A Generalised Difference Equation and Its Dynamics and Solutions
Rational difference equations have a wide range of applications in various fields of science. To illustrate, the equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi>&...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-11-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/12/22/3531 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | Rational difference equations have a wide range of applications in various fields of science. To illustrate, the equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi><msub><mi>x</mi><mi>n</mi></msub></mrow><mrow><mi>c</mi><mo>+</mo><mi>d</mi><msub><mi>x</mi><mi>n</mi></msub></mrow></mfrac></mstyle><mo>,</mo><mspace width="3.33333pt"></mspace><mi>n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo></mrow></semantics></math></inline-formula>, known as the Riccati difference equation, has been applied in the field of optics. In this study, the global asymptotic stability of the difference equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>A</mi><msub><mi>x</mi><mrow><mi>n</mi><mo>−</mo><mfenced separators="" open="[" close="]"><mn>2</mn><mfenced separators="" open="(" close=")"><mi>k</mi><mo>+</mo><mi>j</mi></mfenced><mo>+</mo><mn>1</mn></mfenced></mrow></msub></mrow><mrow><mi>B</mi><mo>+</mo><mi>C</mi><msub><mi>x</mi><mrow><mi>n</mi><mo>−</mo><mo>(</mo><mi>k</mi><mo>+</mo><mi>j</mi><mo>)</mo></mrow></msub><msub><mi>x</mi><mrow><mi>n</mi><mo>−</mo><mfenced separators="" open="[" close="]"><mn>2</mn><mfenced separators="" open="(" close=")"><mi>k</mi><mo>+</mo><mi>j</mi></mfenced><mo>+</mo><mn>1</mn></mfenced></mrow></msub></mrow></mfrac></mstyle><mo>,</mo><mspace width="3.33333pt"></mspace><mi>n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo></mrow></semantics></math></inline-formula>, is proved. The solutions of this difference equation are obtained by applying the standard iteration method, and the periodicity of these solutions is determined. Furthermore, this difference equation represents a generalisation of the results obtained in previous studies. |
|---|---|
| ISSN: | 2227-7390 |