Advertisement

Lithuanian Mathematical Journal

, Volume 50, Issue 4, pp 426–446 | Cite as

Third-order linear differential equation with three additional conditions and formula for Green’s function

  • S. Roman
  • A. Štikonas
Article

Abstract

In this paper, we investigate a third-order linear differential equation with three additional conditions. We find a solution to this problem and give a formula and an existence condition for Green’s function. We compare two Green’s functions for two such problems with different additional conditions: nonlocal and classical boundary conditions. Formula applications are shown by examples.

Keywords

ordinary differential equations linear differential equations Green’s function nonlocal boundary conditions 

MSC

34B05 34B27 34B10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    D.R. Anderson, T.O. Anderson, and M.M. Kleber, Green’s function and existence of solutions for a functional focal differential equation, Electron. J. Differ. Equ., 2006(12):1–14, 2006.MathSciNetGoogle Scholar
  3. 3.
    U.M. Ascher, R.D. Russell, and R.M. Mattheij, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, SIAM, 1995.Google Scholar
  4. 4.
    R.I. Avery, A generalization of the Leggett–Williams fixed point theorem, Math. Sci. Res. Hot-Line, 3:9–14, 1999.MATHMathSciNetGoogle Scholar
  5. 5.
    R.I. Avery and A.C. Peterson, Three positive fixed points of nonlinear operators on ordered Banach spaces, Comput. Math. Appl., 42:313–322, 2001.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Z. Bai, Existence of solutions for some third-order boundary-value problems, Electron. J. Differ. Equ., 2008(25):1–6, 2008.Google Scholar
  7. 7.
    Z. Bai and W. Ge, Existence of three positive solutions for some second-order boundary value problems, Comput. Math. Appl., 48:699–707, 2004.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    A. Cabada, F. Minhós, and A.I. Santos, Solvability for a third order discontinuous fully equation with nonlinear functional boundary conditions, J. Math. Anal. Appl., 322:735–748, 2006.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    R. Čiegis, A. Štikonas, O. Štikonienė, and O. Suboč, Stationary problems with nonlocal boundary conditions, Math. Model. Anal., 6(2):178–191, 2001.MATHMathSciNetGoogle Scholar
  10. 10.
    R. Čiegis, A. Štikonas, O. Štikonienė, and O. Suboč, A monotonic finite-difference scheme for a parabolic problem with nonlocal conditions, Differ. Equ., 38(7):1027–1037, 2002.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    E.A. Coddington and N. Levison, Theory of Ordinary Differential Equations, McGraw–Hill Book Co., Inc., New York, Toronto, London, 1955.MATHGoogle Scholar
  12. 12.
    D. Courant and G.F. Hilbert, Methods of Mathematical Physics, Wiley–Interscience Publications, New York, 1953.Google Scholar
  13. 13.
    D.G. Duffy, Green’s Functions with Applications, Chapman & Hall/CRC Press, 2001.Google Scholar
  14. 14.
    H.M. Fried, Green’s Functions and Ordered Exponentials, Cambridge Univ. Press, 2002.Google Scholar
  15. 15.
    W. Ge, Boundary Value Problems for Ordinary Differential Equations, Science Press, Beijing, 2007.Google Scholar
  16. 16.
    L.-J. Guo, J.-P. Sun, and Y.-H. Zhao, Multiple positive solutions for nonlinear third-order three-point boundary-value problems, Electron. J. Differ. Equ., 2007(112):1–7, 2007.MathSciNetGoogle Scholar
  17. 17.
    G. Infante and J.R.L. Webb, Positive solutions of some nonlocal boundary value problems, Abstr. Appl. Anal., 18:1047–1060, 2003.CrossRefMathSciNetGoogle Scholar
  18. 18.
    G. Infante and J.R.L. Webb, Three-point boundary value problems with solutions that change sign, J. Integral Equations Appl., 15(1):37–57, 2003.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    A.I. Kostrikin, Introduction to Algebra, Springer-Verlag, Berlin, 1982.MATHGoogle Scholar
