Багатоточкові задачі для коливних систем з повільно змінними частотами

Анотація

The authors consider the multipoint problem with parameters, $$dx/d\tau=a(x,\varphi,\tau,╣),\quad d\varphi/d\tau =\omega(\tau)/\varepsilon+b(x,\varphi,\tau,╣),$$ $$F(x\vert \sb {\tau=\tau\sb 1},\dots,x\vert \sb {\tau=\tau\sb r},\varphi\vert \sb {\tau=\tau \sb 1},\dots,\varphi\vert \sb {\tau=\tau\sb r},╣,\varepsilon)=0.$$ Here $x\in{R}\sp n,\ \varphi\in{R}\sp m,\ \tau\in [0,L]\subset{R}, ╣\in{R}\sp s$ is the parameter, $\varepsilon\in(0,\varepsilon\sb 0], F\colon {R}\sp n\times{R}\sp m\times {R}\sp s\times(0,\varepsilon\sb 0]\to{R}\sp {(n+m+s)},\ 0\le\tau\sb 1\le\dots\le \tau\sb r\le L, r\ge2$. The system is averaged with respect to the fast variable $\varphi $. The authors prove the solvability of the problem and obtain the estimate $$\Vert x(\tau,\varepsilon)-y(\tau,\varepsilon)\Vert +\Vert \varphi(\tau,\varepsilon)- \psi(\tau,\varepsilon)\Vert +\Vert ╣(\varepsilon )-╣\sb 0\Vert \le\sigma \varepsilon\sp \alpha$$ for all $(\tau,\varepsilon)\in[0,L]\times(0,\epsilon\sb 0].$ Here $y,\psi$ are the solution of the averaged system, $\sigma\in{R},\ \alpha\in R$.