Method of generalized integral guiding functions in the problem of the existence of periodic solutions for functional-differential inclusions

We suggest new methods for the solution of a periodic problem for a nonlinear object described by the differential inclusion x'(t) ∈ F(t, xt) under the assumption that the multimapping F has convex compact values and satisfies the upper Carath´eodory conditions. We also study the case in which this multimapping is not convex-valued but is normal. The class of normal multimappings includes, for example, bounded almost lower semicontinuous multimappings with compact values and mappings satisfying the Carathéodory conditions. In both cases, a generalized integral guiding function is used to study the problem.