On the Cauchy Problem Set on the Boundary of the Ordinary Differential Equation’s Domain of Definition

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.