Rectifiable oscillations in second-order linear differential equations (Q949637)

The paper under review studies the oscillation and rectifiable property of the second order linear differential equation \[ y''(x)+f(x)y(x)=0, \quad x\in I:=(0,1), \tag{E} \] where \(f\) is a strictly positive continuous map on \(I.\) Under the additional hypothesis \(f\in C^2((0,1])\) and assuming that \(f\) satisfies the Hartman-Wintner condition \[ \lim_{\varepsilon\rightarrow 0} \int_{\varepsilon}^{1} \frac{1}{\root {4}\of{f(x)}} \bigg|\bigg(\frac{1}{\root {4}\of{f(x)}}\bigg)''\bigg|\,dx < \infty, \tag{HW-C} \] in the first part of the paper the authors establish: (1) Equation (E) is rectifiable oscillatory (resp. unrectifiable oscillatory) on \(I\) if and only if (FL) \(\lim_{\varepsilon\rightarrow 0}\int_{\varepsilon}^{1}\root {4}\of{f(x)}\,dx<\infty\) (resp. (IL): \(\lim_{\varepsilon\rightarrow 0}\int_{\varepsilon}^{1}\root {4}\of{f(x)}\,dx=\infty\)); (2) Consider the perturbed equation \[ y''(x)+(f(x)+p(x))y(x)=0 \text{ on } I, \tag{PE} \] where \(\frac{p}{\sqrt{f}}\in L^{1}(I).\) Equation (PE) is rectifiable oscillatory (resp. unrectifiable oscillatory) on \(I\) if and only if condition (FL) (resp. condition (IL)) holds. Recall that an equation is rectifiable oscillatory (resp. unrectifiable oscillatory) on \(I\) if all its solutions are oscillatory and the corresponding graphs have a finite length (resp. an infinite length). It is worth mentioning that the above result \(1.\) improves Theorem 2 of \textit{Wong} [Electronic Journal of Qualitative Theory of Differential Equations 20, 1--12 (2007)]. In the second part of the paper the authors study the values of the upper Minkowski-Bouligand dimension \(\dim_M G(y)\) and the \(s\)-dimensional upper Minkowski content \(M^{s}(G(y))\) of the graphs \(G(y)\) associated to solutions of Eq. (E). Assuming again that \(f\) is of class \(C^2,\) with \(f>0\) on \(I\) and satisfying condition (HW-C), and adding the asymptotical condition \(\lim_{x\rightarrow 0}x^{\alpha}f(x)=\lambda,\) with \(\alpha >2\) and \(\lambda >0\), it is proved: (3) On \(I,\) all the solutions of Eq. (E) verify \(\dim_M G(y)=1\) and \(0<M^{1}(G(y))<\infty\) (resp. \(\dim_M G(y)=s\in [1,2)\) and \(0<M^{s}(G(y))<\infty,\) with \(s=\frac{3}{2}-\frac{2}{\alpha}\)) if \(\alpha\in (2,4)\) (resp. if \(\alpha>4\)). Finally, the four result states that is possible to find an appropriate \(f\) which allows the coexistence of rectifiable and unrectifiable oscillations within the general solution of Eq. (E).