Title: | The Massera-Schäffer problem for a first order linear differential equation (English) |
Author: | Chernyavskaya, Nina A. |
Author: | Shuster, Leonid A. |
Language: | English |
Journal: | Czechoslovak Mathematical Journal |
ISSN: | 0011-4642 (print) |
ISSN: | 1572-9141 (online) |
Volume: | 72 |
Issue: | 2 |
Year: | 2022 |
Pages: | 477-511 |
Summary lang: | English |
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Category: | math |
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Summary: | We consider the Massera-Schäffer problem for the equation $$ -y'(x)+q(x)y(x)=f(x),\quad x\in \mathbb R, $$ where $f\in L_p^{\rm loc}(\mathbb R),$ $p\in [1,\infty )$ and $0\le q\in L_1^{\rm loc}(\mathbb R).$ By a solution of the problem we mean any function $y,$ absolutely continuous and satisfying the above equation almost everywhere in $\mathbb R.$ Let positive and continuous functions $\mu (x)$ and $\theta (x)$ for $x\in \mathbb R$ be given. Let us introduce the spaces \begin {eqnarray*} L_p(\mathbb R,\mu )&=\biggl \{ f\in L_p^{\rm loc}(\mathbb R) \colon \|f\|_{L_p(\mathbb R,\mu )}^p=\int _{-\infty }^\infty |\mu (x)f(x)|^p {\rm d} x<\infty \biggr \},\\ L_p(\mathbb R,\theta )&=\biggl \{f\in L_p^{\rm loc}(\mathbb R) \colon \|f\|_{L_p(\mathbb R,\theta )}^p=\int _{-\infty }^\infty |\theta (x)f(x)|^p {\rm d} x<\infty \biggr \}. \end {eqnarray*} We obtain requirements to the functions $\mu $, $\theta $ and $q$ under which (1) for every function $f\in L_p(\mathbb R,\theta )$ there exists a unique solution $y\in L_p(\mathbb R,\mu )$ of the above equation; (2) there is an absolute constant $c(p)\in (0,\infty )$ such that regardless of the choice of a function $f\in L_p(\mathbb R,\theta )$ the solution of the above equation satisfies the inequality $$\|y\|_{L_p(\mathbb R,\mu )}\le c(p)\|f\|_{L_p(\mathbb R,\theta )}.$$ (English) |
Keyword: | admissible space |
Keyword: | first order linear differential equation |
MSC: | 34A30 |
idZBL: | Zbl 07547216 |
idMR: | MR4412771 |
DOI: | 10.21136/CMJ.2021.0548-20 |
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Date available: | 2022-04-21T19:03:23Z |
Last updated: | 2022-09-08 |
Stable URL: | http://hdl.handle.net/10338.dmlcz/150413 |
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Reference: | [1] Chernyavskaya, N. A.: Conditions for correct solvability of a simplest singular boundary value problem.Math. Nachr. 243 (2002), 5-18. Zbl 1028.34018, MR 1923831, 10.1002/1522-2616(200209)243:1<5::AID-MANA5>3.0.CO;2-B |
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