Homoclinic Solutions for a Class of Perturbed Fractional Hamiltonian Systems with Subquadratic Conditions

Abstract

In this paper, we consider the following perturbed fractional Hamiltonian systems

$$\left\{ {\matrix{{_tD_\infty ^\alpha {(_{ - \infty }}D_t^\alpha u(t)) + L(t)u(t) = {\nabla _u}W(t,u(t)) + {\nabla _u}G(t,u(t)),} \hfill & {t \in \mathbb{R},} \hfill \cr {u \in {H^\alpha }(\mathbb{R},{\mathbb{R}^N}),} \hfill & {} \hfill \cr } } \right.$$

where \(\alpha \in (1/2,1],\,\,L \in C(\mathbb{R},{\mathbb{R}^{N \times N}})\) is symmetric and not necessarily required to be positive definite, \(W \in {C^1}(\mathbb{R}\times {\mathbb{R}^N},\mathbb{R})\) is locally subquadratic and locally even near the origin, and perturbed term \(G \in {C^1}(\mathbb{R} \times {\mathbb{R}^N},\mathbb{R})\) maybe has no parity in u. Utilizing the perturbed method improved by the authors, a sequence of nontrivial homoclinic solutions is obtained, which generalizes previous results.