## Abstract and Applied Analysis

### Multiple Positive Periodic Solutions for Functional Differential Equations with Impulses and a Parameter

Zhenguo Luo

#### Abstract

We apply the Krasnoselskii fixed-point theorem to investigate the existence of multiple positive periodic solutions for a class of impulsive functional differential equations with a parameter; some verifiable sufficient results are established easily. In particular, our results extend and improve some previous results.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 812867, 13 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412607544

Digital Object Identifier
doi:10.1155/2014/812867

Mathematical Reviews number (MathSciNet)
MR3191067

#### Citation

Luo, Zhenguo. Multiple Positive Periodic Solutions for Functional Differential Equations with Impulses and a Parameter. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 812867, 13 pages. doi:10.1155/2014/812867. https://projecteuclid.org/euclid.aaa/1412607544

#### References

• D. Y. Xu and Z. C. Yang, “Impulsive delay differential inequality and stability of neural networks,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 107–120, 2005.
• A. M. Zhao and J. R. Yan, “Asymptotic behavior of solutions of impulsive delay differential equations,” Journal of Mathematical Analysis and Applications, vol. 201, no. 3, pp. 943–954, 1996.
• Y. P. Xing and M. A. Han, “A new approach to stability of impulsive functional differential equations,” Applied Mathematics and Computation, vol. 151, no. 3, pp. 835–847, 2004.
• Z. G. Luo and L. P. Luo, “Global positive periodic solutions of generalized $n$-species competition systems with multiple delays and impulses,” Abstract and Applied Analysis, vol. 2013, Article ID 980974, 12 pages, 2013.
• W. T. Li and H. F. Huo, “Existence and global attractivity of positive periodic solutions of functional differential equations with impulses,” Nonlinear Analysis: Theory, Methods & Applications, vol. 59, no. 6, pp. 857–877, 2004.
• J. S. Yu, “Stability for nonlinear delay differential equations of unstable type under impulsive perturbations,” Applied Mathematics Letters, vol. 14, no. 7, pp. 849–857, 2001.
• X. Y. Li, X. N. Lin, D. Q. Jiang, and X. Y. Zhang, “Existence and multiplicity of positive periodic solutions to functional differential equations with impulse effects,” Nonlinear Analysis: Theory, Methods & Applications, vol. 62, no. 4, pp. 683–701, 2005.
• G. Ballinger and X. Z. Liu, “Existence, uniqueness and boundedness results for impulsive delay differential equations,” Applicable Analysis, vol. 74, no. 1-2, pp. 71–93, 2000.
• J. R. Yan, “Global attractivity for impulsive population dynamics with delay arguments,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 11, pp. 5417–5426, 2009.
• M. Liu and K. Wang, “Asymptotic behavior of a stochastic nonautonomous Lotka-Volterra competitive system with impulsive perturbations,” Mathematical and Computer Modelling, vol. 57, no. 3-4, pp. 909–925, 2013.
• A. M. Samoikleno and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.
• Z. G. Luo, B. X. Dai, and Q. H. Zhang, “Existence of positive periodic solutions for an impulsive semi-ratio-dependent predator-prey model with dispersion and time delays,” Applied Mathematics and Computation, vol. 215, no. 9, pp. 3390–3398, 2010.
• X. Q. Ding, C. W. Wang, and P. Chen, “Permanence for a two-species Gause-type ratio-dependent predator-prey system with time delay in a two-patch environment,” Applied Mathematics and Computation, vol. 219, no. 17, pp. 9099–9105, 2013.
• Y. F. Shao and B. X. Dai, “The dynamics of an impulsive delay predator-prey model with stage structure and Beddington-type functional response,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 3567–3576, 2010.
• X. D. Li and X. L. Fu, “On the global exponential stability of impulsive functional differential equations with infinite delays or finite delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 3, pp. 442–447, 2014.
• Z. G. Luo, L. P. Luo, and Y. H. Zeng, “Positive periodic solutions for impulsive functional differential equations with infinite delay and two parameters,” Journal of Applied Mathematics, vol. 2014, Article ID 751612, 17 pages, 2014.
• Z. J. Zeng, B. Li, and M. Fan, “Existence of multiple positive periodic solutions for functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 1378–1389, 2007.
• N. Zhang, B. X. Dai, and X. Z. Qian, “Periodic solutions for a class of higher-dimension functional differential equations with impulses,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 3, pp. 629–638, 2008.
• D. Q. Jiang, D. O'Regan, R. P. Agarwal, and X. J. Xu, “On the number of positive periodic solutions of functional differential equations and population models,” Mathematical Models & Methods in Applied Sciences, vol. 15, no. 4, pp. 555–573, 2005.
• D. J. Guo, Nonlinear Functional Analysis, Shandong Science and Technology Press, Shandong, China, 2001, (in Chinese).
• V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
• M. A. Krasnoselskii, Positive Solution of Operator Equation, Noordhoff, Gröningen, The Netherlands, 1964.
• K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.
• D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5, Academic Press, Orlando, Fla, USA, 1988.
• Y. H. Fan, W. T. Li, and L. L. Wang, “Periodic solutions of delayed ratio-dependent predator-prey models with monotonic or nonmonotonic functional responses,” Nonlinear Analysis: Real World Applications, vol. 5, no. 2, pp. 247–263, 2004. \endinput