Monotone Semiflows Generated by Neutral Functional Differential Equations With Application to Compartmental Systems

Wu, Jianhong ; Freedman, H. I.

Canadian journal of mathematics, Tome 43 (1991) no. 5, pp. 1098-1120

Résumé

This paper is devoted to the machinery necessary to apply the general theory of monotone dynamical systems to neutral functional differential equations. We introduce an ordering structure for the phase space, investigate its compatibility with the usual uniform convergence topology, and develop several sufficient conditions of strong monotonicity of the solution semiflows to neutral equations. By applying some general results due to Hirsch and Matano for monotone dynamical systems to neutral equations, we establish several (generic) convergence results and an equivalence theorem of the order stability and convergence of precompact orbits. These results are applied to show that each orbit of a closed biological compartmental system is convergent to a single equilibrium.

Détail

DOI : 10.4153/CJM-1991-064-1

Mots-clés : 34K20, 34K25, 34C35, 54H20, 54F25, 92A15

Bibliographie

[1] 1. Alikakos, N.D. and Bates, P.W., Stabilization of solutions for a class of degenerate equations in divergence form in one space dimension, J. Differential Equations 73 (1988), 363–393. Google Scholar

[2] 2. Alikakos, N.D. and Hess, P.H., On the stabilization of discrete monotone dynamical systems, Israel J. Math. 59 (1987), 185–194. Google Scholar

[3] 3. Alikakos, N.D., Hess, P.H. and Matano, H., Discrete ordering-preserving semigroups and stability for periodic parabolic differential equations, J. Differential Eqns. 82 (1989), 322–341. Google Scholar

[4] 4. Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620–709. Google Scholar

[5] 5. Anderson, D.H., Compartmental modeling and tracer kinetics. Lecture Notes in Biomathematics, 56, Springer-Verlag, Heidelberg, 1983. Google Scholar

[6] 6. Araki, M. and Mori, T., On stability criteria for the composite systems including time delays, Trans. SICE 15:2 (1979), 267–268. Google Scholar

[7] 7. Arino, O. and Hasurgui, E., On the asymptotic behavior of solutions of some delay differential systems which has a first integral, J. Math. Anal. Appl. 122 (1987), 36–46. Google Scholar

[8] 8. Arino, O. and Seguier, P., About the behavior at infinity ofsolutions of x(t) = f(t — 1,x(t— 1)j —f(t,x(t)), J. Math. Anal. Appl. 96 (1983), 420–436. Google Scholar

[9] 9. Bellman, R., Topics in pharmacokinetics-I: Concentration-dependent rates, Math. Biosci. 6 (1970), 13–17. Google Scholar

[10] 10. Bellman, R., Topics in pharmacokinetics-H: Identification of time-lag processes, Math. Biosci. 11 (1971), 337- 342. Google Scholar

[11] 11. Bernier, C. and Manitius, A., On semigroups in Rn x LP corresponding to differential equations with delay, Can. J. Math. 30 (1978), 897–914. Google Scholar

[12] 12. Burns, J.A., Herdman, T.L. and Stech, H.W., The Cauchy problem for linear functional differential equations. Proc. Conf. Integral and Functional Differential Equations, (Herdman, T.L., Stech, H.W. and Rankin, S.M. eds.), Marcel Deker, New York, 1981, 1981–139. Google Scholar

[13] 13. Burns, J.A., Linear functional differential equations as semigroups in product spaces, SIAM J. Math. Anal. 14 (1983), 98–116. Google Scholar

[14] 14. Cooke, K.L. and Kaplan, J.L., A periodicity threshold theorem for epidemics and population growth, Math. Biosci. 31 (1976), 87–107. Google Scholar

[15] 15. Cooke, K.L. and Yorke, J., Some equations modelling growth process and gonorrhea epidemics, Math. Biosci 16 (1973), 75–101. Google Scholar

[16] 16. Cruz, M.A. and Hale, J.K., Stability of functional differential equations of neutral type, J. Differential Equations7 (1970), 334–355. Google Scholar

[17] 17. Tongyen, Ding, Asymptotic behavior of solutions of some retarded differential equations, Scientia Sinica (A) 25: 4 (1982), 363–370. Google Scholar

[18] 18. Eisenfeld, J., On approach to equilibrium in nonlinear compartmental systems. Differential Equations and Applications in Ecology, Epidemics and Population Problems , (S.N. Busenberg and K.L. Cooke, eds), Academic Press, New York, 1981, 119–246. Google Scholar

[19] 19. Györi, I., Connections between compartmental systems with pipes and integrodifferential equations, Mathematical Modelling 7 (1987), 1215–1238. Google Scholar

[20] 20. Györi, I. and Eller, J., Compartmental systems with pipes, Math. Biosci. 53(1981 ), 223–247. Google Scholar

[21] 21. Györi, I. and Jianhong Wu, A neutral equation arising from compartmental systems with pipes, J. Dynamics and Differential Eqns. (to appear). Google Scholar

[22] 22. Haddock, J.R., Krisztin, T. and Jianhong Wu, Asymptotic equivalence of neutral equations and infinite delay equations, Nonlin. Anal. TMA 14 (1990), 369–377. Google Scholar

