In this paper, we study the following fractional Navier boundary value problem
$$begin{aligned} left{ begin{array}{lllc} D^{beta }(D^{alpha }u)(x)=u(x)g(u(x)),quad xin (0,1), displaystyle lim _{xlongrightarrow 0}x^{1-beta }D^{alpha }u(x)=-a,quad ,,u(1)=b, end{array} right. end{aligned}$$
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.
