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

Author： Zhou Jianwen  

Publisher： Springer Publishing Company

ISSN： 0167-8019

Source： Acta Applicandae Mathematicae, Vol.117, Iss.1, 2012-02, pp. : 47-69

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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 $mathbb{T}$ 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 $F:[0,T]_{mathbb{T}}times mathbb{R}^{N}rightarrowmathbb{R}$ . 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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