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&...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-11-01
|
| Series: | Entropy |
| Subjects: | |
| Online Access: | https://www.mdpi.com/1099-4300/23/11/1526 |
| _version_ | 1797510405892866048 |
|---|---|
| author | Magdalena Nockowska-Rosiak |
| author_facet | Magdalena Nockowska-Rosiak |
| author_sort | Magdalena Nockowska-Rosiak |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-03-10T05:32:00Z |
| format | Article |
| id | doaj.art-f0be19e74b154aeea8eaec1dd582a2d6 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T05:32:00Z |
| publishDate | 2021-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-f0be19e74b154aeea8eaec1dd582a2d62023-11-22T23:16:15ZengMDPI AGEntropy1099-43002021-11-012311152610.3390/e23111526The Solvability of the Discrete Boundary Value Problem on the Half-LineMagdalena Nockowska-Rosiak0Institute of Mathematics, Lodz University of Technology, ul. Politechniki 8, 93-590 Łódź, PolandThis 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.https://www.mdpi.com/1099-4300/23/11/1526discrete boundary value problem on infinite intervalfixed-point theoremFredholm operator of index 0perturbation technique |
| spellingShingle | Magdalena Nockowska-Rosiak The Solvability of the Discrete Boundary Value Problem on the Half-Line Entropy discrete boundary value problem on infinite interval fixed-point theorem Fredholm operator of index 0 perturbation technique |
| title | 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_short | 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 |
| topic | discrete boundary value problem on infinite interval fixed-point theorem Fredholm operator of index 0 perturbation technique |
| url | https://www.mdpi.com/1099-4300/23/11/1526 |
| work_keys_str_mv | AT magdalenanockowskarosiak thesolvabilityofthediscreteboundaryvalueproblemonthehalfline AT magdalenanockowskarosiak solvabilityofthediscreteboundaryvalueproblemonthehalfline |