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In the paper, we considered the existence and uniqueness of the globalsolution in the space of continuously differentiable functions for a nonlineardifferential equation with the Caputo fractional derivative of general form.Our main method is to derive an integral equation corresponding to the originalnonlinear fractional differential equation and to prove their eqivalence. Oncewe prove the equivalence, then the proof of existence and uniquness of solutioncan be done by standard way of using Banach fixed point theorem.

Existence and Uniqueness of the Solution to a Nonlinear Differential  Equation with Caputo Fractional Derivative in the Space of Continuously  Differentiable Functions