Existence and uniqueness of solution for fractional differential equations with integral boundary conditions

Xiping Liu, University of Shanghai for Science and Technology, Shanghai, P. R. China
Mei Jia, University of Shanghai for Science and Technology, Shanghai, P. R. China
Baofeng Wu, University of Shanghai for Science and Technology, Shanghai, P. R. China

E. J. Qualitative Theory of Diff. Equ., No. 69. (2009), pp. 1-10.

Abstract: This paper is devoted to the existence and uniqueness results of solutions for fractional differential equations with integral boundary conditions.
$$
\left\{
\begin{array}{l}
^C\hspace{-0.2em}D^\alpha x(t)+f(t,x(t),x'(t))=0,\quad t\in(0,1),\\
x(0)=\int^1_0 g_0(s,x(s))\mathrm{d}s ,\\
x(1)=\int^1_0 g_1(s,x(s))\mathrm{d}s ,\\
x^{(k)}(0)=0,\,\ k=2,3,\cdots, [\alpha]-1.
\end{array} \right.
$$
By means of the Banach contraction mapping principle, some new results on the existence and uniqueness are obtained. It is interesting to note that the sufficient conditions for the existence and uniqueness of solutions are dependent on the order $\alpha$.

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