BRICS
Basic Research in Computer Science
A Quantitative Version of
Kirk’s Fixed Point Theorem for
Asymptotic Contractions
Philipp Gerhardy
BRICS Report Series RS-04-32
ISSN 0909-0878 December 2004
BRICS RS-04-32 P. Gerhardy: A Quantitative Version of Kirk’s Fixed Point Theorem for Asymptotic Contractions
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This document in subdirectory RS/04/32/
A quantitative version of Kirk’s fixed point
theorem for asymptotic contractions
Philipp Gerhardy
December, 2004
Abstract
In [J.Math.Anal.App.277(2003) 645-650], W.A.Kirk introduced the
notion of asymptotic contractions and proved a fixed point theorem
for such mappings. Using techniques from proof mining, we develop
a variant of the notion of asymptotic contractions and prove a quan-
titative version of the corresponding fixed point theorem.
1 Introduction
In [3], W.A. Kirk proved a fixed-point theorem for so-called asymptotic
contractions on complete metric spaces, showing that given a continuous
1
asymptotic contraction f for every starting point x the iteration sequence
{f
n
(x)} converges to the unique fixed point of f. The proof is non-elementary,
as it uses an ultrapower construction to establish the fixed point theorem.
Recent alternative proofs by Jachymski and J´o´zwik[2], additionally assum-
ing that f is uniformly continuous, and by Arandelovi´c [1], under the same
assumptions as Kirk, are elementary and avoid ultrapowers, but neither of
the three proofs provides explicit rates of convergence.
1
In [2, 1], it is discussed that the requirement that f is continuous is a necessary
condition for Kirk’s fixed point theorem. By an oversight the requirement was left out in
the original statement of Kirk’s fixed point theorem in [3]
1
Using techniques from proof mining as developed e.g. in [5, 4], we first
derive a suitable generalization of the notion of asymptotic contractivity
and subsequently give an elementary proof of Kirk’s fixed point theorem,
providing an explicit rate of convergence
2
(to the unique fixed point) for
sequences {f
n
(x)}.
In detail, we show that:
• the rate of convergence only depends on the starting point x via a
bound on the iteration sequence {f
n
(x)},
• the rate of convergence only depends on the function f via suitable
moduli expressing its asymptotic contractivity,
• the continuity of f is only necessary to prove the existence of a unique
fixed point, while the convergence to such a fixed point can be proved
without the continuity of f.
2 Preliminaries
In [3], Kirk defines asymptotic contractions as follows:
Definition 2.1 (Kirk[3]). A function f : X → X onametricspace(X, d)
is called an asymptotic contraction with moduli φ, φ
n
:[0,∞)→[0, ∞) if
φ, φ
n
are continuous, φ(s) <sfor all s>0and for all x, y ∈ X
d(f
n
(x),f
n
(y)) ≤ φ
n
(d(x, y))
and moreover φ
n
→ φ uniformly on the range of d.
What is needed to prove the fixed point theorem are not so much the moduli
φ, φ
n
, but instead a function η producing a witness of the inequality φ(s) <s
and a modulus of convergence β for φ
n
yielding a K s.t. for all k ≥ Kφ
k
2
Since an asymptotic contraction need not be non-expansive (cf. Example 2 in [2]),
convergence need not be monotone, and hence an effective rate of convergence can at most
produce a bound M s.t. f
m
(x) is close to the unique fixed point for some m ≤ M .We
will discuss the details later.
2
is close enough to φ and hence f
k
is a contraction. For η it is sufficient to
provide a witness for every interval [l, b], for β it suffices to have uniform
convergence on every interval [l, b], in both cases with 0 <l≤b<∞.
Thus, to give an elementary and effective proof of the fixed point theorem
proved by Kirk, we derive the following generalized definition of asymptotic
contractions:
Definition 2.2. A function f : X → X onametricspace(X, d) is called an
asymptotic contraction if for each b>0there exist moduli η
b
:(0,b]→(0, 1)
and β
b
:(0,b]×(0, ∞) → IN and the following hold:
(1) there exists a sequence of functions φ
n
:(0,∞)→(0, ∞) s.t. for all
x, y ∈ X, for all ε>0and for all n ∈ IN
b ≥ d ( x, y) ≥ ε → d(f
n
(x),f
n
(y)) ≤ φ
n
(ε) · d(x, y),
(2) for each 0 <l≤bthe function β
b
l
:= β
b
(l, ·) is a modulus of uniform
convergence for φ
n
on [l, b], i.e.
∀ε>0∀s∈[l, b] ∀m, n ≥ β
b
l
(ε)(|φ
m
(s)−φ
n
(s)|≤ε),
and (3) defining φ := lim
n→∞
φ
n
, then for each ε>0we have that η
b
(ε) > 0
and φ(s)+η
b
(ε)≤s for each s ∈ [ε, b].
All the relevant information is contained in the moduli η and β and we do
not need to refer to φ, φ
n
at all, as the following proposition shows:
Proposition 2.3. Let (X, d) beametricspace,letfbe an asymptotic con-
traction and let b>0and η
b
,β
b
be given. Then for every ε>0thereisa
K(η
b
,β
b
,ε) s.t. for all k ≥ K, where K = β
b
ε
(
η
b
(ε)
2
),
b ≥ d(x, y) ≥ ε → d(f
k
(x),f
k
(y)) ≤ (1 −
η
b
(ε)
2
) · d(x, y).
Proof: Let K = β
b
ε
(
η
b
(ε)
2
), let a suitable sequence φ
n
be given and let φ :=
lim
n→∞
φ
n
. By the definition of η
b
we have that φ(s)+η
b
(ε)≤1 for s ∈ [ε, b].
By the definition of β
b
the function φ
k
is at least
η
b
(ε)
2
-close to φ for all k ≥ K
and for all s ∈ [ε, b] and hence also φ
k
(s) ≤ 1 −
η
b
(ε)
2
.
3