Second-order differentiability with respect to parameters for differential equations with adaptive delays Authors Yuming Chen Department of Mathematics Wilfrid Laurier University Qingwen Hu Department of Mathematics and Statistics Memorial University of Newfoundland Jianhong Wu Department of Mathematics and Statistics York University Research Article

First Online: 01 April 2010 Received: 15 October 2009 Accepted: 24 January 2010 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 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

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