Existence of solutions for a class of fractional boundary value problems via critical point theory

In this paper, by the critical point theory, a new approach is provided to study the existence of solutions to the following fractional boundary value problem: {ddt(120Dt−β(u′(t))+12tDT−β(u′(t)))+∇F(t,u(t))=0,a.e. t∈[0,T],u(0)=u(T)=0, where 0Dt−β and tDT−β are the left and right Riemann–Liouville fractional integrals of order 0≤β<10≤β<1 respectively, F:[0,T]×RN→R is a given function and ∇F(t,x)∇F(t,x) is the gradient of FF at xx. Our interest in this problem arises from the fractional advection–dispersion equation (see Section 2). The variational structure is established and various criteria on the existence of solutions are obtained.