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 Matematica 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 & 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. doi: 10.1016/j.na.2012.07.030. Google Scholar

[2]

K. Gopalsamy and B. G. Zhang, On delay differential equations with impulses,, J. Math. Anal. Appl., 139 (1989), 110. doi: 10.1016/0022-247X(89)90232-1. Google Scholar

[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. doi: 10.4153/CMB-1996-035-9. Google Scholar

[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. doi: 10.1080/00036810410001724599. Google Scholar

[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. doi: 10.1016/j.na.2005.04.005. Google Scholar

[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. doi: 10.1016/j.na.2005.07.022. Google Scholar

[7]

X. Liu and G. Ballinger, Uniform asymptotic stability of impulsive delay differential equations,, Computers Math. Appl., 41 (2001), 903. doi: 10.1016/S0898-1221(00)00328-X. Google Scholar

[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. doi: 10.1016/S0362-546X(01)00847-1. Google Scholar

[9]

X. Liu and Y. Takeuchi, Periodicity and global dynamics of an impulsive delay Lasota-Wazewska model,, J. Math. Anal. Appl., 327 (2007), 326. doi: 10.1016/j.jmaa.2006.04.026. Google Scholar

[10]

A. Ouahab, Existence and uniqueness results for impulsive functional differential equations with scalar multiple delay and infinite delay,, Nonlinear Anal., 67 (2007), 1027. doi: 10.1016/j.na.2006.06.033. Google Scholar

[11]

S. H. Saker and J. O. Alzabut, On the impulsive delay hematopoiesis model with periodic coefficients,, Rocky Mountain J. Math., 39 (2009), 1657. doi: 10.1216/RMJ-2009-39-5-1657. Google Scholar

[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. doi: 10.1016/j.nonrwa.2006.06.001. Google Scholar

[13]

X. H. Tang, Asymptotic behavior of delay differential equations with instantaneously terms,, J. Math. Anal. Appl., 302 (2005), 342. doi: 10.1016/j.jmaa.2003.12.048. Google Scholar

[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. doi: 10.1017/S0308210500002766. Google Scholar

[15]

M. Wazewska-Czyzewska and A. Lasota, Mathematical problems of the dynamics of red blood cells system,, (Polish) Mat. Stos. (3), 6 (1976), 23. Google Scholar

[16]

J. Yan, Existence and global attractivity of positive periodic solution for an impulsive Lasota-Wazewska model,, J. Math. Anal. Appl., 279 (2003), 111. doi: 10.1016/S0022-247X(02)00613-3. Google Scholar

[17]

J. Yan, Stability for impulsive delay differential equations,, Nonlinear Anal., 63 (2005), 66. doi: 10.1016/j.na.2005.05.001. Google Scholar

[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. doi: 10.1016/j.mcm.2003.12.011. Google Scholar

[19]

R. Ye, Existence of solutions for impulsive partial neutral functional differential equation with infinite delay,, Nonlinear Anal., 73 (2010), 155. doi: 10.1016/j.na.2010.03.008. Google Scholar

[20]

J. S. Yu, Explicit conditions for stability of nonlinear scalar delay differential equations with impulses,, Nonlinear Anal., 46 (2001), 53. doi: 10.1016/S0362-546X(99)00445-9. Google Scholar

[21]

J. S. Yu and B. G. Zhang, Stability theorem for delay differential equations with impulses,, J. Math. Anal. Appl., 199 (1996), 162. doi: 10.1006/jmaa.1996.0134. Google Scholar

[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. doi: 10.1016/j.nonrwa.2007.05.004. Google Scholar

[23]

X. Zhang, Stability on nonlinear delay differential equations with impulses,, Nonlinear Anal., 67 (2007), 3003. doi: 10.1016/j.na.2006.09.051. Google Scholar

[24]

A. Zhao and J. Yan, Asymptotic behavior of solutions of impulsive delay differential equations,, J. Math. Anal. Appl., 201 (1996), 943. doi: 10.1006/jmaa.1996.0293. Google Scholar

show all references

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. doi: 10.1016/j.na.2012.07.030. Google Scholar

