Positive solutions to boundary-value problems of p-Laplacian fractional differential equations with a parameter in the boundary

In this article, we consider the following boundary-value problem of nonlinear fractional differential equation with $p$-Laplacian operator $$displaylines{ D_{0+}^eta(phi_p(D_{0+}^alpha u(t)))+a(t)f(u)=0, quad 01$, $phi_p^{-1}=phi_q$, $1/p+1/q=1$, $0leqslantgamma<1$, $0leqslantxileqslant1$, $lambda>0$ is a parameter, $a:(0,1) o [0,+infty)$ and $f:[0,+infty) o[0,+infty)$ are continuous. By the properties of Green function and Schauder fixed point theorem, several existence and nonexistence results for positive solutions, in terms of the parameter $lambda$ are obtained. The uniqueness of positive solution on the parameter $lambda$ is also studied. Some examples are presented to illustrate the main results.