1. Introduction
The set-valued dynamic system is defined as a pair
(
X
,
T
)
, where
X
is a certain space and
T
is a set-valued map
T
:
X
→
2
X
; here
2
X
denotes the family of all nonempty subsets of the space
X
. For
m
∈
{
0
}
∪
N
, we define
T
[
m
]
=
T
∘
T
∘
⋯
∘
T
(
m
-times) and
T
[
0
]
=
I
X
(an identity map on
X
). By
Fix
(
T
)
and
Per
(
T
)
we denote the sets of all fixed points and periodic points of
T
, respectively; that is,
Fix
(
T
)
=
{
w
∈
X
:
w
∈
T
(
w
)
}
and
Per
(
T
)
=
{
w
∈
X
:
w
∈
T
[
k
]
(
w
)
for some
k
∈
N
}
. A dynamic process or a trajectory starting at
w
0
∈
X
or a motion of the system
(
X
,
T
)
at
w
0
is a sequence
(
w
m
:
m
∈
{
0
}
∪
N
)
defined by
w
m
∈
T
(
w
m
-
1
)
for
m
∈
N
(see, [1–4]).
Recall that a single-valued dynamic system is defined as a pair
(
X
,
T
)
, where
X
is a certain space and
T
is a single-valued map
T
:
X
→
X
; that is,
∀
x
∈
X
{
T
(
x
)
∈
X
}
. By
Fix
(
T
)
and
Per
(
T
)
we denote the sets of all fixed points and periodic points of
T
, respectively; that is,
Fix
(
T
)
=
{
w
∈
X
:
w
=
T
(
w
)
}
and
Per
(
T
)
=
{
w
∈
X
:
w
=
T
[
k
]
(
w
)
for some
k
∈
N
}
. For each
w
0
∈
X
, a sequence
(
w
m
=
T
[
m
]
(
w
0
)
:
m
∈
{
0
}
∪
N
)
is called a Picard iteration starting at
w
0
of the system
(
X
,
T
)
.
Let
X
be a (nonempty) set. A distance on
X
is a map
p
:
X
2
→
[
0
;
∞
)
. The set
X
, together with distances on
X
, is called distance spaces.
The following distance spaces are important for several reasons.
Definition 1.
Let
X
be a (nonempty) set, and let
p
:
X
2
→
[
0
;
∞
)
.
(A)
(
X
,
p
)
is called metric if (i)
∀
u
,
w
∈
X
{
p
(
u
,
w
)
=
0
i
f
f
u
=
w
}
, (ii)
∀
u
,
w
∈
X
{
p
(
u
,
w
)
=
p
(
w
,
u
)
}
, and (iii)
∀
u
,
v
,
w
∈
X
{
p
(
u
,
w
)
≤
p
(
u
,
v
)
+
p
(
v
,
w
)
}
.
(B) (See [5])
(
X
,
p
)
is called ultra metric if (i)
∀
u
,
w
∈
X
{
p
(
u
,
w
)
=
0
i
f
f
u
=
w
}
, (ii)
∀
u
,
w
∈
X
{
p
(
u
,
w
)
=
p
(
w
,
u
)
}
, and (iii)
∀
u
,
v
,
w
∈
X
{
p
(
u
,
w
)
≤
max
{
p
(
u
,
v
)
,
p
(
v
,
w
)
}
}
.
(C) (See [6, 7])
(
X
,
p
)
is called
b
-metric with parameter
C
∈
[
1
;
∞
)
if (i)
∀
u
,
w
∈
X
{
p
(
u
,
w
)
=
0
i
f
f
u
=
w
}
, (ii)
∀
u
,
w
∈
X
{
p
(
u
,
w
)
=
p
(
w
,
u
)
}
, and (iii)
∀
u
,
v
,
w
∈
X
{
p
(
u
,
w
)
≤
C
[
p
(
u
,
v
)
+
p
(
v
,
w
)
]
}
.
(D) (See [8])
(
X
,
p
)
is called partial metric if (i)
∀
u
,
w
∈
X
{
u
=
w
i
f
f
p
(
u
,
u
)
=
p
(
u
,
w
)
=
p
(
w
,
w
)
}
, (ii)
∀
u
,
w
∈
X
{
p
(
u
,
u
)
≤
p
(
u
,
w
)
}
, (iii)
∀
u
,
w
∈
X
{
p
(
u
,
w
)
=
p
(
w
,
u
)
}
, and (iv)
∀
u
,
v
,
w
∈
X
{
p
(
u
,
w
)
≤
p
(
u
,
v
)
+
p
(
v
,
w
)
-
p
(
v
,
v
)
}
.
(E) (See [9])
(
X
,
p
)
is called partial
b
-metric with parameter
C
∈
[
1
;
∞
)
if (i)
∀
u
,
w
∈
X
{
u
=
w
i
f
f
p
(
u
,
u
)
=
p
(
u
,
w
)
=
p
(
w
,
w
)
}
, (ii)
∀
u
,
w
∈
X
{
p
(
u
,
u
)
≤
p
(
u
,
w
)
}
, (iii)
∀
u
,
w
∈
X
{
p
(
u
,
w
)
=
p
(
w
,
u
)
}
, and (iv)
∀
u
,
v
,
w
∈
X
{
p
(
u
,
w
)
≤
C
[
p
(
u
,
v
)
+
p
(
v
,
w
)
]
-
p
(
v
,
v
)
}
.
(F) (See [10])
(
X
,
p
)
is called quasi-metric if (i)
∀
u
,
w
∈
X
{
p
(
u
,
w
)
=
0
i
f
f
u
=
w
}
and (ii)
∀
u
,
v
,
w
∈
X
{
p
(
u
,
w
)
≤
p
(
u
,
v
)
+
p
(
v
,
w
)
}
.
(G)
(
X
,
p
)
is called ultra quasi-metric if (i)
∀
u
,
w
∈
X
{
p
(
u
,
w
)
=
0
i
f
f
u
=
w
}
and (ii)
∀
u
,
v
,
w
∈
X
{
p
(
u
,
w
)
≤
max
{
p
(
u
,
v
)
,
p
(
v
,
w
)
}
}
.
