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

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 $\mathbb{T}$ where u( 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( 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. [ABSTRACT FROM AUTHOR]