Boundedness of solutions of the first-order linear multidimensional difference equations
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914050293628928 |
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| author | Chaikovskyi, Andrii Liubimov, Oleksandr |
| author_facet | Chaikovskyi, Andrii Liubimov, Oleksandr |
| contents | We investigate the boundedness of solutions of the first order linear difference equation of the form $x_{n+1} = Ax_{n} + y_{n}, \; n \geq 1$ where $A$ is a square matrix with complex entries, sequence $\{y_{n}\}_{n\geq 1}$ and initial value $x_1$ are supposed to be known. Firstly, we discuss the one-dimensional case of this equation $x_{n+1} = ax_{n} + y_{n}, \; n \geq 1$ where $a$ is a complex number. In particular, we obtain the sufficient conditions for boundedness or unboundedness of the solutions in case $|a|=1$(the critical case) by considering the exponential sums of the forms $\sum y_{n}e(nφ)$ and $\sum e(f(n))$.
Then we proceed to the investigation of the equation in the multidimensional case and reduce our problem to analysis of the spectrum and Jordan cells of matrix $A$. The problem is especially interesting when spectrum of $A$ contains eigenvalues $λ$ with $|λ|=1$. At the end of the article we obtain a theorem that reveals the connection between equations $x_{n+1} = ax_{n} + y_{n}, \; n \geq 1$ with $|a|=1$ and $x_{n+1} = Jx_{n} + y_{n}, \; n \geq 1$ with $J$ being a Jordan cell of an eigenvalue $λ$, $|λ|=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14842 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Boundedness of solutions of the first-order linear multidimensional difference equations Chaikovskyi, Andrii Liubimov, Oleksandr Dynamical Systems 39A06, 39A45, 11L03, 11L07 We investigate the boundedness of solutions of the first order linear difference equation of the form $x_{n+1} = Ax_{n} + y_{n}, \; n \geq 1$ where $A$ is a square matrix with complex entries, sequence $\{y_{n}\}_{n\geq 1}$ and initial value $x_1$ are supposed to be known. Firstly, we discuss the one-dimensional case of this equation $x_{n+1} = ax_{n} + y_{n}, \; n \geq 1$ where $a$ is a complex number. In particular, we obtain the sufficient conditions for boundedness or unboundedness of the solutions in case $|a|=1$(the critical case) by considering the exponential sums of the forms $\sum y_{n}e(nφ)$ and $\sum e(f(n))$. Then we proceed to the investigation of the equation in the multidimensional case and reduce our problem to analysis of the spectrum and Jordan cells of matrix $A$. The problem is especially interesting when spectrum of $A$ contains eigenvalues $λ$ with $|λ|=1$. At the end of the article we obtain a theorem that reveals the connection between equations $x_{n+1} = ax_{n} + y_{n}, \; n \geq 1$ with $|a|=1$ and $x_{n+1} = Jx_{n} + y_{n}, \; n \geq 1$ with $J$ being a Jordan cell of an eigenvalue $λ$, $|λ|=1$. |
| title | Boundedness of solutions of the first-order linear multidimensional difference equations |
| topic | Dynamical Systems 39A06, 39A45, 11L03, 11L07 |
| url | https://arxiv.org/abs/2509.14842 |