부산대학교 도서관

In this paper, the authors study the possibility of applying the fullaveraging method to ordinary differential inclusions of the form$$\dot x(t) \in F\left(\frac t\varepsilon , x(t)\right),$$where $\varepsilon > 0$ denotes the small perturbation parameter, $t\in [0,L]$ the time variable and $F$ a multifunction with values thatare nonempty compact convex subsets of ${\Bbb R}^d$. They make thefollowing assumptions on the multivalued mapping $F(t,x)$:\roster\item mapping$F(\cdot, x)$ is measurable for all $x \in U$;\item thereexist a locally Lebesgue integrable function $b \: {\Bbb R}_+ \to {\BbbR}_+$ such that $ {\|F(t,x)\|\leq b(t)} $ for all $x\in U$ and $t\in{\Bbb R}_+$, and $\lim_{T\to\infty}\frac 1T\int_0^T b(t)dt =B>0$;\item thereexists a limit, in the sense of the Hausdorff metric $\rho$,$$\lim\limits_{T\to\infty}\frac 1T\int\limits _0^T F(t,x)dt=\overline{F}(x)$$for all $x \in U$;\item thereexists a constant $\lambda > 0$ such that$$\rho\left (\int\limits _{t_1}^{t_2}F(t,u(t))dt, \int\limits_{t_1}^{t_2}F(t,v(t))dt\right)\leq \lambda \int\limits _{t_1}^{t_2}|u(t)-v(t)|dt,$$for any continuous functions $u, v \: {\Bbb R}_+ \to U$ and any real$t_1, t_2 \in {\Bbb R}_+$, $t_1 \leq t_2$.\endroster