Skip to main content
SpringerLink
Log in

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

  • Published:
Lithuanian Mathematical Journal Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D.R. Anderson, Green’s function for a third-order generalized right focal problem, J. Math. Anal. Appl., 288(1):1–14, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  3. U.M. Ascher, R.D. Russell, and R.M. Mattheij, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, SIAM, 1995.

  4. R.I. Avery, A generalization of the Leggett–Williams fixed point theorem, Math. Sci. Res. Hot-Line, 3:9–14, 1999.

    MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. Z. Bai and W. Ge, Existence of three positive solutions for some second-order boundary value problems, Comput. Math. Appl., 48:699–707, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  9. R. Čiegis, A. Štikonas, O. Štikonienė, and O. Suboč, Stationary problems with nonlocal boundary conditions, Math. Model. Anal., 6(2):178–191, 2001.

    MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  11. E.A. Coddington and N. Levison, Theory of Ordinary Differential Equations, McGraw–Hill Book Co., Inc., New York, Toronto, London, 1955.

    MATH  Google Scholar 

  12. D. Courant and G.F. Hilbert, Methods of Mathematical Physics, Wiley–Interscience Publications, New York, 1953.

    Google Scholar 

  13. D.G. Duffy, Green’s Functions with Applications, Chapman & Hall/CRC Press, 2001.

  14. H.M. Fried, Green’s Functions and Ordered Exponentials, Cambridge Univ. Press, 2002.

  15. W. Ge, Boundary Value Problems for Ordinary Differential Equations, Science Press, Beijing, 2007.

    Google Scholar 

  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.

    MathSciNet  Google Scholar 

  17. G. Infante and J.R.L. Webb, Positive solutions of some nonlocal boundary value problems, Abstr. Appl. Anal., 18:1047–1060, 2003.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  19. A.I. Kostrikin, Introduction to Algebra, Springer-Verlag, Berlin, 1982.

    MATH  Google Scholar 

  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. 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. 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.

    MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  24. A.D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002.

    MATH  Google Scholar 

  25. A.D. Pontryagin, Ordinary Differential Equations, Addison–Wesley, Reading, MA, 1962.

    MATH  Google Scholar 

  26. G. Rickayzen, Green’s Functions and Condensed Matter, Academic Press, 1980.

  27. G.F. Roach, Green’s Functions, Cambridge Univ. Press, Cambridge, Great Britain, 1982.

    MATH  Google Scholar 

  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.

  29. S. Roman and A. Štikonas, Green’s functions for stationary problems with nonlocal boundary conditions, Lith. Math. J., 49(2):190–202, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  32. I. Stakgold, Green’s Functions and Boundary Value Problems, Wiley–Interscience Publications, New York, 1979.

    MATH  Google Scholar 

  33. A. Štikonas, The Sturm–Liouville problem with a nonlocal boundary condition, Lith. Math. J., 47(3):336–351, 2007.

    Article  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    MATH  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

  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. Z. Zhao, Positive solutions for singular three-point boundary-value problems, Electron. J. Differ. Equ., 2007:1–8, 2007.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Informatics, Akademijos 4, LT-08663, Vilnius, Lithuania

    S. Roman & A. Štikonas

Authors
  1. S. Roman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Štikonas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Roman.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Roman, S., Štikonas, A. Third-order linear differential equation with three additional conditions and formula for Green’s function. Lith Math J 50, 426–446 (2010). https://doi.org/10.1007/s10986-010-9097-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10986-010-9097-x

Keywords

MSC

Search

Navigation