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]

Nonlinear Anal., 75 (2012), 6570-6587. doi: 10.1016/j.na.2012.07.030.  Google Scholar

[2]

J. Math. Anal. Appl., 139 (1989), 110-122. doi: 10.1016/0022-247X(89)90232-1.  Google Scholar

[3]

Canad. Math. Bull., 39 (1996), 275-283. doi: 10.4153/CMB-1996-035-9.  Google Scholar

[4]

Appl. Anal., 83 (2004), 1279-1290. doi: 10.1080/00036810410001724599.  Google Scholar

[5]

Nonlinear Anal., 62 (2005), 683-701. doi: 10.1016/j.na.2005.04.005.  Google Scholar

[6]

Nonlinear Anal., 64 (2006), 1737-1746. doi: 10.1016/j.na.2005.07.022.  Google Scholar

[7]

Computers Math. Appl., 41 (2001), 903-915. doi: 10.1016/S0898-1221(00)00328-X.  Google Scholar

[8]

Nonlinear Anal., 51 (2002), 633-647. doi: 10.1016/S0362-546X(01)00847-1.  Google Scholar

[9]

J. Math. Anal. Appl., 327 (2007), 326-341. doi: 10.1016/j.jmaa.2006.04.026.  Google Scholar

[10]

Nonlinear Anal., 67 (2007), 1027-1041. doi: 10.1016/j.na.2006.06.033.  Google Scholar

[11]

Rocky Mountain J. Math., 39 (2009), 1657-1688. doi: 10.1216/RMJ-2009-39-5-1657.  Google Scholar

[12]

Nonlinear Anal. RWA, 8 (2007), 1029-1039. doi: 10.1016/j.nonrwa.2006.06.001.  Google Scholar

[13]

J. Math. Anal. Appl., 302 (2005), 342-359. doi: 10.1016/j.jmaa.2003.12.048.  Google Scholar

[14]

Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 951-968. doi: 10.1017/S0308210500002766.  Google Scholar

[15]

(Polish) Mat. Stos. (3), 6 (1976), 23-40.  Google Scholar

[16]

J. Math. Anal. Appl., 279 (2003), 111-120. doi: 10.1016/S0022-247X(02)00613-3.  Google Scholar

[17]

Nonlinear Anal., 63 (2005), 66-80. doi: 10.1016/j.na.2005.05.001.  Google Scholar

[18]

Math. Comput. Modelling, 40 (2004), 509-518. doi: 10.1016/j.mcm.2003.12.011.  Google Scholar

[19]

Nonlinear Anal., 73 (2010), 155-162. doi: 10.1016/j.na.2010.03.008.  Google Scholar

[20]

Nonlinear Anal., 46 (2001), 53-67. doi: 10.1016/S0362-546X(99)00445-9.  Google Scholar

[21]

J. Math. Anal. Appl., 199 (1996), 162-175. doi: 10.1006/jmaa.1996.0134.  Google Scholar

[22]

Nonlinear Anal. RWA, 9 (2008), 1714-1726. doi: 10.1016/j.nonrwa.2007.05.004.  Google Scholar

[23]

Nonlinear Anal., 67 (2007), 3003-3012. doi: 10.1016/j.na.2006.09.051.  Google Scholar

[24]

J. Math. Anal. Appl., 201 (1996), 943-954. doi: 10.1006/jmaa.1996.0293.  Google Scholar

show all references

References:
[1]

Nonlinear Anal., 75 (2012), 6570-6587. doi: 10.1016/j.na.2012.07.030.  Google Scholar

[2]

J. Math. Anal. Appl., 139 (1989), 110-122. doi: 10.1016/0022-247X(89)90232-1.  Google Scholar

[3]

Canad. Math. Bull., 39 (1996), 275-283. doi: 10.4153/CMB-1996-035-9.  Google Scholar

[4]

Appl. Anal., 83 (2004), 1279-1290. doi: 10.1080/00036810410001724599.  Google Scholar

[5]

Nonlinear Anal., 62 (2005), 683-701. doi: 10.1016/j.na.2005.04.005.  Google Scholar

[6]

Nonlinear Anal., 64 (2006), 1737-1746. doi: 10.1016/j.na.2005.07.022.  Google Scholar

[7]

Computers Math. Appl., 41 (2001), 903-915. doi: 10.1016/S0898-1221(00)00328-X.  Google Scholar

[8]

Nonlinear Anal., 51 (2002), 633-647. doi: 10.1016/S0362-546X(01)00847-1.  Google Scholar

[9]

J. Math. Anal. Appl., 327 (2007), 326-341. doi: 10.1016/j.jmaa.2006.04.026.  Google Scholar

[10]

Nonlinear Anal., 67 (2007), 1027-1041. doi: 10.1016/j.na.2006.06.033.  Google Scholar

[11]

Rocky Mountain J. Math., 39 (2009), 1657-1688. doi: 10.1216/RMJ-2009-39-5-1657.  Google Scholar

[12]

Nonlinear Anal. RWA, 8 (2007), 1029-1039. doi: 10.1016/j.nonrwa.2006.06.001.  Google Scholar

[13]

J. Math. Anal. Appl., 302 (2005), 342-359. doi: 10.1016/j.jmaa.2003.12.048.  Google Scholar

[14]

Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 951-968. doi: 10.1017/S0308210500002766.  Google Scholar

[15]

(Polish) Mat. Stos. (3), 6 (1976), 23-40.  Google Scholar

[16]

J. Math. Anal. Appl., 279 (2003), 111-120. doi: 10.1016/S0022-247X(02)00613-3.  Google Scholar

[17]

Nonlinear Anal., 63 (2005), 66-80. doi: 10.1016/j.na.2005.05.001.  Google Scholar

[18]

Math. Comput. Modelling, 40 (2004), 509-518. doi: 10.1016/j.mcm.2003.12.011.  Google Scholar

[19]

Nonlinear Anal., 73 (2010), 155-162. doi: 10.1016/j.na.2010.03.008.  Google Scholar

[20]

Nonlinear Anal., 46 (2001), 53-67. doi: 10.1016/S0362-546X(99)00445-9.  Google Scholar

[21]

J. Math. Anal. Appl., 199 (1996), 162-175. doi: 10.1006/jmaa.1996.0134.  Google Scholar

[22]

Nonlinear Anal. RWA, 9 (2008), 1714-1726. doi: 10.1016/j.nonrwa.2007.05.004.  Google Scholar

[23]

Nonlinear Anal., 67 (2007), 3003-3012. doi: 10.1016/j.na.2006.09.051.  Google Scholar

[24]

J. Math. Anal. Appl., 201 (1996), 943-954. doi: 10.1006/jmaa.1996.0293.  Google Scholar

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