# American Institute of Mathematical Sciences

October  2016, 21(8): 2451-2472. doi: 10.3934/dcdsb.2016055

## On stability for impulsive delay differential equations and application to a periodic Lasota-Wazewska model

 1 Departamento de Matematica and CMAF-CIO, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal 2 CMAT and Departamento de Matemática e Aplicações, Escola de Ciências, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal

Received  July 2015 Revised  June 2016 Published  September 2016

We consider a class of scalar delay differential equations with impulses and satisfying an Yorke-type condition, for which some criteria for the global stability of the zero solution are established. Here, the usual requirements about the impulses are relaxed. The results can be applied to study the stability of other solutions, such as periodic solutions. As an illustration, a very general periodic Lasota-Wazewska model with impulses and multiple time-dependent delays is addressed, and the global attractivity of its positive periodic solution analysed. Our results are discussed within the context of recent literature.
Citation: Teresa Faria, José J. Oliveira. On stability for impulsive delay differential equations and application to a periodic Lasota-Wazewska model. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2451-2472. doi: 10.3934/dcdsb.2016055
##### References:
 [1] T. Faria, M. C. Gadotti and J. J. Oliveira, Stability results for impulsive functional differential equations with infinite delay, Nonlinear Anal., 75 (2012), 6570-6587. doi: 10.1016/j.na.2012.07.030. [2] K. Gopalsamy and B. G. Zhang, On delay differential equations with impulses, J. Math. Anal. Appl., 139 (1989), 110-122. doi: 10.1016/0022-247X(89)90232-1. [3] J. R. Graef, C. Qian and P. W. Spikes, Oscillation and global attractivity in a periodic delay equation, Canad. Math. Bull., 39 (1996), 275-283. doi: 10.4153/CMB-1996-035-9. [4] H.-F. Huo, W.-T. Li and X. Liu, Existence and global attractivity of positive periodic solution of an impulsive delay differential equation, Appl. Anal., 83 (2004), 1279-1290. doi: 10.1080/00036810410001724599. [5] X. Li, X. Lin, D. Jiang and X. Zhang, Existence and multiplicity of positive periodic solutions to functional differential equations with impulse effects, Nonlinear Anal., 62 (2005), 683-701. doi: 10.1016/j.na.2005.04.005. [6] G. Liu, A. Zhao and J. Yan, Existence and global attractivity of unique positive periodic solution for a Lasota-Wazewska model, Nonlinear Anal., 64 (2006), 1737-1746. doi: 10.1016/j.na.2005.07.022. [7] X. Liu and G. Ballinger, Uniform asymptotic stability of impulsive delay differential equations, Computers Math. Appl., 41 (2001), 903-915. doi: 10.1016/S0898-1221(00)00328-X. [8] X. Liu and G. Ballinger, Existence and continuability of solutions for differential equations with delays and state-dependent impulses, Nonlinear Anal., 51 (2002), 