Correct solvability of the Sturm-Liouville equation with delayed argument (Q2630904)

This paper considers the solvability of the equation \[ -y''(x) +q(x)y(x-\varphi(x)) = f(x) \text{ for } x\in \mathbb{R}, \] where \(f\), \(q\) and \(\varphi\) are continuous. A solution of the equation is commonly understood to be a doubly continuously differentiable function that satisfies the equation for all \(x\in\mathbb{R}\). However, the solvability here does not simply mean the existence of any solution, but it means the existence and uniqueness of a certain type solution. For this purpose, the authors introduce the concept of admissible pair of Banach spaces \(B_1\) and \(B_2\) for the equation \(Sy = g, y\in B_1, g\in B_2, S: B_1\to B_2\): for each \(g\in B_2\), there is a unique \(y\in B_1\) satisfying \(Sy = g\) and \(\|y\|_{B_1} \leq c\|g\|_{B_2}\). Then, viewing the differential equation as \(Sy = g\), it is proved that \(\{C^{(2)}_q(\mathbb{R}), C(\mathbb{R})\}\) is admissible if \(q\) and \(\varphi\) meet certain conditions. Although initial and boundary value problems on a finite interval or on a semi-axis have been studied, the feature of the problem and the results presented in this paper seem fresh and interesting.