Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio


IMPACT FACTOR 2018: 0.551

CiteScore 2018: 0.52

SCImago Journal Rank (SJR) 2018: 0.320
Source Normalized Impact per Paper (SNIP) 2018: 0.711

Mathematical Citation Quotient (MCQ) 2018: 0.27

Online
ISSN
1572-9176
See all formats and pricing
More options …
Ahead of print

Issues

On the solvability of a nonlocal problem for the system of Sobolev-type differential equations with integral condition

Anar T. Assanova
  • Corresponding author
  • Institute of Mathematics and Mathematical Modeling, Pushkin Str. 25, 050010 Almaty, Kazakhstan
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2019-02-19 | DOI: https://doi.org/10.1515/gmj-2019-2011

Abstract

Sufficient conditions for the existence and uniqueness of a classical solution to a nonlocal problem for a system of Sobolev-type differential equations with integral condition are established. By introducing a new unknown function, we reduce the considered problem to an equivalent problem consisting of a nonlocal problem for the system of hyperbolic equations of second order with a functional parameter and an integral relation. We propose the algorithm for finding an approximate solution to the investigated problem and prove its convergence.

Keywords: System of Sobolev-type equations; nonlocal problem; integral condition; unique solvability; algorithm

MSC 2010: 35G35; 35G46; 35L53; 35L57

Dedicated to the 100th anniversary of academician Yu. A. Mitropol’skiĭ

References

  • [1]

    A. I. Aristov, On the Cauchy problem for a Sobolev-type equation with a quadratic nonlinearity (in Russian), Izv. Ross. Akad. Nauk Ser. Mat. 75 (2011), no. 5, 3–18; translation in Izv. Math. 75 (2011), no. 5, 871–887. Google Scholar

  • [2]

    A. I. Aristov, On the Cauchy problem for a nonlinear Sobolev type equation (in Russian), Differ. Uravn. 50 (2014), no. 1, 117–120; translation in Differ. Equ. 50 (2014), no. 1, 117–121. Google Scholar

  • [3]

    A. T. Asanova, On solvability of nonlinear boundary value problems with integral condition for the system of hyperbolic equations, Electron. J. Qual. Theory Differ. Equ. 2015 (2015), Paper No. 63. Google Scholar

  • [4]

    A. T. Asanova and D. S. Dzhumabaev, Well-posedness of nonlocal boundary value problems with integral condition for the system of hyperbolic equations, J. Math. Anal. Appl. 402 (2013), no. 1, 167–178. Web of ScienceCrossrefGoogle Scholar

  • [5]

    A. K. Aziz, Periodic solutions of hyperbolic partial differential equations, Proc. Amer. Math. Soc. 17 (1966), 557–566. CrossrefGoogle Scholar

  • [6]

    H. Brill, A semilinear Sobolev evolution equation in a Banach space, J. Differential Equations 24 (1977), no. 3, 412–425. CrossrefGoogle Scholar

  • [7]

    L. Byszewski, Theorem about existence and uniqueness of continuous solution of nonlocal problem for nonlinear hyperbolic equation, Appl. Anal. 40 (1991), no. 2–3, 173–180. CrossrefGoogle Scholar

  • [8]

    L. Byszewski and V. Lakshmikantham, Monotone iterative technique for nonlocal hyperbolic differential problem, J. Math. Phys. Sci. 26 (1992), no. 4, 345–359. Google Scholar

  • [9]

    L. Cesari, Periodic solutions of partial differential equations, Qualitative Methods in the Theory of Non-linear Vibrations, Izdat. Akad. Nauk Ukrain. SSR, Kiev (1961), 440–457. Google Scholar

  • [10]

    D. Colton, Pseudoparabolic equations in one space variable, J. Differential Equations 12 (1972), 559–565. CrossrefGoogle Scholar

  • [11]

    Y. L. Daletskiĭ and M. G. Kreĭn, Stability of Solutions of Differential Equations in Banach Space (in Russian), Nonlinear Anal. Appl. Ser., Izdat. “Nauka”, Moscow, 1970. Google Scholar

  • [12]

    N. D. Golubeva and L. S. Pul’kina, On a nonlocal problem with integral conditions (in Russian), Mat. Zametki 59 (1996), no. 3, 456–458; translation in Math. Notes 59 (1996), no. 3-4, 326–328. Google Scholar

  • [13]

    P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, 1964. Google Scholar

  • [14]

    R. Hryniv and I. Kmit, On a nonclassical problem for some quasilinear hyperbolic equation, Nonlinear Anal. 51 (2002), no. 8, 1405–1419. CrossrefGoogle Scholar

  • [15]

    S. Kharibegashvili, Some multidimensional problems for hyperbolic partial differential equations and systems, Mem. Differ. Equ. Math. Phys. 37 (2006), 1–136. Google Scholar

  • [16]

    S. S. Kharibegashvili, On the well-posedness of some nonlocal problems for the wave equation (in Russian), Differ. Uravn. 39 (2003), no. 4, 539–553, 575; translation in Differ. Equ. 39 (2003), no. 4, 577–592. Google Scholar

  • [17]

    I. Kiguradze and T. Kiguradze, On solvability of boundary value problems for higher order nonlinear hyperbolic equations, Nonlinear Anal. 69 (2008), no. 7, 1914–1933. CrossrefWeb of ScienceGoogle Scholar

  • [18]

    T. Kiguradze, Some boundary value problems for systems of linear partial differential equations of hyperbolic type, Mem. Differ. Equ. Math. Phys. 1 (1994), 1–144. Google Scholar

