P a g e 46 Vol.10 Issue 5(Ver 1.0)September 2010 Global Journal of Science Frontier Research Oscillation criteria for second-order half-linear dynamic equations on time scales Zhenlai Han a,b, Tongxing Li a;b, Shurong Sun a;c, Weisong Chen a GJSFR Classification F (FOR) 010203,010204,010109 Abstract-In this paper, some oscillation criteria are established for second-order half-linear dynamic equations on time scales. The results obtained essentially improve and extend those of Agarwal et al. [R. P. Agarwal, D. O Regan, S. H. Saker, Philostype oscillation criteria for second-order half linear dynamic equations, Rocky Mountain J. Math. 37 (2007) 1085 1104], Saker [S. H. Saker, Oscillation criteria of second order halflinear dynamic equations on time scales, J. Comput. Appl. Math. 177 (2005) 375 387], Hassan [T. S. Hassan, Oscillation criteria for half-linear dynamic equations on time scales, J. Math. Anal. Appl. 345 (2008) 176 185]. Some examples are given to illustrate the main results. Keywords-Oscillation; Half-linear dynamic equations; Time scales Mathematics Subject Classification 2010: 34C10, 34K11, 34N05 I. INTRODUCTION The study of dynamic equations on time scales, which goes back to its founder Stefan Hilger [1], is an area of mathematics that has recently received a lot of attention. Several authors have expounded on various aspects of this new theory, see the survey paper by Agarwal et al. [2] and the references cited therein. For an excellent introduction to the calculus on time scales, see Bohner and Peterson [3]. Further information on working with dynamic equations on time scales can be found in [4].Recently, much attention is attracted by questions of the oscillation and nonoscillation of different classes of dynamic equations on time scales, we refer the reader to the papers [5 17] and the references therein. We are concerned with the oscillation of the second-order half-linear dynamic equations (1) on a time scale T; is a quotient of odd positive integers, r and p are rd-continuous positive functions defined on T: Since we are interested in oscillatory behavior, we assume throughout this paper that the given time scale T is About- Corresponding author: Zhenlai Han, e-mail: hanzhenlai@163.com. This research is supported by the Natural Science Foundation of China (60774004, 60904024), China Postdoctoral Science Foundation funded project (20080441126, 200902564), Shandong Postdoctoral funded project (200802018) and supported by the Natural Science Foundation of Shandong (Y2008A28, ZR2009AL003), also supported by University of Jinan Research Funds for Doctors (XBS0843). 1 unbounded above. We assume t0 2 T and it is convenient to assume t0 > 0: We define the time scale interval of the form Agarwal et al. [5], Grace et al. [8], Hassan [10], Saker et al. [14] studied (1.1), and established some oscillation criteria under the case (2) and the authors obtained some sufficient conditions which guarantee that every solution x of (1.1) oscillates or under the case ( 3) Our aim of this paper is to establish some new oscillation criteria for (1.1) under the case when (1.3). The paper is organized as follows: In Section 2, we shall establish several new oscillation criteria for (1.1). In Section 3, some applications and examples are given to illustrate the main results. II. MAIN RESULTS In this section, by employing a Riccati transformation technique we establish oscillation criteria for Eq. (1.1). To prove our main results, we will use the formula which is a simple consequence of Keller s chain rule [3, Theorem 1.90].In the following, we denote Theorem 2.1-Assume that (1.3) holds, Further, there exists a positive function sufficiently large, such that and (2) then (1.1) is oscillatory. Proof. Let x be a nonoscillatory solution of (1.1). Without loss of generality we assume (1)
Global Journal of Science Frontier Research Vol.10 Issue 5(Ver 1.0),September 2010 P a g e 47 we get (3) Therefore, is an eventually strictly decreasing function, and there exists such that (11) Then is an eventually strictly decreasing function, hence Case 1- Define Integrating it from then (4) By the product rule and then the quotient rule, Taking in the above inequality, we get Thus From (2.4) and (2.3), we obtain By the above inequality and (2.11), (12) In view of Keller s chain rule, we find (5) differentiating (2.11) and by (2.3), we obtain (13) It follows from (2.5) and (2.6) that (6) In view of Keller s chain rule, we see that From (2.3), that (7) Thus (8) Note that we obtain Thus, by (2.7), (2.8) and (2.4), we get On the other hand, from Set (9) From (2.14), we find that get (14) Hence, we Using the inequality we get Noting that By (2.9), it is easy to see that by (2.12), we find that Integrating (2.10) from t1 to t; (10) Hence, by (2.14), which is a contradiction to (2.1). Case2. Define Multiplying (2.15) by we obtain (15)
