부산대학교 도서관

In this paper, the authors study the possibility of applying the full averaging 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 that are nonempty compact convex subsets of ${\Bbb R}^d$. They make the following assumptions on the multivalued mapping $F(t,x)$: \roster \item mapping $F(\cdot, x)$ is measurable for all $x \in U$; \item there exist a locally Lebesgue integrable function $b \: {\Bbb R}_+ \to {\Bbb R}_+$ 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 there exists 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 there exists 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