Admissible spaces for a first order differential equation with delayed argument
Nina A. Chernyavskaya; Lela S. Dorel; Leonid A. Shuster
Czechoslovak Mathematical Journal (2019)
- Volume: 69, Issue: 4, page 1069-1080
- ISSN: 0011-4642
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topChernyavskaya, Nina A., Dorel, Lela S., and Shuster, Leonid A.. "Admissible spaces for a first order differential equation with delayed argument." Czechoslovak Mathematical Journal 69.4 (2019): 1069-1080. <http://eudml.org/doc/294620>.
@article{Chernyavskaya2019,
abstract = {We consider the equation \[ -y^\{\prime \}(x)+q(x)y(x-\varphi (x))=f(x), \quad x \in \mathbb \{R\}, \]
where $\varphi $ and $q$ ($q \ge 1$) 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 \[ \Vert y^\{\prime \}\Vert \_\{C(\mathbb \{R\})\}+\Vert qy\Vert \_\{C(\mathbb \{R\})\}\le c\Vert f\Vert \_\{C(\mathbb \{R\})\}, \]
where the constant $c\in (0,\infty )$ does not depend on the choice of $f$.},
author = {Chernyavskaya, Nina A., Dorel, Lela S., Shuster, Leonid A.},
journal = {Czechoslovak Mathematical Journal},
keywords = {linear differential equation; admissible pair; delayed argument},
language = {eng},
number = {4},
pages = {1069-1080},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Admissible spaces for a first order differential equation with delayed argument},
url = {http://eudml.org/doc/294620},
volume = {69},
year = {2019},
}
TY - JOUR
AU - Chernyavskaya, Nina A.
AU - Dorel, Lela S.
AU - Shuster, Leonid A.
TI - Admissible spaces for a first order differential equation with delayed argument
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 4
SP - 1069
EP - 1080
AB - We consider the equation \[ -y^{\prime }(x)+q(x)y(x-\varphi (x))=f(x), \quad x \in \mathbb {R}, \]
where $\varphi $ and $q$ ($q \ge 1$) 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 \[ \Vert y^{\prime }\Vert _{C(\mathbb {R})}+\Vert qy\Vert _{C(\mathbb {R})}\le c\Vert f\Vert _{C(\mathbb {R})}, \]
where the constant $c\in (0,\infty )$ does not depend on the choice of $f$.
LA - eng
KW - linear differential equation; admissible pair; delayed argument
UR - http://eudml.org/doc/294620
ER -
References
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- Hale, J. K., 10.1007/978-1-4612-9892-2, Applied Mathematical Sciences 3, Springer, New York (1977). (1977) Zbl0352.34001MR0508721DOI10.1007/978-1-4612-9892-2
- Massera, J. L., Schäffer, J. J., Linear Differential Equations and Function Spaces, Pure and Applied Mathematics 21, Academic Press, New York (1966). (1966) Zbl0243.34107MR0212324
- Myshkis, A. D., Linear Differential Equations with Retarded Argument, Nauka, Moskva (1972), Russian. (1972) Zbl0261.34040MR0352648
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