  20. 20.
    K.Q. Lan, Positive characteristic values and optimal constants for three-point boundary value problems, in Differential & Difference Equations and Applications, Hindawi Publ. Corp., New York, 2006, pp. 623–633.Google Scholar
  21. 21.
    S.K. Ntouyas, Nonlocal initial and boundary value problems: A survey, in A. Cañada, P. Drábek, and A. Fonda (Eds.), Handbook of Differential Equations, Ordinary Differential Equations, Vol. 2, Elsevier, North-Holland, 2005, pp. 461–558.Google Scholar
  22. 22.
    A.P. Palamides and A.N. Veloni, A singular third-order 3-point boundary-value problem with nonpositive Green’s function, Electron. J. Differ. Equ., 2007(151):1–13, 2007.MathSciNetGoogle Scholar
  23. 23.
    S. Pečiulytė and A. Štikonas, On positive eigenfunctions of Sturm–Liouville problem with nonlocal two-point boundary condition, Math. Model. Anal., 12(2):215–226, 2007.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    A.D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002.MATHGoogle Scholar
  25. 25.
    A.D. Pontryagin, Ordinary Differential Equations, Addison–Wesley, Reading, MA, 1962.MATHGoogle Scholar
  26. 26.
    G. Rickayzen, Green’s Functions and Condensed Matter, Academic Press, 1980.Google Scholar
  27. 27.
    G.F. Roach, Green’s Functions, Cambridge Univ. Press, Cambridge, Great Britain, 1982.MATHGoogle Scholar
  28. 28.
    S. Roman and A. Štikonas, Green’s functions for stationary problems with four-point nonlocal boundary conditions, in V. Kleiza, S. Rutkauskas, and A. Štikonas (Eds.), Differential Equations and Their Applications (DETA’2009), Kaunas University of Technology, 2009, pp. 123–130.Google Scholar
  29. 29.
    S. Roman and A. Štikonas, Green’s functions for stationary problems with nonlocal boundary conditions, Lith. Math. J., 49(2):190–202, 2009.MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    M. Sapagovas, G. Kairytė, O. Štikonienė, and A. Štikonas, Alternating direction method for a two-dimensional parabolic equation with a nonlocal boundary condition, Math. Model. Anal., 12(1):131–142, 2007.MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    M.P. Sapagovas and A.D. Štikonas, On the structure of the spectrum of a differential operator with a nonlocal condition, Differ. Equ., 41(7):1010–1018, 2005.MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    I. Stakgold, Green’s Functions and Boundary Value Problems, Wiley–Interscience Publications, New York, 1979.MATHGoogle Scholar
  33. 33.
    A. Štikonas, The Sturm–Liouville problem with a nonlocal boundary condition, Lith. Math. J., 47(3):336–351, 2007.CrossRefGoogle Scholar
  34. 34.
    A. Štikonas and S. Roman, Stationary problems with two additional conditions and formulae for Green’s functions, Numer. Funct. Anal. Optim., 30(9):1125–1144, 2009.MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    J.-P. Sun and H.-E. Zhang, Existence of solutions to third-order m-point boundary-value problems, Electron. J. Differ. Equ., 2008(125):1–9, 2008.MATHGoogle Scholar
  36. 36.
    L.X. Truong, L.T.P. Ngoc, and N.T. Long, Positive solutions for an m-point boundary-value problem, Electron. J. Differ. Equ., 2008(111):1–11, 2008.MathSciNetGoogle Scholar
  37. 37.
    J.R.L. Webb, Remarks on positive solutions of some three point boundary value problems, in Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, May 24–27, 2002, Wilmington, NC, USA, 1998, pp. 905–915.Google Scholar
  38. 38.
    B. Yang, Positive solutions of a third-order three-point boundary-value problem, Electron. J. Differ. Equ., 2008(99):1–10, 2008.Google Scholar
  39. 39.
    Z. Zhao, Positive solutions for singular three-point boundary-value problems, Electron. J. Differ. Equ., 2007:1–8, 2007.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsVilniusLithuania

Personalised recommendations