Abstract

In this paper, we investigate the existence of a solution of the Cauchy problem (initial-value problem) with the initial point located on the boundary of the domain of definition of a first-order differential equation. This formulation of the problem differs from the one accepted in classical theory, where the initial point is always an internal point of the domain. Our aim is to find the conditions for the right-hand side of the equation and the boundary that would ensure the existence or absence of a solution to the boundary Cauchy problem. In the previous article devoted to this problem, the authors used the standard Euler polygonal line method to solve this problem and described all the cases when this method was used to get the desired answer. However, the polygonal line method, even given some of its advantages (constructability, the ability to use a computer), requires for its implementation that both the equation and the domain of its definition meet certain restrictions, thus inevitably narrowing the class of acceptable equations. In this paper, we attempt to maximize the results obtained earlier, and for this purpose, we use a completely different approach. The original equation is extended in such a way that the initial boundary value problem becomes the ordinary internal Cauchy problem, for which the standard Peano theorem is applied. In order to answer the question on whether the solution of the modified Cauchy problem is also the solution of the original boundary Cauchy problem, the so-called comparison theorems and differential inequalities are applied. This article is an independent study not based on our previous work. For the completeness of presentation, new proofs are given for the previously obtained results, which are based on the new approach. As a result, we expanded the class of equations under consideration, removed the previous requirements for the convexity and smoothness of the boundary curves, and added the cases that could not be considered using the polygonal line method. This paper aims to fill the gap on the existence or absence of solutions to the boundary Cauchy problem presented in the literature.

中文翻译：

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关于常微分方程定义域边界上的柯西问题集

摘要

在本文中，我们研究了柯西问题（初始值问题）的解的存在性，其初始点位于一阶微分方程定义域的边界上。问题的这种表述不同于经典理论中接受的那种表述，在经典理论中，初始点始终是域的内部点。我们的目的是找到方程右侧和边界的条件，以确保存在或不存在边界柯西问题的解。在专门讨论此问题的上一篇文章中，作者使用标准的Euler多边形线方法解决了此问题，并描述了使用此方法获得所需答案时的所有情况。但是，折线方法即使具有某些优势（可构造性，（使用计算机的能力）要求其实现，即方程式及其定义的域都必须满足某些限制，从而不可避免地缩小了可接受的方程式的范围。在本文中，我们试图最大化先前获得的结果，为此，我们使用了一种完全不同的方法。原始方程式以这样的方式扩展，使得初始边值问题成为普通的内部柯西问题，为此应用了标准Peano定理。为了回答关于改进的柯西问题的解是否也是原始边界柯西问题的解的问题，应用了所谓的比较定理和微分不等式。本文是一项独立研究，并不基于我们之前的工作。为了演示的完整性，基于新方法，为先前获得的结果提供了新的证明。结果，我们扩展了所考虑方程的类别，消除了先前对边界曲线的凸度和平滑度的要求，并增加了使用折线法无法考虑的情况。本文旨在填补文献中提出的边界柯西问题解的存在与否的空白。