- Research Article
- Open Access
Oscillation Criteria for Second-Order Forced Dynamic Equations with Mixed Nonlinearities
© R.P. Agarwal and A. Zafer. 2009
- Received: 21 January 2009
- Accepted: 19 April 2009
- Published: 2 June 2009
We obtain new oscillation criteria for second-order forced dynamic equations on time scales containing mixed nonlinearities of the form , with , , where is a time scale interval with , the functions are right-dense continuous with , is the forward jump operator, , and . All results obtained are new even for and . In the special case when and our theorems reduce to (Y. G. Sun and J. S. W. Wong, Journal of Mathematical Analysis and Applications. 337 (2007), 549–560). Therefore, our results in particular extend most of the related existing literature from the continuous case to arbitrary time scale.
- Dynamic Equation
- Oscillation Criterion
- Jump Operator
- Infinite Interval
- Time Scale Modification
Let be a time scale which is unbounded above and a fixed point. For some basic facts on time scale calculus and dynamic equations on time scales, one may consult the excellent texts by Bohner and Peterson [1, 2].
By a proper solution of (1.1) on we mean a function which is defined and nontrivial in any neighborhood of infinity and which satisfies (1.1) for all , where denotes the set of right-dense continuously differentiable functions from to . As usual, such a solution of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative. The equation is called oscillatory if every proper solution is oscillatory.
for which almost nothing is available in the literature.
We need the following preparatory lemmas. The first two lemmas are given by Wong and Sun as a single lemma [34, Lemma ] for . The proof for the case is exactly the same, in fact one only needs to replace the exponents by in their proof. Lemma 2.3 is the well-known Young inequality.
Lemma 2.3 (Young's Inequality).
then (1.1) is oscillatory.
where stands for the inverse function. In our case, since , dynamic equation (3.15) has a unique solution satisfying . Clearly, the unique solution is . Therefore, on .
Using differential calculus, see , the result follows.
The remainder of the proof is the same as that of Theorem 3.1.
As it is shown in  for the sublinear terms case, we can also remove the sign condition imposed on the coefficients of the sub-half-linear terms to obtain interval criterion which is applicable for the case when some or all of the functions , , are nonpositive. We should note that the sign condition on the coefficients of super-half-linear terms cannot be removed alternatively by the same approach. Furthermore, the function cannot take the value zero on intervals of interest in this case. We have the following theorem.
then (1.1) is oscillatory.
The remainder of the proof is the same as that of Theorem 3.1, hence it is omitted.
4.1. Differential Equations
then (4.1) is oscillatory.
then (4.1) is oscillatory.
4.2. Difference Equations
then (4.13) is oscillatory.
then (4.13) is oscillatory.
then (4.25) is oscillatory.
then (4.25) is oscillatory.
Equation (1.1) has been studied by Sun and Wong  for the case and . Our results in Section 4.1 coincide with theirs when , and therefore the results can be considered as an extension from to . Since the results in  are linked to many well-known oscillation criteria in the literature, the interval oscillation criteria we have obtained provide further extensions of these to time scales.
The results in Sections 4.2 and 4.3 are all new for all values of the parameters. Although there are some results for difference equations in the special case , there is hardly any interval oscillation criteria for the -difference equations case.
Forms Related to (1.1)
It is not difficult to see that time scale modifications of the previous arguments give rise to completely parallel results for the above dynamic equations. For an illustrative example we provide below the version of Theorem 3.1 for (6.4). The other theorems for (6.4), (6.5), and (6.6) can be easily obtained by employing arguments developed for (1.1) in this paper.
An Open Problem
It is of theoretical and practical interest to obtain interval oscillation criteria when there are only sub-half-linear terms in (1.1), that is, when holds for all . Also, the open problems stated in  for the special case with naturally carry over for (1.1).
- Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleGoogle Scholar
- Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.View ArticleMATHGoogle Scholar
- Došlý O, Řehák P: Half-Linear Differential Equations, North-Holland Mathematics Studies. Volume 202. Elsevier; North-Holland, Amsterdam, The Netherlands; 2005:xiv+517.Google Scholar
- Agarwal RP, Grace SR: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2002.Google Scholar
- Agarwal RP, Grace SR, O'Regan D: Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2002:xiv+672.View ArticleMATHGoogle Scholar
- Bohner M, Tisdell CC: Oscillation and nonoscillation of forced second order dynamic equations. Pacific Journal of Mathematics 2007,230(1):59-71. 10.2140/pjm.2007.230.59MathSciNetView ArticleMATHGoogle Scholar
- Došlý O, Marek D: Half-linear dynamic equations with mixed derivatives. Electronic Journal of Differential Equations 2005,2005(90):1-18.Google Scholar
- Erbe L, Peterson A, Saker SH: Hille-Kneser-type criteria for second-order linear dynamic equations. Advances in Difference Equations 2006, 2006:-18.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
