Došlý, Ondřej; Fišnarová, Simona Two-parametric conditionally oscillatory half-linear differential equations. (English) Zbl 1217.34054 Abstr. Appl. Anal. 2011, Article ID 182827, 16 p. (2011). Summary: We study perturbations of the nonoscillatory half-linear differential equation\[ (r(t)\Phi(x'))'+c(t)\Phi(x)=0, \]\(\Phi(x):=|x|^{p-2}x,\) \(p>1\). We find explicit formulas for the functions \(\widehat r\), \(\widehat c\) such that the equation \[ [(r(t)+\lambda\widehat r(t)) \Phi(x')]'+[c(t)+\mu\widehat c(t)] \Phi (x)=0 \]is conditionally oscillatory, that is, there exists a constant \(\gamma\) such that the previous equation is oscillatory if \(\mu-\lambda>\gamma\) and nonoscillatory if \(\mu-\lambda <\gamma\). The obtained results extend previous results concerning two-parametric perturbations of the half-linear Euler differential equation. Cited in 5 Documents MSC: 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations Keywords:nonoscillatory half-linear differential equation; two-parametric perturbations; Euler differential equation PDF BibTeX XML Cite \textit{O. Došlý} and \textit{S. Fišnarová}, Abstr. Appl. Anal. 2011, Article ID 182827, 16 p. (2011; Zbl 1217.34054) Full Text: DOI OpenURL References: [1] E. Hille, “Non-oscillation theorems,” Transactions of the American Mathematical Society, vol. 64, pp. 234-252, 1948. · Zbl 0031.35402 [2] T. Kusano, Y. Naito, and T. Tanigawa, “Strong oscillation and nonoscillation of quasilinear differential equations of second order,” Differential Equations and Dynamical Systems, vol. 2, no. 1, pp. 1-10, 1994. · Zbl 0869.34031 [3] O. Do and H. Haladová, “Half-linear Euler differential equations in the critical case,” to appear in Tatra Mountains Mathematical Publications. [4] O. Do and S. Fi, “Half-linear oscillation criteria: Perturbation in term involving derivative,” Nonlinear Analysis, Theory, Methods and Applications, vol. 73, no. 12, pp. 3756-3766, 2010. · Zbl 1207.34041 [5] P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston, Mass, USA, 2nd edition, 1982. · Zbl 0476.34002 [6] F. Gesztesy and M. Ünal, “Perturbative oscillation criteria and Hardy-type inequalities,” Mathematische Nachrichten, vol. 189, pp. 121-144, 1998. · Zbl 0903.34030 [7] O. Do and M. Ünal, “Conditionally oscillatory half-linear differential equations,” Acta Mathematica Hungarica, vol. 120, no. 1-2, pp. 147-163, 2008. · Zbl 1199.34169 [8] O. Do and P. , Half-Linear Differential Equations, vol. 202 of North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2005. [9] O. Do and S. Fi, “Variational technique and principal solution in half-linear oscillation criteria,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5385-5391, 2011. · Zbl 1217.34053 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.