Research Article

Frontiers of Mathematics in China

, Volume 5, Issue 2, pp 221-286

First online:

Second-order 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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Abstract

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 < ∞).

Keywords

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

MSC

34K05