| Title:
|
A role of the coefficient of the differential term in qualitative theory of half-linear equations (English) |
| Author:
|
Řehák, Pavel |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
135 |
| Issue:
|
2 |
| Year:
|
2010 |
| Pages:
|
151-162 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
half-linear dynamic equation |
| Keyword:
|
time scale |
| Keyword:
|
transformation |
| Keyword:
|
comparison theorem |
| Keyword:
|
oscillation criteria |
| MSC:
|
34C10 |
| MSC:
|
34N05 |
| MSC:
|
39A12 |
| MSC:
|
39A13 |
| idZBL:
|
Zbl 1224.34293 |
| idMR:
|
MR2723082 |
| DOI:
|
10.21136/MB.2010.140692 |
| . |
| Date available:
|
2010-07-20T18:33:15Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140692 |
| . |
| Reference:
|
[1] Agarwal, R. P., Bohner, M.: Quadratic functionals for second order matrix equations on time scales.Nonlinear Anal. 33 (1998), 675-692. Zbl 0938.49001, MR 1634922 |
| Reference:
|
[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 |
| Reference:
|
[3] Bohner, M., Peterson, A. C.: Dynamic Equations on Time Scales: An Introduction with Applications.Birkhäuser, Boston (2001). Zbl 0978.39001, MR 1843232 |
| Reference:
|
[4] Bohner, M., Došlý, O.: The discrete Prüfer transformation.Proc. Amer. Math. Soc. 129 (2001), 2715-2726. Zbl 0980.39006, MR 1838796, 10.1090/S0002-9939-01-05833-6 |
| Reference:
|
[5] Bohner, M., Ünal, M.: Kneser's theorem in $q$-calculus.J. Phys. A: Math. Gen. 38 (2005), 6729-6739. Zbl 1080.39023, MR 2167223, 10.1088/0305-4470/38/30/008 |
| Reference:
|
[6] Došlý, O., Řehák, P.: Half-Linear Differential Equations.Elsevier, North Holland (2005). Zbl 1090.34001, MR 2158903 |
| Reference:
|
[7] Hilger, S.: Analysis on measure chains---a unified approach to continuous and discrete calculus.Res. Math. 18 (1990), 18-56. Zbl 0722.39001, MR 1066641, 10.1007/BF03323153 |
| Reference:
|
[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. Zbl 1176.34059, MR 2411064 |
| Reference:
|
[9] Matucci, S., Řehák, P.: Nonoscillation of half-linear dynamic equations.(to appear) in Rocky Moutain J. Math. MR 2672942 |
| Reference:
|
[10] Řehák, P.: Half-linear dynamic equations on time scales: IVP and oscillatory properties.Nonl. Funct. Anal. Appl. 7 (2002), 361-404. Zbl 1037.34002, MR 1946469 |
| Reference:
|
[11] Řehák, P.: Function sequence technique for half-linear dynamic equations on time scales.Panam. Math. J. 16 (2006), 31-56. Zbl 1102.34020, MR 2186537 |
| Reference:
|
[12] Řehák, P.: How the constants in Hille-Nehari theorems depend on time scales.Adv. Difference Equ. 2006 (2006), 1-15. Zbl 1139.39301, MR 2255171 |
| Reference:
|
[13] Řehák, P.: A critical oscillation constant as a variable of time scales for half-linear dynamic equations.Math. Slov. 60 (2010), 237-256. Zbl 1240.34478, MR 2595363, 10.2478/s12175-010-0009-7 |
| Reference:
|
[14] Řehák, P.: Peculiarities in power type comparison results for half-linear dynamic equations.Submitted. |
| Reference:
|
[15] Řehák, P.: New results on critical oscillation constants depending on a graininess.(to appear) in Dynam. Systems Appl. Zbl 1215.34115, MR 2741922 |
| Reference:
|
[16] : C. A. Swanson.<title>Comparison and Oscillation Theory of Linear Differential Equations</title>Academic Press, New York (1968). MR 0463570 |
| Reference:
|
[17] Willett, D.: Classification of second order linear differential equations with respect to oscillation.Adv. Math. 3 (1969), 594-623. Zbl 0188.40101, MR 0280800, 10.1016/0001-8708(69)90011-5 |
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