We are concerned with a class of singular Hamiltonian systems on time scales. Some results on the existence of periodic solutions are obtained for the system under consideration by means of the variational methods and the critical point theory. 1. Introduction In recent years, dynamic equations on time scales have been studied intensively in the literature [1–7]. Some ideas and methods have been developed to study the existence and multiplicity of solutions for dynamic equations on time scales, for example, the fixed point theory, the method of the upper and lower solutions, the coincidence degree theory, and so on. However, not much work has been seen on the existence of solutions to dynamic equations on time scales through the variational method and the critical point theory; for details see [4–10] and the references therein. For example, authors of [11] give some results on the existence and multiplicity of periodic solutions which are obtained for the Hamiltonian system by means of the saddle point theorem, the least action principle, and the three-critical-point theorem. To the best of our knowledge, it is still worth making an attempt to extend variational methods to study the existence of periodic solutions for various Hamiltonian systems. Naturally, it is interesting and necessary to study the existence of periodic solutions for Hamiltonian systems on time scales. Besides, in [12], using Lyusternik-Schnirelmann theory with classical compact condition, Ambrosetti-Coti Zelati studied the periodic solutions of a fixed energy for Hamiltonian systems with singular potential : After Ambrosetti-Coti Zelati, a lot of mathematicians studied singular Hamiltonian systems (see, e.g., [13–15]). Motivated by the above, in this paper, we consider the following second order Hamiltonian system with a fixed energy on time scale : where denotes the delta (or Hilger) derivative of at , , is the forward jump operator, , and satisfies the following assumption: is measurable in for every and continuously differentiable in for . The paper is organized as follows. In Section 2, we introduce some definitions and make some preparations for later sections. We summarize our main results on the existence of periodic solutions of the second order Hamiltonian system on time scales in Section 3. 2. Preliminaries In this section, we will first recall some fundamental definitions and lemmas which are used in what follows. Definition 1 (see [3]). A time scale is an arbitrary nonempty closed subset of the real set with the topology and ordering inherited from . The forward jump
Z.-C. Hao, T.-J. Xiao, and J. Liang, “Existence of positive solutions for singular boundary value problem on time scales,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 517–528, 2007.
J. W. Zhou and Y. K. Li, “Sobolev's spaces on time scales and its applications to a class of second order Hamiltonian systems on time scales,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 5, pp. 1375–1388, 2010.
J. Zhou and Y. K. Li, “Variational approach to a class of second order Hamiltonian systems on time scales,” Acta Applicandae Mathematicae, vol. 117, pp. 47–69, 2012.
Y.-H. Su and Z. Feng, “A non-autonomous Hamiltonian system on time scales,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 10, pp. 4126–4136, 2012.
Y. Su and Z. Feng, “Homoclinic orbits and periodic solutions for a class of Hamiltonian systems on time scales,” Journal of Mathematical Analysis and Applications, vol. 411, no. 1, pp. 37–62, 2014.
R. P. Agarwal, V. Otero-Espinar, K. Perera, and D. R. Vivero, “Existence of multiple positive solutions for second order nonlinear dynamic BVPs by variational methods,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 1263–1274, 2007.
R. P. Agarwal, V. Otero-Espinar, K. Perera, and D. R. Vivero, “Basic properties of Sobolev's spaces on time scales,” Advances in Difference Equations, vol. 67, pp. 368–381, 2006.
P. H. Rabinowitz, Minimax Method in Critical Point Theory with Applications to Differential Equations, vol. 65 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1986.
A. Ambrosetti and V. Coti Zelati, “Closed orbits of fixed energy for singular Hamiltonian systems,” Archive for Rational Mechanics and Analysis, vol. 112, no. 4, pp. 339–362, 1990.
C. Carminati, é. Séré, and K. Tanaka, “The fixed energy problem for a class of nonconvex singular Hamiltonian systems,” Journal of Differential Equations, vol. 230, no. 1, pp. 362–377, 2006.
L. Pisani, “Periodic solutions with prescribed energy for singular conservative systems involving strong forces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 21, no. 3, pp. 167–179, 1993.
W. B. Gordon, “Conservative dynamical systems involving strong forces,” Transactions of the American Mathematical Society, vol. 204, pp. 113–135, 1975.