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Existence of a periodic solution for subquadratic second-order discrete Hamiltonian system. (English) Zbl 1129.39008
Consider the nonlinear second order discrete Hamiltonian system $$ \Delta^2u(t-1)+\nabla F(t,u(t))=0, \quad \forall t\in Z, $$ where $\Delta u(t)=u(t+1)-u(t)$, $\Delta^2u(t)=\Delta (\Delta u(t))$, $F: Z\times R^N\to R$, $F(t,x)$ is continuously differentiable in $x$ for every $t\in Z$ and $T$-periodic in $t$ for all $x\in R^N$, $T$ is a positive integer, $Z$ is the set of all integers, $\nabla F(t,x)$ denotes the gradient of $F(t,x)$ in $x$. Several criteria on the existence of at least one $T$-periodic solution are presented, which are established by employing minimax methods in critical point theory. The obtained results improve some exisiting ones.

MSC:
39A12Discrete version of topics in analysis
39A11Stability of difference equations (MSC2000)
37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
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References:
[1] Agarwal, R. P.: Difference equations and inequalities. Monographs and textbooks in pure and applied mathematics 228 (2000)
[2] Agarwal, R. P.; Popenda, J.: Periodic solutions of first order linear difference equations. Math. comput. Modelling 22, No. 1, 11-19 (1995) · Zbl 0871.39002
[3] Cerami, G.: An existence criterion for the critical points on unbounded manifolds. Istit. lombardo accad. Sci. lett. Rend. 112, No. 2, 332-336 (1978) · Zbl 0436.58006
[4] Costa, D. G.; Magalhaes, C. A.: Variational elliptic problems which are nonquadratic at infinity. Nonlinear anal. 23, No. 11, 1401-1412 (1994) · Zbl 0820.35059
[5] Gil, M.: Periodic solutions of abstract difference equations. Appl. math. E-notes 1, 18-23 (2001) · Zbl 0981.39008
[6] Guo, Z.; Yu, J.: The existence of periodic and subharmonic solutions of subquadratic second order difference equations. J. London math. Soc. (2) 68, No. 2, 419-430 (2003) · Zbl 1046.39005
[7] Guo, Z.; Yu, J.: Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems. Nonlinear anal. 55, No. 7--8, 969-983 (2003) · Zbl 1053.39011
[8] Guo, Z.; Yu, J.: Existence of periodic and subharmonic solutions for second-order superlinear difference equations. Sci. China ser. A 46, No. 4, 506-515 (2003) · Zbl 1215.39001
[9] Guo, Z.; Yu, J.: Applications of critical theory to difference equations. Differences and differential equations. Fields inst. Commun. 42, 187-200 (2004) · Zbl 1067.39007
[10] Ma, M.; Yu, J.: Existence of multiple positive periodic solutions for nonlinear functional difference equations. J. math. Anal. appl. 305, No. 2, 483-490 (2005) · Zbl 1070.39019
[11] Mawhin, J.; Willem, M.: Critical point theory and Hamiltonian systems. Applied mathematical sciences 74 (1989) · Zbl 0676.58017
[12] Rabinowitz, P. H.: Periodic solutions of Hamiltonian systems. Comm. pure appl. Math. 31, No. 2, 157-184 (1978) · Zbl 0358.70014
[13] Tang, C. -L.: Periodic solutions for nonautonomous second order systems with sublinear nonlinearity. Proc. amer. Math. soc. 126, No. 11, 3263-3270 (1998) · Zbl 0902.34036
[14] Y.-F. Xue, C.-L. Tang, Multiple periodic solutions for superquadratic second-order discrete Hamiltonian systems, preprint · Zbl 1153.39024
[15] Yu, J.; Guo, Z.; Zou, X.: Periodic solutions of second order self-adjoint difference equations. J. London math. Soc. (2) 71, No. 1, 146-160 (2005) · Zbl 1073.39009