Oscillation of perturbed nonlinear dynamic equations on time scales. (English) Zbl 1079.34513
Summary: We consider the oscillatory and asymptotic behavior of bounded solutions of the second-order nonlinear perturbed dynamic equation
\[
\biggl(\alpha (t)\bigl(x^\Delta(t)\bigr)^r\biggr)^\Delta+ F\bigl(t, x^\sigma(t)\bigr)=G\bigl( t,x^\sigma(t),x^\Delta(t)\bigr),\;t\in\mathbb{T},\tag{1}
\]
with \(r=k/l\), and \(k\) even and \(l\) odd positive integer. Some new sufficient conditions are obtained for all bounded solutions of (1) to be oscillatory. Several examples that dwell upon the importance of our results are also included. In particular, our criteria extend some earlier results.
MSC:
| 34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |
| 39A10 | Additive difference equations |
| 34E13 | Multiple scale methods for ordinary differential equations |
References:
| [1] | , , and , Nonoscillation and oscillation theory for functional differential equations, Monographs and textbooks in pure and applied mathematics, Vol. 267 (Marcel Dekker, New York, 2004). |
| [2] | Agarwal, J. Comp. Appl. Math. 141 pp 1– (2002) |
| [3] | Akin, J. Differ. Equations. Appl. 8 pp 389– (2002) |
| [4] | Akin, J. Differ. Equations Appl. 7 pp 793– (2001) |
| [5] | Bohner, Dynam. Systems Appl. 12 pp 45– (2003) |
| [6] | and , Dynamic equations on time scales: an introduction with applications (Birkhäuser Boston, MA, 2001). |
| [7] | and , Advances in dynamic equations on time scales (Birkhäuser Boston, 2003). |
| [8] | Bohner, Math. Comput. Modelling 40(3-4) pp 249– (2004) |
| [9] | Bohner, Rocky Mountain J. Math. 34 pp 1239– (2004) |
| [10] | Dosly, J. Comput. Appl. Math. 141 pp 147– (2002) |
| [11] | Hilger, Results Math. 18 pp 18– (1990) · Zbl 0722.39001 · doi:10.1007/BF03323153 |
| [12] | and , Oscillation of half-linear dynamic equations on time scales, 2003, preprint. |
| [13] | , , and , Dynamic systems on measure chains (Kluwer Academic Publishers, Boston, 1996). · Zbl 0869.34039 |
| [14] | Li, J. Math. Anal. Appl. 217 pp 1– (1998) |
| [15] | Peng, Appl. Math. Comput. 112 pp 103– (2000) |
| [16] | Saker, Appl. Math. Comput. 148 pp 81– (2004) |
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