633-647. doi: 10.1016/S0362-546X(01)00847-1. [9] X. Liu and Y. Takeuchi, Periodicity and global dynamics of an impulsive delay Lasota-Wazewska model, J. Math. Anal. Appl., 327 (2007), 326-341. doi: 10.1016/j.jmaa.2006.04.026. [10] A. Ouahab, Existence and uniqueness results for impulsive functional differential equations with scalar multiple delay and infinite delay, Nonlinear Anal., 67 (2007), 1027-1041. doi: 10.1016/j.na.2006.06.033. [11] S. H. Saker and J. O. Alzabut, On the impulsive delay hematopoiesis model with periodic coefficients, Rocky Mountain J. Math., 39 (2009), 1657-1688. doi: 10.1216/RMJ-2009-39-5-1657. [12] S. H. Saker and J. O. Alzabut, Existence of periodic solutions, global attractivity and oscillation of impulsive delay population model, Nonlinear Anal. RWA, 8 (2007), 1029-1039. doi: 10.1016/j.nonrwa.2006.06.001. [13] X. H. Tang, Asymptotic behavior of delay differential equations with instantaneously terms, J. Math. Anal. Appl., 302 (2005), 342-359. doi: 10.1016/j.jmaa.2003.12.048. [14] X. H. Tang and X. Zou, Stability of scalar delay differential equations with dominant delayed terms, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 951-968. doi: 10.1017/S0308210500002766. [15] M. Wazewska-Czyzewska and A. Lasota, Mathematical problems of the dynamics of red blood cells system, (Polish) Mat. Stos. (3), 6 (1976), 23-40. [16] J. Yan, Existence and global attractivity of positive periodic solution for an impulsive Lasota-Wazewska model, J. Math. Anal. Appl., 279 (2003), 111-120. doi: 10.1016/S0022-247X(02)00613-3. [17] J. Yan, Stability for impulsive delay differential equations, Nonlinear Anal., 63 (2005), 66-80. doi: 10.1016/j.na.2005.05.001. [18] J. Yan, A. Zhao and J. J. Nieto, Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems, Math. Comput. Modelling, 40 (2004), 509-518. doi: 10.1016/j.mcm.2003.12.011. [19] R. Ye, Existence of solutions for impulsive partial neutral functional differential equation with infinite delay, Nonlinear Anal., 73 (2010), 155-162. doi: 10.1016/j.na.2010.03.008. [20] J. S. Yu, Explicit conditions for stability of nonlinear scalar delay differential equations with impulses, Nonlinear Anal., 46 (2001), 53-67. doi: 10.1016/S0362-546X(99)00445-9. [21] J. S. Yu and B. G. Zhang, Stability theorem for delay differential equations with impulses, J. Math. Anal. Appl., 199 (1996), 162-175. doi: 10.1006/jmaa.1996.0134. [22] H. Zhang, L. Chen and J. J. Nieto, A delayed epidemic model with stage-structure and pulses for pest management strategy, Nonlinear Anal. RWA, 9 (2008), 1714-1726. doi: 10.1016/j.nonrwa.2007.05.004. [23] X. Zhang, Stability on nonlinear delay differential equations with impulses, Nonlinear Anal., 67 (2007), 3003-3012. doi: 10.1016/j.na.2006.09.051. [24] A. Zhao and J. Yan, Asymptotic behavior of solutions of impulsive delay differential equations, J. Math. Anal. Appl., 201 (1996), 943-954. doi: 10.1006/jmaa.1996.0293.

