�
99
Fixed-Point Theory for Generalized Metric Spaces
that of spherical completeness, defined next, and that the next two definitions
and the following lemma apply to gums as a special case of d-gums.
4.3.3 Definition Let (X, � , Γ) be a d-gum space. For 0 = γ ∈ Γ and x ∈ X,
the set B γ (x) = {y ∈ X | �(x, y) ≤ γ} is called a (γ-)ball in X with centre or
midpoint x. A d-gum space is called spherically complete if, for any chain C,
with respect to set-inclusion, of non-empty balls in X we have C = ∅.
The stipulation in the definition of spherical completeness that all balls be
non-empty can be dropped when working in a gum rather than in a d-gum,
since in the former case all balls are clearly non-empty.
4.3.4 Definition Let (X, � , Γ) be a d-gum space, and let f : X → X be a
function.
(1) f is called non-expanding if �(f (x), f (y)) ≤ �(x, y) for all x, y ∈ X.
(2) f is called strictly contracting on orbits
11 if �(f
2 (x), f (x)) < �(f (x), x)
for every x ∈ X with x = f (x).
(3) f is called strictly contracting (on X) if �(f (x), f (y)) < �(x, y) for all
x, y ∈ X with x = y.
We will need the following observations, which are well-known for ordinary
ultrametric spaces.
4.3.5 Lemma Let (X, � , Γ) be a d-gum space. For α, β ∈ Γ and x, y ∈ X,
the following statements hold.
(a) If α ≤ β and B α (x) ∩ B β (y) = ∅, then B α (x) ⊆ B β (y).
(b) If B α (x) ∩ B α (y) = ∅, then B α (x) = B α (y). In particular, each element
of a ball is also its centre.
(c) B 1(x,y) (x) = B 1(x,y) (y).
Proof: Let a ∈ B α (x), and let b ∈ B α (x) ∩ B β (y). Then �(a, x) ≤ α and
�(b, x) ≤ α; hence, �(a, b) ≤ α ≤ β. Since �(b, y) ≤ β, we have �(a, y) ≤ β
and, hence, a ∈ B β (y), and this proves the first statement. The second follows
by symmetry and the third by replacing �(x, y) by α and applying (b).
•
The following theorem is the analogue of the Banach contraction mapping
theorem applicable to generalized ultrametrics.
12 It will be proved later by
virtue of proving the more general Theorem 4.5.1.
11 An orbit of f is a subset of X of the form {f n (x) | n ∈ N} for some x ∈ X.
12 Theorem 4.3.6 can be found in [Prieß-Crampe and Ribenboim, 2000c]. An earlier and
less general version appeared in [Prieß-Crampe, 1990].
99
Fixed-Point Theory for Generalized Metric Spaces
that of spherical completeness, defined next, and that the next two definitions
and the following lemma apply to gums as a special case of d-gums.
4.3.3 Definition Let (X, � , Γ) be a d-gum space. For 0 = γ ∈ Γ and x ∈ X,
the set B γ (x) = {y ∈ X | �(x, y) ≤ γ} is called a (γ-)ball in X with centre or
midpoint x. A d-gum space is called spherically complete if, for any chain C,
with respect to set-inclusion, of non-empty balls in X we have C = ∅.
The stipulation in the definition of spherical completeness that all balls be
non-empty can be dropped when working in a gum rather than in a d-gum,
since in the former case all balls are clearly non-empty.
4.3.4 Definition Let (X, � , Γ) be a d-gum space, and let f : X → X be a
function.
(1) f is called non-expanding if �(f (x), f (y)) ≤ �(x, y) for all x, y ∈ X.
(2) f is called strictly contracting on orbits
11 if �(f
2 (x), f (x)) < �(f (x), x)
for every x ∈ X with x = f (x).
(3) f is called strictly contracting (on X) if �(f (x), f (y)) < �(x, y) for all
x, y ∈ X with x = y.
We will need the following observations, which are well-known for ordinary
ultrametric spaces.
4.3.5 Lemma Let (X, � , Γ) be a d-gum space. For α, β ∈ Γ and x, y ∈ X,
the following statements hold.
(a) If α ≤ β and B α (x) ∩ B β (y) = ∅, then B α (x) ⊆ B β (y).
(b) If B α (x) ∩ B α (y) = ∅, then B α (x) = B α (y). In particular, each element
of a ball is also its centre.
(c) B 1(x,y) (x) = B 1(x,y) (y).
Proof: Let a ∈ B α (x), and let b ∈ B α (x) ∩ B β (y). Then �(a, x) ≤ α and
�(b, x) ≤ α; hence, �(a, b) ≤ α ≤ β. Since �(b, y) ≤ β, we have �(a, y) ≤ β
and, hence, a ∈ B β (y), and this proves the first statement. The second follows
by symmetry and the third by replacing �(x, y) by α and applying (b).
•
The following theorem is the analogue of the Banach contraction mapping
theorem applicable to generalized ultrametrics.
12 It will be proved later by
virtue of proving the more general Theorem 4.5.1.
11 An orbit of f is a subset of X of the form {f n (x) | n ∈ N} for some x ∈ X.
12 Theorem 4.3.6 can be found in [Prieß-Crampe and Ribenboim, 2000c]. An earlier and
less general version appeared in [Prieß-Crampe, 1990].