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Keywords:
half-linear dynamic equation; time scale; transformation; comparison theorem; oscillation criteria
half-linear dynamic equation; time scale; transformation; comparison theorem; oscillation criteria
Summary:
The aim of this contribution is to study the role of the coefficient $r$ in the qualitative theory of the equation $(r(t)\Phi (y^{\Delta}))^{\Delta} +p(t)\Phi (y^{\sigma})=0$, where $\Phi (u)=|u|^{\alpha -1}\mathop{\rm sgn}u$ with $\alpha >1$. We discuss sign and smoothness conditions posed on $r$, (non)availability of some transformations, and mainly we show how the behavior of $r$, along with the behavior of the graininess of the time scale, affect some comparison results and (non)oscillation criteria. At the same time we provide a survey of recent results acquired by sophisticated modifications of the Riccati type technique, which are supplemented by some new observations.
References:
[1] Agarwal, R. P., Bohner, M.: Quadratic functionals for second order matrix equations on time scales. Nonlinear Anal. 33 (1998), 675-692. MR 1634922 | Zbl 0938.49001
[2] Agarwal, R. P., Bohner, M., Řehák, P.: Half-linear dynamic equations on time scales: IVP and oscillatory properties. V. Lakshmikantham on his 80th Birthday, R. P. Agarwal, D. O'Regan Nonlinear Analysis and Applications. Vol. I. Kluwer Academic Publishers, Dordrecht (2003), 1-56. MR 2060210
[3] Bohner, M., Peterson, A. C.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001). MR 1843232 | Zbl 0978.39001
[4] Bohner, M., Došlý, O.: The discrete Prüfer transformation. Proc. Amer. Math. Soc. 129 (2001), 2715-2726. DOI 10.1090/S0002-9939-01-05833-6 | MR 1838796 | Zbl 0980.39006
[5] Bohner, M., Ünal, M.: Kneser's theorem in $q$-calculus. J. Phys. A: Math. Gen. 38 (2005), 6729-6739. DOI 10.1088/0305-4470/38/30/008 | MR 2167223 | Zbl 1080.39023
[6] Došlý, O., Řehák, P.: Half-Linear Differential Equations. Elsevier, North Holland (2005). MR 2158903 | Zbl 1090.34001
[7] Hilger, S.: Analysis on measure chains---a unified approach to continuous and discrete calculus. Res. Math. 18 (1990), 18-56. DOI 10.1007/BF03323153 | MR 1066641 | Zbl 0722.39001
[8] Hilscher, R., Tisdell, C. C.: Terminal value problems for first and second order nonlinear equations on time scales. Electron. J. Differential Equations 68 (2008), 21 pp. MR 2411064 | Zbl 1176.34059
[9] Matucci, S., Řehák, P.: Nonoscillation of half-linear dynamic equations. (to appear) in Rocky Moutain J. Math. MR 2672942
[10] Řehák, P.: Half-linear dynamic equations on time scales: IVP and oscillatory properties. Nonl. Funct. Anal. Appl. 7 (2002), 361-404. MR 1946469 | Zbl 1037.34002
[11] Řehák, P.: Function sequence technique for half-linear dynamic equations on time scales. Panam. Math. J. 16 (2006), 31-56. MR 2186537 | Zbl 1102.34020
[12] Řehák, P.: How the constants in Hille-Nehari theorems depend on time scales. Adv. Difference Equ. 2006 (2006), 1-15. MR 2255171 | Zbl 1139.39301
[13] Řehák, P.: A critical oscillation constant as a variable of time scales for half-linear dynamic equations. Math. Slov. 60 (2010), 237-256. DOI 10.2478/s12175-010-0009-7 | MR 2595363 | Zbl 1240.34478
[14] Řehák, P.: Peculiarities in power type comparison results for half-linear dynamic equations. Submitted.
[15] Řehák, P.: New results on critical oscillation constants depending on a graininess. (to appear) in Dynam. Systems Appl. MR 2741922 | Zbl 1215.34115
[16] C. A. Swanson. Comparison and Oscillation Theory of Linear Differential Equations Academic Press, New York (1968). MR 0463570
[17] Willett, D.: Classification of second order linear differential equations with respect to oscillation. Adv. Math. 3 (1969), 594-623. DOI 10.1016/0001-8708(69)90011-5 | MR 0280800 | Zbl 0188.40101
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