Previous |  Up |  Next

Article

Keywords:
half-linear differential equation; oscillation; modified Riccati equation
Summary:
It is shown that oscillation of perturbed second order half-linear differential equations can be derived from oscillation of second order linear differential equations associated with modified Riccati equations. In the main result of the present paper, some of technical assumptions in the known results of this type are removed.
References:
[1] Došlá, Z., Došlý, O.: Principal solution of half-linear differential equation: limit and integral characterization. Electron. J. Qual. Theory Differ. Equ. 2008 (2008), 14 pp., paper No. 7. MR 2509168
[2] Došlý, O.: Perturbations of the half-linear Euler–Weber type differential equation. J. Math. Anal. Appl. 323 (2006), 426–440. DOI 10.1016/j.jmaa.2005.10.051 | MR 2262216
[3] Došlý, O.: Half-linear Euler differential equation and its perturbations. Electron. J. Qual. Theory Differ. Equ. 2016 (2016), 14 pp., paper No. 10. DOI 10.14232/ejqtde.2016.8.10 | MR 3631082
[4] Došlý, O., Elbert, Á.: Integral characterization of the principal solution of half-linear second order differential equations. Studia Sci. Math. Hungar. 36 (2000), 455–469. MR 1798750
[5] Došlý, O., Fišnarová, S.: Half-linear oscillation criteria: Perturbation in term involving derivative. Nonlinear Anal. 73 (2010), 3756–3766. DOI 10.1016/j.na.2010.07.049 | MR 2728552 | Zbl 1207.34041
[6] Došlý, O., Fišnarová, S.: Two-parametric conditionally oscillatory half-linear differential equations. Abstr. Appl. Anal. 2011 (2011), 16 pp., Article ID 182827. DOI 10.1155/2011/182827 | MR 2771241
[7] Došlý, O., Lomtatidze, A.: Oscillation and nonoscillation criteria for half-linear second order differential equations. Hiroshima Math. J. 36 (2006), 203–219. DOI 10.32917/hmj/1166642300 | MR 2259737
[8] Došlý, O., Řehák, P.: Half-Linear Differential Equations. North-Holland Mathematics Studies, vol. 202, Elsevier, Amsterdam, 2005. MR 2158903
[9] Došlý, O., Řezníčková, J.: Regular half-linear second order differential equations. Arch. Math. (Brno) 39 (2003), 233–245. MR 2010724
[10] Dosoudilová, M., Lomtatidze, A., Šremr, J.: Oscillatory properties of solutions to certain two-dimensional systems of non-linear ordinary differential equations. Nonlinear Anal. 120 (2015), 57–75. MR 3348046 | Zbl 1336.34053
[11] Elbert, Á., Schneider, A.: Perturbations of the half-linear Euler differential equation. Results Math. 37 (2000), 56–83. DOI 10.1007/BF03322512 | MR 1742294 | Zbl 0958.34029
[12] Luey, S., Usami, H.: Asymptotic forms of solutions of half-linear ordinary differential equations with integrable perturbations. Hiroshima Math. J. 53 (2023), 171–189. DOI 10.32917/h2022005 | MR 4612154
[13] Naito, M.: Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, I. Opuscula Math. 41 (2021), 71–94. DOI 10.7494/OpMath.2021.41.1.71 | MR 4302442
[14] Naito, M.: Existence and asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations. Opuscula Math. 43 (2023), 221–246. DOI 10.7494/OpMath.2023.43.2.221 | MR 4567780
[15] Naito, M.: Oscillation and nonoscillation for two-dimensional nonlinear systems of ordinary differential equations. Taiwanese J. Math. 27 (2023), 291–319. DOI 10.11650/tjm/221001 | MR 4563521
[16] Naito, M.: Oscillation criteria for perturbed half-linear differential equations. Electron. J. Qual. Theory Differ. Equ. 2024 (2024), 18 pp., paper No. 38. MR 4782772
[17] Naito, M., Usami, H.: On the existence and asymptotic behavior of solutions of half-linear ordinary differential equations. J. Differential Equations 318 (2022), 359–383. DOI 10.1016/j.jde.2022.02.025 | MR 4387287
[18] Řehák, P.: Nonlinear Poincaré–Perron theorem. Appl. Math. Lett. 121 (2021), 7 pp., Article ID 107425. MR 4268643
[19] Řehák, P.: Half-linear differential equations: Regular variation, principal solutions, and asymptotic classes. Electron. J. Qual. Theory Differ. Equ. 2023 (2023), 28 pp., paper No. 1. MR 4541736
[20] Zlámal, M.: Oscillation criterions. Časopis Pěst. Mat. Fys. 75 (1950), 213–218. MR 0042578
Partner of
EuDML logo