Existence theorems of periodic solutions for fourth-order nonlinear functional difference equations

详细信息    查看全文

• 作者：Xia Liu (1)
  Yuanbiao Zhang (2)
  Haiping Shi (3)
  
• 关键词：Existence ; Periodic solutions ; Fourth ; order ; Functional difference equations ; Discrete variational theory ; 39A11
• 刊名：Journal of Applied Mathematics and Computing
• 出版年：2013
• 出版时间：2 - July 2013
• 年：2013
• 卷：42
• 期：1
• 页码：51-67
• 全文大小：549KB
• 参考文献：1. Agarwal, R.P.: Difference Equations and Inequalities: Theory, Methods and Applications. Dekker, New York (1992)
  2. Ahlbrandt, C.D.: Dominant and recessive solutions of symmetric three term recurrences. J. Differ. Equ. 107(2), 238鈥?58 (1994) 10.1006/jdeq.1994.1011">CrossRef
  3. Avery, R.I., Henderson, J.: Existence of three positive pseudo-symmetric solutions for a one dimensional discrete / p-Laplacian. J. Differ. Equ. Appl. 10(6), 529鈥?39 (2004) 10.1080/10236190410001667959">CrossRef
  4. Benci, V., Rabinowitz, P.H.: Critical point theorems for indefinite functionals. Invent. Math. 52(3), 241鈥?73 (1979) 10.1007/BF01389883">CrossRef
  5. Cai, X.C., Yu, J.S., Guo, Z.M.: Existence of periodic solutions for fourth-order difference equations. Comput. Math. Appl. 50(1鈥?), 49鈥?5 (2005) 10.1016/j.camwa.2005.03.004">CrossRef
  6. Cecchi, M., Marini, M., Villari, G.: On the monotonicity property for a certain class of second order differential equations. J. Differ. Equ. 82(2), 15鈥?7 (1989) 10.1016/0022-0396(89)90165-4">CrossRef
  7. Chang, K.C.: Infinite Dimensional Morse Theory and Multiple Solution Problems. Birkh盲user, Boston (1993) 10.1007/978-1-4612-0385-8">CrossRef
  8. Chen, P., Tang, X.H.: Existence of infinitely many homoclinic orbits for fourth-order difference systems containing both advance and retardation. Appl. Math. Comput. 217(9), 4408鈥?415 (2011) 10.1016/j.amc.2010.09.067">CrossRef
  9. Clarke, F.H.: Periodic solutions to Hamiltonian inclusions. J. Differ. Equ. 40(1), 1鈥? (1981) 10.1016/0022-0396(81)90007-3">CrossRef
  10. Cordaro, G.: Existence and location of periodic solution to convex and non coercive Hamiltonian systems. Discrete Contin. Dyn. Syst. 12(5), 983鈥?96 (2005) 10.3934/dcds.2005.12.983">CrossRef
  11. Erbe, L.H., Xia, H., Yu, J.S.: Global stability of a linear nonautonomous delay difference equations. J. Differ. Equ. Appl. 1(2), 151鈥?61 (1995) 10.1080/10236199508808016">CrossRef
  12. Fang, H., Zhao, D.P.: Existence of nontrivial homoclinic orbits for fourth-order difference equations. Appl. Math. Comput. 214(1), 163鈥?70 (2009) 10.1016/j.amc.2009.03.061">CrossRef
  13. Guo, Z.M., Yu, J.S.: Existence of periodic and subharmonic solutions for second-order superlinear difference equations. Sci. China Math. 46(4), 506鈥?15 (2003)
  14. Guo, Z.M., Yu, J.S.: The existence of periodic and subharmonic solutions of subquadratic second order difference equations. J. Lond. Math. Soc. 68(2), 419鈥?30 (2003) 10.1112/S0024610703004563">CrossRef
  15. Guo, Z.M., Yu, J.S.: Applications of critical point theory to difference equations. Fields Inst. Commun. 42, 187鈥?00 (2004)
  16. He, Z.M.: On the existence of positive solutions of / p-Laplacian difference equations. J. Comput. Appl. Math. 161(1), 193鈥?01 (2003) 10.1016/j.cam.2003.08.004">CrossRef
  17. Jiang, D., Chu, J., O鈥橰egan, D., Agarwal, R.P.: Positive solutions for continuous and discrete boundary value problems to the one-dimension / p-Laplacian. Math. Inequal. Appl. 7(4), 523鈥?34 (2004)
  18. Kaplan, J.L., Yorke, J.A.: On the nonlinear differential delay equation / x鈥? / t)=鈭?em class="a-plus-plus">f( / x( / t), / x( / t鈭?)). J.聽Differ. Equ. 23(2), 293鈥?14 (1977) 10.1016/0022-0396(77)90132-2">CrossRef
  19. Kocic, V.L., Ladas, G.: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Kluwer Academic, Dordrecht (1993) 10.1007/978-94-017-1703-8">CrossRef
  20. Li, J.B., He, X.Z.: Proof and generalization of Kaplan-Yorke鈥檚 conjecture on periodic solution of differential delay equations. Sci. China Math. 42(9), 957鈥?64 (1999) 10.1007/BF02880387">CrossRef
  21. Liu, Y.J., Ge, W.G.: Twin positive solutions of boundary value problems for finite difference equations with / p-Laplacian operator. J. Math. Appl. 278(2), 551鈥?61 (2003)
