Compact almost automorphic dynamics of non-autonomous differential equations with exponential dichotomy and applications to biological models with delay
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910423365713920 |
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| author | Chávez, Alan Aragonés, Nelson Pinto, Manuel Zavaleta, Ulices |
| author_facet | Chávez, Alan Aragonés, Nelson Pinto, Manuel Zavaleta, Ulices |
| contents | In the present work, we prove that, if $A(\cdot)$ is a compact almost automorphic matrix and the system $$x'(t) = A(t)x(t)\, ,$$ possesses an exponential dichotomy with Green function $G(\cdot, \cdot)$, then its associated system $$y'(t) = B(t)y(t)\, ,$$ where $B(\cdot) \in H(A)$ (the hull of $A(\cdot)$) also possesses an exponential dichotomy. Moreover, the Green function $G(\cdot, \cdot)$ is compact Bi-almost automorphic in $\mathbb{R}^2$, this implies that $G(\cdot, \cdot)$ is $Δ_2$ - like uniformly continuous, where $Δ_2$ is the principal diagonal of $\mathbb{R}^2$, an important ingredient in the proof of invariance of the compact almost automorphic function space under convolution product with kernel $G(\cdot, \cdot)$. Finally, we study the existence of a positive compact almost automorphic solution of non-autonomous differential equations of biological interest having non-linear harvesting terms and mixed delays. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_16758 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Compact almost automorphic dynamics of non-autonomous differential equations with exponential dichotomy and applications to biological models with delay Chávez, Alan Aragonés, Nelson Pinto, Manuel Zavaleta, Ulices Dynamical Systems In the present work, we prove that, if $A(\cdot)$ is a compact almost automorphic matrix and the system $$x'(t) = A(t)x(t)\, ,$$ possesses an exponential dichotomy with Green function $G(\cdot, \cdot)$, then its associated system $$y'(t) = B(t)y(t)\, ,$$ where $B(\cdot) \in H(A)$ (the hull of $A(\cdot)$) also possesses an exponential dichotomy. Moreover, the Green function $G(\cdot, \cdot)$ is compact Bi-almost automorphic in $\mathbb{R}^2$, this implies that $G(\cdot, \cdot)$ is $Δ_2$ - like uniformly continuous, where $Δ_2$ is the principal diagonal of $\mathbb{R}^2$, an important ingredient in the proof of invariance of the compact almost automorphic function space under convolution product with kernel $G(\cdot, \cdot)$. Finally, we study the existence of a positive compact almost automorphic solution of non-autonomous differential equations of biological interest having non-linear harvesting terms and mixed delays. |
| title | Compact almost automorphic dynamics of non-autonomous differential equations with exponential dichotomy and applications to biological models with delay |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2404.16758 |