Well-posedness and asymptotic behavior of difference equations with a time-dependent delay and applications
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909602325463040 |
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| author | Mazanti, Guilherme Mesquita, Jaqueline G. |
| author_facet | Mazanti, Guilherme Mesquita, Jaqueline G. |
| contents | In this paper, we investigate the well-posedness and asymptotic behavior of difference equations of the form $x(t) = A x(t - τ(t))$, $t \geq 0$, where the unknown function $x$ takes values in $\mathbb R^d$ for some positive integer $d$, $A$ is a $d \times d$ matrix with real coefficients, and $τ\colon [0, +\infty) \to (0, +\infty)$ is a time-dependent delay.
We provide our investigations for three spaces of functions: continuous, regulated, and $L^p$. We compare our results for these three cases, showing how the hypotheses change according to the space that we are treating. Finally, we provide applications of our results to difference equations with state-dependent delays for the cases of continuous and regulated function spaces, as well as to transport equations in one space dimension with time-dependent velocity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03301 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Well-posedness and asymptotic behavior of difference equations with a time-dependent delay and applications Mazanti, Guilherme Mesquita, Jaqueline G. Dynamical Systems 39A06, 39A30, 39A60, 34K40, 34K43, 35Q49 In this paper, we investigate the well-posedness and asymptotic behavior of difference equations of the form $x(t) = A x(t - τ(t))$, $t \geq 0$, where the unknown function $x$ takes values in $\mathbb R^d$ for some positive integer $d$, $A$ is a $d \times d$ matrix with real coefficients, and $τ\colon [0, +\infty) \to (0, +\infty)$ is a time-dependent delay. We provide our investigations for three spaces of functions: continuous, regulated, and $L^p$. We compare our results for these three cases, showing how the hypotheses change according to the space that we are treating. Finally, we provide applications of our results to difference equations with state-dependent delays for the cases of continuous and regulated function spaces, as well as to transport equations in one space dimension with time-dependent velocity. |
| title | Well-posedness and asymptotic behavior of difference equations with a time-dependent delay and applications |
| topic | Dynamical Systems 39A06, 39A30, 39A60, 34K40, 34K43, 35Q49 |
| url | https://arxiv.org/abs/2505.03301 |