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Keywords:
linear differential equation; admissible pair; delayed argument
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 $f$.
References:
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