On approach to the study of the periodic problem for random differential equations

Geometric and topological methods of analysis applied to problems of nonlinear oscillations of dynamical systems go back to the names of A. Poincare, L. Brauer, P.S. Alexandrov, G. Hopf, J. Leray, J. Schauder. Later, these methods were developed and demonstrated their effectiveness in the works of many mathematicians. Note, in particular, an extremely fruitful direction associated with the notion of a guiding function, whose base was laid by M.A. Krasnosel’skii and A.I. Perov. In this paper, to study the periodic problem of random differential equations, we use a modification of the classical notion of a guiding function - a random nonsmooth multivalent guiding function. A significant advantage over the classical approach is the ability to "localize" $ $ the verification of the main condition of "directionality" $ $ on a domain that depends on the guiding function itself and on the domain not of the whole space, but of its subspace of lower dimension. In classical works on the method of guiding functions, as a rule, it is assumed that these functions are smooth over the whole phase space. This condition may seem restrictive, for example, in situations where the guiding potentials are different in different domains of the space. To remove this restriction, the paper considers nonsmooth direction potentials and their generalized gradients.