Boundary Value Problems for Second-Order Differential Equations. Aut. Control Comp. Sci. 52, 797–809 (2018). https://doi.org/10.3103/S0146411618070143

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  • DOI: https://doi.org/10.3103/S0146411618070143

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