Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Translation-invariant monotone systems II: Almost periodic/automorphic case


Authors: Hongxiao Hu and Jifa Jiang
Journal: Proc. Amer. Math. Soc. 138 (2010), 3997-4007
MSC (2010): Primary 37B55, 37C65, 34C27, 92C45
DOI: https://doi.org/10.1090/S0002-9939-2010-10389-1
Published electronically: May 19, 2010
MathSciNet review: 2679621
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper studies almost periodic/automorphic monotone systems with positive translation invariance via skew-product flows. It is proved that every bounded solution of such systems is asymptotically almost periodic/automorphic. Applications are made to a chemical reaction network, especially to enzymatic futile cycles with almost periodic/automorphic reaction coefficients.


References [Enhancements On Off] (What's this?)

  • 1. N. D. Alikakos and P. Hess, On stabilization of discrete monotone dynamical systems, Israel J. Math., 59 (1987), 185-194. MR 920081 (89h:47083)
  • 2. N. D. Alikakos and P. Bates, Stabilization of solutions for a class of degenerate equations in divergence form in one space dimension, J. Differential Equations, 73 (1988), 363-393. MR 943947 (90g:35018)
  • 3. N. D. Alikakos, P. Hess and H. Matano, Discrete order preserving semigroups and stability for periodic parabolic differential equations, J. Differential Equations, 82 (1989), 322-341. MR 1027972 (91i:35016)
  • 4. D. Angeli and E. D. Sontag, Translation-invariant monotone systems, and a global convergence result for enzymatic futile cycles, Nonlinear Anal. Real World Appl., 9 (2008), 128-140. MR 2370168 (2008k:92019)
  • 5. D. Angeli and E. D. Sontag, A note on monotone systems with positive translation invariance, In Control and Automation, 2006. MED '06. 14th Mediterranean Conference on, 28-30 June 2006, pages 1-6.
  • 6. D. Angeli, P. De Leenheer and E. D. Sontag, On the structural monotonicity of chemical networks, in Proc. IEEE Conf. Decision and Control, San Diego, Dec. 2006, pages 7-12.
  • 7. O. Arino, Monotone semi-flows which have a monotone first integral, in ``Delay Differential Equations and Dynamical Systems'', Lecture Notes in Math., Vol. 1475, Springer-Verlag, Berlin/Heidelberg, 1991, pp. 64-75. MR 1132019 (93a:34081)
  • 8. 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. MR 874957 (88b:34115)
  • 9. F. Cao and J. Jiang, On the global attractivity of monotone random dynamical systems, Proc. Amer. Math. Soc., 138 (2010), 891-898.
  • 10. E. N. Dancer and P. Hess, Stability of fixed points for order-preserving discrete-time dynamical systems, J. Reine Angew. Math., 419 (1991), 125-139. MR 1116922 (92i:47088)
  • 11. P. De Leenheer, D. Angeli and E. D. Sontag, Monotone chemical reaction networks, J. Math. Chemistry, 41 (2007), 295-314. MR 2343862 (2009c:92041)
  • 12. J. R. Haddock, M. N. Nkashama and J. Wu, Asymptotic constancy for pseudomonotone dynamical systems on function spaces, J. Differential Equations, 100(1992), 292-311. MR 1194812 (94b:34096)
  • 13. M. W. Hirsch, Systems of differential equations that are competitive or cooperative. II: Convergence almost everywhere, SIAM J. Math. Anal., 16 (1985), 423-439. MR 783970 (87a:58137)
  • 14. M. W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., 383 (1988), 1-53. MR 921986 (89c:58108)
  • 15. M. Hirsch and H. L. Smith, Monotone dynamical systems, in: A. Canada, P. Drabek, A. Fonda (Eds.), Handbook of Differential Equations, Ordinary Differential Equations, second volume, Elsevier, Amsterdam, 2005. MR 2182759 (2006j:37017)
  • 16. M. W. Hirsch, The dynamical systems approach to differential equations, Bull. Amer. Math. Soc., 11 (1984), 1-64. MR 741723 (85m:58060)
  • 17. M. W. Hirsch, Positive equilibria and convergence in subhomogeneous monotone dynamics, pages 169-187, ``Comparison Methods and Stability Theory'' (X. Liu and D. Siegel, eds.), Lecture Notes in Pure and Applied Math, Vol. 162, Marcel Dekker, New York, 1994. MR 1291618 (95f:34090)
  • 18. T. Krisztin and J. Wu, Monotone semiflows generated by neutral equations with different delays in neutral and retarded parts, Acta Math. Univ. Comeniane, 63 (1994), 207-220. MR 1319440 (96b:34097)
  • 19. 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. MR 1383238 (97c:34163)
  • 20. J. Jiang, On strongly monotone flows which are Liapunov stable, Acta Math. Sinica, 33 (1990), 786-790 (Chinese). MR 1090628 (92e:34061)
  • 21. J. Jiang, A note on a global stability theorem of M. W. Hirsch, Proc. Amer. Math. Soc., 112 (1991), 803-806. MR 1043411 (92b:58119)
  • 22. J. Jiang, Periodic time dependent cooperative systems of differential equations with a first integral, Ann. Differential Equations, 8 (1992), 429-437. MR 1215988 (94b:34056)
  • 23. J. Jiang, On the global stability of cooperative systems, Bull. London Math. Soc., 26 (1994), 455-458. MR 1308362 (95i:34089)
  • 24. J. Jiang, Sublinear discrete-time order-preserving dynamical systems, Math. Proc. Camb. Phil. Soc., 119 (1996), 561-574. MR 1357065 (96h:34090)
