R$ and set $B(t,c)\coloneqq \chi(\lVert c\rVert_E)A(t,c)$. Then one can see that $B$ satisfies hypotheses (i) and (ii) of Theorem \[thm:Brezis\]. To see that hypothesis (iii) is also satisfied, it suffices to notice that, by convexity and since $0{\leqslant}\chi{\leqslant}1$, $c+\theta \chi(\lVert c\rVert_E)A(t,c)\in C$ whenever $c+\theta A(t,c)\in C$. Therefore there exists a unique solution $t\mapsto c(t)$ of class $C^1$ of $$\label{eq:182}
\frac{{\mathrm d}}{{\mathrm d}t} c_t=B(t,c_t)\quad\text{in }[0,T],\qquad  c_0=\bar c.$$ Using again that $0{\leqslant}\chi{\leqslant}1$ and , one can see that $$\label{eq:183}
\lVert c_t\rVert_E{\leqslant}\lVert \bar c\rVert_E+MT+M\int_0^T \lVert c_s\rVert_E\,{\mathrm d}s,$$ hence Gronwall’s Lemma implies that $\lVert c_t\rVert_E{\leqslant}\bar R$ for every $t\in[0,T]$. With this, $c_t$ solves . A similar argument shows that any other solution $t\mapsto \hat c_t$ to must satisfy $\lVert \hat c_t\rVert_E{\leqslant}\bar R$ for every $t\in[0,T]$. Thus, uniqueness of solutions for follows from the uniqueness of solutions to . A similar argument also yields .

The abstract model {#sec:abstract_mod}
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The state of our system is described pairs $y\coloneqq(x,\lambda)$. The element $x\in{\mathbb{R}^{d}}$ denotes the position of the agents, whereas the element $\lambda\in{\mathcal P}(U)$ denotes a (probability) distribution over the space $U$, which we assume to be a compact metric space, which can be interpreted as a space of strategies (as in [@AFMS2018], in which case 