Summary: The paper is devoted to the study of a Cauchy-type problem for the nonlinear differential equation of fractional order ,
containing the Marchaud-Hadamard-type fractional derivative , on the half-axis in the space defined for by
Here is the subspace of of functions with compact support on infinity: for large enough . The equivalence of this problem and a nonlinear Volterra integral equation is established. The existence and uniqueness of the solution of the above Cauchy-type problem is proved by using the Banach fixed point theorem. The solution in closed form of the above problem for the linear differential equation with is constructed. The corresponding assertions for the differential equations with the Marchaud-Hadamard fractional derivative are presented. Examples are given.