Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation with p-Laplacian Operator

In this paper, we deal with the following p-Laplacian fractional boundary value problem: \( \phi _p(D_{0+}^\alpha u(t))+f(t,u(t))=0,~0<t<1\), \(u(0)=u'(0)=u'(1)=0, \ \) where \(2<\alpha \leqslant 3\) is a real number. \(D_{0+}^\alpha \) is the standard Riemann–Liouville differentiation, and \(f:[0,1]\times [0,+\infty )\rightarrow [0,+\infty )\) is continuous. By the properties of the Green function and some fixed-point theorems on cone, some existence and multiplicity results of positive solutions are obtained. As applications, examples are presented to illustrate the main results.