Abstract:

In this paper, by using Banach's contraction principle and Schaefer's fixed point theorem, we study the existence and uniqueness of solutions for the boundary value problems of nonlinear fractional differential equations on star graph $\left\{\begin{array}{ll} _{C}D_{0,x}^{\alpha}u_{i}(x)=f_{i}(x,u_{i}(x),\ _{C}D_{0,x}^{\beta}u_{i}(x)), 0<x<l_{i}, &i=1,2,\cdots,k,\\ u_{i}'(0)=u_{i}(1)=0, & i=1,2,\cdots,k, \\ u_{i}''(l_{i})=u_{j}''(l_{j}), & i,j=1,2,\cdots,k, i\neq j, \\ \sum\limits_{i=1}^ku_{i}''(l_{i})=0, & i=1,2,\cdots,k, \end{array} \right.$ where $2<\alpha\leq3, 0<\beta<1,\ _{C}D_{0,x}^{\alpha},\ _{C}D_{0,x}^{\beta}$ are Caputo fractional derivative, $f_{i}, i=1,2,\cdots,k$ with respect to a continuously differentiable function of three variables on $[0,1]\times \mathbb{R}\times \mathbb{R} $.

Key words: Boundary value problem of fractional differential equations, Star graph, Banach's contraction principle, Schaefer's fixed point theorem