Cauchy problem in a scale of Banach spaces

The concept of quasidifferential operator in a scale of Banach spaces is formulated. A theorem of existence and uniqueness of a solution to the Cauchy problem for the equation with a nonlinear quasidifferential operator is proved. As an example of application of the theorem, the correctness of the nonlinear nonlocal problem of plane-parallel unsteady potential motion of a liquid with free boundary is proved.