Existence of Homoclinic Solutions for a Class of Second-Order Hamiltonian Systems with Locally Subquadratic Potentials

In this paper, we mainly consider the existence of homoclinic orbits for the following second-order Hamiltonian systems u(t) - L(t)u(t)+. W t, u(t) = f (t), where L(t) is a positive definite and symmetric matrix for all t. R and the potential functionW(t, u) is locally subquadratic. Here, the coefficient of the upper bound forW is a positive constant, whereas in the previous literature the corresponding coefficient need to be some integrable functions a(t) on R.