title: On the Cauchy problem with an initial point lying on the boundary of ordinary differential equation’s domain of definition
metadata.name: dc.contributor.author
metadata.value: Basov, Vladimir V.
metadata.lang: -
metadata.name: dc.contributor.author
metadata.value: Iljin, Yurii A.
metadata.lang: -
metadata.name: dc.date.accessioned
metadata.value: 2020-12-18T16:52:59Z
metadata.lang: -
metadata.name: dc.date.available
metadata.value: 2020-12-18T16:52:59Z
metadata.lang: -
metadata.name: dc.date.issued
metadata.value: 2020-12
metadata.lang: -
metadata.name: dc.identifier.citation
metadata.value: Basov V. V., Iljin Yu. A. On the Cauchy problem with an initial point lying on the boundary of ordinary differential equation’s domain of definition. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 2020, vol. 7 (65), issue 4, pp. 636–648.
metadata.lang: en_GB
metadata.name: dc.identifier.other
metadata.value: This work is supported by Russian Foundation for Basic Research (grant no. 19-01-00388)
metadata.lang: -
metadata.name: dc.identifier.uri
metadata.value: http://hdl.handle.net/11701/21848
metadata.lang: -
metadata.name: dc.description.abstract
metadata.value: 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.
metadata.lang: en_GB
metadata.name: dc.description.sponsorship
metadata.value: This work is supported by Russian Foundation for Basic Research (grant no. 19-01-00388).
metadata.lang: en_GB
metadata.name: dc.language.iso
metadata.value: ru
metadata.lang: en_GB
metadata.name: dc.publisher
metadata.value: St Petersburg State University
metadata.lang: en_GB
metadata.name: dc.relation.ispartofseries
metadata.value: Vestnik of St Petersburg University. Mathematics. Mechanics. Astronomy;Volume 7 (65); Issue 4
metadata.lang: -
metadata.name: dc.subject
metadata.value: boundary initial-value problem
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: boundary of domain of definition
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: boundary initial point
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: maximal and minimal solutions
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: nonuniqueness sector
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: differential inequality
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: comparison theorems
metadata.lang: en_GB
metadata.name: dc.title
metadata.value: On the Cauchy problem with an initial point lying on the boundary of ordinary differential equation’s domain of definition
metadata.lang: en_GB
metadata.name: dc.type
metadata.value: Article
metadata.lang: en_GB
resources.filename: 636-648.pdf
resources.url: https://dspace.spbu.ru/bitstream/11701/21848/1/636-648.pdf
resources.format: Adobe PDF
resources.size: 849,32 kB