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Modeling and control of local outbreaks of West Nile virus in the United States
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 |
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. |
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), 6570-6587.
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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