Existence and multiplicity results of homoclinic solutions for fractional Hamiltonian systems

In this paper, by the critical point theory, we consider the existence and multiplicity of solutions for the following fractional differential equation tD∞α(−∞Dtαu(t))+L(t)u(t)=∇W(t,u(t)),t∈R, where α∈(12,1], −∞Dtα and tD∞α are left and right Liouville-Weyl fractional derivatives of order α on the whole axis R respectively, u∈Rn, L(t) is positive definite symmetric matrix for all t∈R and W:R×Rn→R is a suitably chosen function.