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Existence results for anti-periodic boundary value problems of fractional differential equations

DOI: 10.1186/1687-1847-2013-53

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Abstract:

In this paper, the author is concerned with the following fractional equation \[ { }^CD_{0+}^\alpha u(t)=f(t,u(t),{}^CD_{0+}^{\alpha _1 } u(t),{ }^CD_{0+}^{\alpha _2 } u(t)),t\in (0,1) \] with the anti-periodic boundary value conditions \[ u(0)=-u(1), \;\; t^{\beta _1 -1}\; {}^CD_{0+}^{\beta _1 } u(t)_{\vert t\to 0}=-t^{\beta _1 -1}\; {}^CD_{0+}^{\beta_1 } u(t)_{\vert t=1}, \] \[ t^{\beta _2 -2}\; {}^CD_{0+}^{\beta _2 } u(t)_{\vert t\to 0} =-t^{\beta _2-2}\; {}^CD_{0+}^{\beta _2 }u(t)_{\vert t=1}, \] where ${ }^CD_{0+}^\gamma $ denotes the Caputo fractional derivative of order $\gamma $, the constants $\alpha ,\alpha _1 ,\alpha _2 ,\beta _1 ,\beta _2 $ satisfy the conditions that $2<\alpha \leq3,0<\alpha _1 \leq1<\alpha _2 \leq2,0<\beta _1 <1<\beta _2 <2$. Differing from the recent researches, the function $f$ involves Caputo fractional derivative ${}^CD_{0+}^{\alpha _1 } u(t)$ and ${}^CD_{0+}^{\alpha _2 } u(t)$. In addition, the author put forward a new anti-periodic boundary value conditions, which are more suitable than that were studied in the recent literature. By applying Banach contraction mapping principle, and Leray-Schauder degree theory, some existence results of solution are obtained.

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