Let $X$ be a non-empty compact convex subset of $\mathbb R^n$, where $n$ is a positive integer. Let $\Gamma$ be a correspondence that maps from $X$ into $X$—meaning that for every $𝑥\in X$, $\Gamma(x)$ is a non-empty subset of $X$—such that
- the graph of $\Gamma$, $\operatorname{Gr}(\Gamma)\equiv\{(x,y)\in X\times X\,|\,y\in\Gamma(x)\}$, is closed in $X\times X$; and
- $\Gamma(x)$ is a convex subset of $X$ for every $x\in X$.
Suppose furthermore that $(f_m)_{m\in\mathbb N}$ and $f$ are such that
- $f_m:X\to X$ is continuous for each $m\in\mathbb N$;
- $f:X\to X$ is such that $f(x)\in\Gamma(x)$ for every $x\in X$; and
- $\lim_{m\to\infty}f_m(x)=f(x)$ for every $x\in X$ pointwise.
For more details on the existence of such $(f_m)_{m\in\mathbb N}$ and $f$, see this.
Let $K\equiv\{x\in X\,|\,x\in\Gamma(x)\}$ denote the set of fixed points of $\Gamma$; that $K$ is not empty follows from Kakutani’s fixed-point theorem. For each $m\in\mathbb N$, let $B_m\equiv\{x\in X\,|\,x=f_m(x)\}$ denote the set of fixed points of $f_m$; that $B_m$ is not empty follows from Brouwer’s fixed-point theorem. Define $$B\equiv\operatorname{closure}\left(\bigcup_{m\in\mathbb N}B_m\right).$$
CONJECTURE: The intersection of $K$ and $B$ is not empty: $K\cap B\neq\varnothing$.
In words: there is at least one fixed point of $\Gamma$ that is arbitrarily close to some fixed points of the functions $(f_m)_{m\in\mathbb N}$. This is plausible, given that $(f_m)_{m\in\mathbb N}$“converges into”$\Gamma$—albeit only pointwise (and not necessarily uniformly), so that elusive convergence behavior making the fixed points of $(f_m)_{m\in\mathbb N}$ elope from the fixed points of $\Gamma$ is also plausible.
MY ATTEMPTS:
- It is straightforward to come up with examples showing that the convex valuedness of $\Gamma$ is indispensable. Even if $K$ is not empty in such cases (which is no longer a guarantee without the assumption of convex valuedness), it can be bounded away from the points in $B$.
- It is also easy to see that $K$ is not necessarily a subset of $B$. That is, not all fixed points of $\Gamma$ can be approximated by fixed points of $(f_m)_{m\in\mathbb N}$. The conjecture is quite weak, stating that at least one fixed point of $\Gamma$ can be thus approximated.
- I have tried hard to come up with counterexamples that are rigged against the conjecture, but it has persisted against my attempts to disprove it. The hypotheses together seem to me to force $K$ to intersect $B$, which makes me suspect that the conjecture is true. The strength of this suspicion is moderate, mixed with skepticism.
- I also sought to establish a positive result. I tried proving the conjecture via separating-hyperplane arguments, but I have kept hitting dead ends so far. The key seems to be to identify a fixed point of $\Gamma$ that is in some vague, to-be-operationalized sense “critical” and is forced to meet a “critical” accumulation point of the set of fixed points of $(f_m)_{m\in\mathbb N}$.
Any insight would be greatly appreciated.