[23] 23. Haddock, J.R. and Terjeki, J., Liapunov-Razumikhin functions and invariance principle for Junctional differential equations, J. Differential Equations 48 (1983), 95–122. Google Scholar

[24] 24. Hale, J.K. Theory of Functional Differential Equations. Springer-Verlag, New York, 1977. Google Scholar

[25] 25. Hale, J.K. and Massatt, P., Asymptotic behavior of gradient-like systems. Dynamical Systems II, (A.R. Bednark and L. Cesari, eds.), 85–102. Google Scholar

[26] 26. Hess, P., On stabilization of discrete strongly order-preserving semigroups and dynamical processes. Proceedings of Trends in Semigroup Theory and Applications, Trieste, September 18-October 2, 1987. Google Scholar

[27] 27. Hirsch, M., The dynamical systems approach to differential equations, Bull. Amer. Math. Sci. 11 (1984), 1–64. Google Scholar

[28] 28. Hirsch, M., Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math. 383(1988), 1–53. Google Scholar

[29] 29. Jacquez, J.A., Compartmental Analysis. Biology and Medicine, Elsevier, Amsterdam, 1972. Google Scholar

[30] 30. Kaplan, J.K., Sorg, M. and York, J., Solutions of x(t) = f(x(t),x(t — L)) have limits whenf is an order relation, Nonlin. Anal., TMA 3:1 (1979), 53–58. Google Scholar

[31] 31. Kunish, K. and Schappacher, W., Order preserving evolution operators of functional differential equations, Bull Un. Mat. Ital. B(6)2(1979), 480–500. Google Scholar

[32] 32. Ladde, G.S., Cellular systems H-Stability of compartmental systems, Math. Biosci. 30 (1976), 1–21. Google Scholar

[33] 33. Lakshmikantham, V. and Leela, S., Differential and Integral Inequalities (II). Academic Press, New York, 1969. Google Scholar

[34] 34. Lewis, R.M. and Anderson, B., Insensitivity of a class of nonlinear compartmental systems to the introduction of arbitrary time delays, IEEE Trans. Circ. Syst. CAS-27(7)(1980), 604–611. Google Scholar

[35] 35. Lopes, O., Forced oscillations in nonlinear neutral differential equations, SIAM J. Appl. Math. 29 (1975), 195–207. Google Scholar

[36] 36. Matano, H., Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto U. 18 (1978), 221–227. Google Scholar

[37] 37. Matano, H., Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci., Koyoto 5 (1979), 410–454. Google Scholar

[38] 38. Matano, H., Existence ofnontrivial unstable sets for equilibriums of strongly order preserving systems, J. Fac. Sci., U. Kyoto, (1983), 645–673. Google Scholar

[39] 39. Maeda, H. and Kodama, S., Qualitative analysis of a class of nonlinear compartmental systems: nonoscillation and asymptotic stability, Math. Biosci. 38 (1978), 35–44. Google Scholar

[40] 40. Martin, R.H., Asymptotic behavior of solutions to a class of quasi-monotone functional differential equations. Abstract Cauchy Problems and Functional Differential Equations, (Kappel, F. and Schappacher, W., eds.) Pitman, New York, 1981. Google Scholar

[41] 41. Martin, R.H. and Smith, H.L., Reaction-diffusion systems with time delays: monotonicity, invariance, comparison and convergence, preprint. Google Scholar

[42] 42. Martin, R.H., Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc. 321 (1990), 1–44. Google Scholar

[43] 43. Mazanov, A., Stability of multi-pool models with lags, J. Theoret. Biol. 59 (1976), 429–442. Google Scholar

[44] 44. Melvin, W.R., A class of neutral functional differential equations,!. Differential Equations 12 (1972), 524- 534. Google Scholar

[45] 45. Ohta, Y., Qualitative analysis of nonlinear quasi-monotone dynamical systems described by functional differential equations, IEEE Trans. Circ. Syst. CAS-28(2)(1981), 138–144. Google Scholar

[46] 46. Salamon, D., Control and Observation of Neutral Systems. Pitman, Boston, 1984. Google Scholar

[47] 47. Seifert, J.G. , Positively invariant closed sets for systems of delay differential equations, J. Differential Equations 22 (1976), 292–304. Google Scholar

[48] 48. Smith, H., Systems of ordinary differential equations which generate an order preserving flow, a survey of results, SIAM Review 30 (1988), 87–113. Google Scholar

[49] 49. Smith, H., Monotone semiflows generated by functional differential equations, J. Differential Equations 66 (1987), 420–442. Google Scholar

[50] 50. Staffans, O.J., A neutral FDE with stable D-operator is retarded, J. Differential Equations 49(1983), 208–217. Google Scholar

[51] 51. Takač, P., Convergence to equilibrium on invariant d-hypersurface for strongly increasing discrete-time semigroup, preprint. Google Scholar

[52] 52. Wu, J., On Haddock's conjecture, Applicable Analysis 33 (1989), 127–137. Google Scholar

[53] 53. Wu, J. , Convergence of monotone dynamical systems with minimal equilibria, Proc. Amer. Math. Soc. (1989), 907–911. Google Scholar