In this work, we consider the existence of solution to the following fractional advection–dispersion equation 0.1 $$\begin{aligned} -\frac{d}{dt} \left( p {_{-\infty }}I_{t}^{\beta }(u'(t)) + q\; {_{t}}I_{\infty }^{\beta }(u'(t))\right) + b(t)u = f(t, u(t)),\;\;t\in \mathbb {R}\end{aligned}$$ - d dt p - ∞ I t β ( u ′ ( t ) ) + q t I ∞ β ( u ′ ( t ) ) + b ( t ) u = f ( t , u ( t ) ) , t ∈ R where $$\beta \in (0,1)$$ β ∈ ( 0 , 1 ) , $$_{-\infty }I_{t}^{\beta }$$ - ∞ I t β and $$_{t}I_{\infty }^{\beta }$$ t I ∞ β denote left and right Liouville–Weyl fractional integrals of order $$\beta $$ β respectively, $$0<p=1-q<1$$ 0 < p = 1 - q < 1 , $$f:\mathbb {R}\times \mathbb {R} \rightarrow \mathbb {R}$$ f : R × R → R and $$b:\mathbb {R} \rightarrow \mathbb {R}^{+}$$ 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.