  22. Matsunaga, H., Hara, T., Sakata, S.: Global attractivity for a nonlinear difference equation with variable delay. Comput. Math. Appl. 41(5鈥?), 543鈥?51 (2001) 10.1016/S0898-1221(00)00297-2">CrossRef
  23. Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Springer, New York (1989) 10.1007/978-1-4757-2061-7">CrossRef
  24. Mickens, R.E.: Difference Equations: Theory and Application. Van Nostrand-Reinhold, New York (1990)
  25. Nussbaum, R.D.: Circulant matrices and differential delay equations. J. Differ. Equ. 60(2), 201鈥?17 (1985) 10.1016/0022-0396(85)90113-5">CrossRef
  26. Pankov, A., Zakhrchenko, N.: On some discrete variational problems. Acta Appl. Math. 65(1鈥?), 295鈥?03 (2001) 10.1023/A:1010655000447">CrossRef
  27. Peterson, A., Ridenhour, J.: The (2,2)-disconjugacy of a fourth order difference equation. J. Differ. Equ. Appl. 1(1), 87鈥?3 (1995) 10.1080/10236199508808009">CrossRef
  28. Popenda, J., Schmeidel, E.: On the solutions of fourth order difference equations. Rocky Mt. J. Math. 25(4), 1485鈥?499 (1995) 10.1216/rmjm/1181072158">CrossRef
  29. Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. Am. Math. Soc., Providence (1986)
  30. Raju, C.K.: Classical time-symmetric electrodynamics. J. Phys. A, Math. Gen. 13(10), 3303鈥?317 (1980) 10.1088/0305-4470/13/10/025">CrossRef
  31. Schulman, L.S.: Some differential-difference equations containing both advance and retardation. J.聽Math. Phys. 15(3), 295鈥?98 (1974) 10.1063/1.1666641">CrossRef
  32. Shi, H.P., Ling, W.P., Long, Y.H., Zhang, H.Q.: Periodic and subharmonic solutions for second order nonlinear functional difference equations. Commun. Math. Anal. 5(2), 50鈥?9 (2008)
  33. Smets, D., Willem, M.: Solitary waves with prescribed speed on infinite lattices. J. Funct. Anal. 149(1), 266鈥?75 (1997) 10.1006/jfan.1996.3121">CrossRef
  34. Thandapani, E., Arockiasamy, I.M.: Fourth-order nonlinear oscillations of difference equations. Comput. Math. Appl. 42(3鈥?), 357鈥?68 (2001) 10.1016/S0898-1221(01)00160-2">CrossRef
  35. Wheeler, J.A., Feynman, R.P.: Classical electrodynamics in terms of direct interparticle action. Rev. Mod. Phys. 21(3), 425鈥?33 (1949) 10.1103/RevModPhys.21.425">CrossRef
  36. Xu, Y.T., 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(1), 189鈥?05 (2001) 10.1006/jmaa.2000.7342">CrossRef
  37. Yan, J., Liu, B.: Oscillatory and asymptotic behavior of fourth order nonlinear difference equations. Acta Math. Sin. 13(1), 105鈥?15 (1997) 10.1007/BF02560530">CrossRef
  38. Yu, J.S., Guo, Z.M.: On boundary value problems for a discrete generalized Emden-Fowler equation. J. Differ. Equ. 231(1), 18鈥?1 (2006) 10.1016/j.jde.2006.08.011">CrossRef
  39. Yu, J.S., Long, Y.H., Guo, Z.M.: Subharmonic solutions with prescribed minimal period of a discrete forced pendulum equation. J. Dyn. Differ. Equ. 16(2), 575鈥?86 (2004) 10.1007/s10884-004-4292-2">CrossRef
  40. Zhou, Z., Zhang, Q.: Uniform stability of nonlinear difference systems. J. Math. Anal. Appl. 225(2), 486鈥?00 (1998) 10.1006/jmaa.1998.6039">CrossRef
  41. Zhou, Z., Yu, J.S., Guo, Z.M.: Periodic solutions of higher-dimensional discrete systems. Proc. R. Soc. Edinb. A 134(5), 1013鈥?022 (2004) 10.1017/S0308210500003607">CrossRef
  42. Zhou, Z., Yu, J.S., Chen, Y.M.: Homoclinic solutions in periodic difference equations with saturable nonlinearity. Sci. China Math. 54(1), 83鈥?3 (2011) 10.1007/s11425-010-4101-9">CrossRef
  
• 作者单位：Xia Liu (1)
  Yuanbiao Zhang (2)
  Haiping Shi (3)
  
  1. Oriental Science and Technology College, Hunan Agricultural University, Changsha, 410128, China
  2. Packaging Engineering Institute, Jinan University, Zhuhai, 519070, China
  3. Basic Courses Department, Guangdong Construction Vocational Technology Institute, Guangzhou, 510450, China
  
• ISSN：1865-2085

文摘

By using of the critical point method, the existence of periodic solutions for fourth-order nonlinear functional difference equations is obtained. The main approaches used in our paper are variational techniques and the Saddle Point Theorem. The problem is to solve the existence of periodic solutions of fourth-order nonlinear functional difference equations. Results obtained generalize and complement the existing one.