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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.

39A12 Discrete version of topics in analysis
39A11 Stability of difference equations (MSC2000)
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
Full Text: DOI
[1] Agarwal, R.P., Difference equations and inequalities, () · Zbl 0339.39003
[2] Agarwal, R.P.; Popenda, J., Periodic solutions of first order linear difference equations, Math. comput. modelling, 22, 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., A 112, 2, 332-336, (1978), (in Italian) · Zbl 0436.58006
[4] Costa, D.G.; Magalhaes, C.A., Variational elliptic problems which are nonquadratic at infinity, Nonlinear anal., 23, 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, 2, 419-430, (2003) · Zbl 1046.39005
[7] Guo, Z.; Yu, J., Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems, Nonlinear anal., 55, 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, 4, 506-515, (2003) · Zbl 1215.39001
[9] Guo, Z.; Yu, J., Applications of critical theory to difference equations. differences and differential equations, (), 187-200 · Zbl 1067.39007
[10] Ma, M.; Yu, J., Existence of multiple positive periodic solutions for nonlinear functional difference equations, J. math. anal. appl., 305, 2, 483-490, (2005) · Zbl 1070.39019
[11] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, () · Zbl 0678.35091
[12] Rabinowitz, P.H., Periodic solutions of Hamiltonian systems, Comm. pure appl. math., 31, 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, 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, 1, 146-160, (2005) · Zbl 1073.39009
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