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

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.