Abstract

The initial-value problem (the Cauchy problem) for an ordinary differential equation of the first order is considered. It is assumed that the right-hand side of the equation is a continuous function defined on a set consisting of a connected open set (a domain) of the two-dimensional Euclidean space, as well as on part of its boundary. It is known that, for any point of the domain, the Peano theorem guarantees the existence of a solution to the Cauchy problem determined on the Peano segment. The sufficient conditions for the existence of a solution to the Cauchy problem set at the boundary point of the domain are formulated, and its existence at some analog of the Peano segment is proved by the Euler polygonal method. Also, the sufficient conditions for the absence of a solution to the Cauchy problem set at the boundary point are presented.

中文翻译：

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关于柯西初始边值问题解的存在性

摘要

考虑一阶常微分方程的初值问题（柯西问题）。假定方程的右手边是在二维欧几里德空间的一个连通的开集（一个域）组成的集合上以及在其边界的一部分上定义的连续函数。众所周知，对于该域的任何一点，皮亚诺定理保证了对在皮亚诺段上确定的柯西问题的解的存在。给出了存在于该区域边界点上的柯西问题的解的存在的充分条件，并通过欧拉多边形方法证明了其在Peano段的某些类似物中的存在。而且，给出了在边界点处没有解决柯西问题的解的充分条件。