Journal of Integral Equations and Applications

Global Dynamics of Strongly Monotone Retarded Equations with Infinite Delay

Jianhong Wu

Full-text: Open access

Article information

Source
J. Integral Equations Applications, Volume 4, Number 2 (1992), 273-306.

Dates
First available in Project Euclid: 5 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1181075685

Digital Object Identifier
doi:10.1216/jiea/1181075685

Mathematical Reviews number (MathSciNet)
MR1172893

Zentralblatt MATH identifier
0758.34059

Citation

Wu, Jianhong. Global Dynamics of Strongly Monotone Retarded Equations with Infinite Delay. J. Integral Equations Applications 4 (1992), no. 2, 273--306. doi:10.1216/jiea/1181075685. https://projecteuclid.org/euclid.jiea/1181075685


Export citation

References

  • D.J. Bradley and R.M. May, Consequences of Helminth aggregation for the dynamics of schistosomiasis, Trans. Roy. Soc. Trop. Med. Hyg. 72 (1978), 262-273.
  • T.A. Burton, Volterra integral and differential equations, Academic Press, New York, 1983.
  • B.D. Coleman and V.J. Mizel, Norms and semigroups in the theory of fading memory, Arch. Rat. Mech. Anal. 23 (1966), 87-123.
  • --------, On the general theory of fading memory, Arch. Rat. Mech. Anal. 29 (1968), 18-31.
  • --------, On the stability of solutions of functional differential equations, Arch. Rat. Mech. Anal. 30 (1968), 173-196.
  • B.D. Coleman and D. Owen, On the initial value problem for a class of functional differential equations, Arch. Rat. Mech. Anal. 55 (1974), 275-299.
  • H. Freedman and J.H. Wu, Monotone semiflows generated by neutral functional differential equations with application to compartmental systems, Canadian J. Math., in press.
  • K. Hadeler and D. Glas, Quasimonotone systems and convergence to equilibrium in a population genetic model, J. Math. Anal. Appl. 95 (1983), 297-303.
  • J.K. Hale, Dynamical systems and stability, J. Math. Anal. Appl. 26 (1969), 39-59.
  • --------, Theory of functional differential equations, Appl. Math. Sci. 3, Springer-Verlag, New York, 1977.
  • --------, Retarded equations with infinite delay, in Proceedings of the summerschool and conference on functional differential equations and approximation of fixed points, (H.O. Peitgen and M.O. Walther, eds.), Bonn, 1978, 157-193.
  • --------, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs 25, AMS, Providence, 1988.
  • J.K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcialaj Ekvac 21 (1978), 11-41.
  • M.W. Hirsch, The dynamical systems approach to differential equations, Bull. Amer. Math. Soc. 11 (1984), 1-64.
  • --------, Systems of differential equations that are competitive or cooperative II: Convergence almost everywhere, SIAM J. Math. Anal. 16 (1985), 423-439.
  • --------, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew Math. 383 (1988), 1-53.
  • F. Kappel and W. Schappacher, Some considerations to the fundamental theory of infinite delay equations, J. Differential Equations 37 (1980), 141-183.
  • F. Kunish and W. Schappacher, Order preserving evolution operators of functional differential equations, Bull. Union Mate. Italiana 16B (1979), 480-500.
  • K.L. Lee and E.R. Lewis, Delay time models of population dynamics with application to schistosomiasis control, IEEE Trans. Biomed. Eng. BME-23 (1976), 225-233.
  • T. Lewis, A model for the parasitic disease bilharziasis, Adv. Appl. Probab. 7 (1975), 673-704.
  • N. MacDonald, Time lags in biological models, Lect. Notes in Biomath. 27, Springer-Verlag, London, 1978.
  • R.H. Martin, Asymptotic behavior of solutions to a class of quasimonotone functional differential equations, in Abstract Cauchy problems and functional differential equations, (F. Kappel and W. Schappacher eds.), Pitman, Boston, 1981, 91-111.
  • H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci. 15 (1979), 401-454.
  • --------, Existence of nontrivial unstable sets for equilibrium of strongly order-preserving systems, J. Fac. Sci. Univ. Tokyo, Sec. t-1A Math. 30 (1983), 645-673.
  • --------, Strongly order-preserving local semi-dynamical systems\emdash/theory and applications, in Proceedings of autumn course on semigroups, (Trieste, Italy, Dec. 1984), Res. Notes in Math., Pitman, New York, 1985.
  • R.M. May, Togetherness among schistosomes, its effects on the dynamics of the infection, Math. Biosci. 35 (1977), 301-343.
  • R.K. Miller, Nonlinear Volterra integral equations, Benjamin, New York, 1971.
  • T. Naito, Integral manifolds for linear functional differential equations on some Banach space, Funkcialaj Ekvac. 13 (1970), 199-213.
  • --------, On linear autonomous retarded equations with an abstract space for infinite delay, J. Differential Equations 33 (1979), 74-91.
  • --------, On autonomous linear functional differential equations with infinite retardations, J. Differential Equations 2 (1976), 297-315.
  • I. Nasell, On eradication of schistosomiasis, Theor. Popul. Biol. 10 (1976), 133-144.
  • I. Nasell and W.M. Hirsch, The transmission dynamics of schistosomiasis, Comm. Pure Appl. Math. 26 (1973), 395-453.
  • R.D. Nussbaum, The radius of the essential spectrum, Duke Math. J. 37 (1970), 473-488.
  • --------, Positive operators and elliptic eigenvalue problems, Math. Zeit. 186 (1984), 247-269.
  • --------, A folk theorem in the spectral theory of $c_0$-semigroups, Pac. J. Math. 113 (1984), 433-449.
  • --------, Hilbert's projective metric and iterated nonlinear maps, Mem. Amer. Math. Soc. 391, 1988.
  • Y. Ohta, Qualitative analysis of nonlinear quasimonotone dynamical systems described by functional differential equations, IEEE Trans. Circuits and Systems 28 (1981), 138-144.
  • K. Schumacher, Existence and continuous dependence for functional-\noindent differential equations with unbounded delay, Arch. Rat. Mech. Anal. 67 (1978), 315-335.
  • --------, Dynamical systems with memory on history-spaces with monotonic seminorms, J. Differential Equations 34 (1979), 440-463.
  • H. Smith, Periodic solutions of periodic competitive and cooperative systems, SIAM J. Math. Anal. 17 (1986), 1289-1318.
  • --------, Invariant curves for mappings, SIAM J. Math. Anal. 17 (1986), 1053-1067.
  • --------, Monotone semiflows generated by functional differential equations, J. Differential Equations 66 (1987), 420-442.
  • J. Wu, N. Liu and S. Zuo, The qualitative analysis of model of the transmission dynamics of Japaneses schistomiasis, Applied Mathematics, a Journal of Chinese Universities 2 (1987), 352-362.
  • J. Wu, Strong monotonicity principles and applications to Volterra integrodifferential equations, Differential Equations: Stability and Control (Saber ed.), Marcel Dekker, 1990, 519-528.
  • --------, Mathematical models for the transmission dynamics of schistosomiasis japonicum, preprint.