P a g e 48 Vol.10 Issue 5(Ver 1.0)September 2010 Global Journal of Science Frontier Research Integrating it from t1 to t; we get Integrating by parts, In view of Keller s chain rule, we obtain (16) (17) (1.1), we obtain (2.3). Therefore, is an eventually strictly decreasing function, and there exists such that, Case 1 Define as (2.4). Proceeding as in the proof of Case 1 in Theorem 2.1, we can obtain a contradiction to (2.1). Case 2- Define as (2.11), that (2.12) and (2.13) hold. differentiating (2.11) and by (2.3), we obtain note that In view of Keller s chain rule, we get By (2.18), (2.17) and (2.16), we obtain that (18) Thus On the other hand, from (22) Define (19) Then by the inequality From (2.22), So, we get Noting that Thus, by (2.20) and (2.19), (20) by (2.12), Therefore, by (2.2), Hence, by (2.22), we get (2.15), then we obtain that (2.16) and (2.17) hold. In view of Keller s chain rule, which contradicts (2.13). This completes the proof. Theorem 2.2-Assume that (1.3) holds, Further, there exists a positive function (2.1) holds and sufficiently large, such that note that we see that From (2.23), (2.17) and (2.16), we obtain that (23) (21) then (1.1) is oscillatory. Proof-Let be a nonoscillatory solution of (1.1). Without loss of generality we assume In view of (24)
Global Journal of Science Frontier Research Vol.10 Issue 5(Ver 1.0),September 2010 P a g e 49 Then by the inequality Case 1- By (2.1), this case is not true. Case 2-When proceeding as in the proof of Case 2 of Theorem 2.1, that (2.15) holds. Multiplying (2.15) by and integrating it from t1 to t; we get Integrating by parts, we see that Thus, by (2.25) and (2.24), (25) in view of Keller s chain rule, we obtain note that It follows from (2.21) that Thus, from (2.27), we get which contradicts (2.13). This completes the proof. Theorem 2.3-Assume that (1.3) holds, Further, there exists a positive function sufficiently large, such that (2.1) holds, and By (2.13), we find that then (28) (26) then (1.1) is oscillatory. Proof-Let be a nonoscillatory solution of (1.1). Without loss of generality we assume As in the proof of Theorem 2.1 or Theorem 2.2, we consider two cases. Noting that we obtain and Thus, from (2.28), we get III. APPLICATIONS AND EXAMPLES which contradicts (2.26). When the proof is similar to the case so we omit it. This completes the proof. Agarwal et al. [5], Grace et al. [8], Hassan [10], Saker [14] considered Eq. (1.1), and established some oscillation criteria for (1.1). We introduce some results as follows. Theorem 3.1-(Saker [14]) Assume (1.3) holds and Furthermore, assume that there exists a positive function such that (2.1) holds, and
P a g e 50 Vol.10 Issue 5(Ver 1.0)September 2010 Global Journal of Science Frontier Research (1) Then every solution of (1.1) oscillates or converges to zero. Theorem 3.2-(Hassan [10]) Assume (1.3) and (3.1) hold, Furthermore, assume that function such that there exists a positive We have (5) (2) Thus, We get that (2.1) holds. On the other hand, Example 3.4-Consider the second-order super-linear dynamic equation. Then every solution of (1.1) oscillates or converges to zero. In the following, we shall give some examples to illustrate the main results. Example 3.1 -Consider the second-order sub-linear dynamic equation (3) We obtain Hence, other hand, (6) We have that (2.1) holds. On the We see that Let : We get that (2.1) holds. On the other hand Hence, from Theorem 2.1, every solution Example 3.2-Consider the second-order super-linear dynamic equation We find Let We get that (2.1) holds. On the other hand, Hence, from Theorem 2.2, every solution of (3.4) oscillates. Example 3.3-Consider the second-order sub-linear dynamic equation (4) Therefore, by Theorem 2.3, every solution of (3.6) oscillates. IV. REFERENCES 1) S. Hilger, Analysis on measure chains a unified approach to continuous and discrete calculus, Results Math. 18 (1990) 18 56. 2) R. P. Agarwal, M. Bohner, D. O Regan, A. Peterson, Dynamic equations on time scales: A survey. J. Comput. Appl. Math. 141 (2002) 1 26. 3) M.Bohner, A. Peterson, Dynamic equations on time scales, an introduction with applications, Birkh auser, Boston, 2001. 4) M.Bohner, A. Peterson, Advances in dynamic equations on time scales, Birkh auser, Boston,2003. 5) R. P. Agarwal, D. O Regan, S. H. Saker, Philostype oscillation criteria for second-order half linear dynamic equations, Rocky Mountain J. Math. 37 (2007) 1085 1104. 6) D.R. Anderson, Oscillation of second-order forced functional dynamic equations with oscillatory potentials, J. Difference Equ. Appl. 13 (2007) 407 421. 7) Erbe, T. S. Hassan, A. Peterson, Oscillation criteria for nonlinear damped dynamic equations on time scales, Appl. Math. Comput. 203 (2008) 343 357. 8) S. R. Grace, M. Bohner, R. P. Agarwal, On the oscillation of second-order half-linear dynamic equations, J. Difference Equ. Appl. 15 (2009) 451 460.
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