Half-linear oscillation criteria: perturbation in term involving derivative. (English) Zbl 1207.34041

Summary: We consider the non-oscillatory half-linear differential equation
\[ (r(t)\Phi(x'))'+c(t)\Phi(x)=0,\quad \Phi(x):=|x|^{p-2}x,\quad p>1, \]
and we study the oscillatory properties of its perturbation
\[ [(r(t)+\widetilde r(t))\Phi(x')]'+(c(t)+\widetilde c(t))\Phi(x)=0.\tag{*} \]
We use the Riccati technique and the relationship between (*) and a certain associated linear equation. The results are applied to a perturbed Euler-type equation.


34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
Full Text: DOI


[1] Díaz, J.I., ()
[2] Yoshida, N., Oscillation theory of partial differential equations, (2008), World Scientific Publishing Co. Pvt. Ltd. Hackensack, NJ · Zbl 1154.35001
[3] Došlý, O., Perturbations of the half-linear euler – weber differential equation, J. math. anal. appl., 323, 426-440, (2006) · Zbl 1107.34030
[4] Došlý, O.; Lomtatidze, A., Oscillation and nonoscillation criteria for half-linear second order differential equations, Hiroshima math. J., 36, 203-219, (2006) · Zbl 1123.34028
[5] Došlý, O.; Peña, S., A linearization method in oscillation theory of half-linear differential equations, J. inequal. appl., 2005, 535-545, (2005) · Zbl 1178.34038
[6] Došlý, O.; Ünal, M., Conditionally oscillatory half-linear differential equations, Acta math. hungar., 120, 147-163, (2008) · Zbl 1199.34169
[7] Elbert, Á.; Schneider, A., Perturbations of the half-linear Euler differential equation, Results math., 37, 56-83, (2000) · Zbl 0958.34029
[8] Jaroš, J.; Kusano, T.; Tanigawa, T., Nonoscillatory half-linear differential equations and generalized karamata functions, Nonlinear anal., 64, 762-787, (2006) · Zbl 1103.34017
[9] Kusano, T.; Manojlović, J.; Tanigawa, T., Existence of regularly varying solutions with nonzero indices of half-linear differential equations with retarded arguments, Comput. math. appl., 59, 411-425, (2010) · Zbl 1189.34121
[10] Pátíková, Z., Asymptotic formulas for nonoscillatory solutions of perturbed half-linear Euler equation, Nonlinear anal., 69, 3281-3290, (2008) · Zbl 1158.34027
[11] Sugie, J.; Yamaoka, N., Comparison theorems for oscillation of second order half-linear differential equations, Acta math. hungar., 111, 165-179, (2006) · Zbl 1116.34030
[12] Yamaoka, N., A nonoscillation theorem for half-linear differential equations with delay nonlinear perturbations, Differ. equ. appl., 1, 209-217, (2009) · Zbl 1187.34087
[13] Krüger, H.; Teschl, G., Effective Prüfer angles and relative oscillation criteria, J. differential equations, 245, 3823-3848, (2008) · Zbl 1167.34009
[14] O. Došlý, P. Hasil, Critical oscillation constant for half-linear differential equations with periodic coefficients, Ann. Mat. Pura Appl., in press (doi:10.1007/s10231-010-0155-0).
[15] Agarwal, R.P.; Grace, R.C.; O’Regan, D., Oscillation theory of second order linear, half-linear, superlinear and sublinear dynamic equations, (2002), Kluwer Academic Publishers Dordrecht, Boston, London · Zbl 1073.34002
[16] Došlý, O.; Řehák, P., ()
[17] Jaroš, J.; Kusano, T., A Picone type identity for half-linear differential equations, Acta math. univ. Comenian., 68, 137-151, (1999) · Zbl 0926.34023
[18] Došlý, O.; Elbert, Á., Integral characterization of principal solution of half-linear differential equations, Studia sci. math. hungar., 36, 455-469, (2000) · Zbl 1012.34029
[19] Došlý, O.; Řezníčková, J., Regular half-linear second order differential equations, Arch. math. (Brno), 39, 233-245, (2003) · Zbl 1119.34029
[20] O. Došlý, S. Fišnarová, Variational technique and principal solution in half-linear oscillation criteria (submitted for publication).
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