cpu \
/
/
\
/
\
/
\
/
/
\
/
\
cp
/
\ \ \ \ \ \ \ \ \
K
B
PCR
/
/
/
/
/
/
/
/
KT
RS
M
dPCR
FIGURE 4.1: Dependencies between fixed-point theorems from Chapters 1
and 4. The lower a theorem is placed in the diagram, the more general it is.
See Table 4.3 for the abbreviations.
113
Fixed-Point Theory for Generalized Metric Spaces
TABLE 4.3: Summary of single-valued fixed-point theorems.
space
name of theorem
reference number symbol
ω-cpo
Kleene
1.1.9
K
cpo
Knaster-Tarski
1.1.10
KT
complete metric Banach
4.2.3
B
compact metric —
4.2.4
cp
gum
Prieß-Crampe and 4.3.6
PCR
Ribenboim
d-metric
Matthews
4.4.6
M
d-gum
—
4.5.1
dPCR
quasimetric
Rutten-Smyth
4.6.3
RS
1
1
1
1
mappings. In fact, we will consider generalizations of several of them to multivalued mappings as well in the later sections of this chapter. Furthermore,
the dependencies between these theorems are depicted in Figure 4.1, where
the letters abbreviate the theorems as listed in Table 4.3. (The abbreviation
“cpu” represents the statement that strictly contracting functions on compact
ultrametric spaces have unique fixed points, which follows immediately from
Theorem 4.2.4.)