In this review we will first look in detail at V.A. Plotnikov's results on the substantiation of full and partial schemes of averaging for differential inclusions in the standard form on final and infinite interval. Then we will consider the algorithms where there is no average, but there is a possibility to find its estimation from below and from above. Such approach is also used when the detection of an average is approximate. This situation is especially typical at consideration of differential inclusions with fast and slow variables. In the last part we will give the results concerning the substantiation of the full and partial averaging method for impulsive differential inclusions on final and infinite intervals.

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We consider a special space of convex compact sets and introduce the notions of derivative and integral for a set-valued mapping that differ from those already known. We also consider a differential equation with set-valued right-hand side satisfying the Carathéodory conditions and prove theorems on the existence and uniqueness of its solutions. This approach, in contrast to the Kaleva approach, enables one to consider fuzzy differential equations as ordinary differential equations with set-valued solutions.The development of the theory of set-valued mappings [1,2] led to the question of the sense in which the derivative and integral of a set-valued mapping should be understood. In 1965, Aumann [3] introduced for the first time the notion of the integral of a set-valued mapping based on the integration of single-valued branches. Later, in 1967, using known approaches for single-valued mappings, Hukuhara [4] introduced the notions of derivative and integral for set-valued mappings. In 1969, de Blasi [5] considered a differential equation with Hukuhara derivative as a generalization of ordinary differential equations. In 1982, Plotnikov [6] considered differential inclusions with Hukuhara derivative. Later, these equations were studied in [7 -17].At the same time, in 1965, Zadeh's work [18] laid the foundation for the development of the theory of fuzzy sets. In 1983, using Hukuhara's approach for α-cuts of fuzzy mappings, Puri and Ralescu [19] introduced the notion of H-derivative and integral for fuzzy mappings. In 1985, Kaleva [20] considered fuzzy differential equations and proved a theorem on the existence and uniqueness of a solution of the Cauchy problem in the case where the right-hand side satisfies the Lipschitz condition. Later, fuzzy differential equations were studied in [21 -25].In the present paper, we introduce a special space of sets Ω, and, for a set-valued mapping whose elements are sets from Ω, we propose definitions of derivative and integral different from those already known. This approach allows one to use these notions in the case of fuzzy mappings as well. It should be noted that, in this case, the complications appearing in using the notions H-derivative and integral from [19] are absent. We also consider a differential equation in the phase space Ω and prove a theorem on the existence of a solution in the case where the right-hand side is a bounded summable function measurable with respect to t and continuous in X and a theorem on the uniqueness of a solution in the case where the right-hand side satisfies, in addition, the Lipschitz condition with respect to X.Consider the ( n + 1 )-dimensional Euclidean space R n + 1 with the norm x x x i i n n = + = + ∑ 2 1 1 .

In this paper the concept of generalized differentiability for set-valued mappings proposed by A.V. Plotnikov, N.V. Skripnik is used. The generalized set-valued differential equations with generalized derivative are considered and the existence and uniqueness theorems are proved.