  • [19]

    T. Kiguradze, On some boundary value problems for nonlinear degenerate hyperbolic equations of higher order, Mem. Differ. Equ. Math. Phys. 31 (2004), 117–122. Google Scholar

  • [20]

    T. Kiguradze, On solvability and well-posedness of boundary value problems for nonlinear hyperbolic equations of the fourth order, Georgian Math. J. 15 (2008), no. 3, 555–569. Google Scholar

  • [21]

    T. Kiguradze and V. Lakshmikantham, On the Dirichlet problem for fourth-order linear hyperbolic equations, Nonlinear Anal. 49 (2002), no. 2, 197–219. CrossrefGoogle Scholar

  • [22]

    T. Kiguradze and V. Lakshmikantham, On the Dirichlet problem in a characteristic rectangle for higher order linear hyperbolic equations, Nonlinear Anal. 50 (2002), no. 8, 1153–1178. CrossrefGoogle Scholar

  • [23]

    T. Kiguradze and V. Lakshmikantham, On initial-boundary value problems in bounded and unbounded domains for a class of nonlinear hyperbolic equations of the third order, J. Math. Anal. Appl. 324 (2006), no. 2, 1242–1261. CrossrefGoogle Scholar

  • [24]

    T. I. Kiguradze and T. Kusano, On ill-posed initial-boundary value problems for higher-order linear hyperbolic equations with two independent variables (in Russian), Differ. Uravn. 39 (2003), no. 10, 1379–1394, 1438–1394; translation in Differ. Equ. 39 (2003), no. 10, 1454–1470. Google Scholar

  • [25]

    T. I. Kiguradze and T. Kusano, On the well-posedness of initial-boundary value problems for higher-order linear hyperbolic equations with two independent variables (in Russian), Differ. Uravn. 39 (2003), no. 4, 516–526, 575; translation in Differ. Equ. 39 (2003), no. 4, 553–563. Google Scholar

  • [26]

    V. Lakshmikantham and S. G. Pandit, Periodic solutions of hyperbolic partial differential equations. Hyperbolic partial differential equations. II, Comput. Math. Appl. 11 (1985), no. 1–3, 249–259. CrossrefGoogle Scholar

  • [27]

    Y. A. Mitropol’skiĭ and L. B. Urmancheva, The two-point problem for systems of hyperbolic equations (in Russian), Ukrain. Mat. Zh. 42 (1990), no. 12, 1657–1663; translation in Ukrainian Math. J. 42 (1990), no. 12, 1492–1498. Google Scholar

  • [28]

    A. M. Nakhushev, Problems with Shift for a Partial Differential Equations (in Russian), Nauka, Moskow, 2006. Google Scholar

  • [29]

    B. I. Ptashnik, Ill-Posed Boundary Value Problems for Partial Differential Equations (in Russian), “Naukova Dumka”, Kiev, 1984. Google Scholar

  • [30]

    A. M. Samoilenko and B. P. Tkach, Numerical-Analytical Methods in the Theory of Periodical Solutions of Equations with Partial Derivatives (in Russian), “Naukova Dumka”, Kiev, 1992. Google Scholar

  • [31]

    Q. Sheng and R. P. Agarwal, Existence and uniqueness of periodic solutions for higher order hyperbolic partial differential equations, J. Math. Anal. Appl. 181 (1994), no. 2, 392–406. CrossrefGoogle Scholar

  • [32]

    R. E. Showalter, Existence and representation theorems for a semilinear Sobolev equation in Banach space, SIAM J. Math. Anal. 3 (1972), 527–543. CrossrefGoogle Scholar

  • [33]

    S. L. Sobolev, On a new problem of mathematical physics (in Russian), Izv. Akad. Nauk SSSR. Ser. Mat. 18 (1954), 3–50. Google Scholar

  • [34]

    B. Soltanalizadeh, H. Roohani Ghehsareh and S. Abbasbandy, A super accurate shifted Tau method for numerical computation of the Sobolev-type differential equation with nonlocal boundary conditions, Appl. Math. Comput. 236 (2014), 683–692. Web of ScienceGoogle Scholar

  • [35]

    B. P. Tkach and L. B. Urmancheva, A numerical-analytic method for finding solutions of systems with distributed parameters and an integral condition (in Russian), Nelīnīĭnī Koliv. 12 (2009), no. 1, 110–119; translation in Nonlinear Oscil. (N. Y.) 12 (2009), no. 1, 113–122. Google Scholar

  • [36]

    G. Vidossich, Periodic solutions of hyperbolic equations using ordinary differential equations, Nonlinear Anal. 17 (1991), no. 8, 703–710. CrossrefGoogle Scholar

  • [37]

    S. V. Zhestkov, The Goursat problem with integral boundary conditions (in Russian), Ukrain. Mat. Zh. 42 (1990), no. 1, 132–135; translation in Ukrainian Math. J. 42 (1990), no. 1, 119–122. Google Scholar

About the article

Received: 2017-02-01

Revised: 2017-07-25

Accepted: 2017-09-07

Published Online: 2019-02-19


Funding Source: Ministry of Education and Science of the Republic of Kazakhstan

Award identifier / Grant number: 0822/ΓΦ4

Award identifier / Grant number: AP05131220

This research is partially supported by Grants No. 0822/ΓΦ4 and No. AP05131220 of Ministry of Education and Science of the Republic Kazakhstan.


Citation Information: Georgian Mathematical Journal, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: https://doi.org/10.1515/gmj-2019-2011.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in