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where D is the Caputo’s fractional derivative of order 0 < α < 1, 0 = t0 < t1 < t2 < · · · < tp < tp+1 = T , f ∈ C([0, T ] × R, R) and Ik ∈ C(R, R) are given functions satisfying some assumptions that will be specified later. ∆x(tk) = x(t + k )−x(t − k ), x(t + k ) and x(t − k ) represent the right and left limits of x(t) at t = tk respectively, and they satisfy that x(t − k ) = x(tk). If Supported by the National Natural Science Foundation of China (11071108), the Provincial Natural Science Foundation of Jiangxi, China (2010GZS0147). Corresponding author. E-mail addresses: xzzhang@yahoo.com.cn (X. Zhang), chuanxizhu@126.com (C. Zhu), wuzhaoqi conquer@163.com (Z. Wu).

The Cauchy problem for a class of fractional impulsive differential equations with delay