Earle–Hamilton fixed-point theorem

In mathematics, the Earle–Hamilton fixed point theorem is a result in geometric function theory giving sufficient conditions for a holomorphic mapping of an open domain in a complex Banach space into itself to have a fixed point. The result was proved in 1968 by Clifford Earle and Richard Hamilton by showing that, with respect to the Carathéodory metric on D, f becomes a contraction mapping to which the Banach fixed-point theorem can be applied.

Statement

Let D be a connected open subset of a complex Banach space X and let f be a holomorphic mapping of D into itself such that:

• the image f(D) is bounded in norm;
• the distance between points f(D) and points in the exterior of D is bounded below by a positive constant.

Then the mapping f has a unique fixed point x in D and if y is any point in D, the iterates fⁿ(y) converge to x.

Proof

Replacing D by an ε-neighbourhood of f(D), it can be assumed that D is itself bounded in norm.

For z in D and v in X, set

$\displaystyle {\alpha (z,v)=\sup _{g}|g^{\prime }(z)v|,}$

where the supremum is taken over all holomorphic functions g on D with |g(z)| < 1.

Define the α-length of a piecewise differentiable curve γ:[0,1] $\rightarrow$ D by

$\displaystyle {\ell (\gamma )=\int _{0}^{1}\alpha (\gamma (t),{\dot {\gamma }}(t))\,dt.}$

The Carathéodory metric is defined by

$\displaystyle {d(x,y)=\inf _{\gamma (0)=x,\gamma (1)=y}\ell (\gamma )}$

for x and y in D.

If the diameter of D is less than R then, by taking suitable holomorphic functions g of the form

$\displaystyle {g(z)=a(z)+b}$

with a in X* and b in C, it follows that

$\displaystyle {\alpha (z,v)\geq \|v\|/R,}$

and hence that

$\displaystyle {d(x,y)\geq \|x-y\|/R.}$

In particular d defines a metric on D.

The chain rule

$\displaystyle {(g\circ f)^{\prime }(z)=g^{\prime }(f(z))f^{\prime }(z)}$

implies that

$\displaystyle {\alpha (f(z),f^{\prime }(z)v)\leq \alpha (z,v)}$

and hence f satisfies the following generalization of the Schwarz-Pick inequality:

$\displaystyle {d(f(x),f(y))\leq d(x,y).}$

For δ sufficiently small and y fixed in D, the same inequality can be applied to the holomorphic mapping

$\displaystyle {f_{\delta ,y}(z)=f(z)+\delta (f(z)-f(x))}$

and yields the improved estimate:

$\displaystyle {d(f(x),f(y))\leq (1+\delta )^{-1}d(x,y).}$

The Banach fixed-point theorem can be applied to the restriction of f to the closure of f(D) on which d defines a complete metric, defining the same topology as the norm.

References

• Earle, Clifford J.; Hamilton, Richard S. (1970), A fixed point theorem for holomorphic mappings, Proc. Sympos. Pure Math., vol. XVI, American Mathemetical Society, pp. 61–65
• Harris, Lawrence A. (2003), "Fixed points of holomorphic mappings for domains in Banach spaces", Abstr. Appl. Anal., 5: 261–274