Locally Lipschitz

Definition

$F (t, ψ) : R \times X \to R^{N}$ is locally Lipschitz in $ψ$ if for each $(t_{0}, ψ_{0}) \in R \times X$ , there exists an $ε_{0} > 0$ and $L > 0$ s.t.

| F (t_{0}, ψ) - F (t_{0}, ψ_{0}) | < L | ψ - ψ_{0} |_{X} for | ψ - ψ_{0} |_{X} < ε_{0} .

人話

$F$ loc. Lip. in $ψ$ 的意思就是隨便給一個 $F$ domain 中的點，都可以找到一個大於 $0$ 的常數 $L$ ，使得只要其它跟它一樣的 $t$ 的點離它夠近，函數值的差就能被它們距離的 $L$ 倍bound住。

Examples

DDE

Consider $\dot{u} (t) = f (t, u (g_{1} (t)), u (g_{2} (t)), \dots, u (g_{m} (t)))$ with $g_{j} (t) \leq t$ for $t \geq t_{0}$ , which can be written as

{\begin{cases} \dot{u} (t) = F (t, u_{t}) \\ u_{t_{0}} = φ \in X \end{cases}

with $F (t, φ) = f (t, φ (g_{1} (t) - t), \dots, φ (g_{m} (t) - t))$ and the phase space $X = C^{0} ([- τ, 0], R^{N})$ . (c.f. A More General Discrete Delay)
Suppose that $f (t, ξ)$ is loc. Lip in $ξ$ . Then $F$ is loc. Lip. in $ψ$ . (c.f. Note)

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