[2]

K. Gopalsamy and B. G. Zhang, On delay differential equations with impulses,, J. Math. Anal. Appl., 139 (1989), 110. doi: 10.1016/0022-247X(89)90232-1. Google Scholar

[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. doi: 10.4153/CMB-1996-035-9. Google Scholar

[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. doi: 10.1080/00036810410001724599. Google Scholar

[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. doi: 10.1016/j.na.2005.04.005. Google Scholar

[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. doi: 10.1016/j.na.2005.07.022. Google Scholar

[7]

X. Liu and G. Ballinger, Uniform asymptotic stability of impulsive delay differential equations,, Computers Math. Appl., 41 (2001), 903. doi: 10.1016/S0898-1221(00)00328-X. Google Scholar

[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. doi: 10.1016/S0362-546X(01)00847-1. Google Scholar

[9]

X. Liu and Y. Takeuchi, Periodicity and global dynamics of an impulsive delay Lasota-Wazewska model,, J. Math. Anal. Appl., 327 (2007), 326. doi: 10.1016/j.jmaa.2006.04.026. Google Scholar

[10]

A. Ouahab, Existence and uniqueness results for impulsive functional differential equations with scalar multiple delay and infinite delay,, Nonlinear Anal., 67 (2007), 1027. doi: 10.1016/j.na.2006.06.033. Google Scholar

[11]

S. H. Saker and J. O. Alzabut, On the impulsive delay hematopoiesis model with periodic coefficients,, Rocky Mountain J. Math., 39 (2009), 1657. doi: 10.1216/RMJ-2009-39-5-1657. Google Scholar

[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. doi: 10.1016/j.nonrwa.2006.06.001. Google Scholar

[13]

X. H. Tang, Asymptotic behavior of delay differential equations with instantaneously terms,, J. Math. Anal. Appl., 302 (2005), 342. doi: 10.1016/j.jmaa.2003.12.048. Google Scholar

[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. doi: 10.1017/S0308210500002766. Google Scholar

[15]

M. Wazewska-Czyzewska and A. Lasota, Mathematical problems of the dynamics of red blood cells system,, (Polish) Mat. Stos. (3), 6 (1976), 23. Google Scholar

[16]

J. Yan, Existence and global attractivity of positive periodic solution for an impulsive Lasota-Wazewska model,, J. Math. Anal. Appl., 279 (2003), 111. doi: 10.1016/S0022-247X(02)00613-3. Google Scholar

[17]

J. Yan, Stability for impulsive delay differential equations,, Nonlinear Anal., 63 (2005), 66. doi: 10.1016/j.na.2005.05.001. Google Scholar

[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. doi: 10.1016/j.mcm.2003.12.011. Google Scholar

[19]

R. Ye, Existence of solutions for impulsive partial neutral functional differential equation with infinite delay,, Nonlinear Anal., 73 (2010), 155. doi: 10.1016/j.na.2010.03.008. Google Scholar

[20]

J. S. Yu, Explicit conditions for stability of nonlinear scalar delay differential equations with impulses,, Nonlinear Anal., 46 (2001), 53. doi: 10.1016/S0362-546X(99)00445-9. Google Scholar

[21]

J. S. Yu and B. G. Zhang, Stability theorem for delay differential equations with impulses,, J. Math. Anal. Appl., 199 (1996), 162. doi: 10.1006/jmaa.1996.0134. Google Scholar

[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. doi: 10.1016/j.nonrwa.2007.05.004. Google Scholar

[23]

X. Zhang, Stability on nonlinear delay differential equations with impulses,, Nonlinear Anal., 67 (2007), 3003. doi: 10.1016/j.na.2006.09.051. Google Scholar

[24]

A. Zhao and J. Yan, Asymptotic behavior of solutions of impulsive delay differential equations,, J. Math. Anal. Appl., 201 (1996), 943. doi: 10.1006/jmaa.1996.0293. Google Scholar

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