Existence of a unique bounded solution to a linear second-order difference equation and the linear first-order difference equation
Abstract We present some interesting facts connected with the following second-order difference equation: x n + 2 − q n x n = f n , n ∈ N 0 , $$x_{n+2}-q_{n}x_{n}=f_{n},\quad n\in \mathbb{N}_{0}, $$ where ( q n ) n ∈ N 0 $(q_{n})_{n\in\mathbb{N}_{0}}$ and ( f n ) n ∈ N 0 $(f_{n})_{n\in\mathbb {N}_{0...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2017-06-01
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| Series: | Advances in Difference Equations |
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| Online Access: | http://link.springer.com/article/10.1186/s13662-017-1227-x |
| _version_ | 1818927718919569408 |
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| author | Stevo Stević |
| author_facet | Stevo Stević |
| author_sort | Stevo Stević |
| collection | DOAJ |
| description | Abstract We present some interesting facts connected with the following second-order difference equation: x n + 2 − q n x n = f n , n ∈ N 0 , $$x_{n+2}-q_{n}x_{n}=f_{n},\quad n\in \mathbb{N}_{0}, $$ where ( q n ) n ∈ N 0 $(q_{n})_{n\in\mathbb{N}_{0}}$ and ( f n ) n ∈ N 0 $(f_{n})_{n\in\mathbb {N}_{0}}$ are given sequences of numbers. We give some sufficient conditions for the existence of a unique bounded solution to the difference equation and present an elegant proof based on a combination of theory of linear difference equations and the Banach fixed point theorem. We also deal with the equation by using theory of solvability of difference equations. A global convergence result of solutions to a linear first-order difference equation is given. Some comments on an abstract version of the linear first-order difference equation are also given. |
| first_indexed | 2024-12-20T03:17:28Z |
| format | Article |
| id | doaj.art-4956196305c846b09cb42d59d6f609cd |
| institution | Directory Open Access Journal |
| issn | 1687-1847 |
| language | English |
| last_indexed | 2024-12-20T03:17:28Z |
| publishDate | 2017-06-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-4956196305c846b09cb42d59d6f609cd2022-12-21T19:55:19ZengSpringerOpenAdvances in Difference Equations1687-18472017-06-012017111310.1186/s13662-017-1227-xExistence of a unique bounded solution to a linear second-order difference equation and the linear first-order difference equationStevo Stević0Mathematical Institute of the Serbian Academy of SciencesAbstract We present some interesting facts connected with the following second-order difference equation: x n + 2 − q n x n = f n , n ∈ N 0 , $$x_{n+2}-q_{n}x_{n}=f_{n},\quad n\in \mathbb{N}_{0}, $$ where ( q n ) n ∈ N 0 $(q_{n})_{n\in\mathbb{N}_{0}}$ and ( f n ) n ∈ N 0 $(f_{n})_{n\in\mathbb {N}_{0}}$ are given sequences of numbers. We give some sufficient conditions for the existence of a unique bounded solution to the difference equation and present an elegant proof based on a combination of theory of linear difference equations and the Banach fixed point theorem. We also deal with the equation by using theory of solvability of difference equations. A global convergence result of solutions to a linear first-order difference equation is given. Some comments on an abstract version of the linear first-order difference equation are also given.http://link.springer.com/article/10.1186/s13662-017-1227-xsecond-order difference equationunique bounded solutionsfixed point theoremlinear first-order difference equation |
| spellingShingle | Stevo Stević Existence of a unique bounded solution to a linear second-order difference equation and the linear first-order difference equation Advances in Difference Equations second-order difference equation unique bounded solutions fixed point theorem linear first-order difference equation |
| title | Existence of a unique bounded solution to a linear second-order difference equation and the linear first-order difference equation |
| title_full | Existence of a unique bounded solution to a linear second-order difference equation and the linear first-order difference equation |
| title_fullStr | Existence of a unique bounded solution to a linear second-order difference equation and the linear first-order difference equation |
| title_full_unstemmed | Existence of a unique bounded solution to a linear second-order difference equation and the linear first-order difference equation |
| title_short | Existence of a unique bounded solution to a linear second-order difference equation and the linear first-order difference equation |
| title_sort | existence of a unique bounded solution to a linear second order difference equation and the linear first order difference equation |
| topic | second-order difference equation unique bounded solutions fixed point theorem linear first-order difference equation |
| url | http://link.springer.com/article/10.1186/s13662-017-1227-x |
| work_keys_str_mv | AT stevostevic existenceofauniqueboundedsolutiontoalinearsecondorderdifferenceequationandthelinearfirstorderdifferenceequation |