(H) The distance
p
is called pseudometric (or the gauge) on
X
if (i)
∀
u
∈
X
{
p
(
u
,
u
)
=
0
}
, (ii)
∀
u
,
w
∈
X
{
p
(
u
,
w
)
=
p
(
w
,
u
)
}
, and (iii)
∀
u
,
v
,
w
∈
X
{
p
(
u
,
w
)
≤
p
(
u
,
v
)
+
p
(
v
,
w
)
}
.
(I) The distance
p
is called quasi-pseudometric (or the quasi-gauge) on
X
if (i)
∀
u
∈
X
{
p
(
u
,
u
)
=
0
}
and (ii)
∀
u
,
v
,
w
∈
X
{
p
(
u
,
w
)
≤
p
(
u
,
v
)
+
p
(
v
,
w
)
}
.
(J) (See [11]) The distance
p
is called ultra quasi-pseudometric (or the ultra quasi-gauge) on
X
if (i)
∀
u
∈
X
{
p
(
u
,
u
)
=
0
}
and (ii)
∀
u
,
v
,
w
∈
X
{
p
(
u
,
w
)
≤
max
{
p
(
u
,
v
)
,
p
(
v
,
w
)
}
}
.
Definition 2 (see [12]).
Let
X
be a (nonempty) set, and let
A
be an index set.
(A) Each family
D
=
{
d
α
:
α
∈
A
}
of pseudometrics
d
α
:
X
2
→
0
,
∞
,
α
∈
A
, is called gauge on
X
. The gauge
D
=
{
d
α
:
α
∈
A
}
on
X
is called separating if
∀
u
,
w
∈
X
{
u
≠
w
⇒
∃
α
∈
A
{
d
α
(
u
,
w
)
>
0
}
}
.
(B) Let the family
D
=
{
d
α
:
α
∈
A
}
be separating gauge on
X
. The topology
T
(
D
)
having as a subbase the family
B
(
D
)
=
{
B
(
u
,
d
α
,
ε
α
)
:
u
∈
X
,
ε
α
>
0
,
α
∈
A
}
of all balls
B
(
u
,
d
α
,
ε
α
)
=
{
v
∈
X
:
d
α
(
u
,
v
)
<
ε
α
}
with
u
∈
X
,
ε
α
>
0
, and
α
∈
A
is called topology induced by
D
on
X
; the topology
T
(
D
)
is Hausdorff.
(C) A topological space
(
X
,
T
)
such that there is a separating gauge
D
on
X
with
T
=
T
(
D
)
is called a gauge space and is denoted by
(
X
,
D
)
.
Definition 3 (see [13]).
Let
X
be a (nonempty) set, and let
A
be an index set.
(A) Each family
P
=
{
p
α
,
α
∈
A
}
of quasi-pseudometrics
p
α
:
X
2
→
0
,
∞
,
α
∈
A
, is called quasi-gauge on
X
.
(B) Let the family
P
=
{
p
α
:
α
∈
A
}
be quasi-gauge on
X
. The topology
T
(
P
)
having as a subbase of the family
B
(
P
)
=
{
B
(
u
,
p
α
,
ε
α
)
:
u
∈
X
,
ε
α
>
0
,
α
∈
A
}
of all balls
B
(
u
,
p
α
,
ε
α
)
=
{
v
∈
X
:
p
α
(
u
,
v
)
<
ε
α
}
with
u
∈
X
,
ε
α
>
0
and
α
∈
A
is called topology induced by
P
on
X
.
(C) A topological space
(
X
,
T
)
such that there is a quasi-gauge
P
on
X
with
T
=
T
(
P
)
is called quasi-gauge space and is denoted by
(
X
,
P
)
.
Remark 4 (see [13, Theorems 4.2 and 2.6]).
Each quasi-uniform space and each topological space is the quasi-gauge space.
There is a growing literature concerning set-valued and single-valued dynamic systems in the above defined distance spaces. These studies contain also various extensions of the Banach [14] and Nadler [15, 16] theorems. Of course, there is a huge literature on this topic. For some such spaces and theorems in these spaces, see, for example, M. M. Deza and E. Deza [17], Kirk and Shahzad [18], and references therein.
Recall that the first convergence, existence, approximation, uniqueness, and fixed point result concerning single-valued contractions in complete metric spaces were obtained by Banach in 1922 [14].
Theorem 5 (see [14]).
Let
(
X
,
d
)
be a complete metric space. If
T
:
X
→
X
and
(1)
∃
0
≤
λ
<
1
∀
x
,
y
∈
X
d
T
x
,
T
y
≤
λ
d
x
,
y
,
then the following are true: (i)
T
has a unique fixed point
w
in
X
(i.e., there exists
w
∈
X
such that
w
=
T
(
w
)
and
Fix
(
T
)
=
{
w
}
)
; and (ii) for each
w
0
∈
X
, the sequence
(
T
[
m
]
(
w
0
)
:
m
∈
N
)
converges to
w
.
The Pompeiu-Hausdorff metric
H
d
on the class of all nonempty closed and bounded subsets
C
B
(
X
)
of the metric space
(
X
,
d
)
is defined as follows:
(2)
H
d
U
,
W
=
max
sup
u
∈
U
d
u
,
W
,
sup
w
∈
W
d
w
,
U
,
U
,
W
∈
C
B
X
,
where for each
x
∈
X
and
V
∈
C
B
(
X
)
,
d
(
x
,
V
)
=
inf
v
∈
V
d
(
x
,
v
)
. Using Pompeiu-Hausdorff metric new contractions were received by Nadler in 1967 and 1969 [15, 16] as a tool to study the existence of fixed points of set-valued maps in complete metric spaces.
Theorem 6 (see [15], [16, Theorem
5
]).
Let
X
,
d
be a complete metric space. If
T
:
X
→
C
B
X
and
(3)
∃
λ
∈
0
;
1
∀
x
,
y
∈
X
H
d
T
x
,
T
y
≤
λ
d
x
,
y
,
then
Fix
(
T
)
≠
⌀
(i.e., there exists
w
∈
X
such that
w
∈
T
(
w
)
).
