On the Method of Guiding Functions in the Problem of the Existence of Bounded Solutions for Differential Inclusions

At the end of the 20th — beginning of the 21st centuries, in connection with the new opportunity of applications to current problems of mathematics, mechanics, control theory, physics and other sciences, the need arose for a significant expansion of the classes of guiding functions under consideration, first introduced by M. A. Krasnosel’skii and A. I. Perov. In particular, for differential equations, a class of guiding functions on a given set and a class of multivalent vector guiding functions were introduced, which were later generalized to the case of differential inclusions. In this paper, along with the classical method of guiding functions, the method of guiding functions on a given set and the method of multivalent vector guiding functions are applied to the problem of the existence of bounded solutions in nonlinear objects described by differential inclusions, the right-hand side of which has convex compact values, satisfies the upper Caratheodory conditions and the sublinear growth condition.