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The existence of periodic and subharmonic solutions to subquadratic discrete Hamiltonian systems

Published online by Cambridge University Press:  17 February 2009

Zhan Zhou ,
Jianshe Yu  and
Zhiming Guo
Zhan Zhou
Affiliation:
School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510405, People's Republic of China; e-mail: zzhou0321@hotmail.com.
Jianshe Yu
Affiliation:
School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510405, People's Republic of China; e-mail: zzhou0321@hotmail.com.
Zhiming Guo
Affiliation:
School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510405, People's Republic of China; e-mail: zzhou0321@hotmail.com.
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Abstract

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In this paper, by using critical point theory, we establish some results for the existence of periodic and subharmonic solutions to subquadratic discrete Hamiltonian systems.

Type
Research Article
Information
The ANZIAM Journal , Volume 47 , Issue 1 , July 2005 , pp. 89 - 102
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Agarwal, R. P., Difference equations and inequalities: theory, methods and applications, 2nd ed. (Marcel Dekker, New York, 2000).CrossRefGoogle Scholar
[2]Agarwal, R. P. and Zhang, W., “Periodic solutions of difference equations with general periodicity”, Comput. Math. Appl. 42 (2001) 719–727CrossRefGoogle Scholar
[3]Ahlbrandt, C. D. and Peterson, A. C., Discrete Hamiltonian systems (Kluwer Academic Publishers, Dordrecht, 1996).CrossRefGoogle Scholar
[4]Benci, V. and Rabinowitz, P. H., “Critical point theorems for indefinite functionals”, Inventiones Math. 52 (1979) 241–273.CrossRefGoogle Scholar
[5]Bohner, M., “Linear Hamiltonian difference systems: disconjugacy and Jacobi-type conditions”, J. Math. Anal. Appl. 199 (1996) 804–826CrossRefGoogle Scholar
[6]Chang, K. C., Critical point theory and its applications (Science and Technical Press, Shanghai, China, 1980).Google Scholar
[7]Chen, S., “Disconjugacy, disfocality, and oscillation of second order difference equations”, J. Differential Equations 107 (1994) 383–394CrossRefGoogle Scholar
[8]Erbe, L. H. and Yan, P., “Disconjugacy for linear Hamiltonian difference systems”, J. Math. Anal. Appl. 167 (1992) 355–367.CrossRefGoogle Scholar
[9]Guo, Z. and Yu, J., “The existence of periodic and subharmonic solutions for second-order super-linear difference equations”, Science in China, Mathemnatics 46 (2005) 506–515.CrossRefGoogle Scholar
[10]Guo, Z. and Yu, J., “Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems”, Nonl. Anal., TMA 55 (2005) 969–983.CrossRefGoogle Scholar
[11]Hartman, P., “Difference equations: disconjugacy, principal solutions, Green's functions, complete monotonicity”, Trans. Amer. Math. Soc. 246 (1978) 1–30.Google Scholar
[12]Mawhin, J. and Willem, M., Critical point theory and Hamiltonian systems (Springer, New York, 1989).CrossRefGoogle Scholar
[13]Rabinowitz, P. H., “Periodic solutions of Hamiltonian systems”, Comm. Pure Appl. Math. 31 (1978) 157–184.CrossRefGoogle Scholar
[14]Rabinowitz, P. H., “On subharmonic solutions of Hamiltonian systems”, Comm. Pure Appl. Math. 33 (1980) 609–633.CrossRefGoogle Scholar
[15]Rabinowitz, P. H., Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics 65 (Published for the Conference Board of the Mathematical Sciences,Washington, DC; by Amer. Math. Soc., Providence, RI, 1986).CrossRefGoogle Scholar
[16]Zhou, Z. and Wu, J., “Attractive periodic orbits in nonlinear delayed discrete-time neural networks with delayed feedback”, J. Differential Equations Appl. 8 (2002) 467–483CrossRefGoogle Scholar
[17]Zhou, Z. and Wu, J., “Attractive periodic orbits for discrete monotone dynamical systems arising from delayed neural networks”, Dyn. Contin. Discrete Impuls. Syst. Ser. B 10 (2005) 95–106Google Scholar