We consider the following semilinear fractional system Dαu=p(t)uavrin (0, 1),Dβv=q(t)usvbin (0, 1),limt→0+t1−αu(t)=limt→0+t1−βv(t)=0, where α, β ∈ (0,1), a, b ∈ (-1, 1), r, s ∈ ℝ such that (1 -|a|)(1 - |b|) - |rs| > 0,Dα,Dβ are the Riemann-Liouville fractional derivatives of orders α, β and the nonlinearities p, q are positive measurable functions on (0, 1). Applying the Schäuder fixed point theorem, we establish the existence and the boundary behaviour of positive solutions in the space of weighted continuous functions.