Use Banach's CMP (Fixed Point Principle) to show existence of a unique fixed point of the $d$-dimensional integral equation $$ x(t) = T[x](t) = x(0) + \int_{0}^{t} f(x(s))\,ds, \quad t \in [0,T] $$ for any finite $T > 0$, any fixed $x(0) \in \mathbb{R}^d$. For this purpose, assume that the function $f$ is continuously differentiable on $\mathbb{R}^d$, $f(0) = 0$ and $f$ satisfies the condition $$ \forall x, y \in \mathbb{R}^d: \langle x - y, f(x) - f(y) \rangle_d \leq 0 \quad (\text{OSL}) $$ where $\langle \cdot , \cdot \rangle_d$ is the Euclidean scalar product of $\mathbb{R}^d$. Check all assumptions of Banach's CMP for operator $T$, set up an appropriate norm and complete metric space carefully. $\textit{Hint: You should analyze}$ $\|T[x](t)\|_2$ $\textit{and}$ $\|T[x](t) - T[y](t)\|_2$ $\textit{on appropriate, closed subsets}:$ $A \subset C^0$ $\textit{or}$ $A \subset C^1$ $\textit{for Euclidean norm}$ $\|\cdot\|_d$.
So I have figured out we need the supremum norm here but I can't really see how else we can proceed to show the map is a contraction mapping, the OSL condition seems absolutely useless.