The global theorem on implicit function and its applications in the theory of ordinary differential equations.

*(Ukrainian. English summary)*Zbl 1007.26009Summary: The classical implicit function theorem (in Italy this theorem is known as the Dini theorem) and its generalizations state the (local) existence of a solution of the equation \(g(t,x)=0\), i.e., they claim the existence of a solution in some neighborhood of a given point. At the same time theorems on global existence of solutions and theorems on existence of solutions on a maximal interval are of great importance in the theory of differential equations, first of all for applications. In this paper we study the problem of existence of global solutions of the equation as well as the problem on extrema of functions defined by a system of ordinary differential equations applied in physics.