Characterizations of multidimensional compact almost automorphic functions and applications to Poissons and heat equations
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912150783524864 |
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| author | Chávez, Alan Castañeda, Jolbyn Carranza, Alexis R. Khalil, Kamal |
| author_facet | Chávez, Alan Castañeda, Jolbyn Carranza, Alexis R. Khalil, Kamal |
| contents | Let \(\mathcal{G}\) be a non-empty subset of the Euclidean space \(\mathbb{R}^m\) (\(m \geq 1\)). This work is dedicated to further exploring the properties of \(\mathcal{G}\)-multi-almost automorphic functions defined on \(\mathbb{R}^m\) with values in a Banach space \(\mathbb{X}\). Using the theory of \(\mathcal{G}\)-multi-almost automorphic functions, we provide two new characterizations of compact almost automorphic functions. In the first characterization, \(\mathcal{G}\) corresponds to the lattice subgroup \(\mathbb{Z}^m \subset \mathbb{R}^m\); in the second, \(\mathcal{G}\) is taken to be a dense subset of \(\mathbb{R}^m\). Furthermore, we establish the invariance of the space of bounded and compactly \(\mathcal{G}\)-multi-almost automorphic functions under integral operators with Bi-almost automorphic kernels. Finally, we present applications to the analysis of the almost automorphic dynamics of Poisson's equation and the heat equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07644 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterizations of multidimensional compact almost automorphic functions and applications to Poissons and heat equations Chávez, Alan Castañeda, Jolbyn Carranza, Alexis R. Khalil, Kamal Dynamical Systems Let \(\mathcal{G}\) be a non-empty subset of the Euclidean space \(\mathbb{R}^m\) (\(m \geq 1\)). This work is dedicated to further exploring the properties of \(\mathcal{G}\)-multi-almost automorphic functions defined on \(\mathbb{R}^m\) with values in a Banach space \(\mathbb{X}\). Using the theory of \(\mathcal{G}\)-multi-almost automorphic functions, we provide two new characterizations of compact almost automorphic functions. In the first characterization, \(\mathcal{G}\) corresponds to the lattice subgroup \(\mathbb{Z}^m \subset \mathbb{R}^m\); in the second, \(\mathcal{G}\) is taken to be a dense subset of \(\mathbb{R}^m\). Furthermore, we establish the invariance of the space of bounded and compactly \(\mathcal{G}\)-multi-almost automorphic functions under integral operators with Bi-almost automorphic kernels. Finally, we present applications to the analysis of the almost automorphic dynamics of Poisson's equation and the heat equation. |
| title | Characterizations of multidimensional compact almost automorphic functions and applications to Poissons and heat equations |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2412.07644 |