Summary: The authors study the following Cauchy-type problem for the nonlinear differential equation of fractional order , ,
containing the Riemann-Liouville fractional derivative , on a finite interval of the real axis in the space of summable functions . An equivalence of this problem and a nonlinear Volterra integral equation are established. The existence and uniqueness of the solution to the above Cauchy-type problem are proved by using the method of successive approximations. Corresponding assertions for the ordinary differential equations are presented. Examples are given.