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Rectifiable oscillations in second-order half-linear differential equations. (English) Zbl 1184.34043

The authors continue their investigation of geometric aspects of oscillation theory of various second order differential equations [see, e.g. M. Pašić and J. S. W. Wong, Differ. Equ. Appl. 1, No. 1, 85–122 (2009; Zbl 1160.26304)]. In this paper, the main attention is devoted to the half-linear differential equation

(Φ(y ' )) ' +f(x)Φ(y)=0,Φ(y):=|y| p-2 y,p>1,(1)

where xI:=(0,1], fC 2 (I), f(x)>0, and lim x0- f(x)=. Integral criteria are established which guarantee that (1) is oscillatory for x0- and that the graph of oscillatory solutions has finite/infinite arclength. Some other geometrical aspects of oscillation of (1), like the fractal dimension of graphs of its solutions, are investigated as well.

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
[1]Acerbi E., Mingione G.: Gradient estimates for the p-Laplacian systems. J. Reine. Angew. Math. 584, 117–148 (2005) · Zbl 1093.76003 · doi:10.1515/crll.2005.2005.584.117
[2]Agarwal R.P., Grace S.R., O’Regan D.: Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations. Kluwer, London (2002)
[3]Brezis H.: Analyse fonctionelle. Théorie et applications. Masson, Paris (1983)
[4]Coppel W.A.: Stability and asymptotic behavior of differential equations. D. C. Heath, Boston (1965)
[5]Došly, O.: Half-Linear differential equations. In: Canada, A., Drabek, P., Fonda, A. (eds.) Handbook of Ordinary Differential Equations, chap. 3, pp. 161–357. Elsevier, Amsterdam (2004)
[6]Elbert A.: A half-linear second order differential equations. Colloq. Math. Soc. Janos Bolyai 30, 158–180 (1979)
[7]Evans L.C., Gariepy R.F.: Measure Theory and Fine Properties of Functions. CRC Press, New York (1999)
[8]Falconer K.: Fractal Geometry. Mathematical Fondations and Applications. Willey, New York (1999)
[9]Hartman, P.: Ordinary Differential Equations, 2nd edn. Birkhauser, Boston (1982)
[10]Kiguradze I.T., Chanturia T.A.: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Kluwer, London (1993)
[11]Kwong M.K., Pašić M., Wong J.S.W.: Rectifiable Oscillations in Second Order Linear Differential Equations. J. Differ. Equ. 245, 2333–2351 (2008) · Zbl 1168.34027 · doi:10.1016/j.jde.2008.05.016
[12]Mattila P.: Geometry of Sets and Measures in Euclidean Spaces. Fractals and rectifiability, Cambridge (1995)
[13]Mirzov, I.D.: Asymptotic Properties of Solutions of Systems of Non-Autonomous Ordinary Differential Equations. Folia Fac. Sc. Natur. Univ. Masaryk. Brun. Math., vol. 14 (2004)
[14]Pašić M.: Minkowski–Bouligand dimension of solutions of the one-dimensional p-Laplacian. J. Differ. Equ. 190, 268–305 (2003) · Zbl 1054.34034 · doi:10.1016/S0022-0396(02)00149-3
[15]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
[16]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
[17]Pašić, M.: Rectifiable and unrectifiable oscillations for a generalization of the Riemann–Weber version of Euler differential equations. Georgian Math. J. (2008, in press)
[18]Peitgen H.O., Jurgens H., Saupe D.: Chaos and Fractals. New Frontiers of Science. Springer, New York (1992)
[19]Rakotoson J.M.: Equivalence between the growth of B(x,r) |u| p dy and T in the equation P(u) = T. J. Differ. Equ. 86, 102–122 (1990) · Zbl 0707.35033 · doi:10.1016/0022-0396(90)90043-O
[20]Rakotoson J.M., Ziemer W.P.: Local behavior of solutions of quasilinear elliptic equations with general structure. Trans. Am. Math. Soc. 319, 747–764 (1990) · Zbl 0708.35023 · doi:10.2307/2001263
[21]Reid W.T.: Sturmian Theory for Ordinary Differential Equations. Springer, New York (1980)
[22]Ruzicka M.: Electro-rheological Fluids: Modeling and Mathematical Theory. Lecture notes in Mathematics, vol. 1748. Springer, New York (2000)
[23]Swanson C.A.: Comparison and Oscillation Theory of Linear Differential Equations. Academic Press, New York (1968)
[24]Tricot C.: Curves and Fractal Dimension. Springer, New York (1995)
[25]Wong J.S.W.: On rectifiable oscillation of Euler type second order linear differential equations E. J. Qual. Theory Differ. Equ. 20, 1–12 (2007)