# Ngô Quốc Anh

## May 26, 2010

### Coupled Fixed-Point Principle

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 20:03

I recently learned from a paper due to Michael Holst et al. published in Comm. Math. Phys. in 2009 [here] the following two fixed-point principles that may be useful elsewhere.

Theorem (Coupled Fixed-Point Principle A). Let $X$ and $Y$ be Banach spaces, and let $Z$ be a Banach space with compact embedding $X: \hookrightarrow Z$. Let $U \subset Z$ be a non-empty, convex, closed, bounded subset, and let

$S : U \to \mathcal R(S) \subset Y,\quad T : U\times \mathcal R(S)\to U \cap X$,

be continuous maps. Then there exist $\varphi \in U \cap X$ and $w \in \mathcal R(S)$ such that

$\varphi = T (\varphi,w)$ and $w = S(\varphi)$.

The proof will be through a standard variation of the Schauder Fixed-PointTheorem.

1. Construction of a non-empty, convex, closed, bounded subset $U \subset Z$.
2. Continuity of a mapping $G : U\subset Z \to U \cap X \subset X$.
3. Compactness of a mapping $F : U \subset Z \to U \subset Z$.
4. Invoking the Schauder Theorem.

In particular, we have the following variant which is useful in practice.

Theorem (Coupled Fixed-Point Principle B). Let $X$ and $Y$ be Banach spaces, and let $Z$ be a real ordered Banach space having the compact embedding $X: \hookrightarrow Z$. Let $[\varphi^-, \varphi^+] \subset Z$ be a nonempty interval which is closed in the topology of $Z$, and set

$U = [\varphi^-, \varphi^+] \cap \overline B_M \subset Z$,

where $\overline B_M$ is the closed ball of finite radius $M >0$ in $Z$ about the origin. Assume $U$ is nonempty, and let the maps

$S : U \to \mathcal R(S) \subset Y,\quad T : U\times \mathcal R(S)\to U \cap X$,

be continuous maps. Then there exist $\varphi \in U \cap X$ and $w \in \mathcal R(S)$ such that

$\varphi = T (\varphi,w)$ and $w = S(\varphi)$.

The proof is just verification of the first theorem. For a proof of all, I refer the reader to the paper mentioned above.