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

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