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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: 2024-07-01
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
Reference: [2] Chernyavskaya, N. A., Shuster, L. A.: Conditions for correct solvability of a simplest singular boundary value problem of general form. I.Z. Anal. Anwend. 25 (2006), 205-235. Zbl 1122.34021, MR 2229446, 10.4171/ZAA/1285
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