In this work, we consider the existence of solution to the following fractional advection–dispersion equation -ddt(p-∞Itβ(u′(t))+qtI∞β(u′(t)))+b(t)u=f(t,u(t)),t∈Rwhere β∈ (0 , 1) , -∞Itβ and tI∞β denote left and right Liouville–Weyl fractional integrals of order β respectively, 0 < p= 1 - q< 1 , f: R× R→ R and b: R→ R+ are continuous functions. Due to the general assumption on the constant p and q, the problem (0.1) does not have a variational structure. Despite that, here we study it performing variational methods, combining with an iterative technique, and give an existence criteria of solution for the problem (0.1) under suitable assumptions.