Geodesic
Admissible spaces for a first order differential equation with delayed argument
Czechoslovak Mathematical Journal, Tome 69 (2019) no. 4, pp. 1069-1080
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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 $f$.
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 $f$.
DOI : 10.21136/CMJ.2019.0062-18
Classification : 34A30, 34B05, 34B40
Keywords: linear differential equation; admissible pair; delayed argument 
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     title = {Admissible spaces for a first order differential equation with delayed argument},
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     pages = {1069--1080},
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Chernyavskaya, Nina A.; Dorel, Lela S.; Shuster, Leonid A. Admissible spaces for a first order differential equation with delayed argument. Czechoslovak Mathematical Journal, Tome 69 (2019) no. 4, pp. 1069-1080. doi: 10.21136/CMJ.2019.0062-18

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