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Abstract In this paper, by using the fibering map and the Nehari manifold, we prove the existence and multiple results of solutions for the following fractional differential equation: { D T α t ( 0 D t α u ) = λ h ( t ) | u | p − 2 u + b ( t ) | u | q − 2 u , t ∈ [ 0 , T ] , u ( 0 ) = u ( T ) = 0 , $$\begin{aligned} \textstyle\begin{cases} {_{t}}D{_{T}^{\alpha}}({_{0}}D{_{t}^{\alpha}}u)=\lambda h(t) \vert u \vert ^{p-2}u+b(t) \vert u \vert ^{q-2}u,\quad t\in[0,T],\\ u(0)=u(T)=0, \end{cases}\displaystyle \end{aligned}$$ where α ∈ ( 1 2 , 1 ) , 0 < p < 2 , q > 2 , λ > 0 $\alpha\in(\frac{1}{2},1), 0< p<2, q>2, \lambda>0$ and h ( t ) , b ( t ) $h(t),b(t)$ are sign-changing continuous functions.