Let $D \subseteq \mathbb{R}^n$ be a bounded domain with smooth boundary and $C^{\alpha}(\overline{D})$, $\alpha \in (0,1)$, the space of Hölder continuous functions with the usual norm $\|\cdot\|_{C^{\alpha}(\overline{D})}$ such that it is a Banach space. Assume $\Lambda\colon C^{\alpha}(\overline{D})\to C^{\alpha}(\overline{D})$ is a map (operator) such that $\Lambda(B) \subseteq B$ for a ball $$B := \{f \in C^{\alpha}(\overline{D}) \colon \, \|f\|_{C^{\alpha}(\overline{D})} \leq R \}$$ of fixed radius $R > 0$.
Lets say I want to find a fixed point $f \in B$ of $\Lambda$, i.e. $\Lambda(f) = f \in B$. In order to apply for instance the Fixed Point Theorem by Schauder, I should search in a (convex) compact subset of $C^{\alpha}(\overline{D})$ instead, and moreover I should first prove that $\Lambda$ is continuous.
I am wondering if it is really necessary to explicitly assume $\Lambda$ being continuous? Could one not just:
- define a sequence $(f_n)_n \subseteq B$ by fixing any $f_0 \in B$ and setting $f_{n+1} := \Lambda(f_n) \in B$,
- and note that by the compact embedding $C^{\alpha}(\overline{D}) \subset \subset C^{\beta}(\overline{D})$ for $\beta \in (0, \alpha)$, the ball $B$ is a compact subset of $C^{\beta}(\overline{D})$,
- and then use this compact embedding to conclude that there is a subsequence $(f_{n_k})_k$ which converges in $C^{\beta}(\overline{D})$ to a function $f \in C^{\beta}(\overline{D})$, $\beta \in (0,\alpha)$,
- and then use $$\|f\|_{C^{\alpha}(\overline{D})} = \sup\limits_{x\neq y} \frac{|f(x)-f(y)|}{|x-y|^{\alpha}} \\ = \sup\limits_{x\neq y} \lim\limits_{k \to +\infty} \frac{|f(x)-f(y)|}{|x-y|^{\alpha}} \\ \leq \sup\limits_{x\neq y} \lim\limits_{k \to +\infty} \frac{|f(x)-f_k(x)| + |f(y)-f_k(y)| + |f_k(x)-f_k(y)|}{|x-y|^{\alpha}} \\ = \sup\limits_{x\neq y} \lim\limits_{k \to \infty} \frac{|f_k(x)-f_k(y)|}{|x-y|^{\alpha}} \leq \sup\limits_{x\neq y} \lim\limits_{k \to \infty} \sup\limits_{x\neq y} \frac{|f_k(x)-f_k(y)|}{|x-y|^{\alpha}} \leq \sup\limits_{x\neq y} \lim\limits_{k \to \infty} R = R.$$
So somehow I could summarize my confusion like this: If I want find a fixed point for an operator on a Hölder space, why do I need more than merely the self-map property for an arbitrarily fixed ball? I see it everywhere that people establish contraction properties or continuity of operators between Hölder spaces, so there must be a crucial misunderstanding/error behind my logic?
Everything you say is technically correct. But if there's no assumption of continuity on $\Lambda$, we can't conclude that $f$ is still a $\Lambda$-fixed point because there's no reason why "$\Lambda(\lim f_n)= \lim \Lambda(f_n)$" then.