  • 25. J. Jiang, Three- and four-dimensional cooperative systems with every equilibrium stable, J. Math. Anal. Appl., 188 (1994), 92-100. MR 1301718 (95i:34049)
  • 26. J. Jiang, Three-dimensional order-preserving discrete time dynamical systems with every fixed point stable, Commun. Appl. Nonlinear Anal., 2 (1995), 85-95. MR 1343599 (96e:58085)
  • 27. J. Jiang, On the analytic order-preserving discrete-time dynamical systems in $ \mathbf{R}^n$ with every fixed point stable, J. London Math. Soc. (2), 53 (1996), 317-324. MR 1373063 (97h:58131)
  • 28. J. Jiang, Periodic monotone systems with an invariant function, SIAM J. Math. Anal., 27 (1996), 1738-1744. MR 1416516 (98h:34089)
  • 29. J. Jiang and X. Zhao, Convergence in monotone and uniformly stable skew-product semiflows with applications, J. Reine Angew. Math., 589 (2005), 21-55. MR 2194677 (2006k:37031)
  • 30. J. Jiang and S. Yu, Stable cycles for attractors of strongly monotone discrete-time dynamical systems, J. Math. Anal. Appl., 202 (1996), 349-362. MR 1402605 (97e:58136)
  • 31. H. Hu and J. Jiang, Translation-invariant monotone systems. I: Autonomous/periodic case, Nonlinear Anal.: Real World Appl., doi:10.016/j.nonrwa.2009.11.015.
  • 32. J. Mierczyński, Strictly cooperative systems with a first integral, SIAM J. Math. Anal., 18 (1987), 642-646. MR 883558 (88e:34093)
  • 33. V. Muñoz-Villarragut, S. Novo and R. Obaya, Neutral functional differential equations with applications to compartmental systems, SIAM J. Math. Anal., 40 (2008), 1003-1028. MR 2452877 (2009i:34170)
  • 34. F. Nakajima, Periodic time-dependent gross-substitute systems, SIAM J. Appl. Math., 36 (1979), 421-427. MR 531605 (80c:93045)
  • 35. S. Novo, R. Obaya and A. M. Sanz, Stability and extensibility results for abstract skew-product semiflows, J. Differential Equations, 235 (2007), 623-646. MR 2317498 (2008h:37018)
  • 36. P. Poláčik, Convergence in smooth strongly monotone flows defined by semilinear parabolic equations, J. Differential Equations, 79 (1989), 89-110. MR 997611 (90f:58025)
  • 37. P. Poláčik and I. Tereščák, Convergence to cycles as a typical asymptotic behavior in smooth strongly monotone discrete-time dynamical systems, Arch. Rational Mech. Anal., 116 (1992), 339-360. MR 1132766 (93b:58088)
  • 38. W. Shen and X.-Q. Zhao, Convergence in almost periodic cooperative systems with a first integral, Proc. Amer. Math. Soc., 133 (2005), 203-212. MR 2085171 (2005d:34076)
  • 39. W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Memoirs of the American Mathematical Society, No. 647, Vol. 136, Providence, RI, 1998. MR 1445493 (99d:34088)
  • 40. H. L. Smith, Cooperative systems of differential equations with concave nonlinearities, Nonlinear Analysis, TMA, 10 (1986), 1037-1052. MR 857738 (87k:58247)
  • 41. H. L. Smith, Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surv. Monogr. 41, Amer. Math. Soc., Providence, RI, 1995. MR 1319817 (96c:34002)
  • 42. P. Takác, Convergence to equilibrium on invariant $ d$-hypersurfaces for strongly increasing discrete-time semigroups, J. Math. Anal. Appl., 148 (1990), 223-244. MR 1052057 (91d:58125)
  • 43. P. Takác, Asymptotic behavior of discrete-time semigroups of sublinear, strongly increasing mappings with applications to biology, Nonlinear Analysis, TMA, 14 (1990), 35-42. MR 1028245 (90j:47088)
  • 44. Y. Wang and X.-Q. Zhao, Convergence in monotone and subhomogeneous discrete dynamical systems on product Banach spaces, Bulletin of the London Math. Soc., 35 (2003), 681-688. MR 1989498 (2004c:35157)
  • 45. J. Wu, Convergence in neutral equations with infinite delay arising from active compartmental systems, ``World Congress of Nonlinear Analysts '92'', Vols. I-IV (Tampa, FL, 1992), 1361-1369, de Gruyter, Berlin, 1996. MR 1389170
  • 46. 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. MR 1138586 (92j:34141)
  • 47. J. Wu, Convergence of monotone dynamical systems with minimal equilibria, Proc. Amer. Math. Soc., 106 (1989), 907-911. MR 1004632 (90j:58130)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37B55, 37C65, 34C27, 92C45

Retrieve articles in all journals with MSC (2010): 37B55, 37C65, 34C27, 92C45


Additional Information

Hongxiao Hu
Affiliation: Department of Mathematics, Tongji University, Shanghai 200029, People’s Republic of China
Email: hhxiao1@126.com

Jifa Jiang
Affiliation: Mathematics and Science College, Shanghai Normal University, Shanghai 200234, People’s Republic of China
Email: jiangjf@shnu.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-2010-10389-1
Keywords: Positive translation invariance, monotonicity, almost periodicity/ automorphy, one-covering property
Received by editor(s): August 3, 2009
Received by editor(s) in revised form: January 17, 2010
Published electronically: May 19, 2010
Additional Notes: The second author is supported partially by Chinese NNSF grant 10671143, Shanghai NSF grant 09ZR1423100, and Innovation Program of Shanghai Municipal Education Commission and RFDP, and is the author to whom correspondence should be addressed.
Communicated by: Yingfei Yi
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society