Asymptotic Behavior of Solutions of Higher-Order Dynamic Equations on Time Scales
© Taixiang Sun et al. 2011
Received: 18 November 2010
Accepted: 23 February 2011
Published: 14 March 2011
We investigate the asymptotic behavior of solutions of the following higher-order dynamic equation , on an arbitrary time scale , where the function is defined on . We give sufficient conditions under which every solution of this equation satisfies one of the following conditions: (1) ; (2) there exist constants with , such that , where are as in Main Results.
Since we are interested in the asymptotic and oscillatory behavior of solutions near infinity, we assume that , and define the time scale interval , where . By a solution of (1.1), we mean a nontrivial real-valued function , which has the property that and satisfies (1.1) on , where is the space of rd-continuous functions. The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solution of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is called nonoscillatory.
The theory of time scales, which has recently received a lot of attention, was introduced by Hilger's landmark paper  in order to create a theory that can unify continuous and discrete analysis. The cases when a time scale is equal to the real numbers or to the integers represent the classical theories of differential and of difference equations. Many other interesting time scales exist, and they give rise to many applications (see ). Not only the new theory of the so-called "dynamic equations" unifies the theories of differential equations and difference equations but also extends these classical cases to cases "in between," for example, to the so-called -difference equations when , which has important applications in quantum theory (see ).
respectively, and established some sufficient conditions for oscillation.
on an arbitrary time scale T. Motivated by the above studies, in this paper, we study (1.1) and give sufficient conditions under which every solution of (1.1) satisfies one of the following conditions: (1) ; (2) there exist constants with , such that , where are as in Section 2.
2. Main Results
To obtain our main results, we need the following lemmas.
Lemma 2.2 (see ).
Lemma 2.3 (see ).
Lemma 2.4 (see ).
Lemma 2.5 (see ).
Lemma 2.6 (see ).
Now, one states and proves the main results.
The rest of the proof is similar to that of Theorem 2.8, and the details are omitted. The proof is completed.
by Example 5.60 in . Thus, it follows from Theorem 2.7 that if is a solution of (3.1) with , then there exist constants with , such that .
This paper was supported by NSFC (no. 10861002) and NSFG (no. 2010GXNSFA013106, no. 2011GXNSFA018135) and IPGGE (no. 105931003060).
- Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results in Mathematics 1990,18(1-2):18-56.MathSciNetView ArticleMATHGoogle Scholar
- Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleMATHGoogle Scholar
- Kac V, Cheung P: Quantum Calculus, Universitext. Springer, New York, NY, USA; 2002:x+112.View ArticleMATHGoogle Scholar
- Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.View ArticleMATHGoogle Scholar
- Bohner M, Saker SH: Oscillation of second order nonlinear dynamic equations on time scales. The Rocky Mountain Journal of Mathematics 2004,34(4):1239-1254. 10.1216/rmjm/1181069797MathSciNetView ArticleMATHGoogle Scholar
- Erbe L: Oscillation results for second-order linear equations on a time scale. Journal of Difference Equations and Applications 2002,8(11):1061-1071. 10.1080/10236190290015317MathSciNetView ArticleMATHGoogle Scholar
- Hassan TS: Oscillation criteria for half-linear dynamic equations on time scales. Journal of Mathematical Analysis and Applications 2008,345(1):176-185. 10.1016/j.jmaa.2008.04.019MathSciNetView ArticleMATHGoogle Scholar
- Agarwal RP, Bohner M, Saker SH: Oscillation of second order delay dynamic equations. The Canadian Applied Mathematics Quarterly 2005,13(1):1-17.MathSciNetMATHGoogle Scholar
- Bohner M, Karpuz B, Öcalan Ö: Iterated oscillation criteria for delay dynamic equations of first order. Advances in Difference Equations 2008, 2008:-12.Google Scholar
- Erbe L, Peterson A, Saker SH: Oscillation criteria for second-order nonlinear delay dynamic equations. Journal of Mathematical Analysis and Applications 2007,333(1):505-522. 10.1016/j.jmaa.2006.10.055MathSciNetView ArticleMATHGoogle Scholar
- Han Z, Shi B, Sun S: Oscillation criteria for second-order delay dynamic equations on time scales. Advances in Difference Equations 2007, 2007:-16.Google Scholar
- Han Z, Sun S, Shi B: Oscillation criteria for a class of second-order Emden-Fowler delay dynamic equations on time scales. Journal of Mathematical Analysis and Applications 2007,334(2):847-858. 10.1016/j.jmaa.2007.01.004MathSciNetView ArticleMATHGoogle Scholar
- Şahiner Y: Oscillation of second-order delay differential equations on time scales. Nonlinear Analysis: Theory, Methods and Applications 2005,63(5–7):e1073-e1080.MATHGoogle Scholar
- Akin-Bohner E, Bohner M, Djebali S, Moussaoui T: On the asymptotic integration of nonlinear dynamic equations. Advances in Difference Equations 2008, 2008:-17.Google Scholar
- Hassan TS: Oscillation of third order nonlinear delay dynamic equations on time scales. Mathematical and Computer Modelling 2009,49(7-8):1573-1586. 10.1016/j.mcm.2008.12.011MathSciNetView ArticleMATHGoogle Scholar
- Grace SR, Agarwal RP, Kaymakçalan B, Sae-jie W: On the oscillation of certain second order nonlinear dynamic equations. Mathematical and Computer Modelling 2009,50(1-2):273-286. 10.1016/j.mcm.2008.12.007MathSciNetView ArticleMATHGoogle Scholar
- Sun T, Xi H, Yu W: Asymptotic behaviors of higher order nonlinear dynamic equations on time scales. Journal of Applied Mathematics and Computing. In pressGoogle Scholar
- Sun T, Xi H, Peng X, Yu W: Nonoscillatory solutions for higher-order neutral dynamic equations on time scales. Abstract and Applied Analysis 2010, 2010:-16.Google Scholar
- Erbe L, Peterson A, Saker SH: Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales. Journal of Computational and Applied Mathematics 2005,181(1):92-102. 10.1016/j.cam.2004.11.021MathSciNetView ArticleMATHGoogle Scholar
- Erbe L, Peterson A, Saker SH: Hille and Nehari type criteria for third-order dynamic equations. Journal of Mathematical Analysis and Applications 2007,329(1):112-131. 10.1016/j.jmaa.2006.06.033MathSciNetView ArticleMATHGoogle Scholar
- Erbe L, Peterson A, Saker SH: Oscillation and asymptotic behavior of a third-order nonlinear dynamic equation. The Canadian Applied Mathematics Quarterly 2006,14(2):129-147.MathSciNetMATHGoogle Scholar
- Karpuz B: Asymptotic behaviour of bounded solutions of a class of higher-order neutral dynamic equations. Applied Mathematics and Computation 2009,215(6):2174-2183. 10.1016/j.amc.2009.08.013MathSciNetView ArticleMATHGoogle Scholar
- Chen D-X: Oscillation and asymptotic behavior for n th-order nonlinear neutral delay dynamic equations on time scales. Acta Applicandae Mathematicae 2010,109(3):703-719. 10.1007/s10440-008-9341-0MathSciNetView ArticleMATHGoogle Scholar
- Bohner M: Some oscillation criteria for first order delay dynamic equations. Far East Journal of Applied Mathematics 2005,18(3):289-304.MathSciNetMATHGoogle Scholar
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