The averaging method and the asymptotic behavior of solutions to differential inclusions (Q1037071)

The authors prove an analog of the first Bogolyubov theorem [\textit{N. N. Bogolyubov} and \textit{Y. A. Mitropol'skii}, Asymptotic methods in the theory of nonlinear oscillations. Fourth edition, revised and augmented (Russian). Nauka, Moscow (1974; Zbl 0303.34043)] for the differential inclusions \[ 0\in x'+\varepsilon F(t,x),\qquad x(0)=k \tag{1} \] and \[ 0\in x'+F_0(x),\qquad x(0)=k.\tag{2} \] The authors assume that the set-valued functions \(F\) and \(F_0\) are upper semicontinuous in \(x\) and satisfy certain other properties, and are related by the ``averaging condition'' \[ \limsup_{n\to \infty} \tfrac1n \int_0^n s(w,F(t,v))dt\leq s(w,F_0(v)) \] where \(s(x^{*},M)=\sup\{<x,x^{*}>, x\in M\}\). Then, it is proven that for any \(\eta>0\) there exists an \(\varepsilon_0 >0\) such that for each \(\varepsilon\in (0,\varepsilon_0]\) and for each solution \(x(t,\varepsilon)\) of (1), there exists a solution \(x_0(t,\varepsilon)\) to (2) such that \(|x(t,\varepsilon)-x_0(t,\varepsilon)|<\eta\) for all \(t \in [0,\frac{L}{\varepsilon}]\), where \(L\) is a constant. Theorems regarding asymptotic behavior and dependence on parameters are also given.