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

See all formats and pricing
More options …
Ahead of print


Construction of Green’s functional for a third order ordinary differential equation with general nonlocal conditions and variable principal coefficient

Kemal Özen
  • Corresponding author
  • Department of Mathematics, Namık Kemal University, Değirmenaltı, 59030, Tekirdağ, Turkey
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2019-02-15 | DOI: https://doi.org/10.1515/gmj-2019-2003


In this work, the solvability of a generally nonlocal problem is investigated for a third order linear ordinary differential equation with variable principal coefficient. A novel adjoint problem and Green’s functional are constructed for a completely nonhomogeneous problem. Several illustrative applications for the theoretical results are provided.

Keywords: Green’s function; nonlocal condition; adjoint problem; variable principal coefficient

MSC 2010: 34B05; 34B10; 34B27; 65N80


  • [1]

    S. S. Akhiev, Representations of solutions of some linear operator equations (in Russian), Dokl. Akad. Nauk SSSR 251 (1980), no. 5, 1037–1040; translation in Sov. Math. Dokl. 21 (1980), no. 2, 555–558. Google Scholar

  • [2]

    S. S. Akhiev, Fundamental solutions of functional-differential equations and their representations (in Russian), Dokl. Akad. Nauk SSSR 275 (1984), no. 2, 273–276; translation in Sov. Math. Dokl. 29 (1984), 180–184. Google Scholar

  • [3]

    S. S. Akhiev, Solvability conditions and Green functional concept for local and nonlocal linear problems for a second order ordinary differential equation, Math. Comput. Appl. 9 (2004), no. 3, 349–358. Google Scholar

  • [4]

    S. S. Akhiev, Green and generalized Green’s functionals of linear local and nonlocal problems for ordinary integro-differential equations, Acta Appl. Math. 95 (2007), no. 2, 73–93. CrossrefGoogle Scholar

  • [5]

    S. S. Akhiev and K. Oruçoğlu, Fundamental solutions of some linear operator equations and applications, Acta Appl. Math. 71 (2002), no. 1, 1–30. CrossrefGoogle Scholar

  • [6]

    D. R. Anderson, Green’s function for a third-order generalized right focal problem, J. Math. Anal. Appl. 288 (2003), no. 1, 1–14. CrossrefGoogle Scholar

  • [7]

    D. R. Anderson and J. M. Davis, Multiple solutions and eigenvalues for third-order right focal boundary value problems, J. Math. Anal. Appl. 267 (2002), no. 1, 135–157. CrossrefGoogle Scholar

  • [8]

    N. Azbelev, V. Maksimov and L. Rakhmatullina, Introduction to the Theory of Linear Functional-differential Equations, Adv. Ser. Math. Sci. Eng. 3, World Federation, Atlanta, 1995. Google Scholar

  • [9]

    A. L. Brown and A. Page, Elements of Functional Analysis, Van Nostrand Reinhold, London, 1970. Google Scholar

  • [10]

    Y. Chen, J. Ren and S. Siegmund, Green’s function for third-order differential equations, Rocky Mountain J. Math. 41 (2011), no. 5, 1417–1448. CrossrefGoogle Scholar

  • [11]

    Z. Cheng and J. Ren, Existence of positive periodic solution for variable-coefficient third-order differential equation with singularity, Math. Methods Appl. Sci. 37 (2014), no. 15, 2281–2289. CrossrefGoogle Scholar

  • [12]

    Z. Cheng and J. Ren, Positive solutions for third-order variable-coefficient nonlinear equation with weak and strong singularities, J. Difference Equ. Appl. 21 (2015), no. 11, 1003–1020. CrossrefGoogle Scholar

  • [13]

    J. R. Graef, L. Kong and B. Yang, Positive solutions for third order multi-point singular boundary value problems, Czechoslovak Math. J. 60(135) (2010), no. 1, 173–182. Google Scholar

  • [14]

