Let $f$ be a continuous function $f: \mathbb{R}\times \mathbb{R^n} \rightarrow \mathbb{R^n}$, $(t,y) \rightarrow f(t,y)$
I do not see well how to write mathematically without error the expression: "$f$ is locally Lipschitz in $y$ uniformly in $t$".
And what would be the difference with the simpler expression: "$f$ is locally Lipschitz in $y$" (but not necessarily uniformly in $t$...)
I'm trying to go back to differential equations and I wouldn't want to start with misinterpretations.
Thanks for any help.
Well, from what I understood from the various documents that I went through (and which do not all give exactly the same formal writings), it seems to me that one could synthesize the mathematical writings of the various expressions in the written forms here below. Would someone like to criticize these formulas and tell me if I was wrong in formalizing the definitions? Thank you
Let $$\begin{cases} I, open \,interval \,of \,\mathbb{R}\\ \Omega, open \,set \,of \,\mathbb{R^n}\\ f, continuous : \underset{(t,y) \rightarrow f(t,y)}{I \times \Omega \rightarrow \mathbb{R^n}} \end{cases}$$
$f$ is said to be locally Lipschitz in $y$ iif:
$$\forall (t,y) \in I \times \Omega, \exists \begin{cases} neighbourhood \,U(t) \subset I\\ neighbourhood \,V(y) \subset \Omega\\ real \,k(t,y) > 0 \end{cases}, \begin{cases} \forall t \in U \\ \forall y_1,y_2 \in V \end{cases}, \lVert f(t,y_1)-f(t,y_2 \rVert \leq k(t,y) \lVert y_1-y_2 \rVert $$
$f$ is said to be locally Lipschitz in $y$ uniformly in $t$ iif:
$$\forall (t,y) \in I \times \Omega, \exists \begin{cases} neighbourhood \,U(t) \subset I\\ neighbourhood \,V(y) \subset \Omega\\ real \,k(y) > 0 \end{cases}, \begin{cases} \forall t \in U \\ \forall y_1,y_2 \in V \end{cases}, \lVert f(t,y_1)-f(t,y_2 \rVert \leq k(y) \lVert y_1-y_2 \rVert $$
$f$ is said to be globally Lipschitz in $y$ iif:
$$\exists k \in \mathcal{C}^0 (I,\mathbb{R}^+), \begin{cases} \forall t \in I \\ \forall y_1,y_2 \in \Omega \end{cases}, \lVert f(t,y_1)-f(t,y_2 \rVert \leq k(t) \lVert y_1-y_2 \rVert $$
$f$ is said to be globally Lipschitz in $y$ uniformly in $t$ iif:
$$\exists k \in \mathbb{R}^+, \begin{cases} \forall t \in I \\ \forall y_1,y_2 \in \Omega \end{cases}, \lVert f(t,y_1)-f(t,y_2 \rVert \leq k \lVert y_1-y_2 \rVert $$
I'm sorry if I seam to break open doors; I would understand that people answer me "you just have to refer to a good book" ...