Positive solutions for boundary value problem of nonlinear fractional functional differential equations

In this paper, we investigate the existence of positive solutions for the nonlinear Captuo fractional order functional differential equation $D_{0+}^{a}u(t)+a(t)f(u_t)=0\text{,}0<t<1\text{,}1<\alpha ⩽2\text{,}$ where $D_{0+}^a$ is the Captuo fractional order derivative, subject to the boundary conditions $\begin{array}{c}‐\mathit{au}(t)+{\mathit{bu}}^{\text{'}}(t)=\xi (t)\text{,}‐\tau ⩽t⩽0\text{,}\hfill \\ \mathit{cu}(t)+{\mathit{du}}^{\text{'}}(t)=\eta (t)\text{,}1⩽t⩽1+\beta \text{,}\hfill \end{array}$ we obtain the existence results of positive solutions by using some fixed point theorems.