American Institute of Mathematical Sciences

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

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.
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

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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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