On the Cauchy problem with an initial point lying on the boundary of ordinary differential equation’s domain of definition

Abstract

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

Keywords:

boundary initial-value problem, boundary of domain of definition, boundary initial point, maximal and minimal solutions, nonuniqueness sector, differential inequality, comparison theorems