doi: 10.3934/dcdsb.2022070
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Positive periodic solutions for systems of impulsive delay differential equations

1. 

Departamento de Matemática and CMAFCIO, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal

2. 

Departamento de Estatística, Análise Matemática e Optimización, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain

*Corresponding author: Teresa Faria

Received  November 2021 Revised  February 2022 Early access March 2022

Fund Project: The first author was supported by FCT-Fundação para a Ciência e a Tecnologia (Portugal) under project UIDB/04561/2020. The paper was partially written during the stay of R. Figueroa in Lisboa, thanks to a José Castillejo 2016 grant of Ministerio de Educación, Cultura y Deporte, Government of Spain

A class of periodic differential $ n $-dimensional systems with patch structure with (possibly infinite) delay and nonlinear impulses is considered. These systems incorporate very general nonlinearities and impulses whose signs may vary. Criteria for the existence of at least one positive periodic solution are presented, extending and improving previous ones established for the scalar case. Applications to systems inspired in mathematical biology models, such as impulsive hematopoiesis and Nicholson-type systems, are also included.

Citation: Teresa Faria, Rubén Figueroa. Positive periodic solutions for systems of impulsive delay differential equations. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022070
References:
[1] R. P. AgarwalM. Meehan and O' Regan, Fixed Point Theory and Applications, Cambridge University Press, 2001.  doi: 10.1017/CBO9780511543005.
[2]

P. Amster and M. Bondorevsky, Persistence and periodic solutions in systems of delay differential equations, Appl. Math. Comput., 403 (2021), Paper No. 126193, 10 pp. doi: 10.1016/j.amc.2021.126193.

[3]

R. Balderrama, New results on the almost periodic solutions for a model of hematopoiesis with an oscillatory circulation loss rate, J. Fixed Point Theory Appl., 22 (2020), Paper No. 42, 18 pp. doi: 10.1007/s11784-020-00776-7.

[4]

M. BenhadriT. Caraballo and H. Zeghdoudi, Existence of periodic positive solutions to a nonlinear Lotka-Volterra competition systems, Opuscula Math., 40 (2020), 341-360.  doi: 10.7494/OpMath.2020.40.3.341.

[5]

L. Berezansky and E. Braverman, Boundedness and persistence of delay differential equations with mixed nonlinearity, Appl. Math. Comp., 279 (2016), 154-169.  doi: 10.1016/j.amc.2016.01.015.

[6]

S. Buedo-Fernández and T. Faria, Periodic solutions for differential equations with infinite delay and nonlinear impulses, Math. Methods Appl. Sci., 43 (2020), 3052-3075.  doi: 10.1002/mma.6100.

[7]

Y. Chen, Periodic solutions of delayed periodic Nicholson blowflies models, Can. Appl. Math. Q., 11 (2003), 23-28. 

[8]

K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.

[9]

H. S. Ding and S. Fu, Periodicity on Nicholson blowflies systems involving patch structure and mortality terms, J. Experimental Theoretical Artificial Intelligence, 32 (2020), 359-371. 

[10]

H. S. DingQ.-L. Liu and J. J. Nieto, Existence of almost periodic solutions to a class of hematopoiesis model, Appl. Math. Model., 40 (2016), 3289-3297.  doi: 10.1016/j.apm.2015.10.020.

[11]

T. Faria, Periodic solutions for a non-monotone family of delayed differential equations with applications to Nicholson systems, J. Differential Equations, 263 (2017), 509-533.  doi: 10.1016/j.jde.2017.02.042.

[12]

T. Faria, Permanence and exponential stability for generalised nonautonomous Nicholson systems, Electron. J. Qual. Theory Differ. Equ., (2021), Paper No. 9, 19 pp.

[13]

T. Faria, Permanence for nonautonomous differential systems with delays in the linear and nonlinear terms, Mathematics, 9 (2021), 263, 20 pp. doi: 10.3390/math9030263.

[14]

T. Faria and J. J. Oliveira, Existence of positive periodic solutions for scalar delay differential equations with and without impulses, J. Dyn. Diff. Equ., 31 (2019), 1223-1245.  doi: 10.1007/s10884-017-9616-0.

[15]

T. Faria and G. Rőst, Persistence, permanence and global stability for an n-dimensional Nicholson system, J. Dyn. Diff. Equ., 26 (2014), 723-744.  doi: 10.1007/s10884-014-9381-2.

