### Abstract:

In this article, we are concerned with the existence of positive solutions for nonlinear fractional differential equation whose nonlinearity contains the first-order derivative,
$$\displaylines{
D_{0^+}^{\alpha}u(t)+f(t,u(t),u'(t))=0,\quad t\in (0,1),\;
n-1<\alpha\leq n,\cr
u^{(i)}(0)=0, \quad i=0,1,2,\dots,n-2,\cr
[D_{0^+}^{\beta}u(t)]_{t=1}=0, \quad 2\leq\beta\leq n-2,
}$$
where $n>4 $ $(n\in\mathbb{N})$, $D_{0^+}^{\alpha}$ is the standard Riemann-Liouville fractional derivative of order $\alpha$ and $f(t,u,u'):[0,1] \times [0,\infty)\times(-\infty,+\infty)
\to [0,\infty)$ satisfies the Caratheodory type condition. Sufficient conditions are obtained for the existence of at least one or two positive solutions by using the nonlinear alternative of the Leray-Schauder type and Krasnosel'skii's fixed point theorem. In addition, several other sufficient conditions are established for the existence of at least triple, $n$ or $2n-1$ positive solutions. Two examples are given to illustrate our theoretical results.