V. Soldatov

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Articles: 3

Approximation properties of multipoint boundary-value problems

H. Masliuk, O. Pelekhata, V. Soldatov

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 26 (2020), no. 2, 119-125

We consider a wide class of linear boundary-value problems for systems of $r$-th order ordinary differential equations whose solutions range over the normed complex space $(C^{(n)})^m$ of $n\geq r$ times continuously differentiable functions $y:[a,b]\to\mathbb{C}^{m}$. The boundary conditions for these problems are of the most general form $By=q$, where $B$ is an arbitrary continuous linear operator from $(C^{(n)})^{m}$ to $\mathbb{C}^{rm}$. We prove that the solutions to the considered problems can be approximated in $(C^{(n)})^m$ by solutions to some multipoint boundary-value problems. The latter problems do not depend on the right-hand sides of the considered problem and are built explicitly.

One-dimensional parameter-dependent boundary-value problems in Hölder spaces

Hanna Masliuk, Vitalii Soldatov

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 24 (2018), no. 2, 143-151

We study the most general class of linear boundary-value problems for systems of $r$-th order ordinary differential equations whose solutions range over the complex H\"older space $C^{n+r,\alpha}$, with $0\leq n\in\mathbb{Z}$ and $0<\alpha\leq1$. We prove a constructive criterion under which the solution to an arbitrary parameter-dependent problem from this class is continuous in $C^{n+r,\alpha}$ with respect to the parameter. We also prove a two-sided estimate for the degree of convergence of this solution to the solution of the corresponding nonperturbed problem.

A criterion for continuity in a parameter of solutions to generic boundary-value problems for higher-order differential systems

Vladimir Mikhailets, Aleksandr Murach, Vitalii Soldatov

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 4, 375-386

We consider the most general class of linear boundary-value problems for ordinary differential systems, of order $r\geq1$, whose solutions belong to the complex space $C^{(n+r)}$, with $0\leq n\in\mathbb{Z}.$ The boundary conditions can contain derivatives of order $l$, with $r\leq l\leq n+r$, of the solutions. We obtain a constructive criterion under which the solutions to these problems are continuous with respect to the parameter in the normed space $C^{(n+r)}$. We also obtain a two-sided estimate for the degree of convergence of these solutions.


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