Positive solutions for a fractional $p$-Laplacian boundary value problem

In this paper we study the existence of positive solutions for the fractional $p$-Laplacian boundary value problem \[\left\{\aligned & D_{0+}^\beta (\phi_p (D_{0+}^\alpha u(t)))=f(t,u(t)), t\in (0,1),\\ & u(0)=u'(0)=0, u'(1)=au'(\xi), D_{0+}^\alpha u(0)=0, D_{0+}^\alpha u(1)= b D_{0+}^\alpha u(\eta), \endaligned \right.\] where $2 1$, $\phi_p^{-1}=\phi_q$, $1/p+1/q=1$, $0<\xi,\eta<1$, $0\le a<\xi^{2-\alpha}$, $0\le b<\eta^{\frac{1-\beta}{p-1}}$ and $f\in C([0,1]\times [0,+\infty),[0,+\infty))$. Using the monotone iterative method and the fixed point index theory in cones, we establish two new existence results when the nonlinearity $f$ is allowed to grow $(p-1)$-sublinearly and $(p-1)$-superlinearly at infinity.