p-Regularity Theory and the Existence of a Solution to a Boundary Value Problem Continuously Dependent on Boundary Conditions
- Authors: Evtushenko Y.G.1,2, Medak B.3, Tret’yakov A.A.1,4,3
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Affiliations:
- Federal Research Center “Computer Science and Control,” Russian Academy of Sciences
- Moscow Institute of Physics and Technology (National Research University)
- Faculty of Exact and Natural Sciences, Siedlce University
- System Research Institute, Polish Academy of Sciences
- Issue: Vol 63, No 6 (2023)
- Pages: 920-936
- Section: ОПТИМАЛЬНОЕ УПРАВЛЕНИЕ
- URL: https://journals.rcsi.science/0044-4669/article/view/136146
- DOI: https://doi.org/10.31857/S0044466923060078
- EDN: https://elibrary.ru/TUDVWN
- ID: 136146
Cite item
Abstract
For a given boundary value problem, the existence of a solution depending continuously on the boundary conditions is analyzed. Previously, such a fact has been known only for the Cauchy problem, which is a classical result in the theory of differential equations. We prove a similar result for boundary value problems in the case when they are p-regular. In the general case, this result does not hold. Several implicit function theorems are proved in the degenerate case, which is a development of p-regularity theory concerning the existence of a solution to nonlinear differential equations. The results are illustrated by an example of a classical boundary value problem, namely, a degenerate Van der Pol equation is considered, for which the existence of a solution depending continuously on the boundary conditions of the perturbed problem is proved.
About the authors
Yu. G. Evtushenko
Federal Research Center “Computer Science and Control,” Russian Academy of Sciences; Moscow Institute of Physics and Technology (National Research University)
Email: yuri-evtushenko@yandex.ru
119333, Moscow, Russia; 141701, Dolgoprudnyi, Moscow oblast, Russia
B. Medak
Faculty of Exact and Natural Sciences, Siedlce University
Email: prof.tretyakov@gmail.com
08-110, Siedlce, Poland
A. A. Tret’yakov
Federal Research Center “Computer Science and Control,” Russian Academy of Sciences; System Research Institute, Polish Academy of Sciences; Faculty of Exact and Natural Sciences, Siedlce University
Author for correspondence.
Email: prof.tretyakov@gmail.com
119333, Moscow, Russia; 01-447, Warsaw, Poland; 08-110, Siedlce, Poland
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