Fixed-Point Theory for Generalized Metric Spaces
131
In fact, g-metrics were called generalized metrics by Khamsi, Kreinovich,
and Misane, but we have changed the terminology since the term “generalized
metric” is, of course, already used differently by us. Actually, we will not work
with g-metrics in general since the closely related generalized ultrametrics will
suffice for our purposes. Indeed, we consider this relationship next, and we
begin by recalling the observations we made in Remark 4.3.2. Thus, let V
denote the set of all expressions of the type 0 or 2
−α , where α > 0 is an
ordinal. An order is defined on V by: 0 ≤ v for every v ∈ V , and 2
−α ≤ 2
−β if
and only if β ≤ α. As a semigroup operation u + v, we will use the maximum
max(u, v). It will be convenient to write
1 2
−α = 2
−(α+1) .
2
The following definition is due to Khamsi, Kreinovich, and Misane, see
[Khamsi et al., 1993].
4.12.6 Definition Assume that α is either a countable ordinal or ω 1 , the first
uncountable ordinal, and that v = (v β ) β<α is a decreasing family of elements
of V . Let X be a g-metric space relative to V , and let (x β ) β<α be a family of
elements of X.
(1) (x β ) is said to v-cluster to x ∈ X if, for all β, we have ρ(x β , x) < v β
whenever β < α.
(2) (x β ) is said to be v-Cauchy if, for all β and γ, we have ρ(x β , x γ ) < v β
whenever β < γ < α.
(3) X is said to be v-complete or just complete if, for every v, every v-Cauchy
family v-clusters to some element in X.
(4) A set Y ⊆ X will be called v-complete or just complete if, for every v,
whenever a v-Cauchy family consists of elements of Y , it v-clusters to
some element of Y .
A close relationship exists between the notion of completeness for g-metrics
and the notion of trans-completeness, Definition 4.3.8, for generalized ultrametrics. Indeed, we show that these notions coincide by showing equivalence
between completeness for g-metrics and spherical completeness for generalized
ultrametrics, see Proposition 4.3.10.
A b
1
4.12.7 Definition A multivalued mapping T : X → P(X) is called a
2
contraction if, for every x ∈ X, for every y ∈ X, and for every a ∈ T (x),
there exists b ∈ T (y) such that ρ(a, b) ≤
1 ρ(x, y).
2
The following theorem was proved by Khamsi, Kreinovich, and Misane in
[Khamsi et al., 1993].
4.12.8 Theorem Let X be a complete g-metric space, let T be a multivalued
A b
1 -contraction defined on X such that T (x) is not empty for some x ∈ X
2
(so that T is not identically empty), and suppose that for every x ∈ X the set
T (x) is complete. Then T has a fixed point.
131
In fact, g-metrics were called generalized metrics by Khamsi, Kreinovich,
and Misane, but we have changed the terminology since the term “generalized
metric” is, of course, already used differently by us. Actually, we will not work
with g-metrics in general since the closely related generalized ultrametrics will
suffice for our purposes. Indeed, we consider this relationship next, and we
begin by recalling the observations we made in Remark 4.3.2. Thus, let V
denote the set of all expressions of the type 0 or 2
−α , where α > 0 is an
ordinal. An order is defined on V by: 0 ≤ v for every v ∈ V , and 2
−α ≤ 2
−β if
and only if β ≤ α. As a semigroup operation u + v, we will use the maximum
max(u, v). It will be convenient to write
1 2
−α = 2
−(α+1) .
2
The following definition is due to Khamsi, Kreinovich, and Misane, see
[Khamsi et al., 1993].
4.12.6 Definition Assume that α is either a countable ordinal or ω 1 , the first
uncountable ordinal, and that v = (v β ) β<α is a decreasing family of elements
of V . Let X be a g-metric space relative to V , and let (x β ) β<α be a family of
elements of X.
(1) (x β ) is said to v-cluster to x ∈ X if, for all β, we have ρ(x β , x) < v β
whenever β < α.
(2) (x β ) is said to be v-Cauchy if, for all β and γ, we have ρ(x β , x γ ) < v β
whenever β < γ < α.
(3) X is said to be v-complete or just complete if, for every v, every v-Cauchy
family v-clusters to some element in X.
(4) A set Y ⊆ X will be called v-complete or just complete if, for every v,
whenever a v-Cauchy family consists of elements of Y , it v-clusters to
some element of Y .
A close relationship exists between the notion of completeness for g-metrics
and the notion of trans-completeness, Definition 4.3.8, for generalized ultrametrics. Indeed, we show that these notions coincide by showing equivalence
between completeness for g-metrics and spherical completeness for generalized
ultrametrics, see Proposition 4.3.10.
A b
1
4.12.7 Definition A multivalued mapping T : X → P(X) is called a
2
contraction if, for every x ∈ X, for every y ∈ X, and for every a ∈ T (x),
there exists b ∈ T (y) such that ρ(a, b) ≤
1 ρ(x, y).
2
The following theorem was proved by Khamsi, Kreinovich, and Misane in
[Khamsi et al., 1993].
4.12.8 Theorem Let X be a complete g-metric space, let T be a multivalued
A b
1 -contraction defined on X such that T (x) is not empty for some x ∈ X
2
(so that T is not identically empty), and suppose that for every x ∈ X the set
T (x) is complete. Then T has a fixed point.