Let $f : \mathbb{R}^n \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ be compact and convex.

For all $p \in \mathbb{R}$, consider $A_p \in \mathbb{R}^{n \times n}$, with $p \mapsto A_p$ continuous. Let $\lim_{p \rightarrow \infty} A_p =: A \in \mathbb{R}^{n \times n}$.

Denote by $\text{Fix}\left[ \cdot \right]$ the set of fixed point, that is, $\text{Fix}\left[ f \right] := \left\{ x \in X \mid x = f(x) \right\} $, $\text{Fix}\left[ A \, f \right] := \left\{ x \in X \mid x = A \, f(x) \right\} $ and $\text{Fix}\left[ A_p \,f \right] := \left\{ x \in X \mid x = A_p \, f(x) \right\} $.

I am wondering about relations between $\lim_{p \rightarrow \infty} \text{Fix}\left[ A_p \,f \right]$ and $\text{Fix}\left[ A \,f \right]$; for instance I am wondering if $$ \limsup_{p \rightarrow \infty} \, \text{Fix}\left[ A_p \,f \right] \subseteq \text{Fix}\left[ A \,f \right]. $$


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