    J. R. Graef and B. Yang, Positive solutions of a third order nonlocal boundary value problem, Discrete Contin. Dyn. Syst. Ser. S 1 (2008), no. 1, 89–97. Google Scholar

  • [15]

    M. Greguš, Third Order Linear Differential Equations, Math. Appl. (East European Series) 22, D. Reidel, Dordrecht, 1987. Google Scholar

  • [16]

    L. V. Kantorovich and G. P. Akilov, Functional Analysis, 2nd ed., Pergamon Press, Oxford, 1982. Google Scholar

  • [17]

    I. T. Kiguradze and T. I. Kiguradze, Conditions for the well-posedness of nonlocal problems for second-order linear differential equations (in Russian), Differ. Uravn. 47 (2011), no. 10, 1400–1411; translation in Differ. Equ. 47 (2011), no. 10, 1414–1425. Google Scholar

  • [18]

    S. G. Kreĭn, Linear Equations in a Banach Space (in Russian), Izdat. “Nauka”, Moscow, 1971. Google Scholar

  • [19]

    S. G. Kreĭn, Linear Equations in Banach Spaces, Birkhäuser, Boston, 1982. Google Scholar

  • [20]

    H. Li, Y. Feng and C. Bu, Non-conjugate boundary value problem of a third order differential equation, Electron. J. Qual. Theory Differ. Equ. 2015 (2015), Paper No. 21. Google Scholar

  • [21]

    S. K. Ntouyas, Nonlocal initial and boundary value problems: a survey, Handbook of Differential Equations: Ordinary Differential Equations. Vol. II, Elsevier, Amsterdam (2005), 461–557. Google Scholar

  • [22]

    K. Özen and K. Oruçoğlu, A representative solution to m-order linear ordinary differential equation with nonlocal conditions by Green’s functional concept, Math. Model. Anal. 17 (2012), no. 4, 571–588. CrossrefGoogle Scholar

  • [23]

    K. Özen and K. Oruçoğlu, A novel approach to construct the adjoint problem for a first-order functional integro-differential equation with general nonlocal condition, Lith. Math. J. 54 (2014), no. 4, 482–502. CrossrefGoogle Scholar

  • [24]

    S. Roman and A. Štikonas, Third-order linear differential equation with three additional conditions and formula for Green’s function, Lith. Math. J. 50 (2010), no. 4, 426–446. CrossrefGoogle Scholar

  • [25]

    Š. Schwabik, M. Tvrdý and O. Vejvoda, Differential and Integral Equations: Boundary Value Problems and Adjoints, D. Reidel, Dordrecht, 1979. Google Scholar

  • [26]

    I. Stakgold, Green’s Functions and Boundary Value Problems, 2nd ed., Pure Appl. Math. (New York), John Wiley & Sons, New York, 1998. Google Scholar

  • [27]

    A. Štikonas, A survey on stationary problems, Green’s functions and spectrum of Sturm–Liouville problem with nonlocal boundary conditions, Nonlinear Anal. Model. Control 19 (2014), no. 3, 301–334. CrossrefGoogle Scholar

  • [28]

    A. Wang, J. Ridenhour and A. Zettl, Construction of regular and singular Green’s functions, Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), no. 1, 171–198. CrossrefGoogle Scholar

  • [29]

    A. Wang and A. Zettl, Green’s function for two-interval Sturm–Liouville problems, Electron. J. Differential Equations 2013 (2013), Paper No. 76. Google Scholar

  • [30]

    A. Zettl, Sturm-Liouville Theory, Math. Surveys Monogr. 121, American Mathematical Society, Providence, 2005. Google Scholar

About the article

Received: 2016-02-19

Revised: 2018-03-04

Accepted: 2018-03-19

Published Online: 2019-02-15

Citation Information: Georgian Mathematical Journal, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: https://doi.org/10.1515/gmj-2019-2003.

Export Citation

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

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Comments (0)

Please log in or register to comment.
Log in