Averaging of differential inclusions with ``fast'' and ``slow'' variables (Q757668)

It is shown that for any solution (x(\(\cdot),y(\cdot))\) to the system of differential inclusions \[ x'(t)\in \mu F(t,x(t),y(t),\mu),\quad y'(t)\in G(t,x(t),y(t),\mu),\quad x(t_ 0)=x_ 0,\quad y(t_ 0)=y_ 0, \] where \(\mu \in (t_ 0,\mu_ 0)\) and \(\mu_ 0\) depends on \(\epsilon\), there exists a solution \(\xi\) (\(\cdot)\) to the average differential inclusion \(\xi '(t)\in \mu F_ 0(\xi),\quad \xi (t_ 0)=x_ 0,\) such that \(| x(t)-\xi (t)| <\epsilon\) for \(t\in [t_ 0,t_ 0+\mu^{-1}]\). Except of some regularity of F, G, \(F_ 0\) it is assumed that for every \(\epsilon >0\) there exists \(s_{\epsilon}>0\) such that for every \(s>s_{\epsilon}\) and every solution \(\bar y(\cdot)\) to the generating inclusion \(\bar y'(t)\in G(t,x_ 0,\bar y(t),0),\quad \bar y(t_ 0)=y_ 0\) we have \[ \frac{1}{s}\int^{t_ 0+s}_{t_ 0}F(t,x_ 0,\bar y(t),0)dt\quad \subset \quad F_ 0^{\epsilon}(x_ 0). \] Some methods for the construction of an averaged inclusion are described.