Markin [19, 20] gave a slighty defferent version of Theorem 6.
Our primary interest is to construct new very general distance spaces, deliver new contractive set-valued and single-valued dynamic systems in these distance spaces, present the new global methods for studying of these dynamic systems in these spaces, and prove new convergence, approximation, existence, uniqueness, periodic point, and fixed point theorems for such dynamic systems.
The goal of the present paper is to introduce and describe the quasi-triangular spaces
(
X
,
P
C
;
A
)
(Section 2) and more general quasi-triangular spaces
(
X
,
P
C
;
A
)
with left (right) families
J
C
;
A
generated by
P
C
;
A
(Sections 3–5). Moreover, we use new methods and adopt ideas of Pompeiu and Hausdorff (Section 7) (see [21] for an excellent introduction to these ideas), to establish in these spaces some versions of Banach and Nadler theorems (Sections 8 and 9). Here studied dynamic systems are left (right)
J
C
;
A
-admissible or left (right)
P
C
;
A
-closed (Section 6). Examples are provided (Sections 10–12) and concluding remarks are given (Section 13).
4. Relations between
J
C
;
A
and
P
C
;
A
Remark 13.
The following result shows that Definition 11 is correct and that
J
(
X
,
P
C
;
A
)
L
∖
{
P
C
;
A
}
≠
⌀
and
J
(
X
,
P
C
;
A
)
R
∖
{
P
C
;
A
}
≠
⌀
.
Theorem 14.
Let
(
X
,
P
C
;
A
)
be the quasi-triangular space. Let
E
⊂
X
be a set containing at least two different points and let
{
μ
α
}
α
∈
A
∈
(
0
;
∞
)
A
where
(14)
∀
α
∈
A
μ
α
≥
δ
α
E
2
C
α
,
∀
α
∈
A
δ
α
E
=
sup
p
α
u
,
w
:
u
,
w
∈
E
.
If
J
C
;
A
=
{
J
α
:
α
∈
A
}
where, for each
α
∈
A
, the distance
J
α
:
X
2
→
[
0
,
∞
)
is defined by
(15)
J
α
u
,
w
=
p
α
u
,
w
i
f
E
∩
u
,
w
=
u
,
w
μ
α
i
f
E
∩
u
,
w
≠
u
,
w
,
then
J
C
;
A
is left and right family generated by
P
C
;
A
.
Proof.
Indeed, we see that condition (
J
1) does not hold only if there exist some
α
0
∈
A
and
u
0
,
v
0
,
w
0
∈
X
such that
(16)
J
α
0
u
0
,
w
0
>
C
α
0
J
α
0
u
0
,
v
0
+
J
α
0
v
0
,
w
0
.
Then (15) implies
{
u
0
,
v
0
,
w
0
}
∩
E
≠
{
u
0
,
v
0
,
w
0
}
and the following Cases 1–4 hold.
Case 1. If
{
u
0
,
w
0
}
⊂
E
, then
v
0
∉
E
and, by (16) and (15),
p
α
0
(
u
0
,
w
0
)
>
2
C
α
0
μ
α
0
. Therefore, by (14),
p
α
0
(
u
0
,
w
0
)
>
2
C
α
0
μ
α
0
≥
δ
α
0
(
E
)
. This is impossible.
Case 2. If
u
0
∈
E
and
w
0
∉
E
, then (16) and (15) give
μ
α
0
>
C
α
0
[
p
α
0
(
u
0
,
v
0
)
+
μ
α
0
]
≥
C
α
0
μ
α
0
whenever
v
0
∈
E
or
μ
α
0
>
C
α
0
[
μ
α
0
+
μ
α
0
]
=
2
C
α
0
μ
α
0
whenever
v
0
∉
E
. This is impossible.
Case 3. If
u
0
∉
E
and
w
0
∈
E
, then (16) and (15) give
μ
α
0
>
C
α
0
[
μ
α
0
+
p
α
0
(
v
0
,
w
0
)
]
≥
C
α
0
μ
α
0
whenever
v
0
∈
E
or
μ
α
0
>
C
α
0
[
μ
α
0
+
μ
α
0
]
=
2
C
α
0
μ
α
0
whenever
v
0
∉
E
. This is impossible.
Case 4. If
u
0
∉
E
and
w
0
∉
E
, then (16) and (15) give
μ
α
0
>
C
α
0
[
μ
α
0
+
μ
α
0
]
=
2
C
α
0
μ
α
0
for
v
0
∈
X
. This is impossible.
Therefore,
∀
α
∈
A
∀
u
,
v
,
w
∈
X
{
J
α
(
u
,
w
)
≤
C
α
[
J
α
(
u
,
v
)
+
J
α
(
v
,
w
)
]
}
; that is, the condition (
J
1) holds.
Assume now that the sequences
(
u
m
:
m
∈
N
)
and
(
w
m
:
m
∈
N
)
in
X
satisfy (8) and (10). Then (12) holds. Indeed, (10) implies
(17)
∀
α
∈
A
∀
0
<
ε
<
μ
α
∃
m
0
=
m
0
α
∈
N
∀
m
≥
m
0
J
α
w
m
,
u
m
<
ε
.
Denoting
m
′
=
min
{
m
0
(
α
)
:
α
∈
A
}
, we see, by (17) and (15), that
∀
m
≥
m
′
{
E
∩
{
w
m
,
u
m
}
=
{
w
m
,
u
m
}
}
. Then, in view of Definition 11(A), (15), and (17), this implies
∀
α
∈
A
∀
0
<
ε
<
μ
α
∃
m
′
∈
N
∀
m
≥
m
′
{
p
α
(
w
m
,
u
m
)
=
J
α
(
w
m
,
u
m
)
<
ε
}
. Hence we obtain that the sequences
(
u
m
:
m
∈
N
)
and
(
w
m
:
m
∈
N
)
satisfy (12). Thus we see that
J
C
;
A
is left family generated by
P
C
;
A
.
In a similar way, we show that (13) holds if
(
u
m
:
m
∈
N
)
and
(
w
m
:
m
∈
N
)
in
X
satisfy (9) and (11). Therefore,
J
C
;
A
is right family generated by
P
C
;
A
. We proved that
J
C
;
A
∈
J
(
X
,
P
C
;
A
)
L
∩
J
(
X
,
P
C
;
A
)
R
holds.
