On parameter-dependent inhomogeneous boundary-value problems in Sobolev spaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910082118189056 |
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| author | Atlasiuk, Olena Mikhailets, Vladimir Taskinen, Jari |
| author_facet | Atlasiuk, Olena Mikhailets, Vladimir Taskinen, Jari |
| contents | We study a wide class of linear inhomogeneous boundary-value problems for $r$th order ODE-systems depending on a parameter $μ$ belonging to a general metric space $\mathcal M$. The solutions belong to the Sobolev spaces $(W^{n+r}_p)^m$, $n\in\mathbb{N}\cup\{0\}$, $m, r \in \mathbb{N}$, $1\leq p\leq \infty$. The boundary conditions are of a most general form $By=c$, where $B$ is an arbitrary continuous operator from $(W^{n+r}_p)^m$ to $\mathbb{C}^{rm}$. Thus, they may contain derivatives of the unknown vector function of integer and/or fractional orders $\geq r$. We find necessary and sufficient conditions for the continuity of solutions with respect to the parameter $μ$. We also prove that the solutions of the original problems can be approximated in the space $(W^{n+r}_p)^m$ by solutions of ODE-systems with polynomial coefficients, right-hand sides of the equation, and multipoint boundary conditions, which are independent of the original problem's right-hand sides. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27345 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On parameter-dependent inhomogeneous boundary-value problems in Sobolev spaces Atlasiuk, Olena Mikhailets, Vladimir Taskinen, Jari Classical Analysis and ODEs 34B05, 34B08, 34B10, 47A531 We study a wide class of linear inhomogeneous boundary-value problems for $r$th order ODE-systems depending on a parameter $μ$ belonging to a general metric space $\mathcal M$. The solutions belong to the Sobolev spaces $(W^{n+r}_p)^m$, $n\in\mathbb{N}\cup\{0\}$, $m, r \in \mathbb{N}$, $1\leq p\leq \infty$. The boundary conditions are of a most general form $By=c$, where $B$ is an arbitrary continuous operator from $(W^{n+r}_p)^m$ to $\mathbb{C}^{rm}$. Thus, they may contain derivatives of the unknown vector function of integer and/or fractional orders $\geq r$. We find necessary and sufficient conditions for the continuity of solutions with respect to the parameter $μ$. We also prove that the solutions of the original problems can be approximated in the space $(W^{n+r}_p)^m$ by solutions of ODE-systems with polynomial coefficients, right-hand sides of the equation, and multipoint boundary conditions, which are independent of the original problem's right-hand sides. |
| title | On parameter-dependent inhomogeneous boundary-value problems in Sobolev spaces |
| topic | Classical Analysis and ODEs 34B05, 34B08, 34B10, 47A531 |
| url | https://arxiv.org/abs/2603.27345 |