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January  2013, 18(1): 185-207. doi: 10.3934/dcdsb.2013.18.185

Direct exponential ordering for neutral compartmental systems with non-autonomous $\mathbf{D}$-operator

Rafael Obaya 1, and  Víctor M. Villarragut 2,

1. 

Departamento de Matemática Aplicada, Escuela de Ingenierías Industriales and member of IMUVA, Instituto de Matemáticas, Universidad de Valladolid, 47011 Valladolid, Spain
2. 

Departamento de Matemática Aplicada, Escuela de Ingenierías Industriales, Universidad de Valladolid, 47011 Valladolid, Spain

Received  May 2012 Revised  July 2012 Published  September 2012

We study closed compartmental systems described by neutral functional differential equations with non-autonomous stable $D$-operator which are monotone for the direct exponential ordering. Under some appropriate conditions on the induced semiflow including uniform stability for the exponential order and the differentiability of the $D$-operator along the base flow, we establish the 1-covering property of omega-limit sets, in order to describe the long-term behavior of the trajectories.
Keywords: neutral functional differential equations, compartmental systems., Non-autonomous dynamical systems, monotone skew-product semiflows, infinite delay.
Mathematics Subject Classification: 37B55, 34K40, 34K1.
Citation: Rafael Obaya, Víctor M. Villarragut. Direct exponential ordering for neutral compartmental systems with non-autonomous $\mathbf{D}$-operator. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 185-207. doi: 10.3934/dcdsb.2013.18.185
References:
[1]

O. Arino and F. Bourad, On the asymptotic behavior of the solutions of a class of scalar neutral equations generating a monotone semiflow, J. Differential Equations, 87 (1990), 84-95.  Google Scholar

[2]

O. Arino and E. Haourigui, On the asymptotic behavior of solutions of some delay differential systems which have a first integral, J. Math. Anal. Appl., 122 (1987), 36-46. doi: 10.1016/0022-247X(87)90342-8.  Google Scholar

[3]

R. Ellis, "Lectures on Topological Dynamics," Benjamin, New York, 1969.  Google Scholar

[4]

A. M. Fink, "Almost Periodic Differential Equations," Lecture Notes in Mathematics Springer-Verlag, Berlin, Heidelberg, New York, 377 (1974), viii+336 pp.  Google Scholar

[5]

I. Gy\Hori, Connections between compartmental systems with pipes and integro-differential equations, Math. Modelling, 7 (1986), 1215-1238. doi: 10.1016/0270-0255(86)90077-1.  Google Scholar

[6]

I. Gy\Hori and J. Eller, Compartmental systems with pipes, Math. Biosci., 53 (1981), 223-247. doi: 10.1016/0025-5564(81)90019-5.  Google Scholar

[7]

I. Gy\Hori and J. Wu, A neutral equation arising from compartmental systems with pipes, J. Dynam. Differential Equations, 3 (1991), 289-311.  Google Scholar

[8]

W. M. Haddad, V. Chellaboina and Q. Hui, "Nonnegative and Compartmental Dynamical Systems," Princeton University Press, 2010.  Google Scholar

[9]

J. K. Hale, "Theory of Functional Differential Equations," Applied Mathematical Sciences vol. 3, Springer-Verlag, Berlin, Heidelberg, New York, 1977.  Google Scholar

[10]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Applied Mathematical Sciences vol. 99, Springer-Verlag, Berlin, Heidelberg, New York, 1993.  Google Scholar

[11]

Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay," Lecture Notes in Math., vol. 1473, Springer-Verlag, Berlin, Heidelberg, 1991.  Google Scholar

[12]

J. A. Jacquez, "Compartmental Analysis in Biology and Medicine," Third Edition, Thomson-Shore Inc., Ann Arbor, Michigan, 1996. Google Scholar

[13]

J. A. Jacquez and C. P. Simon, Qualitative theory of compartmental systems, SIAM Review, 35 (1993), 43-79. doi: 10.1137/1035003.  Google Scholar

[14]

J. Jiang and X.-Q. Zhao, Convergence in monotone and uniformly stable skew-product semiflows with applications, J. Reine Angew. Math., 589 (2005), 21-55. doi: 10.1515/crll.2005.2005.589.21.  Google Scholar

[15]

T. Krisztin and J. Wu, Asymptotic periodicity, monotonicity, and oscillation of solutions of scalar neutral functional differential equations, J. Math. Anal. Appl., 199 (1996), 502-525. doi: 10.1006/jmaa.1996.0158.  Google Scholar

[16]

V. Mu\ noz-Villarragut, S. Novo and R. Obaya, Neutral functional differential equations with applications to compartmental systems, SIAM J. Math. Anal., 40 (2008), 1003-1028. doi: 10.1137/070711177.  Google Scholar

[17]

S. Novo, R. Obaya and A. M. Sanz, Stability and extensibility results for abstract skew-product semiflows, J. Differential Equations, 235 (2007), 623-646.  Google Scholar

[18]

S. Novo, R. Obaya and V. M. Villarragut, Exponential ordering for nonautonomous neutral functional differential equations, SIAM J. Math. Anal., 41 (2009), 1025-1053. doi: 10.1137/080744682.  Google Scholar

[19]

R. Obaya and V. M. Villarragut, Exponential ordering for neutral functional differential equations with non-autonomous linear $D$-operator, J. Dyn. Diff. Equat., 23 (2011), 695-725. doi: 10.1007/s10884-011-9210-9.  Google Scholar

[20]

R. J. Sacker and G. R. Sell, "Lifting Properties in Skew-Products Flows with Applications to Differential Equations," Mem. Amer. Math. Soc., vol. 190, Amer. Math. Soc., Providence, 1977.  Google Scholar

