Infinitely many solutions for nonlinear fractional boundary value problems via variational methods

In this paper, the author considers the following nonlinear fractional boundary value problem: $$ \{ {array}{l} d/dt ( {1/2{ }₀Dₜ-β ({u}'(t))+1/2{ }ₜDT-β ({u}'(t))} )+∇ F(t,u(t))=0, .e. t∈[0,T], \\ u(0)=u(T)=0, {array} . $$ where ${ }₀Dₜ-β $ and ${ }ₜDT-β $ are the left and right Riemann-Liouville fractional integrals of order $0≤β<1$ , respectively, $∇ F(t,x)$ is the gradient of F at x. By applying the variant fountain theorems, the author obtains the existence of infinitely many small or high energy solutions to the above boundary value problem.