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.