Institute of Mathematics

Admissible spaces for a first order differential equation with delayed argument

Nina A. Chernyavskaya, Lela S. Dorel, Leonid A. Shuster

Received January 31, 2018.   Published online June 5, 2019.

Abstract:  We consider the equation $-y'(x)+q(x)y(x-\varphi(x))=f(x), \quad x \in\mathbb R$, where $\varphi$ and $q$ ($q \geq1$) 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 $f$.
Keywords:  linear differential equation; admissible pair; delayed argument
Classification MSC:  34A30, 34B05, 34B40
DOI:  10.21136/CMJ.2019.0062-18

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Affiliations:   Nina A. Chernyavskaya, Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Be'er Sheva, 841 05, Israel, e-mail: nina@math.bgu.ac.il; Lela S. Dorel, Department of Mathematics, Beit Berl College, Kfar Saba, 449 05, Israel, e-mail: lela@post.tau.ac.il; Leonid A. Shuster, Department of Mathematics, Bar-Ilan University, Ramat Gan, 529 00, Israel, e-mail: miriam@macs.biu.ac.il

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