The following is interesting in respect to its use.
Theorem 15.
Let
(
X
,
P
C
;
A
)
be the quasi-triangular space, and let
J
C
;
A
be the left (right) family generated by
P
C
;
A
. If
P
C
;
A
is separating on
X
(i.e., (5) holds), then
J
C
;
A
is separating on
X
; that is,
(18)
∀
u
,
w
∈
X
u
≠
w
⟹
∃
α
∈
A
J
α
u
,
w
>
0
∨
J
α
w
,
u
>
0
holds.
Proof.
We begin by supposing that
u
0
,
w
0
∈
X
,
u
0
≠
w
0
, and
∀
α
∈
A
{
J
α
(
u
0
,
w
0
)
=
0
∧
J
α
(
w
0
,
u
0
)
=
0
}
. Then (
J
1) implies
∀
α
∈
A
{
J
α
(
u
0
,
u
0
)
≤
C
α
[
J
α
(
u
0
,
w
0
)
+
J
α
(
w
0
,
u
0
)
]
=
0
}
or, equivalently,
∀
α
∈
A
{
J
α
(
u
0
,
u
0
)
=
J
α
(
w
0
,
u
0
)
=
0
}
and
∀
α
∈
A
{
J
α
(
u
0
,
u
0
)
=
J
α
(
u
0
,
w
0
)
=
0
}
. Assuming that
u
m
=
u
0
and
w
m
=
w
0
,
m
∈
N
, we conclude that
∀
α
∈
A
{
l
i
m
m
→
∞
s
u
p
n
>
m
J
α
(
u
m
,
u
n
)
=
l
i
m
m
→
∞
J
α
(
w
m
,
u
m
)
=
0
}
and
∀
α
∈
A
{
l
i
m
m
→
∞
s
u
p
n
>
m
J
α
(
u
n
,
u
m
)
=
l
i
m
m
→
∞
J
α
(
u
m
,
w
m
)
=
0
}
. Therefore, it is not hard to see that (8)–(11) hold and, by (
J
2
), the above considerations lead to the following conclusion:
u
0
≠
w
0
∧
∀
α
∈
A
{
l
i
m
m
→
∞
p
α
(
w
m
,
u
m
)
=
l
i
m
m
→
∞
p
α
(
u
m
,
w
m
)
=
0
}
or, equivalently,
u
0
≠
w
0
∧
∀
α
∈
A
{
p
α
(
w
0
,
u
0
)
=
p
α
(
u
0
,
w
0
)
=
0
}
. However,
P
C
;
A
is separating. A contradiction. Therefore,
J
C
;
A
is separating.
8. Convergence, Existence, Approximation, and Periodic Point Theorem of Nadler Type for Left (Right) Set-Valued Quasi-Contractions
The following result extends Theorem 6 to spaces
(
X
,
P
C
;
A
)
.
Theorem 26.
Let
(
X
,
P
C
;
A
)
be the quasi-triangular space, and let
(
X
,
T
)
be the set-valued dynamic system,
T
:
X
→
2
X
. Let
η
∈
{
1,2
,
3
}
, and let
λ
=
λ
α
α
∈
A
∈
[
0
;
1
)
A
.
Assume that there exist a left (right) family
J
C
;
A
generated by
P
C
;
A
and a point
w
0
∈
X
with the following properties.
(A1)
(
X
,
T
)
is left
(
D
η
;
2
X
L
-
J
C
;
A
,
λ
)
-quasi-contraction (right
(
D
η
;
2
X
R
-
J
C
;
A
,
λ
)
-quasi-contraction) on
X
.
(A2)
(
X
,
T
)
is left (right)
J
C
;
A
-admissible in
w
0
.
(A3) For every
x
∈
X
and for every
β
=
β
α
α
∈
A
∈
(
0
;
∞
)
A
there exists
y
∈
T
(
x
)
such that
(26)
∀
α
∈
A
J
α
x
,
y
<
J
α
x
,
T
x
+
β
α
,
(27)
∀
α
∈
A
J
α
y
,
x
<
J
α
T
x
,
x
+
β
α
.
Then the following hold.
(B1) There exist a dynamic process
(
w
m
:
m
∈
{
0
}
∪
N
)
of the system
(
X
,
T
)
starting at
w
0
,
∀
m
∈
{
0
}
∪
N
{
w
m
+
1
∈
T
(
w
m
)
}
, and a point
w
∈
X
such that
(
w
m
:
m
∈
{
0
}
∪
N
)
is left (right)
P
C
;
A
-convergent to
w
.
(B2) If the set-valued dynamic system
(
X
,
T
[
k
]
)
is left (right)
P
C
;
A
-closed on
X
for some
k
∈
N
, then
Fix
(
T
[
k
]
)
≠
⌀
and there exist a dynamic process
(
w
m
:
m
∈
{
0
}
∪
N
)
of the system
(
X
,
T
)
starting at
w
0
,
∀
m
∈
{
0
}
∪
N
{
w
m
+
1
∈
T
(
w
m
)
}
, and a point
w
∈
Fix
(
T
[
k
]
)
such that
(
w
m
:
m
∈
{
0
}
∪
N
)
is left (right)
P
C
;
A
-convergent to
w
.
Proof.
We prove only the case when
J
C
;
A
is a left family generated by
P
C
;
A
,
(
X
,
T
)
is left
J
C
;
A
-admissible in a point
w
0
∈
X
, and
(
X
,
T
[
k
]
)
is left
P
C
;
A
-closed on
X
. The case of “right” will be omitted, since the reasoning is based on the analogous technique.
Part 1. Assume that (A1)–(A3) hold.
By (21) and the fact that
J
α
:
X
2
→
0
;
∞
,
α
∈
A
, we choose
(28)
r
=
r
α
α
∈
A
∈
0
;
∞
A
such that
(29)
∀
α
∈
A
J
α
w
0
,
T
w
0
<
1
-
λ
α
C
α
r
α
.
