Existence of Positive Solutions for Fractional Differential Equation with Fractional Differential Boundary Value Condition

The existence of positive solutions for the fractional differential equation with fractional differential boundary value condition $\left\{\begin{array}{l}D_{0+}^v u(t)+h(t) f(t, u(t))=0 \quad(0<t<1, n-1<v \leqslant n), \\u(0)=u^{\prime}(0)=u^{\prime \prime}(0)=\cdots=u^{(n-2)}(0)=0 \quad(n \geqslant 3), \\\left(D_{0+}^\alpha u(t)\right)_{t=1}=\sum\limits_{i=1}^{m-2} \beta_i\left(D_{0+}^{\alpha_i} u(t)\right)_{t=\eta_i}\left(1 \leqslant \alpha_i \leqslant \alpha \leqslant n-2\right)\end{array}\right.$ is considered under some conditions, where D₀₊^v is Rimann-Liouvile fractional differential, η_i∈(0, 1), 0 < η₁ < η₂ < … < η_m-2 < 1, β_i∈[0, ∞). Firstly, the Green function for the above fractional differential equation is constructed. The properties of the Green's function are obtained. Secondly, by using the fixed point index theorem on convex functional to calculate the fixed point index, the conclusion that there is at least one positive solution to the above boundary value problem is obtained. Finally, an example is given to illustrate the application of the main theorem.