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A problem with integral conditions with respect to time for Gårding hyperbolic equations

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Ukrainian Mathematical Journal Aims and scope

In a domain specified in the form of a Cartesian product of a segment [0, T] and the space \( {{\mathbb{R}}^p} \), we study a problem with integral conditions with respect to the time variable for Gårding hyperbolic equations with constant coefficients in the class of functions almost periodic in the space variables. A criterion for the unique solvability of this problem and sufficient conditions for the existence of its solution are established in different function spaces. To solve the problem of small denominators arising in the construction of solutions of the posed problem, we use the metric approach.

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References

  1. B. V. Dmitriev, “Nonlocal problem with nonlinear integral conditions for a hyperbolic equation,” Vestn. Samar. Gos. Univ., No. 1(18), 26–32 (2009).

  2. V. S. Il’kiv and T. V. Maherovs’ka, “Problem with integral conditions for a second-order partial differential equation,” Visn. Nats. Univ. “L’vivs’ka Politekhnika,” Fiz.-Mat. Nauk., No. 625, 12–19 (2008).

  3. G. A. Lukina, “Boundary-value problems with integral boundary conditions for the linearized Korteweg–de Vries equation,” Vestn. Yuzhno-Ural. Gos. Univ., Ser. Mat. Model Program., Issue 8, No. 17, 52–61 (2011).

  4. O. M. Medvid’ and M. M. Symotyuk, “Diophantine approximations of the characteristic determinant of an integral problem for a linear partial differential equation,” Nauk. Visn. Cherniv. Nats. Univ., Ser. Mat., Issue 228, 74–85 (2004).

  5. O. M. Medvid’ and M. M. Symotyuk, “Integral problem for linear partial differential equations,” Mat. Stud., 28, No. 2, 115–141 (2007).

    MATH  MathSciNet  Google Scholar 

  6. A. M. Kuz’ and B. I. Ptashnyk, “Problem with integral conditions for the Klein–Gordon equation in the class of functions almost periodic in the space variables,” Prikl. Probl. Mekh. Mat., Issue 8, 41–53 (2010).

    Google Scholar 

  7. L. S. Pul’kina, “Nonlocal problem with integral conditions for a hyperbolic equation,” Differents. Uravn., 40, No. 7, 887–892 (2004).

    MathSciNet  Google Scholar 

  8. B. I. Ptashnyk, V. S. Il’kiv, I. Ya. Kmit’, and V. M. Polishchuk, Nonlocal Boundary-Value Problems for Partial Differential Equations [in Ukrainian], Naukova Dumka, Kyiv (2002).

  9. M. M. Symotyuk and O. M. Medvid’, “Problem with integral conditions for linear partial differential equations with constant coefficients,” Mat. Met. Fiz.-Mekh. Polya, 46, No. 4, 98–107 (2003).

    Google Scholar 

  10. P. I. Shtabalyuk, “On almost periodic solutions of one problem with nonlocal conditions,” Visn. Derzh. Univ. “L’vivs’ka Politekhnika,” Differents. Uravn. Zastos., No. 286 (1995), pp. 153–165.

  11. G. Avalishvili, M. Avalishvili, and D. Gordeziani, “On integral nonlocal boundary-value problems for some partial differential equations,” Bull. Georg. Nat. Acad. Sci., 5, No. 1, 31–37 (2011).

    MATH  MathSciNet  Google Scholar 

  12. S. Mesluob and A. Bouziani, “Mixed problem with integral conditions for a certain class of hyperbolic equations,” J. Appl. Math., No. 3, 107–116 (2001).

    Google Scholar 

  13. R. S. Guter, L. D. Kudryavtsev, and B. V. Levitan, Elements of the Theory of Functions [in Russian], Fizmatgiz, Moscow (1963).

    Google Scholar 

  14. A. S. Besicovitch, Almost Periodic Functions, Dover, Cambridge (1954).

    Google Scholar 

  15. M. A. Shubin, “Almost periodic functions and differential operators with partial derivatives,” Usp. Mat. Nauk, 33, No. 2, 3–47 (1978).

    MATH  Google Scholar 

  16. V. I. Gorbachuk and M. L. Gorbachuk, Boundary-Value Problems for Differential-Operator Equations [in Russian], Naukova Dumka, Kiev (1984).

    Google Scholar 

  17. I. M. Gel’fand and G. E. Shilov, Generalized Functions. Some Problems of the Theory of Differential Equations. Issue 3 [in Russian], Fizmatgiz, Moscow (1958).

  18. D. K. Faddeev and I. S. Somins’kyi, A Collection of Problems on Higher Algebra [in Ukrainian], Vyshcha Shkola, Kyiv (1971).

    Google Scholar 

  19. L. Hörmander, The Analysis of Linear Partial Differential Operators. Vol. 2. Differential Operators with Constant Coefficients, Springer, Berlin (1983).

    Google Scholar 

  20. Ya. D. Tamarkin, Some General Problems of the Theory of Ordinary Differential Equations and Expansion of Arbitrary Functions in Series [in Russian], Petrograd (1917).

  21. P. Lancaster, Theory of Matrices, Academic Press, New York (1969).

    MATH  Google Scholar 

  22. V. G. Sprindzhuk, Metric Theory of Diophantine Approximations [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  23. N. N. Bogolyubov, Yu. A. Mitropol’skii, and A. M. Samoilenko, Method of Accelerated Convergence in Nonlinear Mechanics [in Russian], Naukova Dumka, Kiev (1969).

    Google Scholar 

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 2, pp. 252–265, February, 2013.

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Kuz’, A.M., Ptashnyk, B.I. A problem with integral conditions with respect to time for Gårding hyperbolic equations. Ukr Math J 65, 277–293 (2013). https://doi.org/10.1007/s11253-013-0777-7

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