Existence of Nonoscillatory Solutions of a Class of Nonlinear Dynamic Equations with a Forced Term

Affiliation(s)

College of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao, China.

College of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao, China.

ABSTRACT

In this paper, we consider the following forced higher-order nonlinear neutral dynamic equation on time scales. By using Banach contraction principle, we obtain sufficient conditions for the existence of nonoscillatory solutions for general and which means that we allow oscillatory and . We give some examples to illustrate the obtained results.

In this paper, we consider the following forced higher-order nonlinear neutral dynamic equation on time scales. By using Banach contraction principle, we obtain sufficient conditions for the existence of nonoscillatory solutions for general and which means that we allow oscillatory and . We give some examples to illustrate the obtained results.

Cite this paper

S. Zhu and X. Zhang, "Existence of Nonoscillatory Solutions of a Class of Nonlinear Dynamic Equations with a Forced Term,"*Open Journal of Discrete Mathematics*, Vol. 2 No. 4, 2012, pp. 173-178. doi: 10.4236/ojdm.2012.24035.

S. Zhu and X. Zhang, "Existence of Nonoscillatory Solutions of a Class of Nonlinear Dynamic Equations with a Forced Term,"

References

[1] S. Hilger, “Analysis on Measure Chains—A Unified Approach to Continuous and Discrete Calculus,” Results in Mathematics, Vol. 18, No. 1-2, 1990, pp. 18-56.

[2] R. Agarwal, M. Bohner, D. O’Regan and A. Peterson, “Dynamic Equations on Time Scales: A Survey,” Journal of Computational and Applied Mathematics, Vol. 141, No. 1-2, 2002, pp. 1-26. doi:10.1016/S0377-0427(01)00432-0

[3] M. Bohner and A. Peterson, “Dynamic Equations on Time Scales: An Introduction with Applications,” Birkh?user, Boston, 2001.

[4] M. Bohner and A. Peterson, “Advances in Dynamic Equations on Time Scales,” Birkh?user, Boston, 2003. doi:10.1007/978-0-8176-8230-9

[5] B. G. Zhang and S. L. Zhu, “Oscillation of Second-Order Nonlinear Delay Dynamic Equations on Time Scales,” Computers & Mathematics with Applications, Vol. 49, No. 4, 2005, pp. 599-609.

[6] Q. L. Li and Z. Zhang, “Existence of Solutions to Nth Order Neutral Dynamic Equations on Time Scale,” Electronic Journal of Differential Equations, Vol. 2010, No. 151, 2010, pp. 1-8. http://ejde.math.txstate.edu or http://ejde.math.unt.edu, ftp ejde.math.txstate.edu

[7] D. X. Chen, “Oscillation and Asymptotic Behavior for Nth-Order Nonlinear Neutral Delay Dynamic Eqautions on Time Scales,” Acta Applicandae Mathematicae, Vol. 109, No. 3, 2010, pp. 703-719. doi:10.1007/s10440-008-9341-0

[8] T. S. Hassan, “Oscillation of Third Order Nonlinear Delay Dynamic Equations on Time Scales,” Mathematical and Computer Modelling, Vol. 49, No. 7-8, 2009, pp. 1573-1586. doi:10.1016/j.mcm.2008.12.011

[9] Z. Q. Zhu and Q. R. Wang, “Existence of Nonoscillatory Solutions to Neutral Dynamic Equations on Time Scales,” Journal of Mathematical Analysis and Applications, Vol. 335, No. 2, 2007, pp. 751-762. doi:10.1016/j.jmaa.2007.02.008

[10] T. Li, Z. Han, S. Sun and D. Yang, “Existence of Nonoscillatory Solutions to Second-Order Neutral Delay Dynamic Equations on Time Scales,” Advances in Difference Equations, Vol. 2009, 2009, pp. 1-10. doi:10.1155/2009/562329

[11] T. X. Sun, H. Xi, X. Peng and W. Yu, “Nonoscillatory Solutions for Higher-Order Neutral Dynamic Equations on Time Scales,” Abstract and Applied Analysis, Vol. 2010, 2010, pp. 1-16. doi:10.1155/2010/428963

[12] B. G. Zhang and X. H. Deng, “Oscillation of Delay Differential Equations on Time Scales,” Mathematical and Computer Modelling, Vol. 36, No. 11-13, 2002, pp. 1307-1318. doi:10.1016/S0895-7177(02)00278-9

[13] B. G. Zhang and Y. J. Sun, “Existence of Nonoscillatory Solutions of a Class of Nonlinear Difference Equations with a Forced Term,” Mathematica Bohemica, Vol. 126, No. 3, 2001, pp. 639-647.

