Mutual \(\epsilon\)-approximation of solutions of a system of differential inclusions and of the averaged inclusion
The paper is a continuation of the above reviewed paper. It is proved that not only for any solution (x(\(\cdot),y(\cdot))\) to the system of differential inclusions \[ (1)\quad x'(t)\in \mu F(t,x(t),y(t),\mu),\quad y'(t)\in G(t,x(t),y(t),\mu),\quad x(t_ 0)=x_ 0,\quad y(t_ 0)=y_ 0, \] where \(\mu \in (t_ 0,\mu_ 0)\) and \(\mu_ 0\) depends on \(\epsilon\), there exists a solution \(\xi\) (\(\cdot)\) to the average differential inclusion \((2)\quad \xi '(t)\in \mu F_ 0(\xi),\quad \xi (t_ 0)=x_ 0,\) such that \((3)\quad | x(t)-\xi (t)| <\epsilon \text{ for } t\in [t_ 0,t_ 0+\mu^{-1}],\) but also for any solution \(\xi\) (\(\cdot)\) to (2) there exists a solution (x(\(\cdot),y(\cdot))\) to (1) such that (3) holds true.
scientific article; zbMATH DE number 4192144