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Notes on periodic solutions of discrete steady state systems. (English) Zbl 1141.39011

The authors investigate the existence of a nontrivial periodic solution to a discrete steady state system \[ X_{n+1}+X_{n-1}-f(n, X_n)=0, n\in {\mathbb Z} \]
where \(f=(f_1, f_2, \dots, f_k)^{T}\in C({\mathbb Z},\times {\mathbb R}^{k}, {\mathbb R}^{k})\) and there is a positive integer \(\omega\) such that \(f(t+\omega, U)=f(t, U)\). By using a special version of the mountain pass theorem in the critical point theory, the authors obtain some sufficient conditions to ensure the existence of nontrivial periodic solutions. Two examples are given to illustrate the effectiveness of these results. Also they show that the results obtained in this paper are different from those Z. Zhou, J. S. Yu and Z. M. Guo [Proc. R. Soc. Edinb., Sect. A, Math. 134, No. 5, 1013–1022 (2004; Zbl 1073.39010)].

MSC:

39A11 Stability of difference equations (MSC2000)

Citations:

Zbl 1073.39010
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References:

[1] Cheng, S.S. - Partial Difference Equations, Taylor and Francis, 2003. · Zbl 1016.39001
[2] Agarwal, R.P. - Difference Equations and Inequalities: Theory, Methods and Applications, Marcel Dekker, 2000. · Zbl 0952.39001
[3] Guo, Z.M. and Yu, J.S. - Existence of periodic and subharmonic solutions for second-order superlinear difference equations, Science in China (Series A), 46 (2003), 506-515. · Zbl 1215.39001 · doi:10.1007/BF02884022
[4] Guo, Z.M. and Yu, J.S. - The existence of periodic and subharmonic solutions to subquadratic second-order difference equations, J. London Math. Soc., 68 (2003), 419-430. · Zbl 1046.39005 · doi:10.1112/S0024610703004563
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