I've been studying set-valued analysis and I realize that I haven't been paying very close attention to some of the language being used, and I could use some clarification.
Here is an example from Discontinuous Dynamical Systems: A Tutorial on Solutions, Nonsmooth Analysis, and Stability by Jorge Corte:
Proposition S2: Let $F:[0,\infty )\times\mathbb{R}^d\to\mathfrak{B}(\mathbb{R}^d)$ be locally bounded and take nonempty, compact, and convex values. Assume that, for each $t\in\mathbb{R}$, the set-valued map $x\mapsto F(t,x)$ is upper semicontinuous, and, for each $x\in\mathbb{R}^d$, the set-valued map $t\mapsto F(t,x)$ is measurable. Then, for all $(t_0,x_0)\in[0,\infty )\times\mathbb{R}^d$, there exists a Caratheodory solution of $\dot{x}(t)=X(t,x(t))$ with initial condition $x(t_0)=x_0$.
So, when Cortes says, "...take nonempty, compact, and convex values", what does he mean specifically?
The codomain of $F$ is a set of subsets of $\mathbb{R}^d$. Then
means that the sets that turn up in the image of $F$ have those properties.
Your guess that this phrase describes the input to $F$ is wrong.