多点边值问题解的逼近性质
Approximation properties of solutions to multipoint boundary-value problems
摘要：
我们考虑一类广泛的线性边值问题系统的$m$常微分方程的$r$阶，被称为一般边值问题。它们的解$y:[a,b]\to \mathbb{C}^{m}$属于Sobolev空间$(W_1^{r})^m$，边界条件以$By=q$的形式给出，其中$B:(C^{(r-1)})^{m}\to\mathbb{C}^{rm}$是任意的连续线性算子.我们证明了这样一个问题的解决方案可以近似与任意精度$(W_1^{r})^m$的解决方案，多点边值问题具有相同的右手边。这些多点问题是明确建立的，不依赖于一般边值问题的右侧。对于这些问题，我们得到了在赋范空间$(W_1^{r})^m$和$(C^{(r-1)})^{m}$中解的误差估计。
英文摘要：
We consider a wide class of linear boundary-value problems for systems of $m$ ordinary differential equations of order $r$, known as general boundary-value problems. Their solutions $y:[a,b]\to \mathbb{C}^{m}$ belong to the Sobolev space $(W_1^{r})^m$, and the boundary conditions are given in the form $By=q$ where $B:(C^{(r-1)})^{m}\to\mathbb{C}^{rm}$ is an arbitrary continuous linear operator. We prove that a solution to such a problem can be approximated with an arbitrary precision in $(W_1^{r})^m$ by solutions to multipoint boundary-value problems with the same right-hand sides. These multipoint problems are built explicitly and do not depend on the right-hand sides of the general boundary-value problem. For these problems, we obtain estimates of errors of solutions in the normed spaces $(W_1^{r})^m$ and $(C^{(r-1)})^{m}$.
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