Approximation from Above of Systems of Differential Inclusions with Non-Lipschitzian Right-Hand Side

E. V. Sokolovskaya ; O. P. Filatov

Matematičeskie zametki, Tome 78 (2005) no. 5, pp. 763-772

Résumé

Suppose that $\mathbb R^n$ is the $p$-dimensional space with Euclidean norm ${\|\cdot\|}$, $K(\mathbb R^p)$ is the set of nonempty compact sets in $\mathbb R^p$, $\mathbb R_+=0,+\infty)$, $D=\mathbb R_+\times\mathbb R^m\times\mathbb R^n\times[0,a]$, $D_0=\mathbb R_+\times\mathbb R^m$, $F_0\colon D_0\to K(\mathbb R^m)$, and $\operatorname{co}F_0$ is the convex cover of the mapping $F_0$. We consider the Cauchy problem for the system of differential inclusions $$ \dot x\in\mu F(t,x,y,\mu),\quad \dot y\in G(t,x,y,\mu),\qquad x(0)=x_0,\quad y(0)=y_0 $$ with slow $x$ and fast $y$ variables; here $F\colon D\to K(\mathbb R^m)$, $G\colon D\to K(\mathbb R^n)$, and $\mu\in[0,a]$ is a small parameter. It is assumed that this problem has at least one solution on $[0,1/\mu]$ for all sufficiently small $\mu\in[0,a]$. Under certain conditions on $F$, $G$, and $F_0$, comprising both the usual conditions for approximation problems and some new ones (which are weaker than the Lipschitz property), it is proved that, for any $\varepsilon>0$, there is a $\mu_0>0$ such that for any $\mu\in(0,\mu_0]$ and any solution $(x_\mu(t),y_\mu(t))$ of the problem under consideration, there exists a solution $u_\mu(t)$ of the problem $\dot u\in\mu\operatorname{co}F_0(t,u)$, $u(0)=x_0$ for which the inequality $\|x_\mu(t)-u_\mu(t)\|\varepsilon$ holds for each $t\in[0,1/\mu]$.