Results in Mathematics

, Volume 48, Issue 3–4, pp 310–325 | Cite as

Positive Periodic Solutions of Systems of First Order Ordinary Differential Equations



Consider the n-dimensional nonautonomous system ẋ(t) = A(t)G(x(t)) − B(t)F(x(t − τ(t))) Let u = (u 1,…,u n ), \(f^{i}_{0}={\rm lim}_{\|{\rm u}\|\rightarrow 0}{f^{i}(\rm u)\over \|u\|}\), \(f^{i}_{\infty}={\rm lim}_{\|{\rm u}\|\rightarrow \infty}{f^{i}(\rm u)\over \|u\|}\), i = l,…,n, F = (f 1…,f n ), \({\rm F_{0}}={\rm max}_{i=1,\ldots,n}{f^{i}_{0}}\) and \({\rm F_{\infty}}={\rm max}_{i=1,\ldots,n}{f^{i}_{\infty}}\). Under some quite general conditions, we prove that either F0 = 0 and F = ∞, or F0 = ∞ and F = 0, guarantee the existence of positive periodic solutions for the system for all λ > 0. Furthermore, we show that F0 = F = 0, or F = F = ∞ guarantee the multiplicity of positive periodic solutions for the system for sufficiently large, or small λ, respectively. We also establish the nonexistence of the system when either F0 and F > 0, or F0 and F, < for sufficiently large, or small λ, respectively. We shall use fixed point theorems in a cone.

2000 Mathematics Subject Classification



positive periodic solutions existence fixed point theorem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S-N. Chow, Existence of periodic solutions of autonomous functional differential equations, J. Differential Equations 15(1974), 350–378.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    K, Deimling, “Nonlinear Functional Analysis,” Springer, Berlin, 1985.MATHCrossRefGoogle Scholar
  3. [3]
    D. Guo and V. Lakshmikantham, “Nonlinear Problems in Abstract Cones,” Academic Press, Orlando, FL, 1988.Google Scholar
  4. [4]
    W.S. Gurney, S.P. Blythe and R.N. Nisbet, Nicholson’s blowflies revisited, Nature, 287(1980), 17–21.CrossRefGoogle Scholar
  5. [5]
    K. P. Hadeler and J. Tomiuk, Periodic solutions of difference differential equations, Arch. Rat. Mech. Anal. 65 (1977), 87–95.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    D. Jiang, J. Wei and B. Zhang, Positive periodic solutions of functional differential equations and population models. Electron. J. Differential Equations 2002, No. 71, 1-13.Google Scholar
  7. [7]
    M. Krasnoselskii, Positive solutions of operator equations, Noordhoff, Groningen, 1964.Google Scholar
  8. [8]
    Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.MATHGoogle Scholar
  9. [9]
    Y. Kuang, Global attractivity and periodic solutions in delay-differential equations related to models in physiology and population biology. Japan J. Indust. Appl. Math. 9 (1992), 205–238.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Y. Kuang and H. L. Smith, Periodic solutions of differential delay equations with threshold-type delays, Oscillations and Dynamics in Delay Equations, Contemp. Math. 129 (1992), 153–176.MathSciNetCrossRefGoogle Scholar
  11. [11]
    M.C. Mackey and L. Glass, Oscillations and chaos in physiological control systems, Science, 197 ( 1997), 287–289.CrossRefGoogle Scholar
  12. [12]
    B. Tang and Y. Kuang, Existence, uniqueness and asymptotic stability of periodic solutions of periodic functional-differential systems. Tohoku Math. J. (2) 49 (1997), 217-239Google Scholar
  13. [13]
    H. Wang, On the number of positive solutions of nonlinear systems, J. Math. Anal. Appl., 281(2003) 287–306.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    H. Wang, Positive periodic solutions of functional differential equations, Journal of Differential Equations 202 (2004), 354–366.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    H. Wang, Y. Kuang, M. Fen, Periodic Solutions of Systems of Delay Differential Equations.Google Scholar
  16. [16]
    M. Wazewska-Czyzewska and A. Lasota, Mathematical problems of the dynamics of a system of red blood cells. (Polish) Mat. Stos. 6 (1976), 23–40.MathSciNetMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag, Basel 2005

Authors and Affiliations

  1. 1.Department of MathematicsNational University of IrelandGalwayIreland
  2. 2.Department of Mathematical Sciences and Applied ComputingArizona State UniversityPhoenixUSA

Personalised recommendations