带有分数阶导数边值条件的分数阶微分方程正解的存在性

讨论了一类带有分数阶导数边值条件的分数阶微分方程 $\left\{\begin{array}{l}D_{0+}^v u(t)+h(t) f(t, u(t))=0 \quad(00+v是Rimann-Liouvile分数阶导数，η_i∈(0, 1), 0 < η₁ < η₂ < … < η_m-2 < 1, β_i∈0, ∞)。文中给出其格林函数及相关性质，运用凸泛函上的不动点指数定理来计算不动点指数，从而得到了上述边值问题至少存在一个正解的结论。最后通过一个例子说明定理的具体应用。

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(00+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.