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Existence and uniqueness of solution for fractional differential equations with integral boundary conditions. (English) Zbl 1201.34010

Summary: This paper is devoted to existence and uniqueness of solutions for fractional differential equations with integral boundary conditions.
\[ \begin{cases} ^CD^\alpha x(t)+f(t,x(t),x'(t))=0,\quad & t\in(0,1),\\ x(0)=\int^1_0 g_0(s,x(s))\,\mathrm{d}s ,\\ x(1)=\int^1_0 g_1(s,x(s))\,\mathrm{d}s ,\\ x^{(k)}(0)=0,\,\;k=2,3,\dots, [\alpha]-1.\end{cases} \]
By means of the Banach contraction mapping principle, some new results on existence and uniqueness are obtained. It is interesting to note that the sufficient conditions for existence and uniqueness of solutions depend on the order \(\alpha\).

MSC:

34A08 Fractional ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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