Abstract
Second-order half-linear differential equation (H): \({(\Phi (y'))'+f(x)\Phi (y)=0}\) on the finite interval I = (0,1] will be studied, where \({\Phi (u)=|u|^{p-2}u}\) , p > 1 and the coefficient f(x) > 0 on I, \({f\in C^{2}((0,1])}\) , and \({\lim_{x\rightarrow0}f(x)=\infty }\) . In case when p = 2, the equation (H) reduces to the harmonic oscillator equation (P): y′′ + f(x)y = 0. In this paper, we study the oscillations of solutions of (H) with special attention to some geometric and fractal properties of the graph \({G(y)=\{(x,y(x)):0\leq x\leq 1\}\subseteq {\bf {R}}^{2}}\) . We establish integral criteria necessary and sufficient for oscillatory solutions with graphs having finite and infinite arclength. In case when \({f(x)\sim \lambda x^{-\alpha}}\), λ > 0, α > p, we also determine the fractal dimension of the graph G(y) of the solution y(x). Finally, we study the L p nonintegrability of the derivative of all solutions of the equation (H).
Article PDF
Similar content being viewed by others
References
Acerbi E., Mingione G.: Gradient estimates for the p-Laplacian systems. J. Reine. Angew. Math. 584, 117–148 (2005)
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)
Brezis H.: Analyse fonctionelle. Théorie et applications. Masson, Paris (1983)
Coppel W.A.: Stability and asymptotic behavior of differential equations. D. C. Heath, Boston (1965)
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)
Elbert A.: A half-linear second order differential equations. Colloq. Math. Soc. Janos Bolyai 30, 158–180 (1979)
Evans L.C., Gariepy R.F.: Measure Theory and Fine Properties of Functions. CRC Press, New York (1999)
Falconer K.: Fractal Geometry. Mathematical Fondations and Applications. Willey, New York (1999)
Hartman, P.: Ordinary Differential Equations, 2nd edn. Birkhauser, Boston (1982)
Kiguradze I.T., Chanturia T.A.: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Kluwer, London (1993)
Kwong M.K., Pašić M., Wong J.S.W.: Rectifiable Oscillations in Second Order Linear Differential Equations. J. Differ. Equ. 245, 2333–2351 (2008)
Mattila P.: Geometry of Sets and Measures in Euclidean Spaces. Fractals and rectifiability, Cambridge (1995)
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)
Pašić M.: Minkowski–Bouligand dimension of solutions of the one-dimensional p-Laplacian. J. Differ. Equ. 190, 268–305 (2003)
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)
Pašić M.: Fractal oscillations for a class of second-order linear differential equations of Euler type. J. Math. Anal. Appl. 341, 211–223 (2008)
Pašić, M.: Rectifiable and unrectifiable oscillations for a generalization of the Riemann–Weber version of Euler differential equations. Georgian Math. J. (2008, in press)
Peitgen H.O., Jűrgens H., Saupe D.: Chaos and Fractals. New Frontiers of Science. Springer, New York (1992)
Rakotoson J.M.: Equivalence between the growth of \({\int_{B(x,r)}|\nabla u|^{p}dy}\) and T in the equation P(u) = T. J. Differ. Equ. 86, 102–122 (1990)
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)
Reid W.T.: Sturmian Theory for Ordinary Differential Equations. Springer, New York (1980)
Ruzicka M.: Electro-rheological Fluids: Modeling and Mathematical Theory. Lecture notes in Mathematics, vol. 1748. Springer, New York (2000)
Swanson C.A.: Comparison and Oscillation Theory of Linear Differential Equations. Academic Press, New York (1968)
Tricot C.: Curves and Fractal Dimension. Springer, New York (1995)
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)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pašić, M., Wong, J.S.W. Rectifiable oscillations in second-order half-linear differential equations. Annali di Matematica 188, 517–541 (2009). https://doi.org/10.1007/s10231-008-0087-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-008-0087-0
Keywords
- Oscillations
- Nonlinear equations
- Graph
- Rectifiability
- Fractal dimension
- Minkowski content
- Asymptotics
- Perturbation