In this paper, we study the following fractional Navier boundary value problem
$$\begin{aligned} \left\{ \begin{array}{lllc} D^{\beta }(D^{\alpha }u)(x)=u(x)g(u(x)),\quad x\in (0,1), \\ \displaystyle \lim _{x\longrightarrow 0}x^{1-\beta }D^{\alpha }u(x)=-a,\quad \,\,u(1)=b, \end{array} \right. \end{aligned}$$
where
\(\alpha ,\beta \in (0,1]\) such that
\(\alpha +\beta >1\),
\(D^{\beta }\) and
\(D^{\alpha }\) stand for the standard Riemann–Liouville fractional derivatives and
a,
b are nonnegative constants such that
\(a+b>0\). The function
g is a nonnegative continuous function in
\(0,\infty )\) that is required to satisfy some suitable integrability condition. Using estimates on the Green’s function and a perturbation argument, we prove the existence of a unique positive continuous solution, which behaves like the unique solution of the homogeneous problem.