Previous |  Up |  Next

Article

Keywords:
half-linear differential equation; Hartman-Wintner criterion; Riccati equation; principal solution
Summary:
We establish Hartman-Wintner type criteria for the half-linear second order differential equation \[ \left(r(t)\Phi (x^{\prime })\right)^{\prime }+c(t)\Phi (x)=0,\quad \Phi (x)=|x|^{p-2}x,\ p>1, \] where this equation is viewed as a perturbation of another equation of the same form.
References:
[1] T. Chantladre, N. Kandelaki, A. Lomtatidze: Oscillation and nonoscillation criteria for a second order linear equation. Georgian Math. J. 6 (1999), 401–414. MR 1692963
[2] O. Došlý: Half-Linear Differential Equations. A. Cañada, P. Drábek, A. Fonda (eds.), Handbook of Differential Equations: Ordinary Differential Equations, Vol. I, Elsevier, Amsterdam, 2004, pp. 161–357. MR 2166491 | Zbl 1090.34027
[3] O. Došlý, A. Lomtatidze: Oscillation and nonoscillation criteria for half-linear second order differential equations. (to appear). MR 2259737
[4] O. Došlý, Z. Pátíková: Hille-Wintner type comparison criteria for half-linear second order differential equations. Arch. Math., Brno 42 (2006), 185–194. MR 2240356
[5] O. Došlý, P. Řehák: Half-Linear Differential Equations. North Holland Mathematics Studies 202, Elsevier, Amsterdam, 2005. MR 2158903
[6] O. Došlý, J. Řezníčková: Regular half-linear second order differential equations. Arch. Math. 39 (2003), 233–245. MR 2010724
[7] P. Hartman: Ordinary Differential Equations. SIAM, Philadelphia, 2002. MR 1929104 | Zbl 1009.34001
[8] Á. Elbert, T. Kusano: Principal solutions of nonoscillatory half-linear differential equations. Adv. Math. Sci. Appl. 18 (1998), 745–759. MR 1657164
[9] Á. Elbert, A. Schneider: Perturbations of the half-linear Euler differential equation. Result. Math. 37 (2000), 56–83. MR 1742294
[10] J. Jaroš, T. Kusano: A Picone type identity for half-linear differential equations. Acta Math. Univ. Comenianea 68 (1999), 137–151. MR 1711081
[11] N. Kandelaki, A. Lomtatidze, D. Ugulava: On oscillation and nonoscillation of a second order half-linear equation. Georgian Math. J. 2 (2000), 329–346. MR 1779555
[12] H. J. Li, C. C. Yeh: Oscillations of half-linear second order differential equations. Hiroshima Math. J. 25 (1995), 585–594. MR 1364076
[13] J. D. Mirzov: Analogue of the Hartman theorem. Diff. Urav. 25 (1989), 216–222. (Russian) MR 0994702
[14] J. D. Mirzov: Asymptotic Properties of Solutions of Systems of Nonlinear Nonautonomous Ordinary Differential Equations. Masaryk University Press, Brno, 2004. MR 2144761 | Zbl 1154.34300
[15] J. D. Mirzov: Principal and nonprincipal solutions of a nonoscillatory system. Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 31 (1988), 100–117. MR 1001343
[16] J. Řezníčková: Half-linear Hartman-Wintner theorems. Stud. Univ. Žilina Math. Ser. 15 (2002), 56–66. MR 1980763 | Zbl 1051.34026
Partner of
EuDML logo