[16]

I. GyőriF. Hartung and N. A. Mohamady, Boundedness of positive solutions of a system of nonlinear delay equations, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 809-836.  doi: 10.3934/dcdsb.2018044.

[17]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41. 

[18]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[19]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432.

[20]

C. Huang, J. Wang and L. Huang, Asymptotically almost periodicity of delayed Nicholson-type system involving patch structure, Electron. J. Differ. Equ., 61 (2020), Paper No. 61, 17 pp.

[21]

D. Jiang and J. Wei, Existence of positive periodic solutions for Volterra integro-differential equations, Acta Math. Sci., 21 (2001), 553-560.  doi: 10.1016/S0252-9602(17)30445-9.

[22]

F. KongJ. J. Nieto and X. Fu, Stability analysis of anti-periodic solutions of the time-varying delayed hematopoiesis model with discontinuous harvesting terms, Acta Appl. Math., 170 (2020), 141-162.  doi: 10.1007/s10440-020-00328-8.

[23]

W.-T. Li and Y.-H. Fan, Existence and global attractivity of positive periodic solutions for the impulsive delay Nicholson's blowflies model, J. Comput. Appl. Math., 201 (2007), 55-68.  doi: 10.1016/j.cam.2006.02.001.

[24]

X. LiX. LinD. 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.

[25]

Y. K. Li, Periodic solutions for delay Lotka-Volterra competition systems, J. Math. Anal. Appl., 246 (2000), 230-244.  doi: 10.1006/jmaa.2000.6784.

[26]

B. Liu and S. Gong, Periodic solution for impulsive cellar neural networks with time-varying delays in the leakage terms, Abstr. Appl. Anal., 2013 (2013), Art. ID 701087, 10 pp. doi: 10.1155/2013/701087.

[27]

G. LiuJ. Yan and F. Zhang, Existence and global attractivity of unique positive periodic solution for a model of hematopoiesis, J. Math. Anal. Appl., 334 (2007), 157-171.  doi: 10.1016/j.jmaa.2006.12.015.

[28]

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.

[29]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control system, Science, 197 (1977), 287-289. 

[30]

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.

[31]

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.

[32]

A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. doi: 10.1142/9789812798664.

[33]

X. H. Tang and X. Zou, On positive periodic solution of Lotka-Volterra competition systems with deviating arguments, Proc. Amer. Math. Soc., 134 (2006), 2967-2974.  doi: 10.1090/S0002-9939-06-08320-1.

[34]

X. H. Tang and X. Zou, The existence and global exponential stability of a periodic solution of a class of delay differential equations, Nonlinearity, 22 (2009), 2423-2442.  doi: 10.1088/0951-7715/22/10/007.

[35]

L. Troib, Periodic solutions of Nicholson-type delay differential systems, Funct. Diff. Equ., 21 (2014), 171-187. 

[36]

A. WanD. Jiang and X. Xu, A new existence theory for positive periodic solutions to functional differential equations, Comput. Math. Appl., 47 (2004), 1257-1262.  doi: 10.1016/S0898-1221(04)90120-4.

[37]

W. WangF. Liu and W. Chen, Exponential stability of pseudo almost periodic delayed Nicholson-type system with patch structure, Math. Methods Appl. Sci., 42 (2019), 592-604.  doi: 10.1002/mma.5364.

[38]

Y. Xu, New stability theorem for periodic Nicholson's model with mortality term, Appl. Math. Lett., 94 (2019), 59-65.  doi: 10.1016/j.aml.2019.02.021.

[39]

J. Yan, Stability for impulsive delay differential equations, Nonlinear Anal., 63 (2005), 66-80.  doi: 10.1016/j.na.2005.05.001.

[40]

J. Yan, Existence of positive periodic solutions of impulsive functional differential equations with two parameters, J. Math. Anal. Appl., 327 (2007), 854-868.  doi: 10.1016/j.jmaa.2006.04.018.

[41]

R. Zhang, Y. Huang and T. Wei, Positive periodic solution for Nicholson-type delay systems with impulsive effects, Adv. Difference Equ., 2015 (2015), 371, 16 pp. doi: 10.1186/s13662-015-0705-2.

