Positive Solutions for a Fractional $p$--Laplacian Boundary Value Problem

Jiafa Xu, Donal O'Regan

In this paper we study the existence of positive solutions for the fractional $p$-Laplacian boundary value problem \[ \begin{cases} D^\beta_{0+}(hi_p(D^lpha_{0+}u(t)))=f(t,u(t)), tı(0,1) u(0)=u'(0)=0, u'(1)=au'(\xi), D^lpha_{0+}u(0)=0, D^lpha_{0+}u(1)=bD^lpha_{0+}u(\eta), \end{cases} \] where $2<\alpha\leq3$, $1<\beta\leq2$, $D^\alpha_{0+},D^\beta_{0+}$ are the standard Riemann-Liouville fractional derivatives, $\phi_p(s)=|s|^{p-2}s$, $p>1$, $\phi_p^{-1}=\phi_q$, $1/p+1/q=1$, $0<\xi$, $\eta<1$, $0\leq a<\xi^{2-\alpha}$, $0\leq b<\eta^{\frac{1-\beta}{p-1}}$ 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.