Skip to main content
SpringerLink
Log in

Problem with nonlocal conditions for weakly nonlinear hyperbolic equations

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

For weakly nonlinear hyperbolic equations of order n, n≥3, with constant coefficients in the linear part of the operator, we study a problem with nonlocal two-point conditions in time and periodic conditions in the space variable. Generally speaking, the solvability of this problem is connected with the problem of small denominators whose estimation from below is based on the application of the metric approach. For almost all (with respect to the Lebesgue measure) coefficients of the equation and almost all parameters of the domain, we establish conditions for the existence of a unique classical solution of the problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. A. Artem’ev, “Periodic solutions of one class of partial differential equations,” Izv. Akad. Nauk SSSR, Ser. Mat., No. 1, 15–50 (1937).

    Google Scholar 

  2. O. Vejvoda, L. Harrmann, V. Lovicar, et al., Partial Differential Equations: Time-Periodic Solutions, Noordhoff, Alphen an den Rijn-Sijthoff (1981).

    Google Scholar 

  3. B. I. Ptashnik, Ill-Posed Boundary-Value Problems for Partial Differential Equations [in Russian], Naukova Dumka, Kiev (1984).

    Google Scholar 

  4. O. Vejvoda and M. Shtedry, “Existence of classical periodic solutions of the wave equation. Relationship between the number-theoretic nature of the period and geometric properties of solutions,” Differents. Uravn., 20, No. 10, 1733–1739 (1984).

    MathSciNet  Google Scholar 

  5. L. G. Tret’yakova, “On the problem of 2π-periodic solutions of the equation of oscillations of a nonlinear string,” Vestn. Belorus. Univ., Ser. 1, No. 3, 49–51 (1986).

    MathSciNet  Google Scholar 

  6. E. Sinestrari and G. F. Webb, “Nonlinear hyperbolic systems with nonlocal boundary conditions,” J. Math. Anal. Appl., 121, No. 2, 449–464 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  7. V. V. Marinets, “On some problems for systems of nonlinear partial differential equations with nonlocal boundary conditions,” Differents. Uravn., 24, No. 8, 1393–1397 (1988).

    MathSciNet  Google Scholar 

  8. P. I. Plotnikov and L. N. Yungerman, “Periodic solutions of a weakly nonlinear wave equation with irrational ratio of the period to the length of the interval,” Differents. Uravn., 24, No. 9, 1599–1607 (1988).

    MathSciNet  Google Scholar 

  9. L. Byszewski, “Existence and uniqueness of solutions of nonlocal problems for hyperbolic equation u ix =F(x, t, u, u x ),” J. Appl. Math. Stochast. Anal., 3, No. 3, 163–168 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  10. L. Byszewski and V. Lakshmikanthan, “Monotone iterative technique for nonlocal hyperbolic differential problems,” J. Math. Phys. Sci., 26, No. 4, 345–359 (1992).

    MATH  MathSciNet  Google Scholar 

  11. S. V. Zhestkov, “On the existence of ω-periodic solutions of quasilinear wave systems with n-spatial variables,” Vest. Akad. Navuk Belarus., Ser. Fiz.-Mat. Navuk, No. 2, 5–11 (1994).

    MathSciNet  Google Scholar 

  12. I.Ya. Kmit’, “On a nonlocal problem for a quasilinear hyperbolic system of the first order with two independent variables,” Ukr. Mat. Zh., 45, No. 9, 1307–1313 (1993).

    MathSciNet  Google Scholar 

  13. I. Ya. Kmit’, “On a problem with nonlocal (in time) conditions for hyperbolic systems,” Mat. Met. Fiz.-Mekh. Polya, Issue 37, 21–25 (1994).

    Google Scholar 

  14. T. I. Kiguradze, “One boundary-value problem for hyperbolic systems,” Dokl. RAN, Mat., 328, No. 2, 135–138 (1993).

    Google Scholar 

  15. T. I. Kiguradze, “On bounded and periodic (in a strip) solutions of quasilinear hyperbolic systems,” Differents. Uravn., 30, No. 10, 1760–1773 (1994).

    MathSciNet  Google Scholar 

  16. Yu. A. Mitropol’skii and L. B. Urmancheva, “On a two-point problem for systems of hyperbolic equations,” Ukr. Mat. Zh., 42, No. 2, 1657–1663 (1990).

    MathSciNet  Google Scholar 

  17. Yu. A. Mitropol’skii, G. P. Khoma, and M. I. Gromyak, Asymptotic Methods for the Investigation of Quasiwave Hyperbolic Equations [in Russian], Naukova Dumka, Kiev (1990).

    Google Scholar 

  18. Yu. A. Mitropol’skii and G. P. Khoma, “On periodic solutions of the second-order wave equations. V,” Ukr. Mat. Zh., 45, No. 8, 1115–1121 (1993).

    MathSciNet  Google Scholar 

  19. Yu. A. Mitropol’skii and N. G. Khoma, “Periodic solutions of quasilinear second-order hyperbolic equations,” Ukr. Mat. Zh., 47, No. 10, 1370–1375 (1995).

    MathSciNet  Google Scholar 

  20. V. I. Arnol’d, “Small denominators. I. On the mapping of a circle onto itself,” Izv. Akad. Nauk SSSR, Ser. Mat., 25, No. 1, 21–86 (1961).

    MathSciNet  Google Scholar 

  21. L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow (1977).

    Google Scholar 

Download references

Authors
  1. T. P. Goi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. I. Ptashnik
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L’viv. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 186–195, February, 1997.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Goi, T.P., Ptashnik, B.I. Problem with nonlocal conditions for weakly nonlinear hyperbolic equations. Ukr Math J 49, 204–215 (1997). https://doi.org/10.1007/BF02486436

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02486436

Keywords

Search

Navigation