Put
(30)
∀
α
∈
A
β
α
0
=
1
-
λ
α
C
α
r
α
-
J
α
w
0
,
T
w
0
.
In view of (28) and (29) this implies
β
(
0
)
=
β
α
0
α
∈
A
∈
(
0
;
∞
)
A
and we apply (26) to find
w
1
∈
T
(
w
0
)
such that
(31)
∀
α
∈
A
J
α
w
0
,
w
1
<
J
α
w
0
,
T
w
0
+
β
α
0
.
We see from (30) and (31) that
(32)
∀
α
∈
A
J
α
w
0
,
w
1
<
1
-
λ
α
C
α
r
α
.
Put now
(33)
∀
α
∈
A
β
α
1
=
λ
α
C
α
1
-
λ
α
C
α
r
α
-
J
α
w
0
,
w
1
.
Then, in view of (32), we get
β
(
1
)
=
{
β
α
(
1
)
}
α
∈
A
∈
0
;
∞
A
and applying again (26) we find
w
2
∈
T
(
w
1
)
such that
(34)
∀
α
∈
A
J
α
w
1
,
w
2
<
J
α
w
1
,
T
w
1
+
β
α
1
.
Observe that
(35)
∀
α
∈
A
J
α
w
1
,
w
2
<
λ
α
C
α
1
-
λ
α
C
α
r
α
.
Indeed, from (34), Definition 23(A), and using (33), in the event that
η
=
1
or
η
=
2
or
η
=
3
, we get
(36)
∀
α
∈
A
J
α
w
1
,
w
2
<
J
α
w
1
,
T
w
1
+
β
α
1
≤
sup
J
α
u
,
T
w
1
:
u
∈
T
w
0
+
β
α
1
≤
D
η
;
2
X
;
α
L
-
J
C
;
A
T
w
0
,
T
w
1
+
β
α
1
≤
λ
α
C
α
J
α
w
0
,
w
1
+
β
α
1
=
λ
α
C
α
1
-
λ
α
C
α
r
α
.
Thus (35) holds.
Next define
(37)
∀
α
∈
A
β
α
2
=
λ
α
C
α
λ
α
C
α
1
-
λ
α
C
α
r
α
-
J
α
w
1
,
w
2
.
Then, in view of (35),
β
(
2
)
=
β
α
2
α
∈
A
∈
0
;
∞
A
. Applying (26) in this situation, we conclude that there exists
w
3
∈
T
(
w
2
)
such that
(38)
∀
α
∈
A
J
α
w
2
,
w
3
<
J
α
w
2
,
T
w
2
+
β
α
2
.
We seek to show that
(39)
∀
α
∈
A
J
α
w
2
,
w
3
<
λ
α
C
α
2
1
-
λ
α
C
α
r
α
.
By (38), Definition 23(A), and using (37), in the event that
η
=
1
or
η
=
2
or
η
=
3
, it follows that
(40)
∀
α
∈
A
J
α
w
2
,
w
3
<
J
α
w
2
,
T
w
2
+
β
α
2
⩽
sup
u
∈
T
w
1
J
α
u
,
T
w
2
+
β
α
2
≤
D
η
;
2
X
;
α
L
-
J
C
;
A
T
w
1
,
T
w
2
+
β
α
2
≤
λ
α
C
α
J
α
w
1
,
w
2
+
β
α
2
=
λ
α
C
α
2
1
-
λ
α
C
α
r
α
.
Thus (39) holds.
Proceeding as before, using Definition 23(A), we get that there exists a sequence
(
w
m
:
m
∈
N
)
in
X
satisfying
(41)
∀
m
∈
N
w
m
+
1
∈
T
w
m
and for calculational purposes, upon letting
∀
m
∈
N
{
β
(
m
)
=
{
β
α
(
m
)
}
α
∈
A
}
where
(42)
∀
α
∈
A
∀
m
∈
N
β
α
m
=
λ
α
C
α
·
λ
α
C
α
m
-
1
1
-
λ
α
C
α
r
α
-
J
α
w
m
-
1
,
w
m
we observe that
∀
m
∈
N
{
β
(
m
)
∈
(
0
;
∞
)
A
}
,
(43)
∀
α
∈
A
∀
m
∈
N
J
α
w
m
,
w
m
+
1
<
J
α
w
m
,
T
w
m
+
β
α
m
,
(44)
∀
α
∈
A
∀
m
∈
N
J
α
w
m
,
w
m
+
1
<
λ
α
C
α
m
1
-
λ
α
C
α
r
α
.
Let now
m
<
n
. Using (
J
1), we get
(45)
∀
α
∈
A
J
α
w
m
,
w
n
⩽
C
α
J
α
w
m
,
w
m
+
1
+
C
α
2
J
α
w
m
+
1
,
w
m
+
2
+
⋯
+
C
α
n
-
m
-
1
J
α
w
n
-
2
,
w
n
-
1
+
C
α
n
-
m
-
1
J
α
w
n
-
1
,
w
n
=
∑
j
=
0
n
-
m
-
2
C
α
j
+
1
J
α
w
m
+
j
,
w
m
+
j
+
1
+
C
α
n
-
m
-
1
J
α
w
n
-
1
,
w
n
.
Hence, by (44), for each
α
∈
A
,
(46)
J
α
w
m
,
w
n
<
1
-
λ
α
C
α
·
r
α
∑
j
=
0
n
-
m
-
2
C
α
j
+
1
λ
α
C
α
m
+
j
+
C
α
n
-
m
-
1
λ
α
C
α
n
-
2
=
1
-
λ
α
C
α
·
r
α
C
α
λ
α
C
α
m
∑
j
=
0
n
-
m
-
2
λ
α
j
+
C
α
λ
α
2
λ
α
n
C
α
m
.
This and (41) mean that
(47)
∃
w
m
:
m
∈
N
∀
m
∈
0
∪
N
w
m
+
1
∈
T
w
m
and since
m
<
n
implies
λ
α
n
≤
λ
α
m
,
(48)
∀
α
∈
A
lim
m
→
∞
sup
n
>
m
J
α
w
m
,
w
n
≤
lim
m
→
∞
sup
n
>
m
1
-
λ
α
C
α
·
r
α
C
α
λ
α
C
α
m
1
-
λ
α
-
1
+
C
α
λ
α
2
λ
α
n
C
α
m
≤
l
i
m
m
→
∞
1
-
λ
α
C
α
r
α
C
α
λ
α
C
α
m
1
-
λ
α
-
1
+
C
α
λ
α
2
λ
α
C
α
m
=
0
.