[21]

W. X. Shen and Y. F. Yi, "Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows," Mem. Amer. Math. Soc., 136 (1998), x+93 pp.  Google Scholar

[22]

Z. Wang and J. Wu, Neutral functional differential equations with infinite delay, Funkcial. Ekvac., 28 (1985), 157-170.  Google Scholar

[23]

J. Wu, Unified treatment of local theory of NFDEs with infinite delay, Tamkang J. Math., 22 (1991), 51-72.  Google Scholar

[24]

J. Wu and H. I. Freedman, Monotone semiflows generated by neutral functional differential equations with application to compartmental systems, Can. J. Math., 43 (1991), 1098-1120. doi: 10.4153/CJM-1991-064-1.  Google Scholar

show all references

References:
[1]

O. Arino and F. Bourad, On the asymptotic behavior of the solutions of a class of scalar neutral equations generating a monotone semiflow, J. Differential Equations, 87 (1990), 84-95.  Google Scholar

[2]

O. Arino and E. Haourigui, On the asymptotic behavior of solutions of some delay differential systems which have a first integral, J. Math. Anal. Appl., 122 (1987), 36-46. doi: 10.1016/0022-247X(87)90342-8.  Google Scholar

[3]

R. Ellis, "Lectures on Topological Dynamics," Benjamin, New York, 1969.  Google Scholar

[4]

A. M. Fink, "Almost Periodic Differential Equations," Lecture Notes in Mathematics Springer-Verlag, Berlin, Heidelberg, New York, 377 (1974), viii+336 pp.  Google Scholar

[5]

I. Gy\Hori, Connections between compartmental systems with pipes and integro-differential equations, Math. Modelling, 7 (1986), 1215-1238. doi: 10.1016/0270-0255(86)90077-1.  Google Scholar

[6]

I. Gy\Hori and J. Eller, Compartmental systems with pipes, Math. Biosci., 53 (1981), 223-247. doi: 10.1016/0025-5564(81)90019-5.  Google Scholar

[7]

I. Gy\Hori and J. Wu, A neutral equation arising from compartmental systems with pipes, J. Dynam. Differential Equations, 3 (1991), 289-311.  Google Scholar

[8]

W. M. Haddad, V. Chellaboina and Q. Hui, "Nonnegative and Compartmental Dynamical Systems," Princeton University Press, 2010.  Google Scholar

[9]

J. K. Hale, "Theory of Functional Differential Equations," Applied Mathematical Sciences vol. 3, Springer-Verlag, Berlin, Heidelberg, New York, 1977.  Google Scholar

[10]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Applied Mathematical Sciences vol. 99, Springer-Verlag, Berlin, Heidelberg, New York, 1993.  Google Scholar

[11]

Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay," Lecture Notes in Math., vol. 1473, Springer-Verlag, Berlin, Heidelberg, 1991.  Google Scholar

[12]

J. A. Jacquez, "Compartmental Analysis in Biology and Medicine," Third Edition, Thomson-Shore Inc., Ann Arbor, Michigan, 1996. Google Scholar

[13]

J. A. Jacquez and C. P. Simon, Qualitative theory of compartmental systems, SIAM Review, 35 (1993), 43-79. doi: 10.1137/1035003.  Google Scholar

[14]

J. Jiang and X.-Q. Zhao, Convergence in monotone and uniformly stable skew-product semiflows with applications, J. Reine Angew. Math., 589 (2005), 21-55. doi: 10.1515/crll.2005.2005.589.21.  Google Scholar

[15]

T. Krisztin and J. Wu, Asymptotic periodicity, monotonicity, and oscillation of solutions of scalar neutral functional differential equations, J. Math. Anal. Appl., 199 (1996), 502-525. doi: 10.1006/jmaa.1996.0158.  Google Scholar

[16]

V. Mu\ noz-Villarragut, S. Novo and R. Obaya, Neutral functional differential equations with applications to compartmental systems, SIAM J. Math. Anal., 40 (2008), 1003-1028. doi: 10.1137/070711177.  Google Scholar

[17]

S. Novo, R. Obaya and A. M. Sanz, Stability and extensibility results for abstract skew-product semiflows, J. Differential Equations, 235 (2007), 623-646.  Google Scholar

[18]

S. Novo, R. Obaya and V. M. Villarragut, Exponential ordering for nonautonomous neutral functional differential equations, SIAM J. Math. Anal., 41 (2009), 1025-1053. doi: 10.1137/080744682.  Google Scholar

[19]

R. Obaya and V. M. Villarragut, Exponential ordering for neutral functional differential equations with non-autonomous linear $D$-operator, J. Dyn. Diff. Equat., 23 (2011), 695-725. doi: 10.1007/s10884-011-9210-9.  Google Scholar

[20]

R. J. Sacker and G. R. Sell, "Lifting Properties in Skew-Products Flows with Applications to Differential Equations," Mem. Amer. Math. Soc., vol. 190, Amer. Math. Soc., Providence, 1977.  Google Scholar

[21]

W. X. Shen and Y. F. Yi, "Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows," Mem. Amer. Math. Soc., 136 (1998), x+93 pp.  Google Scholar

[22]

Z. Wang and J. Wu, Neutral functional differential equations with infinite delay, Funkcial. Ekvac., 28 (1985), 157-170.  Google Scholar

[23]

J. Wu, Unified treatment of local theory of NFDEs with infinite delay, Tamkang J. Math., 22 (1991), 51-72.  Google Scholar

[24]

J. Wu and H. I. Freedman, Monotone semiflows generated by neutral functional differential equations with application to compartmental systems, Can. J. Math., 43 (1991), 1098-1120. doi: 10.4153/CJM-1991-064-1.  Google Scholar

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