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Rectifiable oscillations in second-order linear differential equations. (English) Zbl 1168.34027

The paper under review studies the oscillation and rectifiable property of the second order linear differential equation

y '' (x)+f(x)y(x)=0,xI:=(0,1),(E)

where f is a strictly positive continuous map on I·

Under the additional hypothesis fC 2 ((0,1]) and assuming that f satisfies the Hartman-Wintner condition

lim ε0 ε 1 1 f(x) 4|1 f(x) 4 '' |dx<,( 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 ε0 ε 1 f(x) 4dx< (resp. (IL): lim ε0 ε 1 f(x) 4dx=);

(2) Consider the perturbed equation

y '' (x)+(f(x)+p(x))y(x)=0onI,( PE )

where p fL 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 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 x0 x α f(x)=λ, with α>2 and λ>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))< (resp. dim M G(y)=s[1,2) and 0<M s (G(y))<, with s=3 2-2 α) if α(2,4) (resp. if α>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).

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34A30Linear ODE and systems, general
28A75Length, area, volume, other geometric measure theory
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