Now, since
(
X
,
T
)
is left
J
C
;
A
-admissible in
w
0
∈
X
, by Definition 20(A), properties (47) and (48) imply that there exists
w
∈
X
such that
(49)
∀
α
∈
A
lim
m
→
∞
J
α
w
,
w
m
=
0
.
Next, defining
u
m
=
w
m
and
w
m
=
w
for
m
∈
N
, by (48) and (49) we see that conditions (8) and (10) hold for the sequences
(
u
m
:
m
∈
N
)
and
(
w
m
:
m
∈
N
)
in
X
. Consequently, by (
J
2
), we get (12) which implies that
(50)
∀
α
∈
A
lim
m
→
∞
p
α
w
,
w
m
=
lim
m
→
∞
p
α
w
m
,
u
m
=
0
and so in particular we see that
w
∈
L
I
M
(
w
m
:
m
∈
N
)
L
-
P
C
;
A
.
Part 2. Assume that (A1)–(A3) hold and that, for some
k
∈
N
,
(
X
,
T
[
k
]
)
is left
P
C
;
A
-closed on
X
.
By Part 1,
L
I
M
(
w
m
:
m
∈
{
0
}
∪
N
)
L
-
P
C
;
A
≠
⌀
and since, by (47),
w
(
m
+
1
)
k
∈
T
[
k
]
(
w
m
k
)
for
m
∈
{
0
}
∪
N
, thus defining
(
x
m
=
w
m
-
1
+
k
:
m
∈
N
)
, we see that
(
x
m
:
m
∈
N
)
⊂
T
[
k
]
(
X
)
,
L
I
M
(
x
m
:
m
∈
{
0
}
∪
N
)
L
-
P
C
;
A
=
L
I
M
(
w
m
:
m
∈
{
0
}
∪
N
)
L
-
P
C
;
A
≠
⌀
, the sequences
(
v
m
=
w
(
m
+
1
)
k
:
m
∈
N
)
⊂
T
[
k
]
(
X
)
and
(
u
m
=
w
m
k
:
m
∈
N
)
⊂
T
[
k
]
(
X
)
satisfy
∀
m
∈
N
{
v
m
∈
T
[
k
]
(
u
m
)
}
and, as subsequences of
(
x
m
:
m
∈
{
0
}
∪
N
)
, are left
P
C
;
A
-converging to each point of the set
L
I
M
(
w
m
:
m
∈
{
0
}
∪
N
)
L
-
P
C
;
A
. Moreover, by Remark 18,
L
I
M
(
w
m
:
m
∈
N
)
L
-
P
C
;
A
⊂
L
I
M
(
v
m
:
m
∈
N
)
L
-
P
C
;
A
and
L
I
M
(
w
m
:
m
∈
N
)
L
-
P
C
;
A
⊂
L
I
M
(
u
m
:
m
∈
N
)
L
-
P
C
;
A
. By the above and by Definition 22, since
T
[
k
]
is left
P
C
;
A
-closed, we conclude that there exist
w
∈
L
I
M
(
w
m
:
m
∈
{
0
}
∪
N
)
L
-
P
C
;
A
=
L
I
M
(
x
m
:
m
∈
N
)
L
-
P
C
;
A
such that
w
∈
T
[
k
]
(
w
)
.
Part 3. The result now follows at once from Parts 1 and 2.
9. Theorem of Banach Type in Quasi-Triangular Spaces
(
X
,
P
C
;
A
)
In this section, in the quasi-triangular spaces
(
X
,
P
C
;
A
)
, using left (right) families
J
C
;
A
generated by
P
C
;
A
, we construct two types of left (right) single-valued quasi-contractions
T
:
X
→
X
, and convergence, existence, approximation, uniqueness, periodic point, and fixed point theorem for such quasi-contractions is also proved.
The following Definition 27 can be stated as a single-valued version of Definition 23.
Definition 27.
Let
(
X
,
P
C
;
A
)
be the quasi-triangular space, and let
J
C
;
A
be the left (right) family generated by
P
C
;
A
. Let
(
X
,
T
)
be the single-valued dynamic system, let
λ
=
λ
α
α
∈
A
∈
[
0
;
1
)
A
, and let
η
∈
{
1,2
}
.
(A) If
J
C
;
A
∈
J
(
X
,
P
C
;
A
)
L
, then we define the left
D
X
,
η
L
-
J
C
;
A
-quasi-distance on
X
by
D
X
,
η
L
-
J
C
;
A
=
{
D
η
;
X
;
α
L
-
J
C
;
A
:
X
2
→
[
0
;
∞
)
,
α
∈
A
}
where
(51)
∀
α
∈
A
∀
u
,
w
∈
X
D
1
;
X
;
α
L
-
J
C
;
A
u
,
w
=
max
J
α
u
,
w
,
J
α
w
,
u
,
∀
α
∈
A
∀
u
,
w
∈
X
D
2
;
X
;
α
L
-
J
C
;
A
u
,
w
=
J
α
u
,
w
.
One says that
(
X
,
T
)
is left
(
D
X
,
η
L
-
J
C
;
A
,
λ
)
-quasi-contraction on
X
if
(52)
∀
α
∈
A
∀
x
,
y
∈
X
C
α
·
D
η
;
X
;
α
L
-
J
C
;
A
T
x
,
T
y
≤
λ
α
J
α
x
,
y
.
(B) If
J
C
;
A
∈
J
(
X
,
P
C
;
A
)
R
, then one defines the right
D
X
,
η
R
-
J
C
;
A
-quasi-distance on
X
by
D
X
,
η
R
-
J
C
;
A
=
{
D
η
;
X
;
α
R
-
J
C
;
A
:
X
2
→
[
0
;
∞
)
,
α
∈
A
}
where
(53)
∀
α
∈
A
∀
u
,
w
∈
X
D
1
;
X
;
α
R
-
J
C
;
A
u
,
w
=
max
J
α
u
,
w
,
J
α
w
,
u
,
∀
α
∈
A
∀
u
,
w
∈
X
D
2
;
X
;
α
R
-
J
C
;
A
u
,
w
=
J
α
u
,
w
.
