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.

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