Differentiability of Solutions with Respect to the Initial Data in Differential Equations with State-dependent Delays

Nonlinear Variation of Constants Formula for Differential Equations with State-Dependent Delays

“…In Sect. 4 we extend the results of [21] concerning the differentiability of the solutions wrt the initial time using weaker assumptions. In this manuscript we unify and simplify the sufficient conditions for differentiability.…”

“…The next result about the well-posedness of solutions is proved in [21] under a weaker assumption when only Lipschitz continuity of f and τ is assumed wrt the second and third, and wrt the second arguments, respectively, instead of the continuous differentiability. The notations and estimates introduced in the next theorem will be essential in the following sections.…”

“…Differentiability of the solutions of SD-DDEs wrt to the initial function (and other parameters) was studied in [10,19,21,22,24,35,36]. The only paper which proves differentiability of solutions of SDDDEs wrt the initial time σ is [21]. In Sect.…”

An analysis on the stability of a state dependent delay differential equation

“…!T / C p /. Moreover, the singular cases of Rekasius transform (5) are satisfied for equation (16) and equation (21). C p //; p 2 Z are obtained under the condition jA 0 j < A 1 .…”

“…There are some of the earliest studies on SDDE in [9][10][11]. Moreover, many researches on stability, bifurcations and existence of solutions of SDDE have been done so far, for example, [3,4,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. Especially, [32] can be seen as a detailed review on DDE and SDDE and related studies.…”