Positive Solutions for a Fractional p-Laplacian Boundary Value Problem

Abstract

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 $2<\alpha \le 3$, $1<\beta\le 2$, $D_{0+}^\alpha,
D_{0+}^\beta$ are the standard Riemann-Liouville fractional
derivatives, $\phi_p(s)=|s|^{p-2}s,p>1$, $\phi_p^{-1}=\phi_q$,
$1/p+1/q=1$,
$0<\xi,\eta<1$, $0\le
a<\xi^{2-\alpha}$, $0\le b<\eta^{\frac{1-\beta}{p-1}}$ 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.