Article
| 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 |
| 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 |
| Reference: | [3] Chernyavskaya, N. A., Shuster, L. A.: Spaces admissible for the Sturm-Liouville equation.Commun. Pure Appl. Anal. 17 (2018), 1023-1052. Zbl 1397.34052, MR 3809112, 10.3934/cpaa.2018050 |
| Reference: | [4] Courant, R.: Differential and Integral Calculus. I.Blackie & Son, Glasgow (1945). Zbl 0018.30001, MR 1009558, 10.1002/9781118033241 |
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| Reference: | [6] Lukachev, M., Shuster, L.: On uniqueness of the solution of a linear differential equation without boundary conditions.Funct. Differ. Equ. 14 (2007), 337-346. Zbl 1148.34303, MR 2323215 |
| Reference: | [7] Massera, J. L., Schäffer, J. J.: Linear Differential Equations and Function Spaces.Pure and Applied Mathematics 21. Academic Press, New York (1966). Zbl 0243.34107, MR 0212324 |
| Reference: | [8] Mynbaev, K. T., Otelbaev, M. O.: Weighted Function Spaces and the Spectrum of Differential Operators.Nauka, Moskva (1988), Russian. Zbl 0651.46037, MR 0950172 |
| Reference: | [9] Titchmarsh, E. C.: The Theory of Functions.Clarendon Press, Oxford (1932). Zbl 0005.21004, MR 3728294 |
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