Asymptotic behavior of second-order nonlinear neutral dynamic equations
© Jing et al.; licensee Springer. 2013
Received: 31 July 2013
Accepted: 10 September 2013
Published: 8 November 2013
This paper is concerned with oscillation and asymptotic behavior of a second-order neutral delay dynamic equation on an arbitrary time scale. We obtain two theorems which guarantee that every solution of the studied equation oscillates or converges to zero. These results improve and complement some known results given in the literature.
MSC:34K11, 34N05, 39A10, 39A12, 39A13, 39A21.
on an arbitrary time scale , where γ is a quotient of odd positive integers, r and p are positive rd-continuous functions on , . Also, we assume that are rd-continuous, , , , for all , and there exists a positive rd-continuous function q defined on such that .
The theory of dynamic equations on time scales, which goes back to its founder Hilger , has recently attracted attention of researchers. Several authors have expounded on various aspects of this new theory; see the survey paper written by Agarwal et al.  and the references cited therein. The books on the subject of time scales, by Bohner and Peterson [3, 4], present much of time scale calculus.
Since we are interested in oscillatory and asymptotic properties, we assume throughout this paper that the given time scale is unbounded above. We assume that , and it is convenient to assume that , and define the time scale interval of the form by . Throughout, we use the notation . By a solution of equation (1.1), we mean a non-trivial real-valued function , which has the property that z and are defined and Δ-differentiable for and satisfies equation (1.1) on . The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solution x of equation (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is called nonoscillatory.
In 2007, Saker et al.  posed an open problem as follows: How to establish oscillation criteria for equation (1.1) when condition (1.5) holds? Assuming (1.5), Zhang et al. [21, 22] obtained some sufficient conditions which insure that all solutions of equation (1.1) are oscillatory.
The purpose of this paper is to present some asymptotic tests for equation (1.1) in the case where (1.5) holds. This paper is organized as follows: In the next section, we shall establish the main results. In Section 3, two examples are provided to illustrate the results obtained.
In the sequel, when we write a functional inequality without specifying its domain of validity, we assume that it holds for all sufficiently large t.
2 Main results
and, for sufficiently large , .
In order to prove our main results, we will use the following result; see [, Theorem 2.1].
Then every solution x of equation (1.1) is oscillatory.
then every solution x of equation (1.1) is oscillatory or .
Therefore, is strictly decreasing, and there exists a such that or for . We consider each of two cases separately.
Case 1. Assume that for . As in the proof of [, Theorem 2.1], we can obtain a contradiction to (2.1).
which yields , this is a contradiction. Hence, . By virtue of , . The proof is complete. □
Next, we establish another criterion which improves Theorem 2.2.
then every solution x of equation (1.1) is oscillatory or .
Proof Let x be a nonoscillatory solution of equation (1.1). Without loss of generality, we assume that , , and for . Then for . In view of (1.1), we get (2.3). Thus, is strictly decreasing, and there exists a such that or for . We consider each of two cases separately.
Case 1. Assume that for . Similarly to the proof of [, Theorem 2.1], we can obtain a contradiction to (2.1).
which implies that , this is a contradiction. Hence, . By , . This completes the proof. □
Remark 2.1 When , Theorems 2.2 and 2.3 improve results of Han et al. [, Theorems 2.1 and 2.2] since our results do not require condition (1.3).
In this section, we give two examples to illustrate applications of results in the previous section.
Thus, we have by Theorem 2.2 that every solution x of (3.1) is oscillatory or .
Hence, by Theorem 2.3, every solution x of (3.2) is oscillatory or .
This research is supported by the National Key Basic Research Program of P.R. China (2013CB035604) and the NNSF of P.R. China (Grant Nos. 61034007, 51277116, and 51107069).
