Existence and uniqueness of positive solutions for integral boundary problems of nonlinear fractional differential equations with p -Laplacian operator

Abstract

In this paper, we deal with the following integral boundary problem of nonlinear fractional differential equations with $p$-Laplacian operator \begin{displaymath} \begin{array}{lll} D_{0+}^{\gamma}(\phi_p(D_{0+}^{\alpha}u(t))) + f(t,u(t))=0, \quad 0 \lt t \lt 1,\\ u(0) = u'(0)=0, \quad u'(1) = \int_0^\eta u(s)\,ds,\quad D_{0+}^{\alpha}u(t)|_{t=0}=0, \end{array} \end{displaymath} where $0 \lt \gamma \lt 1$, $2 \lt \alpha \lt 3$, $D_{0+}^{\alpha}$ is the standard Riemann-Liouville fractional derivative, $\phi_p(s)=|s|^{p-2}s, p>1$, $(\phi_p)^{-1}=\phi_q$, ${1}/{p}+{1}/{q}=1$. By the properties of the Green function, the lower and upper solution method and fixed-point theorem in partially ordered sets, some new existence and uniqueness of positive solutions to the above boundary value problem are established. As applications, examples are presented to illustrate the main results.