existence and uniqueness theorem, analytic approximate solution, movable singular point, priori uncertainty, complex region.

Nonlinear differential equations have features that do not allow the use of the classical theory of linear differential equations. The presence of movable singular points actualize the development of the theory. In this respect, there are publications devoted to the major problems (theorems of existence and uniqueness of nonlinear differential equations’ solutions), and a new approach to their proof, which allows the use of these results in the solution of all the problems arising in the analytical method of nonlinear differential equations’ approximate solution. This paper presents a generalization to the complex domain, previous results in the real domain.

Orlov Viktor Nikolaevich, Dr. Sci. Phys. & Math., Theory and Methods of Teaching Mathematics, Humanitarian and Pedagogical Academy (branch) of V. I. Vernadsky Crimean Federal University, Yalta, Russia; Professor at the Department of Mathematical Analysis, Algebra and Geometry of

I. Y. Yakovlev Cheboksary State Pedagogical University, Cheboksary, Russia.

Hmara Pavlina Vasil’evna, 2nd Year Graduate Student, Humanitarian and Pedagogical Academy (branch) of V. I. Vernadsky Crimean Federal University, Yalta, Russia.