Frontiers of Mathematics in China

, Volume 5, Issue 2, pp 221–286 | Cite as

Second-order differentiability with respect to parameters for differential equations with adaptive delays

  • Yuming Chen
  • Qingwen Hu
  • Jianhong WuEmail author
Research Article


In this paper, we study the second-order differentiability of solutions with respect to parameters in a class of delay differential equations, where the evolution of the delay is governed explicitly by a differential equation involving the state variable and the parameters. We introduce the notion of locally complete triple-normed linear space and obtain an extension of the well-known uniform contraction principle in such spaces. We then apply this extended principle and obtain the second-order differentiability of solutions with respect to parameters in the W 1,p -norm (1 ⩽ p < ∞).


Delay differential equation adaptive delay differentiability of solution state-dependent delay uniform contraction principle locally complete triple-normed linear space 




Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aiello W G, Freedman H I, Wu J. Analysis of a model representing stage-structured population growth with state-dependent time delay. SIAM J Appl Math, 1992, 52: 855–869zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ait Dads E H, Ezzinbi K. Boundedness and almost periodicity for some state-dependent delay differential equations. Electron J Differential Equations, 2002, 67: 1–13MathSciNetGoogle Scholar
  3. 3.
    Al-Omari J F M, Gourley S A. Dynamics of a stage-structured population model incorporating a state-dependent maturation delay. Nonlinear Anal Real World Appl, 2005, 6: 13–33zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Arino O, Hadeler K P, Hbid M L. Existence of periodic solutions for delay differential equations with state dependent delay. J Differential Equations, 1998, 144: 263–301zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Arino O, Hbid M L, Bravo de la Parra R. A mathematical model of growth of population of fish in the larval stage: density-dependence effects. Math Biosci, 1998, 150: 1–20zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Arino O, Sánchez E. Delays included in population dynamics. Mathematical Modeling of Population Dynamics, Banach Center Publications, 2004, 63: 9–146Google Scholar
  7. 7.
    Arino O, Sánchez E, Fathallah A. State-dependent delay differential equations in population dynamics: modeling and analysis. Fields Inst Commun, 2001, 29: 19–36Google Scholar
  8. 8.
    Bartha M. Periodic solutions for differential equations with state-dependent delay and positive feedback. Nonlinear Anal, 2003, 53: 839–857zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bélair J. Population models with state-dependent delays. Lecture Notes in Pure and Appl Math, 1991, 131: 165–176Google Scholar
  10. 10.
    Bélair J. Stability analysis of an age-structured model with a state-dependent delay. Canad Appl Math Quart, 1998, 6: 305–319zbMATHMathSciNetGoogle Scholar
  11. 11.
    Brokate M, Colonius F. Linearizing equations with state-dependent delays. Appl Math Optim, 1990, 21: 45–52zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Cao Y, Fan J, Gard T C. The effect of state-dependent delay on a stage-structured population growth model. Nonlinear Anal, 1992, 19: 95–105zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Cooke K L, Huang W Z. On the problem of linearization for state-dependent delay differential equations. Proc Amer Math Soc, 1996, 124: 1417–1426zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Györi I, Hartung F. On the exponential stability of a state-dependent delay equation. Acta Sci Math (Szeged), 2000, 66: 71–84zbMATHMathSciNetGoogle Scholar
  15. 15.
    Hale J K, Ladeira L A C. Differentiability with respect to delays. J Differential Equations, 1991, 92: 14–26zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hartung F. On Differentiability of SolutionsWith State-dependent Delay Equations. Ph D dissertation, University of Texas at Dallas, 1995Google Scholar
  17. 17.
    Hartung F. Linearized stability in periodic functional differential equations with state-dependent delays. J Comput Appl Math, 2005, 174: 201–211zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Hartung F, Krisztin T, Walther H -O, Wu J. Functional Differential Equations with State-dependent Delays: Theory and Applications. In: Canada A, Drabek P, Fonda A, eds. Handbook of Differential Equations: Ordinary Differential Equations, Vol 3. Amsterdam: Elsevier, North Holland, 2006Google Scholar
  19. 19.
    Hartung F, Turi J. On differentiability of solutions with respect to parameters in state-dependent delay equations. J Differential Equations, 1997, 135: 192–237zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Hartung F, Turi J. Linearized stability in functional differential equations with statedependent delays. Discrete Contin Dynam Systems, 2001, Added Volume: 416–425Google Scholar
  21. 21.
    Kalton N. Quasi-Banach Spaces. Handbook of the Geometry of Banach Spaces, Vol 2. Amsterdam: North-Holland, 2003, 1099–1130CrossRefGoogle Scholar
  22. 22.
    Krisztin T. A local unstable manifold for differential equations with state-dependent delay. Discrete Contin Dyn Syst, 2003, 9: 993–1028zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Krisztin T, Arino O. The two-dimensional attractor of a differential equation with state-dependent delay. J Dynam Differential Equations, 2001, 13: 453–522zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Li Y, Kuang Y. Periodic solutions in periodic state-dependent delay equations and population models. Proc Amer Math Soc, 2002, 130: 1345–1353zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Magal P, Arino O. Existence of periodic solutions for a state dependent delay differential equation. J Differential Equations, 2000, 165: 61–95zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Mahaffy J M, Bélair J, Mackey M C. Hematopoietic model with moving boundary condition and state-dependent delay: application in erythropoiesis. J Theor Biol, 1998, 190: 135–146CrossRefGoogle Scholar
  27. 27.
    Niri K. Oscillations in differential equations with state-dependent delays. Nelīnīinī Koliv, 2003, 6: 252–259zbMATHMathSciNetGoogle Scholar
  28. 28.
    Ouifki R, Hbid M L. Periodic solutions for a class of functional differential equations with state-dependent delay close to zero. Math Models Methods Appl Sci, 2003, 13: 807–841zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Rai S, Robertson R L. Analysis of a two-stage population model with space limitations and state-dependent delay. Canad Appl Math Quart, 2000, 8: 263–279zbMATHMathSciNetGoogle Scholar
  30. 30.
    Rai S, Robertson R L. A stage-structured population model with state-dependent delay. Int J Differ Equ Appl, 2002, 6: 77–91zbMATHMathSciNetGoogle Scholar
  31. 31.
    Smith H L. Some results on the existence of periodic solutions of state-dependent delay differential equations. Pitman Res Notes Math Ser, 1992, 272: 218–222Google Scholar
  32. 32.
    Walther H -O. Stable periodic motion of a system with state dependent delay. Differential Integral Equations, 2002, 15: 923–944zbMATHMathSciNetGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Department of MathematicsWilfrid Laurier UniversityWaterlooCanada
  2. 2.Department of Mathematics and StatisticsMemorial University of NewfoundlandSt. John’sCanada
  3. 3.Department of Mathematics and StatisticsYork UniversityTorontoCanada

Personalised recommendations