
Existence results for antiperiodic boundary value problems of fractional differential equationsDOI: 10.1186/16871847201353 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 antiperiodic boundary value conditions \[ u(0)=u(1), \;\; t^{\beta _1 1}\; {}^CD_{0+}^{\beta _1 } u(t)_{\vert t\to 0}=t^{\beta _1 1}\; {}^CD_{0+}^{\beta_1 } u(t)_{\vert t=1}, \] \[ t^{\beta _2 2}\; {}^CD_{0+}^{\beta _2 } u(t)_{\vert t\to 0} =t^{\beta _22}\; {}^CD_{0+}^{\beta _2 }u(t)_{\vert t=1}, \] 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 antiperiodic boundary value conditions, which are more suitable than that were studied in the recent literature. By applying Banach contraction mapping principle, and LeraySchauder degree theory, some existence results of solution are obtained.
