 The Existence of Solutions to a Class of Multi-point Boundary Value Problem of Fractional Differential Equation
Received:November 26, 2011  Revised:October 09, 2012
Key Words: fractional differential equation   multi-point boundary value problem   coincidence degree.
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.11071001), the Natural Science Foundation of Anhui Province (Grant No.\,1208085MA13) and 211 Project of Anhui University (Grant No.KJTD002B).
 Author Name Affiliation Xiaohong HAO Department of Mathematical Sciences, Anhui University, Anhui 230601, P. R. China Zongfu ZHOU Department of Mathematical Sciences, Anhui University, Anhui 230601, P. R. China
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In this paper, we consider the following multi-point boundary value problem of fractional differential equation \align &D^{\alpha}_{0+}u(t) = f(t, u(t),~D^{\alpha-1}_{0+}u(t), D^{\alpha-2}_{0+}u(t), D^{\alpha-3}_{0+}u(t)),~~t\in(0,1), \\&I^{4-\alpha}_{0+}u(0) = 0, ~D^{\alpha-1}_{0+}u(0)=\displaystyle{\sum_{i=1}^{m}}\alpha_{i}D^{\alpha-1}_{0+}u(\xi_{i}),\\&D^{\alpha-2}_{0+}u(1)=\sum\limits_ {j=1}^{n}\beta_{j} D^{\alpha-2}_{0+}u(\eta_{j}),~D^{\alpha-3}_{0+}u(1)-D^{\alpha-3}_{0+}u(0)=D^{\alpha-2}_{0+}u(\frac{1}{2}),\endalign where $3<\alpha \leq 4$ is a real number. By applying Mawhin coincidence degree theory and constructing suitable operators, some existence results of solutions can be established.