A More General Discrete Delay

Time delays are not necessarily constants. They can hide in functions like $g (t)$ that satisfies $g (t) \leq t$ for all admissible $t$ .

In that case, when we want to define the interval where our functions live, we usually take $[t_{0} - τ, \dots ⟩$ , where we wish $τ$ satisfies

t - τ \leq inf {g (t) : t \in [t_{0}, \dots ⟩},

which is equivalent to

τ \geq sup {t - g (t) : t \in [t_{0, \dots} ⟩} .

τ \geq sup_{j = 1, \dots, m} {t - g_{j} (t) : t \in [t_{0, \dots} ⟩} .

for more general case.

(c.f. NTHU MATH 526500 DDE Note week 3)