Acta Applicandae Mathematicae

, Volume 110, Issue 1, pp 131–152 | Cite as

Existence and Multiplicity of Solutions of Second-Order Difference Boundary Value Problems



This paper concerns the existence and multiplicity of solutions of second-order difference boundary value problems. Under the assumptions which guarantee the existence of at least one nontrivial solution of the homogeneous problem, we obtain the existence of exactly three solutions of the nonhomogeneous problem with some other suitable conditions.


Second-order difference boundary value problems Existence and multiplicity Three-critical-point theorem Isolated zero point 

Mathematics Subject Classification (2000)



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agarwal, R.P.: Difference Equations and Inequalities. Marcel Dekker, New York (1998) Google Scholar
  2. 2.
    Agarwal, R.P., O’Regan, D.: Boundary value problems for discrete equations. Appl. Math. Lett. 10, 83–89 (1997) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Agarwal, R.P., O’Regan, D.: A fixed-point approach for nonlinear discrete boundary value problems. Comput. Math. Appl. 36, 115–121 (1998) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Agarwal, R.P., O’Regan, D.: Nonpositive discrete boundary value problems. Nonlinear Anal. 39, 207–215 (2000) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Agarwal, R.P., Perera, K., O’Regan, D.: Multiple positive solutions of singular and nonsingular discrete problems via variational methods. Nonlinear Anal. 58, 69–73 (2004) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Atici, F.M., Cabada, A.: Existence and uniqueness results for discrete second-order periodic boundary value problems. Comput. Math. Appl. 45, 417–427 (2003) CrossRefMathSciNetGoogle Scholar
  7. 7.
    Atici, F.M., Guseinov, Sh.G.: Positive periodic solutions for nonlinear difference equations with periodic coefficients. J. Math. Anal. Appl. 232, 166–182 (1999) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Aykut, N.: Existence of positive solutions for boundary value problems of second-order functional difference equations. Comput. Math. Appl. 48, 517–527 (2004) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bin, H.H., Yu, J.S., Guo, Z.M.: Nontrivial periodic solutions for asymptotically linear resonant difference problem. J. Math. Anal. Appl. 322, 477–488 (2006) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Cabada, A., Otero-Espinar, V., Vivero, D.R.: Optimal conditions to ensure the stability of periodic solutions of first order difference equations lying between lower and upper solutions. Comput. Math. Appl. 176, 45–57 (2005) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Cai, X.C., Yu, J.S.: Existence theorems for second-order discrete boundary value problems. J. Math. Anal. Appl. 320, 649–661 (2006) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Castro, A., Lazer, A.C.: Critical point theory and the number of solutions of a nonlinear Dirichlet problem. Ann. Mat. Pura Appl. 120, 113–137 (1979) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Chang, K.C.: Infinite-Dimensional Morse Theory and Multiple Solution Problems. Birkhäuser, Boston (1993) MATHGoogle Scholar
  14. 14.
    Guo, D.J.: Nonlinear Functional Analysis. Shandong Publishing House of Science and Technology, Jinan (2001) (in Chinese) Google Scholar
  15. 15.
    Guo, Z.M., Yu, J.S.: Existence of periodic and subharmonic solutions for second order superlinear difference equations. Sci. Sinica Ser. A 46, 506–515 (2003) MathSciNetGoogle Scholar
  16. 16.
    Guo, Z.M., Yu, J.S.: The existence of periodic and subharmonic solutions to subquadratic second order difference equations. J. Lond. Math. Soc. 68, 419–430 (2003) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Guo, Z.M., Yu, J.S.: Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems. Nonlinear Anal. 55, 969–983 (2003) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Guo, Z.M., Yu, J.S.: Multiplicity results for periodic solutions to second-order difference equations. J. Dyn. Differ. Equ. 18, 943–960 (2006) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Hale, J.K.: Ordinary Differential Equations. Wiley Interscience, New York (1969) MATHGoogle Scholar
  20. 20.
    Henderson, J., Thompson, H.B.: Existence of multiple solutions for second-order discrete boundary value problems. Comput. Math. Appl. 43, 1239–1248 (2002) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Jiang, L.Q., Zhou, Z.: Existence of nontrivial solutions for discrete nonlinear two point boundary value problems. Appl. Math. Comput. 180, 318–329 (2006) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Liang, H.H., Weng, P.X.: Existence and multiple solutions for a second-order difference boundary value problem via critical point theory. J. Math. Anal. Appl. 326, 511–520 (2007) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Springer, New York (1989) MATHGoogle Scholar
  24. 24.
    Roger, A.H., Charles, R.J.: Matrix Analysis. Cambridge University Press, Cambridge (1985) MATHGoogle Scholar
  25. 25.
    Sharkovsky, A.N., Maistrenko, Y.L., Romanenko, E.Y.: Difference Equations and Their Applications. Kluwer Academic, Dordrecht (1993) Google Scholar
  26. 26.
    Thompson, H.B., Tisdell, C.: Systems of difference equations associated with boundary value problems for second order systems of ordinary differential equations. J. Math. Anal. Appl. 248, 333–347 (2000) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Vainberg, M.M.: Variational Methods for the Study of Nonlinear Operators. Holden-Day, San Francisco (1964) MATHGoogle Scholar
  28. 28.
    Walter, W.: Ordinary Differential Equations. Springer, New York (1998) MATHGoogle Scholar
  29. 29.
    Yu, J.S., Bin, H.H., Guo, Z.M.: Multiple periodic solutions for discrete Hamiltonian systems. Nonlinear Anal. 66, 1498–1512 (2007) MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Yu, J.S., Zheng, B.: Multiplicity of periodic solutions for second-order discrete Hamiltonian systems with a small forcing term. Nonlinear Anal. 69, 3016–3029 (2008) MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Zhou, Z., Yu, J.S., Guo, Z.M.: Periodic solutions of high-dimensional discrete systems. Proc. R. Soc. Edinb. Sect. A Math. 134, 1013–1022 (2004) MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Zhou, Z., Yu, J.S., Guo, Z.M.: The existence of periodic and subharmonic solutions to subquadratic discrete Hamiltonian systems. Aust. New Zealand Ind. Appl. Math. J. 47, 89–102 (2005) MATHMathSciNetGoogle Scholar
  33. 33.
    Zhuang, W., Cheng, S.S.: Monotone methods for a discrete boundary value problem. Comput. Math. Appl. 32, 41–49 (1996) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.College of Mathematics and Information SciencesGuangzhou UniversityGuangzhouPeople’s Republic of China

Personalised recommendations