Existence of periodic solutions for second-order nonlinear difference equations
- Department of Information Engineering, Jieyang Vocational and Technical College, Jieyang 522000, China.
- Quality Control Office, Zhongshan Torch College, Zhongshan 528436, China.
- Modern Business and Management Department, Guangdong Construction Vocational Technology Institute, Guangzhou 510440, China.
By using the critical point method, the existence of periodic solutions for second-order nonlinear difference equations is obtained. The proof is based on the Saddle Point Theorem in combination with variational
technique. The problem is to solve the existence of periodic solutions of second-order nonlinear difference
equations. One of our results obtained complements the result in the literature.
- periodic solutions
- nonlinear difference equations
- discrete variational theory.
R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods and Applications, Marcel Dekker, New York (2000)
R. P. Agarwal, P. J. Y. Wong , Advanced Topics in Difference Equations, Kluwer Academic Publishers, Dordrecht (1997)
C. D. Ahlbrandt, A. C. Peterson, Discrete Hamiltonian Systems: Difference Equations, Continued Fraction and Riccati Equations, Kluwer Academic Publishers, Dordrecht (1996)
X. C. Cai, J. S. Yu, Existence theorems of periodic solutions for second-order nonlinear difference equations, Adv. Difference Equ., 2008 (2008), 11 pages.
A. Castro, R. Shivaji, Nonnegative solutions to a semilinear Dirichlet problem in a ball are positive and radially symmetric, Comm. Partial Differential Equations, 14 (1989), 1091-1100.
S. Z. Chen, Disconjugacy, disfocality, and oscillation of second order difference equation, J. Differential Equations, 107 (1994), 383-394.
J. R. Esteban, J. L. Vázquez, On the equation of turbulent filtration in one-dimensional porous media, Nonlinear Anal., 10 (1986), 1303-1325.
Z. M. Guo, J. S. Yu , Existence of periodic and subharmonic solutions for second-order superlinear difference equations, Sci. China Math., 46 (2003), 506-515.
Z. M. Guo, J. S. Yu, The existence of periodic and subharmonic solutions of subquadratic second order difference equations, J. London Math. Soc., 68 (2003), 419-430.
Z. M. Guo, J. S. Yu, Applications of critical point theory to difference equations, Fields Inst. Commun., 42 (2004), 187-200.
H. G. Kaper, M. Knaap, M. K. Kwong, Existence theorems for second order boundary value problems, Differential Integral Equations, 4 (1991), 543-554.
Y. J. Liu, W. G. Ge, Twin positive solutions of boundary value problems for finite difference equations with p-Laplacian operator, J. Math. Anal. Appl., 278 (2003), 551-561.
H. Matsunaga, T. Hara, S. Sakata, Global attractivity for a nonlinear difference equation with variable delay, Computers Math. Appl., 41 (2001), 543-551.
H. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York (1989)
M. S. Peng, Q. L. Xu, L. H. Huang, W. G. Ge, Asymptotic and oscillatory behavior of solutions of certain second order nonlinear difference equations, Comput. Math. Appl., 37 (1999), 9-18.
P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157-184.
H. P. Shi, W. P. Ling, Y. H. Long, H. Q. Zhang, Periodic and subharmonic solutions for second order nonlinear functional difference equations, Commun. Math. Anal., 5 (2008), 50-59.
D. Smets, M. Willem, Solitary waves with prescribed speed on infinite lattices, J. Funct. Anal., 149 (1997), 266-275.
Y. T. Xu, Z. M. Guo, 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.
J. S. Yu, Z. M. Guo, On boundary value problems for a discrete generalized Emden-Fowler equation, J. Differential Equations, 231 (2006), 18-31.
J. S. Yu, Z. M. Guo, X. F. Zou, Periodic solutions of second order self-adjoint difference equations, J. London Math. Soc., 71 (2005), 146-160.
R. Y. Zhang, Z. C. Wang, J. S. Yu, Necessary and sufficient conditions for the existence of positive solutions of nonlinear difference equations, Fields Inst. Commun., 42 (2004), 385-396.
Z. Zhou, J. S. Yu, Y. M. Chen, Homoclinic solutions in periodic difference equations with saturable nonlinearity, Sci. China Math., 54 (2011), 83-93.
Z. Zhou, J. S. Yu, Z. M. Guo , Periodic solutions of higher-dimensional discrete systems, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 1013-1022.
Z. Zhou, Q. Q. Zhang, Uniform stability of nonlinear difference systems, J. Math. Anal. Appl., 225 (1998), 486-500.