Homoclinic Orbits for a Class of Subquadratic Second Order Hamiltonian Systems

The existence and multiplicity of homoclinic orbits are considered for a class of subquadratic second order Hamiltonian systems . Recent results from the literature are generalized and significantly improved. Examples are also given in this paper to illustrate our main results. 1. Introduction and Main Results In this paper, we will study the existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems of the type: where is a symmetric matrix valued function and . As usual, we say that is a nontrivial homoclinic orbit (to 0) if and as (see [1]). In the following, denotes the standard inner product in and is the induced norm. Hamiltonian system theory, a classical as well as a modern study area, widely consists in mathematical sciences, life sciences, and various aspects of social science. Lots of mechanical and field theory models even exist in the form of Hamiltonian system. Solutions of Hamiltonian system can be divided into periodic solution, subharmonic solution, homoclinic orbit, heteroclinic orbit, and so on. Homoclinic orbit was discovered among the models of nonlinear dynamical system and affects the nature of the whole system significantly. Starting from the Poincaré era, the study of homoclinic orbits in the nonlinear dynamical system exploits Perturbation method mainly. It is not until the recent decade that variational principle has been widely used. The existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems have been extensively investigated in many recent papers (see [1–16]). The main feature of the problem is the lack of global compactness due to unboundedness of domain. To overcome the difficulty, many authors have considered the periodic case, autonomous case, or asymptotically periodic case (see [1–4]). Some papers treat the coercive case (see [5–8]). Recently, the symmetric case has been dealt with (see [9–11]). Compared with the superquadratic case, the case that is subquadratic as has been considered only by a few authors. As far as the author is aware, the author in [5] first discussed the subquadratic case. Later, the authors in [12] dealt with this case by use of a standard minimizing argument, which is the following theorem. Theorem 1 (see [12]). Assume that and satisfy the following conditions: (A1) is a symmetric and positive definite matrix for all and there is a continuous function such that and as ;(A2) , where is a continuous function such that and is a constant. Then, (1) has at least one nontrivial homoclinic orbit. Subsequently, the condition ( ) was