129
Fixed-Point Theory for Generalized Metric Spaces
4.11 Metrics and Multivalued Mappings
We discuss here a result established by M.A. Khamsi, V. Kreinovich, and
D. Misane, see [Khamsi et al., 1993], which is a multivalued version of the
Banach contraction mapping theorem, Theorem 4.2.3.
4.11.1 Definition Let (X, d) be a metric space. A multivalued mapping T :
X → P(X) is called a contraction if there exists a non-negative real number
λ < 1 such that for every x ∈ X, for every y ∈ X, and for all a ∈ T (x) there
exists b ∈ T (y) such that d(a, b) ≤ λd(x, y).
The result we wish to state is as follows; a proof of it will be given in
Section 4.13.
4.11.2 Theorem (Banach multivalued) Let X be a complete metric
space, and suppose that T is a multivalued contraction on X such that, for
every x ∈ X, the set T (x) is closed and non-empty. Then T has a fixed point.
This theorem was also established with a specific objective in view, namely,
to show the existence of answer sets for disjunctive logic programs which are
countably stratified, again see [Khamsi et al., 1993].
4.12 Generalized Ultrametrics and Multivalued
Mappings
We next turn our attention to multivalued versions of the Prieß-Crampe
and Ribenboim theorem, Theorem 4.3.6.
4.12.1 Definition Let (X, �, Γ) be a generalized ultrametric space. A multivalued mapping T defined on X is called strictly contracting (on X) (respectively, non-expanding (on X)) if, for all x, y ∈ X with x = y and for
every a ∈ T (x), there exists an element b ∈ T (y) such that �(a, b) < �(x, y)
(�(a, b) ≤ �(x, y)). Furthermore, the mapping T is called strictly contracting
on orbits if, for every x ∈ X and for every a ∈ T (x) with a = x, there exists
an element b ∈ T (a) such that �(a, b) < �(a, x).
For T : X → P(X), let Π x = {�(x, y) | y ∈ T (x)}, and, for a subset Δ ⊆ Γ,
denote by Min Δ the set of all minimal elements of Δ.
Note that these definitions collapse to those already considered for singlevalued mappings if, in fact, T is single valued, meaning that T (x) is a singleton
set for each x ∈ X.
Fixed-Point Theory for Generalized Metric Spaces
4.11 Metrics and Multivalued Mappings
We discuss here a result established by M.A. Khamsi, V. Kreinovich, and
D. Misane, see [Khamsi et al., 1993], which is a multivalued version of the
Banach contraction mapping theorem, Theorem 4.2.3.
4.11.1 Definition Let (X, d) be a metric space. A multivalued mapping T :
X → P(X) is called a contraction if there exists a non-negative real number
λ < 1 such that for every x ∈ X, for every y ∈ X, and for all a ∈ T (x) there
exists b ∈ T (y) such that d(a, b) ≤ λd(x, y).
The result we wish to state is as follows; a proof of it will be given in
Section 4.13.
4.11.2 Theorem (Banach multivalued) Let X be a complete metric
space, and suppose that T is a multivalued contraction on X such that, for
every x ∈ X, the set T (x) is closed and non-empty. Then T has a fixed point.
This theorem was also established with a specific objective in view, namely,
to show the existence of answer sets for disjunctive logic programs which are
countably stratified, again see [Khamsi et al., 1993].
4.12 Generalized Ultrametrics and Multivalued
Mappings
We next turn our attention to multivalued versions of the Prieß-Crampe
and Ribenboim theorem, Theorem 4.3.6.
4.12.1 Definition Let (X, �, Γ) be a generalized ultrametric space. A multivalued mapping T defined on X is called strictly contracting (on X) (respectively, non-expanding (on X)) if, for all x, y ∈ X with x = y and for
every a ∈ T (x), there exists an element b ∈ T (y) such that �(a, b) < �(x, y)
(�(a, b) ≤ �(x, y)). Furthermore, the mapping T is called strictly contracting
on orbits if, for every x ∈ X and for every a ∈ T (x) with a = x, there exists
an element b ∈ T (a) such that �(a, b) < �(a, x).
For T : X → P(X), let Π x = {�(x, y) | y ∈ T (x)}, and, for a subset Δ ⊆ Γ,
denote by Min Δ the set of all minimal elements of Δ.
Note that these definitions collapse to those already considered for singlevalued mappings if, in fact, T is single valued, meaning that T (x) is a singleton
set for each x ∈ X.