Parameter-dependent inhomogeneous boundary-value problems in spaces of continuously differentiable functions
Abstract
We investigate a broad class of linear inhomogeneous boundary-value problems of order r for systems of ordinary differential equations depending on a parameter μ belonging to an abstract metric space ℳ. The solutions belong to the spaces of continuously differentiable vector-valued functions (C(n+r))m, n∈ℕ∪{0}, m, r ∈ ℕ. The boundary conditions have the most general form By=c, where B is an arbitrary continuous linear operator from (C(n+r))m to ℂrm. Thus, they may contain derivatives of the unknown vector-valued function of integer and/or fractional orders up to n+r inclusive. We establish necessary and sufficient conditions for the continuous dependence of solutions on the parameter μ. Furthermore, we prove that the solutions of the original problems can be approximated in the space (C(n+r))m by solutions of systems of ordinary differential equations with polynomial coefficients, polynomial right-hand sides, and multipoint boundary conditions, where the approximating problems are constructed independently of the right-hand sides of the original problem.
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BibTeX
@misc{atlasiuk2026parameterdependent,
title = {{Parameter-dependent inhomogeneous boundary-value problems in spaces of continuously differentiable functions}},
author = {O. M. Atlasiuk and V. A. Mikhailets},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07991},
eprint = {2611.07991},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07991},
}
RIS
TY - UNPB
TI - Parameter-dependent inhomogeneous boundary-value problems in spaces of continuously differentiable functions
AU - O. M. Atlasiuk
AU - V. A. Mikhailets
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07991
UR - https://arxiv.org/abs/2611.07991
AB - We investigate a broad class of linear inhomogeneous boundary-value problems of order r for systems of ordinary differential equations depending on a parameter μ belonging to an abstract metric space ℳ. The solutions belong to the spaces of continuously differentiable vector-valued functions (C^(n+r))^m, n∈ℕ∪{0}, m, r ∈ ℕ. The boundary conditions have the most general form By=c, where B is an arbitrary continuous linear operator from (C^(n+r))^m to ℂ^rm. Thus, they may contain derivatives of the unknown vector-valued function of integer and/or fractional orders up to n+r inclusive. We establish necessary and sufficient conditions for the continuous dependence of solutions on the parameter μ. Furthermore, we prove that the solutions of the original problems can be approximated in the space (C^(n+r))^m by solutions of systems of ordinary differential equations with polynomial coefficients, polynomial right-hand sides, and multipoint boundary conditions, where the approximating problems are constructed independently of the right-hand sides of the original problem.
ER -