Frontiers of Mathematics in China >

2010 , Vol. 5 >Issue 2: 221 - 286

DOI: https://doi.org/10.1007/s11464-010-0005-9

Research articles

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

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  • 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;

Published date: 05 Jun 2010

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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 W1,p-norm (1≤p<∞).

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

Cite this article

Yuming CHEN, Qingwen HU, Jianhong WU, . Second-order differentiability with respect to parameters for differential equations with adaptive delays[J]. Frontiers of Mathematics in China, 2010 , 5(2) : 221 -286 . DOI: 10.1007/s11464-010-0005-9

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