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Periodic and subharmonic solutions for second order \(\boldsymbol{p}\)-Laplacian difference equations

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Abstract

In this paper, some sufficient conditions for the existence and multiplicity of periodic and subharmonic solutions to second order p-Laplacian difference equations are obtained by using the critical point theory. The proof is based on the Linking theorem in combination with variational technique.

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References

  1. Agarwal R P, Difference Equations and Inequalities: Theory, Methods and Applications (New York: Marcel Dekker) (1992)

    MATH  Google Scholar 

  2. Agarwal R P, Perera K and O’regan D, Multiple positive solutions of singular and nonsingular discrete problems via variational methods, Nonlinear Anal. 58 (2004) 69–73

    Article  MathSciNet  MATH  Google Scholar 

  3. Agarwal R P, Perera K and O’regan D, Multiple positive solutions of singular discrete p-Laplacian problems via variational methods, Adv. Differ. Equ. 2005 (2005) 93–99

    MathSciNet  MATH  Google Scholar 

  4. Anderson D R, Avery R I and Henderson J, Existence of solutions for a one dimensional p-Laplacian on time-scales, J. Differ. Equ. Appl. 10 (2006) 889–896

    Article  MathSciNet  Google Scholar 

  5. Avery R I and Henderson J, Existence of three positive pseudo-symmetric solutions for a one dimensional discrete p-Laplacian, J. Differ. Equ. Appl. 10 (2004) 529–539

    Article  MathSciNet  MATH  Google Scholar 

  6. Avery R I and Pererson A C, Three positive fixed points of nonlinear operators on ordered Banach space, Comput. Math. Appl. 42 (2001) 313–322

    Article  MathSciNet  MATH  Google Scholar 

  7. Chang K C, Infinite Dimensional Morse Theory and Multiple Solution Problems (Boston: Birkhäuser) (1993)

    MATH  Google Scholar 

  8. Chen P and Fang H, Existence of periodic and subharmonic solutions for second-order p-Laplacian difference equations, Adv. Differ. Equ. 2007 (2007) 1–9

    Article  MathSciNet  Google Scholar 

  9. Erbe L H, Xia H and Yu J S, Global stability of a linear nonautonomous delay difference equations, J. Differ. Equ. Appl. 1 (1995) 151–161

    Article  MathSciNet  MATH  Google Scholar 

  10. Guo D J, Nonlinear Functional Analysis (Jinan: Shandong Scientific Press) (1985)

    Google Scholar 

  11. Guo Z M and Yu J S, Applications of critical point theory to difference equations, Fields Inst. Commun. 42 (2004) 187–200

    MathSciNet  Google Scholar 

  12. Guo Z M and Yu J S, The existence of periodic and subharmonic solutions for second-order superlinear difference equations, Sci. China A46 (2003) 506–515

    MathSciNet  Google Scholar 

  13. Guo Z M and Yu J S, The existence of periodic and subharmonic solutions of subquadratic second order difference equations, J. London Math. Soc. 68 (2003) 419– 430

    Article  MathSciNet  MATH  Google Scholar 

  14. He Z, On the existence of positive solutions of p-Laplacian difference equations, J. Comput. Appl. Math. 161 (2003) 193–201

    Article  MathSciNet  MATH  Google Scholar 

  15. Jiang D, Chu J, O’regan D and Agarwal R P, Positive solutions for continuous and discrete boundary value problems to the one-dimension p-Laplacian, Math. Inequal. Appl. 7 (2004) 523–534

    MathSciNet  MATH  Google Scholar 

  16. Kocic V L and Ladas G, Global Behavior of Nonlinear Difference Equations of High Order with Applications (Boston: Kluwer Academic Publishers) (1993)

    Google Scholar 

  17. Li Y and Lu L, Existence of positive solutions of p-Laplacian difference equations, Appl. Math. Lett. 19 (2006) 1019–1023

    Article  MathSciNet  MATH  Google Scholar 

  18. Liu Y and Ge W G, Twin positive solutions of boundary value problems for finite difference equations with p-Laplacian operator, J. Math. Appl. 278 (2003) 551–561

