Frontiers of Mathematics in China
, Volume 5, Issue 2, pp 221286
First online:
Secondorder differentiability with respect to parameters for differential equations with adaptive delays
 Yuming ChenAffiliated withDepartment of Mathematics, Wilfrid Laurier University
 , Qingwen HuAffiliated withDepartment of Mathematics and Statistics, Memorial University of Newfoundland
 , Jianhong WuAffiliated withDepartment of Mathematics and Statistics, York University Email author
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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 < ∞).
Keywords
Delay differential equation adaptive delay differentiability of solution statedependent delay uniform contraction principle locally complete triplenormed linear spaceMSC
34K05 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
 201006
 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