УДК 517.9 Розглянуто різні означення похідної множиннозначного відображеннята їхні властивості. Вивчається лінійне множиннозначне диференціальне рівняння та досліджується існування розв'язків цього рівняння з похідною Хукухари, PS-похідноюта BG-похідною. Отримані результати проілюстровано на модельних прикладах.

We discuss various definitions and properties of the derivatives of set-valued mappings. We also consider a linear set-valued differential equation and investigate the problem of existence of solutions of this equation with Hukuhara derivative, PS-derivative, and BG-derivative. The obtained results are illustrated by model examples.

In this article one optimal control problem when the system behavior is described by linear fuzzy differential equations is considered. The conditions of its solvability are formulated, and the optimal time and optimal controls are obtained.

The article discusses various definitions of the derivative of a set-valued mapping and their properties. Also, a linear set-valued differential equation is considered and the existence of solutions for this equation with Hukuhara derivative, Plotnikov-Skripnik derivative and Bede-Gal derivative is investigated.

The article presents some definitions of derivatives for set-valued mappings and their properties. A linear set-valued differential equation is considered and conditions for the existence of basic solutions are given. Subsequently, one optimal control problem is considered, when the system behavior is described by linear set-valued differential equations.

We consider a multivalued discrete system and study its properties and the existence of its solution.

In the introduction of this article, we gave a brief overview of publications on the theory of set-valued equations. Next, we considered slow-fast systems of set-valued differential equations and substantiated the possibility of applying the averaging method for approximate resolution or investigation of the properties of solutions of such systems. c ©2019 World Academic Press, UK. All rights reserved.

In the introduction of the article we given an overview of the results for set-valued equations. Further we considered the set-valued discrete-time dynamical systems and substantiates the averaging method for nonlinear set-valued discrete-time systems with a small parameter.

Recently, many authors have considered the existence, uniqueness and properties of solutions of set-valued differential and integral-differential equations, higher-order equations and investigated impulsive and control systems within the framework of the theory of set-valued equations. Obviously, obtaining all these results would be impossible without the development of the theory of set-valued analysis. In particular, when considering set-valued differential equations, when the right-hand side satisfies Caratheodory conditions, absolutely continuous set-valued mappings are considered as solutions. The article show that absolutely continuous set-valued mappings (under the existing concepts of the derivative and integral) do not satisfy those properties that are satisfied by single-valued absolutely continuous functions and therefore it is proposed to introduce additionally the concept of a integrally absolutely continuous set-valued mapping.

In this article we consider the averaging method for differential inclusions with fuzzy right-hand side for the case when the limit of a method of an average does not exist.

In the given article we consider a problem of a meeting of fuzzy linear objects and we receive a necessary condition of an optimality.

We consider the applicability of the scheme of complete averaging to control problems with terminal quality criterion in the case where the behavior of the system is described by a controlled fuzzy differential equation.

We propose and justify algorithms for partial and complete averaging of systems of discrete equations and inclusions. On the basis of the averaging schemes obtained, we construct algorithms for the numerical asymptotic solution of problems of optimal control for discrete systems.

We justify partial averaging schemes for quasidifferential equations that generalize the corresponding results for differential equations and inclusions.

We justify a method of complete and partial averaging on finite and infinite intervals for differential inclusions with many-valued pulses.

IT HAS been shown previously that the method of polymerization by irradiation of solid materials in the presence of monomers in the vapour state extends the possibilities of this process considerably and enables graft polymerization to be achieved not only on organic, but also on mineral substrates [1-2]. :New organo-mineral ion-exchange materials [3] and mechanically strong semiconductors [4] have been produced by this means. A process of a similar nature, based on irradiation of mineral powders in the presence of a two-component gas mixture has been applied to the synthesis of higher carboxylic acids [5]. The process of polymerization from the vapour phase on mineral surfaces has however not been studied sufficiently, in particular the fundamental kinetic relationships have not been elucidated, and knowledge of these is necessary both for selection of the optimal technological conditions and for understanding the peculiarities and mechanism of the process. From the kinetic point of view this process is of particular interest because essentially we have here a new case of heterogeneous polymerization with the distinctive feature that active eentres initiating polymerization are generated continuously during the course of the reaction throughout the entire volume of the solid phase, both in the original mineral particles and in the newly formed surface layer of polymer. This investigation is devoted to elucidation of the main kinetic relationships of radiation graft polymerization from the vapour phase on mineral surfaces. The materials selected for study were methyl methacrylate (MMA) and styrene (St) polymerized on magnesium oxide and Aerosi]. With the four systems a study was made of the dependence of the reaction rate on dose rate, temperature, vapour pressure of monomer and the degree of adsorption of the latter on the irradiated substrate. We also studied the dependence of the reaction rate on the quantity of polymer formed on the mineral powder up to formation on the mineral particles of fairly thick polymer layers, corresponding to the thickness of the grafted layers in the organo-mineral graft materials mentioned above.