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 and be Banach spaces, and let be a Banach space with compact embedding . Let be a non-empty, convex, closed, bounded subset, and let,

be continuous maps. Then there exist and such that

and .

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

- Construction of a non-empty, convex, closed, bounded subset .
- Continuity of a mapping .
- Compactness of a mapping .
- Invoking the Schauder Theorem.

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

Theorem(Coupled Fixed-Point Principle B). Let and be Banach spaces, and let be a real ordered Banach space having the compact embedding . Let be a nonempty interval which is closed in the topology of , and set,

where is the closed ball of finite radius in about the origin. Assume is nonempty, and let the maps

,

be continuous maps. Then there exist and such that

and .

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