# Variational Approach to a Class of Second Order Hamiltonian Systems on Time Scales

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## Abstract

In this paper, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for the second order Hamiltonian system on time scale $$\mathbb{T}$$

$$\left\{\begin{array}{l@{\quad}l}u^{\Delta^{2}}(t)+A(\sigma(t))u(\sigma(t))+\nabla F(\sigma(t),u(\sigma(t)))=0,& \hbox{\ \Delta-a.e. t\in [0,T]_{_{\mathbb{T}}}^{\kappa},} \\u(0)-u(T)=0,\qquad u^{\Delta}(0)-u^{\Delta}(T)=0,& \hbox{}\end{array}\right.$$

where u Δ(t) denotes the delta (or Hilger) derivative of u at t, $$u^{\Delta^{2}}(t)=(u^{\Delta})^{\Delta}(t)$$, σ is the forward jump operator, T is a positive constant, A(t)=[d ij (t)] is a symmetric N×N matrix-valued function defined on $$[0,T]_{\mathbb{T}}$$ with $$d_{ij}\in L^{\infty}([0,T]_{\mathbb{T}},\mathbb{R})$$ for all i,j=1,2,…,N, and $$F:[0,T]_{\mathbb{T}}\times \mathbb{R}^{N}\rightarrow\mathbb{R}$$. By establishing a proper variational setting, two existence results and two multiplicity results are obtained. Finally, three examples are presented to illustrate the feasibility and effectiveness of our results.

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Correspondence to Yongkun Li.

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 10971183.

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Zhou, J., Li, Y. Variational Approach to a Class of Second Order Hamiltonian Systems on Time Scales. Acta Appl Math 117, 47–69 (2012). https://doi.org/10.1007/s10440-011-9649-z

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• DOI: https://doi.org/10.1007/s10440-011-9649-z