[14] Y. Zhou and B. G. Zhang, “Existence of Nonoscillatory Solutions of Higher-Order Neutral Delay Difference Equations,” Computers & Mathematics with Applications, Vol. 45, No. 6-9, 2003, pp. 991-1000. doi:10.1016/S0898-1221(03)00074-9

[15] W. D. Lu, “Existence of Nonoscillatory Solutions of First Order Nonlinear Neutral Equations,” Journal of the Australian Mathematical Society Series B, Vol. 32, No. 2, 1990, pp. 180-192.

[1] S. Hilger, “Analysis on Measure Chains—A Unified Approach to Continuous and Discrete Calculus,” Results in Mathematics, Vol. 18, No. 1-2, 1990, pp. 18-56.

[2] R. Agarwal, M. Bohner, D. O’Regan and A. Peterson, “Dynamic Equations on Time Scales: A Survey,” Journal of Computational and Applied Mathematics, Vol. 141, No. 1-2, 2002, pp. 1-26. doi:10.1016/S0377-0427(01)00432-0

[3] M. Bohner and A. Peterson, “Dynamic Equations on Time Scales: An Introduction with Applications,” Birkh?user, Boston, 2001.

[4] M. Bohner and A. Peterson, “Advances in Dynamic Equations on Time Scales,” Birkh?user, Boston, 2003. doi:10.1007/978-0-8176-8230-9

[5] B. G. Zhang and S. L. Zhu, “Oscillation of Second-Order Nonlinear Delay Dynamic Equations on Time Scales,” Computers & Mathematics with Applications, Vol. 49, No. 4, 2005, pp. 599-609.

[6] Q. L. Li and Z. Zhang, “Existence of Solutions to Nth Order Neutral Dynamic Equations on Time Scale,” Electronic Journal of Differential Equations, Vol. 2010, No. 151, 2010, pp. 1-8. http://ejde.math.txstate.edu or http://ejde.math.unt.edu, ftp ejde.math.txstate.edu

[7] D. X. Chen, “Oscillation and Asymptotic Behavior for Nth-Order Nonlinear Neutral Delay Dynamic Eqautions on Time Scales,” Acta Applicandae Mathematicae, Vol. 109, No. 3, 2010, pp. 703-719. doi:10.1007/s10440-008-9341-0

[8] T. S. Hassan, “Oscillation of Third Order Nonlinear Delay Dynamic Equations on Time Scales,” Mathematical and Computer Modelling, Vol. 49, No. 7-8, 2009, pp. 1573-1586. doi:10.1016/j.mcm.2008.12.011

[9] Z. Q. Zhu and Q. R. Wang, “Existence of Nonoscillatory Solutions to Neutral Dynamic Equations on Time Scales,” Journal of Mathematical Analysis and Applications, Vol. 335, No. 2, 2007, pp. 751-762. doi:10.1016/j.jmaa.2007.02.008

[10] T. Li, Z. Han, S. Sun and D. Yang, “Existence of Nonoscillatory Solutions to Second-Order Neutral Delay Dynamic Equations on Time Scales,” Advances in Difference Equations, Vol. 2009, 2009, pp. 1-10. doi:10.1155/2009/562329

[11] T. X. Sun, H. Xi, X. Peng and W. Yu, “Nonoscillatory Solutions for Higher-Order Neutral Dynamic Equations on Time Scales,” Abstract and Applied Analysis, Vol. 2010, 2010, pp. 1-16. doi:10.1155/2010/428963

[12] B. G. Zhang and X. H. Deng, “Oscillation of Delay Differential Equations on Time Scales,” Mathematical and Computer Modelling, Vol. 36, No. 11-13, 2002, pp. 1307-1318. doi:10.1016/S0895-7177(02)00278-9

[13] B. G. Zhang and Y. J. Sun, “Existence of Nonoscillatory Solutions of a Class of Nonlinear Difference Equations with a Forced Term,” Mathematica Bohemica, Vol. 126, No. 3, 2001, pp. 639-647.

[14] Y. Zhou and B. G. Zhang, “Existence of Nonoscillatory Solutions of Higher-Order Neutral Delay Difference Equations,” Computers & Mathematics with Applications, Vol. 45, No. 6-9, 2003, pp. 991-1000. doi:10.1016/S0898-1221(03)00074-9

[15] W. D. Lu, “Existence of Nonoscillatory Solutions of First Order Nonlinear Neutral Equations,” Journal of the Australian Mathematical Society Series B, Vol. 32, No. 2, 1990, pp. 180-192.