Method of non-smooth integral guiding functions in periodic solutions problem for inclusions with causal multioperators

As it is known, differential inclusions are very useful mathematical tools to describe nonlinear control systems with feedback, automatic control systems, discontinuous systems, impulse response and other objects of modern engineering, mechanics, physics. In the present paper the new method to solving the problem of periodic oscillations of controlled systems described by a differential inclusion with a causal multioperator is introduced. Firstly differential equations with causal operator, or Volterra type equations where considered by L. Tonelli and A.N. Tikhonov. A.N. Tikhonov used them as the model in study of some thermal conductivity problems, in particular the problem of body coding when there is radiation from its surface. At first we consider the case when the multioperator is closed and convex-valued. Then the case of a non-convex-valued and lower semicontinuous right-hand part is considered. As the main research tool of the problem in both cases a modified method of the classical guiding function is applied. Namely, the method of nonsmooth integral guiding function is considered. Application of topological degree theory and this method allows to establish the solvability of periodic problem in each of the cases.