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) is an element of F(t, x(t)) under the assumption that the multimapping F has convex compact values and satisfies the upper Caratheodory 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 Caratheodory conditions. In both cases, a generalized integral guiding function is used to study the problem.