    MathSciNet  MATH  Google Scholar 

  19. Matsunaga H, Hara T and Sakata S, Global attractivity for a nonlinear difference equation with variable delay, Comput. Math. Appl. 41 (2001) 543–551

    Article  MathSciNet  MATH  Google Scholar 

  20. Mawhin J and Willem M, Critical Point Theory and Hamiltonian Systems (New York: Springer) (1989)

    MATH  Google Scholar 

  21. Moser J, Stable and Radom Motions in Dynamical Systems (Princeton: Princeton University Press) (1973)

    Google Scholar 

  22. Pankov A and Zakhrchenko N, On some discrete variational problems, Acta Appl. Math. 65 (2001) 295–303

    Article  MathSciNet  MATH  Google Scholar 

  23. Rabinowitz P H, Minimax Methods in Critical Point Theory with Applications to Differential Equations (Providence, RI, New York: Amer. Math. Soc.) (1986)

    Google Scholar 

  24. Shi H P, Ling W P, Long Y H and Zhang H Q, Periodic and subharmonic solutions for second order nonlinear functional difference equations, Commun. Math. Anal. 5(2) (2008) 50–59

    MathSciNet  MATH  Google Scholar 

  25. Smets D and Willem M, Solitary waves with prescribed speed on infinite lattices, J. Funct. Anal. 149 (1997) 266–275

    Article  MathSciNet  MATH  Google Scholar 

  26. Tian Y and Ge W G, The existence of solutions for a second-order discrete Neumann problem with a p-Laplacian, J. Appl. Math. Comput. 26 (2008) 333–340

    Article  MathSciNet  MATH  Google Scholar 

  27. Tian Y and Ge W G, Existence results for discrete Sturm-Liuville problem via variational methods, J. Differ. Equ. Appl. 13 (2007) 467–478

    Article  MathSciNet  MATH  Google Scholar 

  28. Xu Y T and Guo Z M, Applications of a Z p index theory to periodic solutions for a class of functional differential equations, J. Math. Anal. Appl. 257 (2001) 189–205

    Article  MathSciNet  MATH  Google Scholar 

  29. Yu J S, Long Y H and Guo Z M, Subharmonic solutions with prescribed minimal period of a discrete forced pendulum equation, J. Dyn. Differ. Equ. 16 (2004) 575–586

    Article  MathSciNet  MATH  Google Scholar 

  30. Yu J S and Guo Z M, On boundary value problems for a discrete generalized Emden-Fowler equation, J. Differ. Equ. 231 (2006) 18–31

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhou Z, Yu J S and Guo Z M, Periodic solutions of higher-dimensional discrete systems, Proc. R. Soc. Edinb. A134 (2004) 1013–1022

    Article  MathSciNet  Google Scholar 

  32. Zhou Z and Zhang Q, Uniform stability of nonlinear difference systems, J. Math. Anal. Appl. 225 (1998) 486–500

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Oriental Science and Technology College, Hunan Agricultural University, Changsha, 410128, People’s Republic of China

    XIA LIU

  2. Packaging Engineering Institute, Jinan University, Zhuhai, 519070, People’s Republic of China

    YUANBIAO ZHANG

  3. College of Mathematics and Information Sciences, Guangzhou University, Guangzhou, 510006, People’s Republic of China

    BO ZHENG

  4. Basic Courses Department, Guangdong Construction Vocational Technology Institute, Guangzhou, 510450, China

    HAIPING SHI

Authors
  1. XIA LIU
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  2. YUANBIAO ZHANG
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  3. BO ZHENG
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  4. HAIPING SHI
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Correspondence to HAIPING SHI.

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LIU, X., ZHANG, Y., ZHENG, B. et al. Periodic and subharmonic solutions for second order \(\boldsymbol{p}\)-Laplacian difference equations. Proc Math Sci 121, 457–468 (2011). https://doi.org/10.1007/s12044-011-0046-3

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  • DOI: https://doi.org/10.1007/s12044-011-0046-3

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