- Hilger S: Analysis on measure chains-a unified approach to continuous and discrete calculus. Results Math. 1990, 18: 18–56. 10.1007/BF03323153MathSciNetView ArticleMATHGoogle Scholar
- Agarwal RP, Bohner M, O’Regan D, Peterson A: Dynamic equations on time scales: a survey. J. Comput. Appl. Math. 2002, 141: 1–26. 10.1016/S0377-0427(01)00432-0MathSciNetView ArticleMATHGoogle Scholar
- Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston; 2001.View ArticleGoogle Scholar
- Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston; 2003.View ArticleMATHGoogle Scholar
- Baculíková B, Džurina J: Oscillation of third-order neutral differential equations. Math. Comput. Model. 2010, 52: 215–226. 10.1016/j.mcm.2010.02.011View ArticleMATHGoogle Scholar
- Han Z, Li T, Sun S, Sun Y: Remarks on the paper [Appl. Math. Comput. 207 (2009) 388–396]. Appl. Math. Comput. 2010, 215: 3998–4007. 10.1016/j.amc.2009.12.006MathSciNetView ArticleMATHGoogle Scholar
- Agarwal RP, O’Regan D, Saker SH: Oscillation criteria for second-order nonlinear neutral delay dynamic equations. J. Math. Anal. Appl. 2004, 300: 203–217. 10.1016/j.jmaa.2004.06.041MathSciNetView ArticleMATHGoogle Scholar
- Erbe L, Hassan TS, Peterson A: Oscillation criteria for nonlinear functional neutral dynamic equations on time scales. J. Differ. Equ. Appl. 2009, 15: 1097–1116. 10.1080/10236190902785199MathSciNetView ArticleMATHGoogle Scholar
- Şahiner Y: Oscillation of second order neutral delay and mixed type dynamic equations on time scales. Adv. Differ. Equ. 2006, 2006: 1–9.Google Scholar
- Saker SH: Oscillation of second-order nonlinear neutral delay dynamic equations on time scales. J. Comput. Appl. Math. 2006, 187: 123–141. 10.1016/j.cam.2005.03.039MathSciNetView ArticleMATHGoogle Scholar
- Saker SH, Agarwal RP, O’Regan D: Oscillation results for second-order nonlinear neutral delay dynamic equations on time scales. Appl. Anal. 2007, 86: 1–17. 10.1081/00036810601091630MathSciNetView ArticleMATHGoogle Scholar
- Saker SH, O’Regan D: New oscillation criteria for second-order neutral functional dynamic equations via the generalized Riccati substitution. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 423–434. 10.1016/j.cnsns.2009.11.032MathSciNetView ArticleMATHGoogle Scholar
- Chen DX: Oscillation of second-order Emden-Fowler neutral delay dynamic equations on time scales. Math. Comput. Model. 2010, 51: 1221–1229. 10.1016/j.mcm.2010.01.004View ArticleMATHGoogle Scholar
- Zhang SY, Wang QR: Oscillation of second-order nonlinear neutral dynamic equations on time scales. Appl. Math. Comput. 2010, 216: 2837–2848. 10.1016/j.amc.2010.03.134MathSciNetView ArticleMATHGoogle Scholar
- Wu HW, Zhuang RK, Mathsen RM: Oscillation criteria for second-order nonlinear neutral variable delay dynamic equations. Appl. Math. Comput. 2006, 178: 321–331. 10.1016/j.amc.2005.11.049MathSciNetView ArticleMATHGoogle Scholar
- Li T, Han Z, Sun S, Yang D: Existence of nonoscillatory solutions to second-order neutral delay dynamic equations on time scales. Adv. Differ. Equ. 2009, 2009: 1–10.MathSciNetGoogle Scholar
- Candan T: Oscillation criteria for second-order nonlinear neutral dynamic equations with distributed deviating arguments on time scales. Adv. Differ. Equ. 2013, 2013: 1–8. 10.1186/1687-1847-2013-1MathSciNetView ArticleGoogle Scholar
- Karpuz B: Necessary and sufficient conditions on the asymptotic behavior of second-order neutral delay dynamic equations with positive and negative coefficients. Math. Methods Appl. Sci. 2013. 10.1002/mma.2884Google Scholar
- Li T, Agarwal RP, Bohner M: Some oscillation results for second-order neutral dynamic equations. Hacet. J. Math. Stat. 2012, 41: 715–721.MathSciNetMATHGoogle Scholar
- Yang J: Oscillation criteria for certain third-order variable delay functional dynamic equations on time scales. J. Appl. Math. Comput. 2013, 43: 445–466. 10.1007/s12190-013-0672-2View ArticleMATHGoogle Scholar
- Zhang C, Agarwal RP, Bohner M, Li T: New oscillation results for second-order neutral delay dynamic equations. Adv. Differ. Equ. 2012, 2012: 1–14. 10.1186/1687-1847-2012-1MathSciNetView ArticleMATHGoogle Scholar
- Zhang, C, Agarwal, RP, Bohner, M, Li, T: Oscillation of second-order nonlinear neutral dynamic equations with noncanonical operators. Bull. Malays. Math. Sci. Soc. (2013, in press)Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.