| 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 |
| . |
| Category:
|
math |
| . |
| 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 |
| . |
| Date available:
|
2022-04-21T19:03:23Z |
| Last updated:
|
2024-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/150413 |
| . |
| 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:
|
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| Reference:
|
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| Reference:
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| Reference:
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| 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:
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| Reference:
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| . |
