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On constants in nonoscillation criteria for half-linear differential equations. (English) Zbl 1232.34052
Summary: We study the half-linear differential equation $$(r(t)\Phi(x'))' + c(t)\Phi(x) = 0 \text{, where }\Phi(x) = |x|^{p-2}, p > 1.$$ Using the modified Riccati technique, we derive new nonoscillation criteria for this equation. The results are closely related to the classical Hille-Nehari criteria and allow to replace the fixed constants in known nonoscillation criteria by a certain one-parametric expression.

MSC:
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
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References:
[1] O. Do\vslý and P. \vRehák, Half-Linear Differential Equations, vol. 202 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2005.
[2] O. Do\vslý and J. \vRezní\vcková, “Oscillation and nonoscillation of perturbed half-linear Euler differential equations,” Publicationes Mathematicae Debrecen, vol. 71, no. 3-4, pp. 479-488, 2007. · Zbl 1164.34012
[3] O. Do\vslý, “Perturbations of the half-linear Euler-Weber type differential equation,” Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 426-440, 2006. · Zbl 1107.34030 · doi:10.1016/j.jmaa.2005.10.051
[4] S. Fi\vsnarová and R. Ma\vrík, “Half-linear ODE and modified Riccati equation: comparison theorems, integral characterization of principal solution,” Nonlinear Analysis: Theory, Methods and Applications, vol. 74, no. 17, pp. 6427-6433, 2011. · Zbl 1229.34048 · doi:10.1016/j.na.2011.06.025
[5] Á. Elbert and A. Schneider, “Perturbations of the half-linear Euler differential equation,” Results in Mathematics, vol. 37, no. 1-2, pp. 56-83, 2000. · Zbl 0958.34029 · doi:10.1007/BF03322512
[6] O. Do\vslý and M. Ünal, “Conditionally oscillatory half-linear differential equations,” Acta Mathematica Hungarica, vol. 120, no. 1-2, pp. 147-163, 2008. · Zbl 1199.34169 · doi:10.1007/s10474-007-7120-4
[7] N. Kandelaki, A. Lomtatidze, and D. Ugulava, “On oscillation and nonoscillation of a second order half-linear equation,” Georgian Mathematical Journal, vol. 7, no. 2, pp. 329-346, 2000. · Zbl 0957.34032 · eudml:48668
[8] O. Do\vslý, “A remark on the linearization technique in half-linear oscillation theory,” Opuscula Mathematica, vol. 26, no. 2, pp. 305-315, 2006. · Zbl 1135.34314
[9] O. Do\vslý and S. Fi\vsnarová, “Half-linear oscillation criteria: perturbation in term involving derivative,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 12, pp. 3756-3766, 2010. · Zbl 1207.34041 · doi:10.1016/j.na.2010.07.049
[10] O. Do\vslý and M. Ünal, “Half-linear differential equations: linearization technique and its application,” Journal of Mathematical Analysis and Applications, vol. 335, no. 1, pp. 450-460, 2007. · Zbl 1128.34017 · doi:10.1016/j.jmaa.2007.01.080