We consider $\mathcal{F} : \mathbb{R}^d \rightarrow \mathcal{P}_c(\mathbb{R}^d)$ taking values in the set $\mathcal{P}_c(\mathbb{R}^d)$ of compact subsets of $\mathbb{R}^d$, and the differential inclusion problem: $$\left\{\begin{array}{l} y'(t) \in \mathcal{F}(y(t)) \\ y(0) = y_{\text{init}} \end{array}\right.$$ Show that if for every compact $K \subset \mathbb{R}^d$, there exists $C_K > 0$ such that $\mathcal{F}$ satisfies: $$\forall x, y \in K, \forall v_x \in \mathcal{F}(x), \forall v_y \in \mathcal{F}(y), \quad \langle v_x - v_y, x - y \rangle \leqslant C_K \|x - y\|^2$$ then problem (2) admits at most one maximal solution. (Hint: You may look at $\|X(t) - Y(t)\|^2$ for $X$ and $Y$ two solutions.)
We consider $\mathcal{F} : \mathbb{R}^d \rightarrow \mathcal{P}_c(\mathbb{R}^d)$ taking values in the set $\mathcal{P}_c(\mathbb{R}^d)$ of compact subsets of $\mathbb{R}^d$, and the differential inclusion problem:
$$\left\{\begin{array}{l} y'(t) \in \mathcal{F}(y(t)) \\ y(0) = y_{\text{init}} \end{array}\right.$$
Show that if for every compact $K \subset \mathbb{R}^d$, there exists $C_K > 0$ such that $\mathcal{F}$ satisfies:
$$\forall x, y \in K, \forall v_x \in \mathcal{F}(x), \forall v_y \in \mathcal{F}(y), \quad \langle v_x - v_y, x - y \rangle \leqslant C_K \|x - y\|^2$$
then problem (2) admits at most one maximal solution. (Hint: You may look at $\|X(t) - Y(t)\|^2$ for $X$ and $Y$ two solutions.)