中文摘要：

研究如下的Caputo分数阶微分积分方程初值问题：{（cDαa＋g）（x）=f（x,cDβa＋g（x））∫＋xaK（x,t,cDβa＋g（t））dt,g（k）（a）=η（k）,n-1〈β〈α〈n,k=0,1,2,…,n,其中：f：[a,b]×R→R是一个连续可微函数,且K：[a,b]×[a,b]×R→R是一个连续函数.方程中的非齐次项含有较低阶的Caputo分数阶导数.在几组不同的充分条件下,分别运用Leray-Schauder非线性选择定理和Banach压缩映射原理证明了这类方程初值问题解的存在性和唯一性.

英文摘要：

The studied initial value problem of fractional integro-differential equations with Caputo＇s fractional derivative is presented such as：{（cD αa＋g）（x）=f（x,cDβa＋g（x））＋∫xaK（x,t,cDβa＋g（t））dt,g（k）（a）=η（k）,n-1βαn,k=0,1,2,…,n,where f：×R→R is continuously differentiable,and K：××R→R is continuous.The inhomogeneous term of the equation includes the fractional derivative of lower orders.Under several types of sufficient conditions,the existence and uniqueness of solutions to this kind of fractional differential equation is proved by the Leray-Schauder theorem and the Banach contraction principle.