Lithuanian Mathematical Journal

, Volume 57, Issue 1, pp 109–127 | Cite as

Generalized Green’s functions for mth-order discrete nonlocal problems

  • Gailė Paukštaitė
  • Artūras Štikonas


In this paper, we consider mth-order linear discrete problems with m nonlocal conditions. We investigate a generalized Green’s function, describing the minimum norm least squares solution, and present its properties, which resemble properties of an ordinary Green’s function.


discrete problem nonlocal conditions generalized Green’s function ordinary Green’s function least squares solution Moore–Penrose inverse 


15A09 65Q10 65N21 65N80 


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  1. 1.
    D.R. Anderson, Green’s function for a third-order generalized right focal problem, J. Math. Anal. Appl., 288(1):1–14, 2003, 10.1016/S0022-247X(03)00132-X.
  2. 2.
    D.R. Anderson, T.O. Anderson, and M.M. Kleber, Green’s function ad existence of solutions for a functional focal differential equation, Electron. J. Differ. Equ., 2006(12):1–14, 2006.MATHGoogle Scholar
  3. 3.
    Z. Bai, Existence of solutions for some third-order boundary-value problems, Electron. J. Differ. Equ., 2008(25):1–6, 2008.MathSciNetGoogle Scholar
  4. 4.
    A. Ben-Israel and T.N.E. Greville, Generalized Inverses. Theory and Applications, Springer, New York, 2003.Google Scholar
  5. 5.
    A.A. Boichuk and A.M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, VSP, Utrecht, Boston, 2004, 10.1515/9783110944679.
  6. 6.
    K. Ghanbary, Similarities of discrete and continuous Sturm–Liouville problems, Electron. J. Differ. Equ., 172:1–8, 2007.MathSciNetGoogle Scholar
  7. 7.
    J.R. Graef, L. Kong, and B. Yang, Positive solutions for a nonlinear higher order boundary-value problem, Discrete Contin. Dyn. Syst., 2009(spec. issue):276–285, 2009.Google Scholar
  8. 8.
    L.-J. Guo, J.-P. Sun, and Y.-H. Sun, Multiple positive solutions of nonlinear third-order three-point boundary value problems, Electron. J. Differ. Equ., 2007(112):1–7, 2007.MathSciNetGoogle Scholar
  9. 9.
    J. Henderson and S.K. Ntouyas, Positive solutions for systems of nth order three-point nonlocal boundary value problems, Electron. J. Qual. Theory Differ. Equ., 2007(18):1–12, 2007, 10.14232/ejqtde.2007.1.18.
  10. 10.
    S. Hoffmann, G. Plonka, and J. Weickert, Discrete Green’s functions for harmonic and biharmonic in-painting with sparse atoms, in X.-C. Tai, E. Bae, T.F. Chan, and M. Lysaker (Eds.), Energy Min-imization Methods in Computer Vision and Pattern Recognition. 10th International Conference, EMM-CVPR 2015, Hong Kong, China, January 13–16, 2015. Proceedings, Springer, Cham, 2015, pp. 169–182, 10.1007/978-3-319-14612-6_13.
  11. 11.
    Y. Li and Z.-L. Zhang, Digraph Laplacian and the degree of asymmetry, Internet Math., 8(4):381–401, 2012, 10.1080/15427951.2012.708890.
  12. 12.
    Y. Liu and O’Regan, Multiplicity results using bifurcation techniques for a class of fourth-order m-point boundary value problems, Bound. Value Probl., 2009:970135, 2009, 10.1155/2009/970135.
  13. 13.
    J. Locker, The generalized Green’s function for an nth order linear differential operator, Trans. Am. Math. Soc., 228:243–268, 1977, 10.2307/1998529.
  14. 14.
    C.D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia, PA, 2004.Google Scholar
  15. 15.
    G. Paukštaitė and A. Štikonas, Generalized Green’s functions for second-order discrete boundary-value problems with nonlocal boundary conditions, Liet. Matem. Rink., Proc. Lith. Math. Soc., Ser. A, 53:96–101, 2012.Google Scholar
  16. 16.
    G. Paukštaitė and A. Štikonas, Generalized Green’s functions for the second-order discrete problems with nonlocal conditions, Lith. Math. J., 54(2):203–219, 2014, 10.1007/s10986-014-9238-8.
  17. 17.
    G. Paukštaitė and A. Štikonas, Ordinary and generalized Green’s functions for the second order discrete nonlocal problems, Bound. Value Probl., 2015:207, 2015, 10.1186/s13661-015-0474-6.
  18. 18.
    R. Penrose, A generalized inverse for matrices, Proc. Camb. Philos. Soc., 51:406–413, 1955, 10.1017/S0305004100030401.
  19. 19.
    G. Plonka, S. Hoffmann, and J. Weickert, Pseudo-inverses of difference matrices and their application to sparse signal approximation, 2015, arXiv:1504.04266v1.Google Scholar
  20. 20.
    S. Roman, Green’s functions for boundary-value problems with nonlocal boundary conditions, PhD thesis, Vilnius University, 2011,
  21. 21.
    S. Roman, Linear differential equation with additional conditions and formulae for Green’s function, Math. Model. Anal., 16(3):401–417, 2011, 10.3846/13926292.2011.602125.
  22. 22.
    S. Roman and A. Štikonas, Green’s function for discrete problems with nonlocal boundary conditions, Liet. Matem. Rink., LMD Darbai, 52:291–296, 2011.Google Scholar
  23. 23.
    I. Stakgold and M. Holst, Green’s Functions and Boundary Value Problems, 3rd ed., John Wiley & Sons, Hoboken, NJ, 2011, 10.1002/9780470906538.
  24. 24.
    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(3):301–334, 2014,
  25. 25.
    A. Štikonas and S. Roman, Green’s function for discrete mth-order problems, Lith. Math. J., 52(3):334–351, 2012, 10.1007/s10986-012-9177-1.

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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Faculty of Mathematics and InformaticsVilnius UniversityVilniusLithuania
  2. 2.Institute of Mathematics and InformaticsVilnius UniversityVilniusLithuania

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