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

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.
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.

[1]

Sylvia Novo, Carmen Núñez, Rafael Obaya, Ana M. Sanz. Skew-product semiflows for non-autonomous partial functional differential equations with delay. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4291-4321. doi: 10.3934/dcds.2014.34.4291

[2]

Juan A. Calzada, Rafael Obaya, Ana M. Sanz. Continuous separation for monotone skew-product semiflows: From theoretical to numerical results. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 915-944. doi: 10.3934/dcdsb.2015.20.915

[3]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703

[4]

Saša Kocić. Reducibility of skew-product systems with multidimensional Brjuno base flows. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 261-283. doi: 10.3934/dcds.2011.29.261

[5]

David Cheban, Cristiana Mammana. Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 499-515. doi: 10.3934/dcds.2007.18.499

[6]

Tomás Caraballo, Alexandre N. Carvalho, Henrique B. da Costa, José A. Langa. Equi-attraction and continuity of attractors for skew-product semiflows. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 2949-2967. doi: 10.3934/dcdsb.2016081

[7]

Rafael Obaya, Ana M. Sanz. Persistence in non-autonomous quasimonotone parabolic partial functional differential equations with delay. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3947-3970. doi: 10.3934/dcdsb.2018338

[8]

Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure and Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281

[9]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087

[10]

Grzegorz Łukaszewicz, James C. Robinson. Invariant measures for non-autonomous dissipative dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4211-4222. doi: 10.3934/dcds.2014.34.4211

[11]

Michael Zgurovsky, Mark Gluzman, Nataliia Gorban, Pavlo Kasyanov, Liliia Paliichuk, Olha Khomenko. Uniform global attractors for non-autonomous dissipative dynamical systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 2053-2065. doi: 10.3934/dcdsb.2017120

[12]

Michael Dellnitz, Christian Horenkamp. The efficient approximation of coherent pairs in non-autonomous dynamical systems. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3029-3042. doi: 10.3934/dcds.2012.32.3029

[13]

Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure and Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809

[14]

Bixiang Wang. Multivalued non-autonomous random dynamical systems for wave equations without uniqueness. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 2011-2051. doi: 10.3934/dcdsb.2017119

[15]

Fuke Wu, Shigeng Hu. The LaSalle-type theorem for neutral stochastic functional differential equations with infinite delay. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 1065-1094. doi: 10.3934/dcds.2012.32.1065

[16]

Eduardo Hernández, Donal O'Regan. $C^{\alpha}$-Hölder classical solutions for non-autonomous neutral differential equations. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 241-260. doi: 10.3934/dcds.2011.29.241

[17]

Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195

[18]

Birgit Jacob, Hafida Laasri. Well-posedness of infinite-dimensional non-autonomous passive boundary control systems. Evolution Equations and Control Theory, 2021, 10 (2) : 385-409. doi: 10.3934/eect.2020072

[19]

Iacopo P. Longo, Sylvia Novo, Rafael Obaya. Topologies of continuity for Carathéodory delay differential equations with applications in non-autonomous dynamics. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5491-5520. doi: 10.3934/dcds.2019224

[20]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (68)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]