�.
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Fixed-Point Theory for Generalized Metric Spaces
We note finally that the constructions used for casting domains into generalized ultrametrics as in Section 4.8.2 and for casting generalized ultrametrics
into chain-complete partial orders as in Section 4.8.3 are not inverses of each
other, and the exact relationship between these processes remains to be determined.
4.8.4 GUMS and d-GUMS
We move next to study relationships between gums and d-gums and provide results somewhat parallel to those of Section 4.8.1, where we contrasted
metrics and d-metrics. Indeed, our main objective here is to investigate the
relationship between the Prieß-Crampe and Ribenboim theorem, Theorem
4.3.6, and its dislocated version, Theorem 4.5.2.
4.8.18 Proposition Let (X, �, Γ) be a dislocated generalized ultrametric
space, and define d : X × X → Γ by setting d(x, y) = �(x, y) for x = y
and setting d(x, x) = 0 for all x ∈ X. Then d is a generalized ultrametric.
Proof: The proof is straightforward following Proposition 4.8.8.
•
4.8.19 Definition The generalized ultrametric d just defined from the dgeneralized ultrametric � is called the generalized ultrametric associated with
4.8.20 Proposition Let (X, �, Γ) be a dislocated generalized ultrametric
space, and let d denote the generalized ultrametric associated with �. If d is
spherically complete, then � is spherically complete. If f is strictly contracting
relative to �, then f is strictly contracting relative to d.
Proof: We first show that non-empty balls in � contain all their midpoints.
So let {y | �(x, y) ≤ α} be some non-empty ball in � with midpoint x. Then
there is some z ∈ {y | �(x, y) ≤ α}, and we obtain �(x, x) ≤ �(x, z) by (U4).
Since �(x, z) ≤ α, we have x ∈ {y | �(x, y) ≤ α}. Hence, every non-empty ball
in � is also a ball with respect to d.
Now let B be a chain of non-empty balls in �. Then B is also a chain of
balls in d and has non-empty intersection by spherical completeness of d, as
required.
Let x, y ∈ X with x = y, and assume �(f (x), f (y)) < �(x, y). If f (x) =
f (y), then d(f (x), f (y)) = 0, and hence d(f (x), f (y)) < d(x, y). If f (x) = f (y),
then x = y, and so d(f (x), f (y)) = �(f (x), f (y)) < �(x, y) = d(x, y), as
required.
•
123
Fixed-Point Theory for Generalized Metric Spaces
We note finally that the constructions used for casting domains into generalized ultrametrics as in Section 4.8.2 and for casting generalized ultrametrics
into chain-complete partial orders as in Section 4.8.3 are not inverses of each
other, and the exact relationship between these processes remains to be determined.
4.8.4 GUMS and d-GUMS
We move next to study relationships between gums and d-gums and provide results somewhat parallel to those of Section 4.8.1, where we contrasted
metrics and d-metrics. Indeed, our main objective here is to investigate the
relationship between the Prieß-Crampe and Ribenboim theorem, Theorem
4.3.6, and its dislocated version, Theorem 4.5.2.
4.8.18 Proposition Let (X, �, Γ) be a dislocated generalized ultrametric
space, and define d : X × X → Γ by setting d(x, y) = �(x, y) for x = y
and setting d(x, x) = 0 for all x ∈ X. Then d is a generalized ultrametric.
Proof: The proof is straightforward following Proposition 4.8.8.
•
4.8.19 Definition The generalized ultrametric d just defined from the dgeneralized ultrametric � is called the generalized ultrametric associated with
4.8.20 Proposition Let (X, �, Γ) be a dislocated generalized ultrametric
space, and let d denote the generalized ultrametric associated with �. If d is
spherically complete, then � is spherically complete. If f is strictly contracting
relative to �, then f is strictly contracting relative to d.
Proof: We first show that non-empty balls in � contain all their midpoints.
So let {y | �(x, y) ≤ α} be some non-empty ball in � with midpoint x. Then
there is some z ∈ {y | �(x, y) ≤ α}, and we obtain �(x, x) ≤ �(x, z) by (U4).
Since �(x, z) ≤ α, we have x ∈ {y | �(x, y) ≤ α}. Hence, every non-empty ball
in � is also a ball with respect to d.
Now let B be a chain of non-empty balls in �. Then B is also a chain of
balls in d and has non-empty intersection by spherical completeness of d, as
required.
Let x, y ∈ X with x = y, and assume �(f (x), f (y)) < �(x, y). If f (x) =
f (y), then d(f (x), f (y)) = 0, and hence d(f (x), f (y)) < d(x, y). If f (x) = f (y),
then x = y, and so d(f (x), f (y)) = �(f (x), f (y)) < �(x, y) = d(x, y), as
required.
•