[42]

X. Zhang and M. Feng, Multi-parameter, impulsive effects and positive periodic solutions of first-order functional differential equations, Bound. Value Probl., 2015 (2015), 137, 22 pp. doi: 10.1186/s13661-015-0401-x.

[43]

X. ZhangJ. Yan and A. Zhao, Existence of positive periodic solutions for an impulsive differential equation, Nonlinear Anal., 68 (2008), 3209-3216.  doi: 10.1016/j.na.2007.03.014.

[44]

X.-Q. Zhao, Permanence implies the existence of interior periodic solutions for FDEs, Qual. Theory Differ. Equ. Appl., 2 (2008), 125-137. 

show all references

References:
[1] R. P. AgarwalM. Meehan and O' Regan, Fixed Point Theory and Applications, Cambridge University Press, 2001.  doi: 10.1017/CBO9780511543005.
[2]

P. Amster and M. Bondorevsky, Persistence and periodic solutions in systems of delay differential equations, Appl. Math. Comput., 403 (2021), Paper No. 126193, 10 pp. doi: 10.1016/j.amc.2021.126193.

[3]

R. Balderrama, New results on the almost periodic solutions for a model of hematopoiesis with an oscillatory circulation loss rate, J. Fixed Point Theory Appl., 22 (2020), Paper No. 42, 18 pp. doi: 10.1007/s11784-020-00776-7.

[4]

M. BenhadriT. Caraballo and H. Zeghdoudi, Existence of periodic positive solutions to a nonlinear Lotka-Volterra competition systems, Opuscula Math., 40 (2020), 341-360.  doi: 10.7494/OpMath.2020.40.3.341.

[5]

L. Berezansky and E. Braverman, Boundedness and persistence of delay differential equations with mixed nonlinearity, Appl. Math. Comp., 279 (2016), 154-169.  doi: 10.1016/j.amc.2016.01.015.

[6]

S. Buedo-Fernández and T. Faria, Periodic solutions for differential equations with infinite delay and nonlinear impulses, Math. Methods Appl. Sci., 43 (2020), 3052-3075.  doi: 10.1002/mma.6100.

[7]

Y. Chen, Periodic solutions of delayed periodic Nicholson blowflies models, Can. Appl. Math. Q., 11 (2003), 23-28. 

[8]

K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.

[9]

H. S. Ding and S. Fu, Periodicity on Nicholson blowflies systems involving patch structure and mortality terms, J. Experimental Theoretical Artificial Intelligence, 32 (2020), 359-371. 

[10]

H. S. DingQ.-L. Liu and J. J. Nieto, Existence of almost periodic solutions to a class of hematopoiesis model, Appl. Math. Model., 40 (2016), 3289-3297.  doi: 10.1016/j.apm.2015.10.020.

[11]

T. Faria, Periodic solutions for a non-monotone family of delayed differential equations with applications to Nicholson systems, J. Differential Equations, 263 (2017), 509-533.  doi: 10.1016/j.jde.2017.02.042.

[12]

T. Faria, Permanence and exponential stability for generalised nonautonomous Nicholson systems, Electron. J. Qual. Theory Differ. Equ., (2021), Paper No. 9, 19 pp.

[13]

T. Faria, Permanence for nonautonomous differential systems with delays in the linear and nonlinear terms, Mathematics, 9 (2021), 263, 20 pp. doi: 10.3390/math9030263.

[14]

T. Faria and J. J. Oliveira, Existence of positive periodic solutions for scalar delay differential equations with and without impulses, J. Dyn. Diff. Equ., 31 (2019), 1223-1245.  doi: 10.1007/s10884-017-9616-0.

[15]

T. Faria and G. Rőst, Persistence, permanence and global stability for an n-dimensional Nicholson system, J. Dyn. Diff. Equ., 26 (2014), 723-744.  doi: 10.1007/s10884-014-9381-2.

[16]

I. GyőriF. Hartung and N. A. Mohamady, Boundedness of positive solutions of a system of nonlinear delay equations, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 809-836.  doi: 10.3934/dcdsb.2018044.

[17]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41. 

[18]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[19]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432.

[20]

C. Huang, J. Wang and L. Huang, Asymptotically almost periodicity of delayed Nicholson-type system involving patch structure, Electron. J. Differ. Equ., 61 (2020), Paper No. 61, 17 pp.

