ABSTRACT. An apparatus for proving existence theorems for periodic solutions of equations with discontinuous right-hand side and differential inclusions is developed.KEY WORDS: differential equations and inclusions, topology of the solution space, existence theorems, Cauchy problem, periodic solutions.In this paper, we explain how to extend the applicability area of the approach [1] to the existence of periodic solutions for wider classes of ordinary differential equations and inclusions.We shall use the ideas developed in [2][3][4]. Part of the notation from [2] will be used without additional explanations (see also [3,4]).Earlier, similar results were proved for equations with a continuous right-hand side or equations whose right-hand side satisfies the Carath~odory condition. As a consequence of the Cauchy problem theory that emerges in the context of our general approach (see [2][3][4][5]), our results turn out to be applicable to equations and inclusions with complicated discontinuities in their right-hand sides, including discontinuities in the space variables.w The length of the existence interval for solutions of the Cauchy problemWe assume that the right-hand side of the differential inclusion in question, Stated in this way, the uniqueness of the zero solution turns out to be similar to the following condition:any solution of inclusion (1) whose domain touches the segment [to, tl] can be continued to the entire segment It0, C1].Indeed, under natural restrictions on (1), the violation of this condition implies that inclusion (1) has a solution z(C), C0 _< t < C1, in the domain U0 such that [Iz(c)l[ --* cc as c -, tl or a solution z(t),Therefore, one can apply the considerations used in [6] to study the length of the existence interval for the solution. In order to simplify formulations, we shall not strive for the greatest possible generality.Recall some key notations from [2-4].The symbol Ri(U) denotes the set of all Z C Cs(U) that satisfy the condition if z E Z and a segment I lies in the domain ~r(z) of the function z, then zli E Z.