- Kartsatos AG: Maintenance of oscillations under the effect of a periodic forcing term. Proceedings of the American Mathematical Society 1972, 33: 377-383. 10.1090/S0002-9939-1972-0330622-0MathSciNetView ArticleMATHGoogle Scholar
- Manojlović JV: Oscillation criteria for second-order half-linear differential equations. Mathematical and Computer Modelling 1999,30(5-6):109-119. 10.1016/S0895-7177(99)00151-XMathSciNetView ArticleMATHGoogle Scholar
- Řehák P: Half-linear dynamic equations on time scales: IVP and oscillatory properties. Nonlinear Functional Analysis and Applications 2002,7(3):361-403.MathSciNetMATHGoogle Scholar
- Saker SH: Oscillation criteria of second-order half-linear dynamic equations on time scales. Journal of Computational and Applied Mathematics 2005,177(2):375-387. 10.1016/j.cam.2004.09.028MathSciNetView ArticleMATHGoogle Scholar
- Sun YG, Agarwal RP:Forced oscillation of th-order nonlinear differential equations. Functional Differential Equations 2004,11(3-4):587-596.MathSciNetMATHGoogle Scholar
- Sun YG, Saker SH: Forced oscillation of higher-order nonlinear differential equations. Applied Mathematics and Computation 2006,173(2):1219-1226. 10.1016/j.amc.2005.04.065MathSciNetView ArticleMATHGoogle Scholar
- Sun YG, Wong JSW:Note on forced oscillation of th-order sublinear differential equations. Journal of Mathematical Analysis and Applications 2004,298(1):114-119. 10.1016/j.jmaa.2004.03.076MathSciNetView ArticleMATHGoogle Scholar
- Teufel H Jr.: Forced second order nonlinear oscillation. Journal of Mathematical Analysis and Applications 1972, 40: 148-152. 10.1016/0022-247X(72)90037-6MathSciNetView ArticleGoogle Scholar
- Wang Q-R, Yang Q-G: Interval criteria for oscillation of second-order half-linear differential equations. Journal of Mathematical Analysis and Applications 2004,291(1):224-236. 10.1016/j.jmaa.2003.10.028MathSciNetView ArticleMATHGoogle Scholar
- Wong JSW: Second order nonlinear forced oscillations. SIAM Journal on Mathematical Analysis 1988,19(3):667-675. 10.1137/0519047MathSciNetView ArticleMATHGoogle Scholar
- El-Sayed MA: An oscillation criterion for a forced second order linear differential equation. Proceedings of the American Mathematical Society 1993,118(3):813-817.MathSciNetMATHGoogle Scholar
- Anderson DR: Oscillation of second-order forced functional dynamic equations with oscillatory potentials. Journal of Difference Equations and Applications 2007,13(5):407-421. 10.1080/10236190601116209MathSciNetView ArticleMATHGoogle Scholar
- Anderson DR, Zafer A: Interval criteria for second-order super-half-linear functional dynamic equations with delay and advanced arguments. to appear in Journal of Difference Equations and ApplicationsGoogle Scholar
- Li W-T: Interval oscillation of second-order half-linear functional differential equations. Applied Mathematics and Computation 2004,155(2):451-468. 10.1016/S0096-3003(03)00790-2MathSciNetView ArticleMATHGoogle Scholar
- Li W-T, Cheng SS: An oscillation criterion for nonhomogenous half-linear differential equations. Applied Mathematics Letters 2002,15(3):259-263. 10.1016/S0893-9659(01)00127-6MathSciNetView ArticleMATHGoogle Scholar
- Nasr AH: Sufficient conditions for the oscillation of forced super-linear second order differential equations with oscillatory potential. Proceedings of the American Mathematical Society 1998,126(1):123-125. 10.1090/S0002-9939-98-04354-8MathSciNetView ArticleMATHGoogle Scholar
- Sun YG, Ou CH, Wong JSW: Interval oscillation theorems for a second-order linear differential equation. Computers & Mathematics with Applications 2004,48(10-11):1693-1699. 10.1016/j.camwa.2003.08.012MathSciNetView ArticleMATHGoogle Scholar
- Sun YG: A note Nasr's and Wong's papers. Journal of Mathematical Analysis and Applications 2003,286(1):363-367. 10.1016/S0022-247X(03)00460-8MathSciNetView ArticleMATHGoogle Scholar
- Wong JSW: Oscillation criteria for a forced second-order linear differential equation. Journal of Mathematical Analysis and Applications 1999,231(1):235-240. 10.1006/jmaa.1998.6259MathSciNetView ArticleMATHGoogle Scholar
- Yang Q: Interval oscillation criteria for a forced second order nonlinear ordinary differential equations with oscillatory potential. Applied Mathematics and Computation 2003,135(1):49-64. 10.1016/S0096-3003(01)00307-1MathSciNetView ArticleMATHGoogle Scholar
- Zafer A: Interval oscillation criteria for second order super-half-linear functional differential equations with delay and advanced arguments. to appear in Mathematische NachrichtenGoogle Scholar
- Güvenilir AF, Zafer A: Second-order oscillation of forced functional differential equations with oscillatory potentials. Computers & Mathematics with Applications 2006,51(9-10):1395-1404. 10.1016/j.camwa.2006.02.002MathSciNetView ArticleMATHGoogle Scholar
- Řehák P: Hardy inequality on time scales and its applications to half-linear dynamic equations. Journal of Inequalities and Applications 2005,2005(7):495-507.MATHGoogle Scholar
- Řehák P: On certain comparison theorems for half-linear dynamic equations on time scales. Abstract and Applied Analysis 2004,2004(7):551-565. 10.1155/S1085337504306251View ArticleMATHGoogle Scholar
- Sun YG, Wong JSW: Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities. Journal of Mathematical Analysis and Applications 2007,334(1):549-560. 10.1016/j.jmaa.2006.07.109MathSciNetView ArticleMATHGoogle Scholar
- Beckenbach EF, Bellman R: Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F.. Volume 30. Springer, Berlin, Germany; 1961:xii+198.Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.