Wintner-type nonoscillation theorems for conformable linear Sturm-Liouville differential equations (Q6597986)

Let continuous functions \(\kappa_0, \kappa_1 : [0, 1] \times \mathbb{R} \to [0, \infty)\) be such that \N\[ \lim_{\alpha \to 0^+} \kappa_0 (\alpha, t) = 0, \qquad \lim_{\alpha \to 0^+} \kappa_1 (\alpha, t) = 1, \] \N\[ \lim_{\alpha \to 1^-} \kappa_0 (\alpha, t) = 1, \qquad \lim_{\alpha \to 1^-} \kappa_1 (\alpha, t) = 0, \] \N\[ \kappa_0 (\alpha, t) \ne 0, \quad \alpha \in (0, 1], \qquad \kappa_1 (\alpha, t) \ne 0, \quad \alpha \in [0, 1).\] \N\NFor such functions, we define the differential operator \N\[ D^\alpha f (t) = \kappa_1 (\alpha, t) f(t) + \kappa_0 (\alpha, t) f^\prime(t). \] \NThe author analyses conformable linear Sturm-Liouville differential equations in the form \N\[ D^{\alpha} \left(r(t) D^{\alpha} x \right) + c(t) x = 0, \tag{1}\]\Nwhere \(\alpha \in (0, 1]\) and \(r : [t_0, \infty) \to (0, \infty)\), \( c : [t_0, \infty) \to \mathbb{R}\) are continuous functions for some \(t_0 \ge 0\). Note that, putting \(\alpha = 1\) in Eq. (1), we obtain the classical second order linear differential equations. Applying the Riccati technique, the author proves non-oscillation criteria for Eq. (1), where the main result is formulated as Theorem~3.1 (the Wintner-type non-oscillation theorem). The main result is illustrated by two non-trivial examples.