Poster Session

Admissible Spaces for a First Order Equation with Delayed Argument

Lea Dorel
Beit Berl College
Israel
Co-Author(s):    N. Chernyavskaya, L. Shuster
Abstract:
We consider the equation \begin{align}\label{0.11} -y`(x)+q(x)y(x-\varphi(x))=f(x), \quad x \in \mathbb R, \end{align} where $f \in C(\mathbb R)$ and \begin{align}\label{0.12} 0\leq \varphi\in C^{\rm loc}(\mathbb R),\ \ \ 1\leq q \in C^{\rm loc}(\mathbb R). \end{align} Here $C^{\rm loc}(\mathbb R)$ is the set of functions continuous in every point of the number axis $\mathbb R$. By a solution of (\ref{0.11}) we mean any function $y$, continuously differentiable everywhere in $\mathbb R$, which satisfies the equation (\ref{0.11}) for all $x \in \mathbb R$. We show that under certain conditions on the functions $\varphi$ and $q$ in addition to (\ref{0.12}), the equation (\ref{0.11}) 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 $f\in {C(\mathbb R)}$.