One says that
(
X
,
T
)
is right
(
D
X
,
η
R
-
J
C
;
A
,
λ
)
-quasi-contraction on
X
if
(54)
∀
α
∈
A
∀
x
,
y
∈
X
C
α
·
D
η
;
X
;
α
R
-
J
C
;
A
T
x
,
T
y
≤
λ
α
J
α
x
,
y
.
Remark 28.
Observe that (52) and (54) extend (1).
The following terminologies will be much used in the sequel.
Definition 29.
Let
(
X
,
P
C
;
A
)
be the quasi-triangular space, and let
J
C
;
A
be the left (right) family generated by
P
C
;
A
. Let
(
X
,
T
)
be the single-valued dynamic system,
T
:
X
→
X
.
(A) Given
w
0
∈
X
, One says that
(
X
,
T
)
is left (right)
J
C
;
A
-admissible in
w
0
if, for the sequence
(
w
m
=
T
[
m
]
(
w
0
)
:
m
∈
{
0
}
∪
N
)
,
L
I
M
(
w
m
:
m
∈
{
0
}
∪
N
)
L
-
J
C
;
A
≠
⌀
(
L
I
M
(
w
m
:
m
∈
{
0
}
∪
N
)
R
-
J
C
;
A
≠
⌀
)
whenever
(55)
∀
α
∈
A
lim
m
→
∞
sup
n
>
m
J
α
w
m
,
w
n
=
0
∀
α
∈
A
lim
m
→
∞
sup
n
>
m
J
α
w
n
,
w
m
=
0
.
(B) We say that
(
X
,
T
)
is left (right)
J
C
;
A
-admissible on
X
, if
(
X
,
T
)
is left (right)
J
C
;
A
-admissible in each point
w
0
∈
X
.
Remark 30.
Let
(
X
,
P
C
;
A
)
be the quasi-triangular space, and let
J
C
;
A
be the left (right) family generated by
P
C
;
A
. Let
(
X
,
T
)
be the single-valued dynamic system on
X
. If
(
X
,
P
C
;
A
)
is left (right)
J
C
;
A
-sequentially complete, then
(
X
,
T
)
is left (right)
J
C
;
A
-admissible on
X
.
We can define the following generalization of continuity.
Definition 31.
Let
(
X
,
P
C
;
A
)
be the quasi-triangular space. Let
(
X
,
T
)
be the single-valued dynamic system,
T
:
X
→
X
, and let
k
∈
N
. The single-valued dynamic system
(
X
,
T
[
k
]
)
is said to be a left (right)
P
C
;
A
-closed on
X
if for each sequence
(
x
m
:
m
∈
N
)
in
T
[
k
]
(
X
)
, left (right)
P
C
;
A
-converging in
X
(thus
L
I
M
(
x
m
:
m
∈
N
)
L
-
P
C
;
A
≠
⌀
(
L
I
M
(
x
m
:
m
∈
N
)
R
-
P
C
;
A
≠
⌀
)
)
and having subsequences
(
v
m
:
m
∈
N
)
and
(
u
m
:
m
∈
N
)
satisfying
∀
m
∈
N
{
v
m
=
T
[
k
]
(
u
m
)
}
; the following property holds: there exists
x
∈
L
I
M
(
x
m
:
m
∈
N
)
L
-
P
C
;
A
(
x
∈
L
I
M
(
x
m
:
m
∈
N
)
R
-
P
C
;
A
)
such that
x
=
T
[
k
]
(
x
)
(
x
=
T
[
k
]
(
x
)
)
.
The following result extends Theorem 5 to spaces
(
X
,
P
C
;
A
)
.
Theorem 32.
Let
(
X
,
P
C
;
A
)
be the quasi-triangular space, and let
(
X
,
T
)
be the single-valued dynamic system,
T
:
X
→
2
X
. Let
η
∈
{
1,2
}
, and let
λ
=
λ
α
α
∈
A
∈
0
;
1
A
.
Assume that there exist a left (right) family
J
C
;
A
generated by
P
C
;
A
and a point
w
0
∈
X
with the following properties.
(A1)
(
X
,
T
)
is left
(
D
X
,
η
L
-
J
C
;
A
,
λ
)
-quasi-contraction (right
(
D
X
,
η
R
-
J
C
;
A
,
λ
)
-quasi-contraction) on
X
.
(A2)
(
X
,
T
)
is left (right)
J
C
;
A
-admissible in a point
w
0
∈
X
.
Then the following hold.
(B1) There exists a point
w
∈
X
such that the sequence
(
w
m
=
T
[
m
]
(
w
0
)
:
m
∈
{
0
}
∪
N
)
starting at
w
0
is left (right)
P
C
;
A
-convergent to
w
.
(B2) If the single-valued dynamic system
(
X
,
T
[
k
]
)
is left (right)
P
C
;
A
-closed on
X
for some
k
∈
N
, then
Fix
(
T
[
k
]
)
≠
⌀
, there exists a point
w
∈
Fix
(
T
[
k
]
)
such that the sequence
(
w
m
=
T
[
m
]
(
w
0
)
:
m
∈
{
0
}
∪
N
)
starting at
w
0
is left (right)
P
C
;
A
-convergent to
w
, and
(56)
∀
α
∈
A
∀
v
∈
Fix
T
k
J
α
v
,
T
v
=
J
α
T
v
,
v
=
0
.
(B3) If the family
P
C
;
A
=
{
p
α
,
α
∈
A
}
is separating on
X
and if the single-valued dynamic system
(
X
,
T
[
k
]
)
is left (right)
P
C
;
A
-closed on
X
for some
k
∈
N
, then there exists a point
w
∈
X
such that
(57)
Fix
T
k
=
Fix
T
=
w
,
the sequence
(
w
m
=
T
[
m
]
(
w
0
)
:
m
∈
{
0
}
∪
N
)
starting at
w
0
is left (right)
P
C
;
A
- convergent to
w
, and
(58)
∀
α
∈
A
J
α
w
,
w
=
0
.
