Solution of Tikhonov’s Motion-Separation Problem Using the Modified Newton–Kantorovich Theorem

Abstract

The paper presents a new way to prove the existence of a solution of the well-known Tikhonov’s problem on systems of ordinary differential equations in which one part of the variables performs “fast” motions and the other part, “slow” motions. Tikhonov’s problem has been the subject of a large number of works in connection with its applications to a wide range of mathematical models in natural science and economics. Only a short list of publications, which present the proof of the existence of solutions in this problem, is cited. The aim of the paper is to demonstrate the possibility of applying the modified Newton–Kantorovich theorem to prove the existence of a solution in Tikhonov’s problem. The technique proposed can be used to prove the existence of solutions of other classes of problems with a small parameter.