zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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).

MSC:
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34A30Linear ODE and systems, general
28A75Length, area, volume, other geometric measure theory
References:
[1]Coppel, W. A.: Stability and asymptotic behavior of differential equations, (1965) · Zbl 0154.09301
[2]Evans, L. C.; Gariepy, R. F.: Measure theory and fine properties of functions, (1999)
[3]Falconer, K. J.: On the Minkowski measurability of fractals, Proc. amer. Math. soc. 123, 1115-1124 (1995) · Zbl 0838.28006 · doi:10.2307/2160708
[4]Falconer, K. J.: Fractal geometry. Mathematical foundations and applications, (1999)
[5]Hartman, P.: Ordinary differential equations, (1982) · Zbl 0476.34002
[6]Jaffard, S.; Meyer, Y.: Wavelet methods for pointwise regularity and local oscillations of functions, Mem. amer. Math. soc. 123, 1-110 (1996) · Zbl 0873.42019
[7]Jaroš, J.; Takasi, K.: On black hole solutions of second order differential equations with a singular nonlinearity in the differential operator, Funkcial. ekvac. 43, 491-509 (2000) · Zbl 1155.34321
[8]Lapidus, M. L.: Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl – Berry conjecture, Trans. amer. Math. soc. 325, 465-529 (1991) · Zbl 0741.35048 · doi:10.2307/2001638
[9]Lapidus, M. L.; Van Frankenhuysen, M.: Fractal geometry and number theory, complex dimensions of fractal strings and zeros of zeta functions, (2000) · Zbl 0981.28005
[10]Mattila, P.: Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability, (1995)
[11]Pašić, M.: Minkowski – bouligand dimension of solutions of the one-dimensional p-Laplacian, J. differential equations 190, 268-305 (2003) · Zbl 1054.34034 · doi:10.1016/S0022-0396(02)00149-3
[12]Pašić, M.: Rectifiable and unrectifiable oscillations for a class of second-order linear differential equations of Euler type, J. math. Anal. appl. 335, 724-738 (2007) · Zbl 1126.34023 · doi:10.1016/j.jmaa.2007.01.099
[13]M. Pašić, Rectifiable and unrectifiable oscillations for a generalization of the Riemann – Weber version of Euler differential equations, Georgian Math. J., in press · Zbl 1172.34025 · doi:http://www.heldermann.de/GMJ/GMJ15/GMJ154/gmj15060.htm
[14]Pašić, M.: Fractal oscillations for a class of second-order linear differential equations of Euler type, J. math. Anal. appl. 341, 211-223 (2008) · Zbl 1145.34022 · doi:10.1016/j.jmaa.2007.09.068
[15]Peitgen, H. O.; Jürgens, H.; Saupe, D.: Chaos and fractals. New frontiers of science, (1992) · Zbl 0779.58004
[16]Rakotoson, J. M.: Equivalence between the growth of B(x,r)|&nabla;u|pdy and T in the equation P(u)=T, J. differential equations 86, 102-122 (1990)
[17]Reid, W. T.: Sturmian theory for ordinary differential equations, (1980)
[18]Swanson, C. A.: Comparison and oscillation theory of linear differential equations, (1968) · Zbl 0191.09904
[19]Tricot, C.: Curves and fractal dimension, (1995)
[20]Wong, J. S. W.: On rectifiable oscillation of Euler type second order linear differential equations, Electron. J. Qual. theory differ. Equ. 20, 1-12 (2007) · Zbl 1182.34049 · doi:emis:journals/EJQTDE/2007/200720.html
[21]Žubrinić, D.; Županović, V.: Fractal analysis of spiral trajectories of some planar vector fields, Bull. sci. Math. 129, 457-485 (2005) · Zbl 1076.37015 · doi:10.1016/j.bulsci.2004.11.007