On the differentiability with respect to the initial condition of a solution to an ordinary differential equation

In this article a version of the theorem on differentiability of an implicit function is proved, and is then applied to establish a theorem on the differentiability with respect to the initial condition of a solution to an ordinary differential equation in a locally convex space obtained by introducing into a Banach space a locally convex topology satisfying certain conditions. An example is exhibited which cannot be studied by previously known theorems of this type.
Bibliography: 13 titles.