| Summary: | In this article, we consider the following boundary-value problem of nonlinear fractional differential equation with $p$-Laplacian operator $$displaylines{ D_{0+}^eta(phi_p(D_{0+}^alpha u(t)))+a(t)f(u)=0, quad 0<t<1, cr u(0)=gamma u(xi)+lambda, quad phi_p(D_{0+}^alpha u(0))=(phi_p(D_{0+}^alpha u(1)))' =(phi_p(D_{0+}^alpha u(0)))''=0, }$$ where $0<alphaleqslant1$, $2<etaleqslant 3$ are real numbers, $D_{0+}^alpha, D_{0+}^eta$ are the standard Caputo fractional derivatives, $phi_p(s)=|s|^{p-2}s$, $p>1$, $phi_p^{-1}=phi_q$, $1/p+1/q=1$, $0leqslantgamma<1$, $0leqslantxileqslant1$, $lambda>0$ 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.
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