The Solvability of the Discrete Boundary Value Problem on the Half-Line

The Solvability of the Discrete Boundary Value Problem on the Half-Line

This paper provides conditions for the existence of a solution to the second-order nonlinear boundary value problem on the half-line of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ&...

Descripción completa

Guardado en:
Detalles Bibliográficos
Autor principal: Magdalena Nockowska-Rosiak
Formato: article
Lenguaje:EN
Publicado: MDPI AG 2021
Materias:
discrete boundary value problem on infinite interval
fixed-point theorem
Fredholm operator of index 0
perturbation technique
Science
Q
Astrophysics
QB460-466
Physics
QC1-999
Acceso en línea:https://doaj.org/article/f0be19e74b154aeea8eaec1dd582a2d6
Etiquetas: Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
id oai:doaj.org-article:f0be19e74b154aeea8eaec1dd582a2d6
record_format dspace
spelling oai:doaj.org-article:f0be19e74b154aeea8eaec1dd582a2d62021-11-25T17:30:32ZThe Solvability of the Discrete Boundary Value Problem on the Half-Line10.3390/e231115261099-4300https://doaj.org/article/f0be19e74b154aeea8eaec1dd582a2d62021-11-01T00:00:00Zhttps://www.mdpi.com/1099-4300/23/11/1526https://doaj.org/toc/1099-4300This paper provides conditions for the existence of a solution to the second-order nonlinear boundary value problem on the half-line of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><mfenced separators="" open="(" close=")"><mi>a</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>Δ</mo><mi>x</mi><mo>(</mo><mi>n</mi><mo>)</mo></mfenced><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><mo>Δ</mo><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow><mo>,</mo><mspace width="1.em"></mspace><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi><mo>∪</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mi>x</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>+</mo><mi>β</mi><mi>a</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>Δ</mo><mi>x</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mspace width="1.em"></mspace><mi>x</mi><mo>(</mo><mo>∞</mo><mo>)</mo><mo>=</mo><mi>d</mi><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. To achieve our goal, we use Schauder’s fixed-point theorem and the perturbation technique for a Fredholm operator of index 0. Moreover, we construct the necessary condition for the existence of a solution to the considered problem.Magdalena Nockowska-RosiakMDPI AGarticlediscrete boundary value problem on infinite intervalfixed-point theoremFredholm operator of index 0perturbation techniqueScienceQAstrophysicsQB460-466PhysicsQC1-999ENEntropy, Vol 23, Iss 1526, p 1526 (2021)
institution DOAJ
collection DOAJ
language EN
topic discrete boundary value problem on infinite interval
fixed-point theorem
Fredholm operator of index 0
perturbation technique
Science
Q
Astrophysics
QB460-466
Physics
QC1-999
spellingShingle discrete boundary value problem on infinite interval
fixed-point theorem
Fredholm operator of index 0
perturbation technique
Science
Q
Astrophysics
QB460-466
Physics
QC1-999
Magdalena Nockowska-Rosiak
The Solvability of the Discrete Boundary Value Problem on the Half-Line
description This paper provides conditions for the existence of a solution to the second-order nonlinear boundary value problem on the half-line of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><mfenced separators="" open="(" close=")"><mi>a</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>Δ</mo><mi>x</mi><mo>(</mo><mi>n</mi><mo>)</mo></mfenced><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><mo>Δ</mo><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow><mo>,</mo><mspace width="1.em"></mspace><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi><mo>∪</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mi>x</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>+</mo><mi>β</mi><mi>a</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>Δ</mo><mi>x</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mspace width="1.em"></mspace><mi>x</mi><mo>(</mo><mo>∞</mo><mo>)</mo><mo>=</mo><mi>d</mi><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. To achieve our goal, we use Schauder’s fixed-point theorem and the perturbation technique for a Fredholm operator of index 0. Moreover, we construct the necessary condition for the existence of a solution to the considered problem.
format article
author Magdalena Nockowska-Rosiak
author_facet Magdalena Nockowska-Rosiak
author_sort Magdalena Nockowska-Rosiak
title The Solvability of the Discrete Boundary Value Problem on the Half-Line
title_short The Solvability of the Discrete Boundary Value Problem on the Half-Line
title_full The Solvability of the Discrete Boundary Value Problem on the Half-Line
title_fullStr The Solvability of the Discrete Boundary Value Problem on the Half-Line
title_full_unstemmed The Solvability of the Discrete Boundary Value Problem on the Half-Line
title_sort solvability of the discrete boundary value problem on the half-line
publisher MDPI AG
publishDate 2021
url https://doaj.org/article/f0be19e74b154aeea8eaec1dd582a2d6
work_keys_str_mv AT magdalenanockowskarosiak thesolvabilityofthediscreteboundaryvalueproblemonthehalfline
AT magdalenanockowskarosiak solvabilityofthediscreteboundaryvalueproblemonthehalfline
_version_ 1718412320992919552

Ejemplares similares