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Riemann-Weber half-linear equation; Riccati technique; power comparison theorem; perturbation principle; principal solution
We consider the half-linear second order differential equation which is viewed as a perturbation of the so-called Riemann-Weber half-linear differential equation. We present a comparison theorem with respect to the power of the half-linearity in the equation under consideration. Our research is motivated by the recent results published by J. Sugie, N. Yamaoka, Acta Math. Hungar. 111 (2006), 165–179.
[1] R. P. Agarwal, S. R. Grace, D. O’Regan: Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations. Kluwer Academic Publishers, Dordrecht, 2002. MR 2091751
[2] O. Došlý: 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 | Zbl 1107.34030
[3] O. Došlý, A. Lomtatidze: 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
[4] O. Došlý, P. Řehák: Half-Linear Differential Equations. North-Holland Mathematics Studies 202, Elsevier, Amsterdam, 2005. MR 2158903
[5] Á. Elbert: A half-linear second order differential equation. Colloq. Math. Soc. János Bolyai 30 (1979), 153–180. MR 0680591
[6] Á. Elbert: Oscillation and nonoscillation theorems for some non-linear ordinary differential equations. Lect. Notes Math. 964 (1982), 187–212. DOI 10.1007/BFb0064999
[7] Á. Elbert, T. Kusano: Principal solutions of nonoscillatory half-linear differential equations. Adv. Math. Sci. Appl. 18 (1998), 745–759.
[8] Á. Elbert, A. Schneider: Perturbations of the half-linear Euler differential equation. Result. Math. 37 (2000), 56–83. DOI 10.1007/BF03322512 | MR 1742294
[9] J. D. Mirzov: On some analogs of Sturm’s and Kneser’s theorem for nonlinear systems. J. Math. Anal. Appl. 53 (1976), 418–425. DOI 10.1016/0022-247X(76)90120-7 | MR 0402184
[10] J. D. Mirzov: On the principal and nonprincipal solutions of a nonoscillatory system. Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 31 (1988), 100–117. MR 1001343
[11] P. Řehák: On certain comparison theorems for half-linear dynamic equations on time scales. Abstr. Appl. Anal. 7 (2004), 551–564. MR 2084935 | Zbl 1106.34019
[12] J. Sugie, N. Yamaoka: Growth conditions and oscillation of nonlinear differential equations with $p$-Laplacian. J. Math. Anal. Appl. 305 (2005), 18–34. MR 2132886
[13] J. Sugie, N. Yamaoka: Comparison theorems for oscillation of second-order half-linear differential equations. Acta Math. Hungar. 111 (2006), 165–179. DOI 10.1007/s10474-006-0029-5 | MR 2188979
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