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01 Apr 2010
Secondorder differentiability with respect to parameters for differential equations with adaptive delays
 Yuming Chen,
 Qingwen Hu,
 Jianhong Wu
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In this paper, we study the secondorder 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 triplenormed linear space and obtain an extension of the wellknown uniform contraction principle in such spaces. We then apply this extended principle and obtain the secondorder differentiability of solutions with respect to parameters in the W ^{1,p }norm (1 ⩽ p < ∞).
 Aiello W G, Freedman H I, Wu J. Analysis of a model representing stagestructured population growth with statedependent time delay. SIAM J Appl Math, 1992, 52: 855–869 CrossRef
 Ait Dads E H, Ezzinbi K. Boundedness and almost periodicity for some statedependent delay differential equations. Electron J Differential Equations, 2002, 67: 1–13
 AlOmari J F M, Gourley S A. Dynamics of a stagestructured population model incorporating a statedependent maturation delay. Nonlinear Anal Real World Appl, 2005, 6: 13–33 CrossRef
 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: 263–301 CrossRef
 Arino O, Hbid M L, Bravo de la Parra R. A mathematical model of growth of population of fish in the larval stage: densitydependence effects. Math Biosci, 1998, 150: 1–20 CrossRef
 Arino O, Sánchez E. Delays included in population dynamics. Mathematical Modeling of Population Dynamics, Banach Center Publications, 2004, 63: 9–146
 Arino O, Sánchez E, Fathallah A. Statedependent delay differential equations in population dynamics: modeling and analysis. Fields Inst Commun, 2001, 29: 19–36
 Bartha M. Periodic solutions for differential equations with statedependent delay and positive feedback. Nonlinear Anal, 2003, 53: 839–857 CrossRef
 Bélair J. Population models with statedependent delays. Lecture Notes in Pure and Appl Math, 1991, 131: 165–176
 Bélair J. Stability analysis of an agestructured model with a statedependent delay. Canad Appl Math Quart, 1998, 6: 305–319
 Brokate M, Colonius F. Linearizing equations with statedependent delays. Appl Math Optim, 1990, 21: 45–52 CrossRef
 Cao Y, Fan J, Gard T C. The effect of statedependent delay on a stagestructured population growth model. Nonlinear Anal, 1992, 19: 95–105 CrossRef
 Cooke K L, Huang W Z. On the problem of linearization for statedependent delay differential equations. Proc Amer Math Soc, 1996, 124: 1417–1426 CrossRef
 Györi I, Hartung F. On the exponential stability of a statedependent delay equation. Acta Sci Math (Szeged), 2000, 66: 71–84
 Hale J K, Ladeira L A C. Differentiability with respect to delays. J Differential Equations, 1991, 92: 14–26 CrossRef
 Hartung F. On Differentiability of SolutionsWith Statedependent Delay Equations. Ph D dissertation, University of Texas at Dallas, 1995
 Hartung F. Linearized stability in periodic functional differential equations with statedependent delays. J Comput Appl Math, 2005, 174: 201–211 CrossRef
 Hartung F, Krisztin T, Walther H O, Wu J. Functional Differential Equations with Statedependent 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, 2006
 Hartung F, Turi J. On differentiability of solutions with respect to parameters in statedependent delay equations. J Differential Equations, 1997, 135: 192–237 CrossRef
 Hartung F, Turi J. Linearized stability in functional differential equations with statedependent delays. Discrete Contin Dynam Systems, 2001, Added Volume: 416–425
 Kalton N. QuasiBanach Spaces. Handbook of the Geometry of Banach Spaces, Vol 2. Amsterdam: NorthHolland, 2003, 1099–1130 CrossRef
 Krisztin T. A local unstable manifold for differential equations with statedependent delay. Discrete Contin Dyn Syst, 2003, 9: 993–1028 CrossRef
 Krisztin T, Arino O. The twodimensional attractor of a differential equation with statedependent delay. J Dynam Differential Equations, 2001, 13: 453–522 CrossRef
 Li Y, Kuang Y. Periodic solutions in periodic statedependent delay equations and population models. Proc Amer Math Soc, 2002, 130: 1345–1353 CrossRef
 Magal P, Arino O. Existence of periodic solutions for a state dependent delay differential equation. J Differential Equations, 2000, 165: 61–95 CrossRef
 Mahaffy J M, Bélair J, Mackey M C. Hematopoietic model with moving boundary condition and statedependent delay: application in erythropoiesis. J Theor Biol, 1998, 190: 135–146 CrossRef
 Niri K. Oscillations in differential equations with statedependent delays. Nelīnīinī Koliv, 2003, 6: 252–259
 Ouifki R, Hbid M L. Periodic solutions for a class of functional differential equations with statedependent delay close to zero. Math Models Methods Appl Sci, 2003, 13: 807–841 CrossRef
 Rai S, Robertson R L. Analysis of a twostage population model with space limitations and statedependent delay. Canad Appl Math Quart, 2000, 8: 263–279
 Rai S, Robertson R L. A stagestructured population model with statedependent delay. Int J Differ Equ Appl, 2002, 6: 77–91
 Smith H L. Some results on the existence of periodic solutions of statedependent delay differential equations. Pitman Res Notes Math Ser, 1992, 272: 218–222
 Walther H O. Stable periodic motion of a system with state dependent delay. Differential Integral Equations, 2002, 15: 923–944
 Title
 Secondorder differentiability with respect to parameters for differential equations with adaptive delays
 Journal

Frontiers of Mathematics in China
Volume 5, Issue 2 , pp 221286
 Cover Date
 20100601
 DOI
 10.1007/s1146401000059
 Print ISSN
 16733452
 Online ISSN
 16733576
 Publisher
 SP Higher Education Press
 Additional Links
 Topics
 Keywords

 Delay differential equation
 adaptive delay
 differentiability of solution
 statedependent delay
 uniform contraction principle
 locally complete triplenormed linear space
 34K05
 Authors

 Yuming Chen ^{(1)}
 Qingwen Hu ^{(2)}
 Jianhong Wu ^{(3)}
 Author Affiliations

 1. Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, N2L 3C5, Canada
 2. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S7, Canada
 3. Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3, Canada