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

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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

Abstract
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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 and 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.
[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.
[3]	R. Ellis, "Lectures on Topological Dynamics," Benjamin, New York, 1969.
[4]	A. M. Fink, "Almost Periodic Differential Equations," Lecture Notes in Mathematics Springer-Verlag, Berlin, Heidelberg, New York, 377 (1974), viii+336 pp.
[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.
[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.
[7]	I. Gy\Hori and J. Wu, A neutral equation arising from compartmental systems with pipes, J. Dynam. Differential Equations, 3 (1991), 289-311.
[8]	W. M. Haddad, V. Chellaboina and Q. Hui, "Nonnegative and Compartmental Dynamical Systems," Princeton University Press, 2010.
[9]	J. K. Hale, "Theory of Functional Differential Equations," Applied Mathematical Sciences vol. 3, Springer-Verlag, Berlin, Heidelberg, New York, 1977.
[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.
[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.
[12]	J. A. Jacquez, "Compartmental Analysis in Biology and Medicine," Third Edition, Thomson-Shore Inc., Ann Arbor, Michigan, 1996.
[13]	J. A. Jacquez and C. P. Simon, Qualitative theory of compartmental systems, SIAM Review, 35 (1993), 43-79. doi: 10.1137/1035003.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[22]	Z. Wang and J. Wu, Neutral functional differential equations with infinite delay, Funkcial. Ekvac., 28 (1985), 157-170.
[23]	J. Wu, Unified treatment of local theory of NFDEs with infinite delay, Tamkang J. Math., 22 (1991), 51-72.
[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.

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.
[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.
[3]	R. Ellis, "Lectures on Topological Dynamics," Benjamin, New York, 1969.
[4]	A. M. Fink, "Almost Periodic Differential Equations," Lecture Notes in Mathematics Springer-Verlag, Berlin, Heidelberg, New York, 377 (1974), viii+336 pp.
[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.
[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.
[7]	I. Gy\Hori and J. Wu, A neutral equation arising from compartmental systems with pipes, J. Dynam. Differential Equations, 3 (1991), 289-311.
[8]	W. M. Haddad, V. Chellaboina and Q. Hui, "Nonnegative and Compartmental Dynamical Systems," Princeton University Press, 2010.
[9]	J. K. Hale, "Theory of Functional Differential Equations," Applied Mathematical Sciences vol. 3, Springer-Verlag, Berlin, Heidelberg, New York, 1977.
[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.
[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.
[12]	J. A. Jacquez, "Compartmental Analysis in Biology and Medicine," Third Edition, Thomson-Shore Inc., Ann Arbor, Michigan, 1996.
[13]	J. A. Jacquez and C. P. Simon, Qualitative theory of compartmental systems, SIAM Review, 35 (1993), 43-79. doi: 10.1137/1035003.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[22]	Z. Wang and J. Wu, Neutral functional differential equations with infinite delay, Funkcial. Ekvac., 28 (1985), 157-170.
[23]	J. Wu, Unified treatment of local theory of NFDEs with infinite delay, Tamkang J. Math., 22 (1991), 51-72.
[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.

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