Proofs of averaging theorems for differential inclusions (Q2780459)

The author considers the differential inclusion with slow and fast variables NEWLINE\[NEWLINE \dot{x}\in\mu F(t,x,y,\mu), \;\;\dot{y}\in G(t,x,y,\mu), NEWLINE\]NEWLINE where \(\mu\) is a small parameter. In the earlier papers [the author, Differ. Equations 31, 47--54 (1995); translation from Differ. Uravn. 31, 54--62 (1995; Zbl 0852.34015); the author and \textit{M. M. Khapaev}, Math. Notes 47, 596--601 (1990); translation from Mat. Zametki 47, No. 6, 102--109 (1990; Zbl 0723.34010) etc., see also more detailed discussion in the book by \textit{O. P. Filatov} and \textit{M. M. Khapaev}, Averaging of systems of differential inclusions (in Russian), Moscow: Moscow University Press (1998)], the author stated and studied the problem of approximation of the slow variables \(x\) by the averaged differential inclusion on the asymptotically large time interval \(O(\mu^{-1})\). It was shown that there are three different types of approximation: the upper approximation, lower one, and reciprocal one.NEWLINENEWLINENEWLINEHere, new proofs of the averaging theorems are given for approximating inclusions in all three types mentioned above. These new proofs are simpler than the previous ones. Simplification is accomplished by using the lemma on continuous dependence of the solution of the differential inclusion from the right-hand part and initial conditions. This important lemma is also completely proved in this paper.