The method of generalized integral guiding function in the periodic problem of differential inclusions

In the present paper we consider new methods for solving the periodic problem for a nonlinear system governed by a differential inclusion of the following form:
$$x'(t)\in F(t,x(t)).$$
In the first part of the article we assume that the multivalued map $F:\mathbb{R} \times \mathbb{R}^n \multimap \mathbb{R}^n$ has convex compact values, satisfies the upper Caratheodory conditions, sublinear growth condition and $T$-periodic in the first argument. Under the above assumptions the closed multivalued superposition operator $P_F\,:\,C([0,T];\mathbb{R}^n)\rightarrow P(L^1([0,T];\mathbb{R}^n))$, assigning to each function $x(\cdot)$ the set of all integrable selections of the multifunction $F(t,x(t))$ is well defined. In the second part of the article we assume that the multivalued map $F:\mathbb{R} \times \mathbb{R}^n \multimap \mathbb{R}^n$ is regular with compact values satisfying the $T$-periodicity condition in the first argument. Notice that the class of regular multimaps is broad enough. It includes, in particular, bounded almost lower semicontinuous multimaps with compact values. In both cases for the study of the periodic problem the generalized integral guiding function method is applied. An essential development of the concept of the guiding function is the fact that the basic condition is assumed to hold, firstly, in an integral form; secondly, in the domain defined by the guiding function; and at last, not necessarily for all integrable selections of the superposition multioperator. Application of the coincidence degree theory and the multivalued maps theory allows to establish the solvability of the periodic problem.