Upper and Lower Solutions Method for a Class of Singular Fractional Boundary Value Problems with p-Laplacian Operator

Abstract

The upper and lower solutions method is used to study the $p$-Laplacian fractional boundary value problem $D_{0+}^{\gamma }\left({\varphi }_p\right(D_{0+}^{\alpha }u\left(t\right)\left)\right)=f(t,u(t\left)\right)$, $0<t<1$, $u\left(0\right)=0$, $u\left(1\right)=au\left(\xi \right)$, $D_{0+}^{\alpha }u\left(0\right)=0$, and $D_{0+}^{\alpha }u\left(1\right)=b{D}_{0+}^{\alpha }u\left(\eta \right)$, where $1<\alpha ,\gamma ⩽2,0⩽a,b⩽1,0<\xi ,\eta <1$. Some new results on the existence of at least one positive solution are obtained. It is valuable to point out that the nonlinearity $f$ can be singular at $t=\mathrm{0,1}$ or $u=0$.