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##### References:
 [1] T. Faria, M. C. Gadotti and J. J. Oliveira, Stability results for impulsive functional differential equations with infinite delay, Nonlinear Anal., 75 (2012), 6570-6587. doi: 10.1016/j.na.2012.07.030. [2] K. Gopalsamy and B. G. Zhang, On delay differential equations with impulses, J. Math. Anal. Appl., 139 (1989), 110-122. doi: 10.1016/0022-247X(89)90232-1. [3] J. R. Graef, C. Qian and P. W. Spikes, Oscillation and global attractivity in a periodic delay equation, Canad. Math. Bull., 39 (1996), 275-283. doi: 10.4153/CMB-1996-035-9. [4] H.-F. Huo, W.-T. Li and X. Liu, Existence and global attractivity of positive periodic solution of an impulsive delay differential equation, Appl. Anal., 83 (2004), 1279-1290. doi: 10.1080/00036810410001724599. [5] X. Li, X. Lin, D. Jiang and X. Zhang, Existence and multiplicity of positive periodic solutions to functional differential equations with impulse effects, Nonlinear Anal., 62 (2005), 683-701. doi: 10.1016/j.na.2005.04.005. [6] G. Liu, A. Zhao and J. Yan, Existence and global attractivity of unique positive periodic solution for a Lasota-Wazewska model, Nonlinear Anal., 64 (2006), 1737-1746. doi: 10.1016/j.na.2005.07.022. [7] X. Liu and G. Ballinger, Uniform asymptotic stability of impulsive delay differential equations, Computers Math. Appl., 41 (2001), 903-915. doi: 10.1016/S0898-1221(00)00328-X. [8] X. Liu and G. Ballinger, Existence and continuability of solutions for differential equations with delays and state-dependent impulses, Nonlinear Anal., 51 (2002), 633-647. doi: 10.1016/S0362-546X(01)00847-1. [9] X. Liu and Y. Takeuchi, Periodicity and global dynamics of an impulsive delay Lasota-Wazewska model, J. Math. Anal. Appl., 327 (2007), 326-341. doi: 10.1016/j.jmaa.2006.04.026. [10] A. Ouahab, Existence and uniqueness results for impulsive functional differential equations with scalar multiple delay and infinite delay, Nonlinear Anal., 67 (2007), 1027-1041. doi: 10.1016/j.na.2006.06.033. [11] S. H. Saker and J. O. Alzabut, On the impulsive delay hematopoiesis model with periodic coefficients, Rocky Mountain J. Math., 39 (2009), 1657-1688. doi: 10.1216/RMJ-2009-39-5-1657. [12] S. H. Saker and J. O. Alzabut, Existence of periodic solutions, global attractivity and oscillation of impulsive delay population model, Nonlinear Anal. RWA, 8 (2007), 1029-1039. doi: 10.1016/j.nonrwa.2006.06.001. [13] X. H. Tang, Asymptotic behavior of delay differential equations with instantaneously terms, J. Math. Anal. Appl., 302 (2005), 342-359. doi: 10.1016/j.jmaa.2003.12.048. [14] X. H. Tang and X. Zou, Stability of scalar delay differential equations with dominant delayed terms, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 951-968. doi: 10.1017/S0308210500002766. [15] M. Wazewska-Czyzewska and A. Lasota, Mathematical problems of the dynamics of red blood cells system, (Polish) Mat. Stos. (3), 6 (1976), 23-40. [16] J. Yan, Existence and global attractivity of positive periodic solution for an impulsive Lasota-Wazewska model, J. Math. Anal. Appl., 279 (2003), 111-120. doi: 10.1016/S0022-247X(02)00613-3. [17] J. Yan, Stability for impulsive delay differential equations, Nonlinear Anal., 63 (2005), 66-80. doi: 10.1016/j.na.2005.05.001. [18] J. Yan, A. Zhao and J. J. Nieto, Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems, Math. Comput. Modelling, 40 (2004), 509-518. doi: 10.1016/j.mcm.2003.12.011. [19] R. Ye, Existence of solutions for impulsive partial neutral functional differential equation with infinite delay, Nonlinear Anal., 73 (2010), 155-162. doi: 10.1016/j.na.2010.03.008. [20] J. S. Yu, Explicit conditions for stability of nonlinear scalar delay differential equations with impulses, Nonlinear Anal., 46 (2001), 53-67. doi: 10.1016/S0362-546X(99)00445-9. [21] J. S. Yu and B. G. Zhang, Stability theorem for delay differential equations with impulses, J. Math. Anal. Appl., 199 (1996), 162-175. doi: 10.1006/jmaa.1996.0134. [22] H. Zhang, L. Chen and J. J. Nieto, A delayed epidemic model with stage-structure and pulses for pest management strategy, Nonlinear Anal. RWA, 9 (2008), 1714-1726. doi: 10.1016/j.nonrwa.2007.05.004. [23] X. Zhang, Stability on nonlinear delay differential equations with impulses, Nonlinear Anal., 67 (2007), 3003-3012. doi: 10.1016/j.na.2006.09.051. [24] A. Zhao and J. Yan, Asymptotic behavior of solutions of impulsive delay differential equations, J. Math. Anal. Appl., 201 (1996), 943-954. doi: 10.1006/jmaa.1996.0293.
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