多点边值问题的逼近性质
Approximation properties of multipoint boundary-value problems
摘要：
本文考虑一类$r$阶常微分方程组的线性边值问题，其解的范围在$r$ ×连续可微函数$y:[a,b]\to\mathbb{C}^{m}$的赋范复空间$(C^{(n)})^m$上.这些问题的边界条件是最一般的形式$By=q$，其中$B$是从$(C^{(n)})^{m}$到$\mathbb{C}^{rm}$的任意连续线性算子。我们证明了所考虑的问题的解决方案可以近似在$(C^{(n)})^m$的解决方案，一些多点边值问题。后者的问题不依赖于所考虑的问题的右手边，并明确建立。
英文摘要：
We consider a wide class of linear boundary-value problems for systems of $r$-th order ordinary differential equations whose solutions range over the normed complex space $(C^{(n)})^m$ of $n\geq r$ times continuously differentiable functions $y:[a,b]\to\mathbb{C}^{m}$. The boundary conditions for these problems are of the most general form $By=q$, where $B$ is an arbitrary continuous linear operator from $(C^{(n)})^{m}$ to $\mathbb{C}^{rm}$. We prove that the solutions to the considered problems can be approximated in $(C^{(n)})^m$ by solutions to some multipoint boundary-value problems. The latter problems do not depend on the right-hand sides of the considered problem and are built explicitly.
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