Frontiers of Mathematics in China

, Volume 5, Issue 2, pp 221–286

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

Research Article

DOI: 10.1007/s11464-010-0005-9

Cite this article as:
Chen, Y., Hu, Q. & Wu, J. Front. Math. China (2010) 5: 221. doi:10.1007/s11464-010-0005-9

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 W1,p-norm (1 ⩽ p < ∞).

Keywords

Delay differential equationadaptive delaydifferentiability of solutionstate-dependent delayuniform contraction principlelocally complete triple-normed linear space

MSC

34K05

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