| Title:
|
Admissible spaces for a first order differential equation with delayed argument (English) |
| Author:
|
Chernyavskaya, Nina A. |
| Author:
|
Dorel, Lela S. |
| Author:
|
Shuster, Leonid A. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
69 |
| Issue:
|
4 |
| Year:
|
2019 |
| Pages:
|
1069-1080 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider the equation $$ -y'(x)+q(x)y(x-\varphi (x))=f(x), \quad x \in \mathbb R, $$ where $\varphi $ and $q$ ($q \geq 1$) are positive continuous functions for all $ x\in \mathbb R $ and $f \in C(\mathbb R)$. By a solution of the equation we mean any function $y$, continuously differentiable everywhere in $\mathbb R$, which satisfies the equation for all $x \in \mathbb R$. We show that under certain additional conditions on the functions $\varphi $ and $q$, the above equation has a unique solution $y$, satisfying the inequality $$ \|y'\|_{C(\mathbb R)}+\|qy\|_{C(\mathbb R)}\leq c\|f\|_{C(\mathbb R)}, $$ where the constant $c\in (0,\infty )$ does not depend on the choice of $f$. (English) |
| Keyword:
|
linear differential equation |
| Keyword:
|
admissible pair |
| Keyword:
|
delayed argument |
| MSC:
|
34A30 |
| MSC:
|
34B05 |
| MSC:
|
34B40 |
| idZBL:
|
07144876 |
| idMR:
|
MR4039621 |
| DOI:
|
10.21136/CMJ.2019.0062-18 |
| . |
| Date available:
|
2019-11-28T08:51:30Z |
| Last updated:
|
2022-01-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147915 |
| . |
| Reference:
|
[1] Azbelev, N. V., Kultyshev, S. Yu., Tsalynk, V. Z.: Functional Differential Equations and Variational Problems.R & C Dynamics, Moskva (2006), Russian. MR 1190052 |
| Reference:
|
[2] Chernyavskaya, N., Shuster, L.: Correct solvability of the Sturm-Liouville equation with delayed argument.J. Differ. Equations 261 (2016), 3247-3267. Zbl 1348.34118, MR 3527629, 10.1016/j.jde.2016.05.027 |
| Reference:
|
[3] El'sgol'ts, L. È., Norkin, S. B.: Introduction to the Theory of Differential Equations with Deviating Argument.Nauka, Moskva (1971), Russian. Zbl 0224.34053, MR 0352646 |
| Reference:
|
[4] Hale, J. K.: Theory of Functional Differential Equations.Applied Mathematical Sciences 3, Springer, New York (1977). Zbl 0352.34001, MR 0508721, 10.1007/978-1-4612-9892-2 |
| Reference:
|
[5] 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:
|
[6] Myshkis, A. D.: Linear Differential Equations with Retarded Argument.Nauka, Moskva (1972), Russian. Zbl 0261.34040, MR 0352648 |
| . |