[21]

D. Jiang and J. Wei, Existence of positive periodic solutions for Volterra integro-differential equations, Acta Math. Sci., 21 (2001), 553-560.  doi: 10.1016/S0252-9602(17)30445-9.

[22]

F. KongJ. J. Nieto and X. Fu, Stability analysis of anti-periodic solutions of the time-varying delayed hematopoiesis model with discontinuous harvesting terms, Acta Appl. Math., 170 (2020), 141-162.  doi: 10.1007/s10440-020-00328-8.

[23]

W.-T. Li and Y.-H. Fan, Existence and global attractivity of positive periodic solutions for the impulsive delay Nicholson's blowflies model, J. Comput. Appl. Math., 201 (2007), 55-68.  doi: 10.1016/j.cam.2006.02.001.

[24]

X. LiX. LinD. 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.

[25]

Y. K. Li, Periodic solutions for delay Lotka-Volterra competition systems, J. Math. Anal. Appl., 246 (2000), 230-244.  doi: 10.1006/jmaa.2000.6784.

[26]

B. Liu and S. Gong, Periodic solution for impulsive cellar neural networks with time-varying delays in the leakage terms, Abstr. Appl. Anal., 2013 (2013), Art. ID 701087, 10 pp. doi: 10.1155/2013/701087.

[27]

G. LiuJ. Yan and F. Zhang, Existence and global attractivity of unique positive periodic solution for a model of hematopoiesis, J. Math. Anal. Appl., 334 (2007), 157-171.  doi: 10.1016/j.jmaa.2006.12.015.

[28]

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.

[29]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control system, Science, 197 (1977), 287-289. 

[30]

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.

[31]

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.

[32]

A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. doi: 10.1142/9789812798664.

[33]

X. H. Tang and X. Zou, On positive periodic solution of Lotka-Volterra competition systems with deviating arguments, Proc. Amer. Math. Soc., 134 (2006), 2967-2974.  doi: 10.1090/S0002-9939-06-08320-1.

[34]

X. H. Tang and X. Zou, The existence and global exponential stability of a periodic solution of a class of delay differential equations, Nonlinearity, 22 (2009), 2423-2442.  doi: 10.1088/0951-7715/22/10/007.

[35]

L. Troib, Periodic solutions of Nicholson-type delay differential systems, Funct. Diff. Equ., 21 (2014), 171-187. 

[36]

A. WanD. Jiang and X. Xu, A new existence theory for positive periodic solutions to functional differential equations, Comput. Math. Appl., 47 (2004), 1257-1262.  doi: 10.1016/S0898-1221(04)90120-4.

[37]

W. WangF. Liu and W. Chen, Exponential stability of pseudo almost periodic delayed Nicholson-type system with patch structure, Math. Methods Appl. Sci., 42 (2019), 592-604.  doi: 10.1002/mma.5364.

[38]

Y. Xu, New stability theorem for periodic Nicholson's model with mortality term, Appl. Math. Lett., 94 (2019), 59-65.  doi: 10.1016/j.aml.2019.02.021.

[39]

J. Yan, Stability for impulsive delay differential equations, Nonlinear Anal., 63 (2005), 66-80.  doi: 10.1016/j.na.2005.05.001.

[40]

J. Yan, Existence of positive periodic solutions of impulsive functional differential equations with two parameters, J. Math. Anal. Appl., 327 (2007), 854-868.  doi: 10.1016/j.jmaa.2006.04.018.

[41]

R. Zhang, Y. Huang and T. Wei, Positive periodic solution for Nicholson-type delay systems with impulsive effects, Adv. Difference Equ., 2015 (2015), 371, 16 pp. doi: 10.1186/s13662-015-0705-2.

[42]

X. Zhang and M. Feng, Multi-parameter, impulsive effects and positive periodic solutions of first-order functional differential equations, Bound. Value Probl., 2015 (2015), 137, 22 pp. doi: 10.1186/s13661-015-0401-x.

[43]

X. ZhangJ. Yan and A. Zhao, Existence of positive periodic solutions for an impulsive differential equation, Nonlinear Anal., 68 (2008), 3209-3216.  doi: 10.1016/j.na.2007.03.014.

[44]

X.-Q. Zhao, Permanence implies the existence of interior periodic solutions for FDEs, Qual. Theory Differ. Equ. Appl., 2 (2008), 125-137. 

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