Proof.
By Theorem 26, we prove only (56)–(58) and only in the case of “left.” We omit the proof in the case of “right,” which is based on the analogous technique.
Part 1. Property (56) holds. Suppose that
∃
α
0
∈
A
∃
v
∈
Fix
(
T
[
k
]
)
{
J
α
0
(
v
,
T
(
v
)
)
>
0
}
. Of course,
v
=
T
[
k
]
(
v
)
=
T
[
2
k
]
(
v
)
,
T
(
v
)
=
T
[
2
k
]
(
T
(
v
)
)
and, for
η
∈
{
1,2
}
, by Definition 27(A),
(59)
0
<
J
α
0
v
,
T
v
=
J
α
0
T
2
k
v
,
T
2
k
T
v
≤
D
η
;
X
;
α
0
L
-
J
C
;
A
T
2
k
v
,
T
2
k
T
v
≤
λ
α
0
C
α
0
J
α
0
T
2
k
-
1
v
,
T
2
k
-
1
T
v
≤
λ
α
0
C
α
0
·
D
η
;
X
;
α
0
L
-
J
C
;
A
T
2
k
-
1
v
,
T
2
k
-
1
T
v
≤
λ
α
0
C
α
0
2
J
α
0
T
2
k
-
2
v
,
T
2
k
-
2
T
v
≤
⋯
≤
λ
α
0
C
α
0
2
k
J
α
0
v
,
T
v
<
J
α
0
v
,
T
v
,
which is impossible. Therefore,
(60)
∀
α
∈
A
∀
v
∈
Fix
T
k
J
α
v
,
T
v
=
0
.
Suppose now that
∃
α
0
∈
A
∃
v
∈
Fix
(
T
[
k
]
)
{
J
α
0
(
T
(
v
)
,
v
)
>
0
}
. Then, by Definition 27(A) and property (60), using the fact that
v
=
T
[
k
]
(
v
)
=
T
[
2
k
]
(
v
)
, we get, for
η
∈
{
1,2
}
, that
(61)
0
<
J
α
0
T
v
,
v
=
J
α
0
T
k
+
1
v
,
T
2
k
v
≤
∑
m
=
1
k
-
2
C
α
0
m
J
α
0
T
k
+
m
v
,
T
k
+
m
+
1
v
+
C
α
0
k
-
2
J
α
0
T
2
k
-
1
v
,
T
2
k
v
≤
∑
m
=
1
k
-
2
C
α
0
m
·
D
η
;
X
;
α
0
L
-
J
C
;
A
T
k
+
m
v
,
T
k
+
m
+
1
v
+
C
α
0
k
-
2
·
D
η
;
X
;
α
0
L
-
J
C
;
A
T
2
k
-
1
v
,
T
2
k
v
≤
∑
m
=
1
k
-
2
C
α
0
m
λ
α
0
C
α
0
k
+
m
J
α
0
v
,
T
v
+
C
α
0
k
-
2
λ
α
0
C
α
0
2
k
-
1
J
α
0
v
,
T
v
=
0
,
which is impossible. Therefore,
(62)
∀
α
∈
A
∀
v
∈
Fix
T
k
J
α
T
v
,
v
=
0
.
We see that (56) is a consequence of (60) and (62).
Part 2. Properties (57) and (58) hold. We first observe that
(63)
∀
v
∈
Fix
T
k
T
v
=
v
;
in other words,
Fix
(
T
[
k
]
)
=
F
i
x
(
T
)
. In fact, if
v
∈
Fix
(
T
[
k
]
)
and
T
(
v
)
≠
v
, then, since the family
P
C
;
A
=
{
p
α
,
α
∈
A
}
is separating on
X
, we get that
T
(
v
)
≠
v
⇒
∃
α
∈
A
{
p
α
(
T
(
v
)
,
v
)
>
0
∨
p
α
(
v
,
T
(
v
)
)
>
0
}
. In view of Theorem 15 this implies
T
(
v
)
≠
v
⇒
∃
α
∈
A
{
J
α
(
T
(
v
)
,
v
)
>
0
∨
J
α
(
v
,
T
(
v
)
)
>
0
}
. However, by property (56), this is impossible.
Next we see that
∀
v
∈
Fix
(
T
k
)
=
Fix
(
T
)
{
J
α
(
v
,
v
)
=
0
}
. In fact, by Definition 11(A) and property (56), we conclude that
∀
α
∈
A
∀
v
∈
Fix
(
T
[
k
]
)
{
J
α
(
v
,
v
)
≤
C
α
[
J
α
(
v
,
T
(
v
)
)
+
J
α
(
T
(
v
)
,
v
)
]
=
0
}
.
Finally, suppose that
u
,
w
∈
F
i
x
(
T
)
and
u
≠
w
. Then, since the family
P
C
;
A
=
{
p
α
,
α
∈
A
}
is separating on
X
, we get
∃
α
0
∈
A
{
p
α
0
(
u
,
w
)
>
0
∨
p
α
0
(
w
,
u
)
>
0
}
. By applying Theorem 15, this implies
∃
α
0
∈
A
{
J
α
0
(
u
,
w
)
>
0
∨
J
α
0
(
w
,
u
)
>
0
}
. Consequently, for
η
∈
{
1,2
}
, by Definition 27(A), we conclude that
(64)
∃
α
0
∈
A
J
α
0
u
,
w
>
0
,
J
α
0
u
,
w
=
J
α
0
T
u
,
T
w
≤
D
η
;
X
;
α
0
L
-
J
C
;
A
T
u
,
T
w
≤
λ
α
0
C
α
0
J
α
0
u
,
w
<
J
α
0
u
,
w
or
J
α
0
w
,
u
>
0
,
J
α
0
w
,
u
=
J
α
0
T
w
,
T
u
≤
D
η
;
X
;
α
0
L
-
J
C
;
A
T
w
,
T
u
≤
λ
α
0
C
α
0
J
α
0
w
,
u
<
J
α
0
w
,
u
,
which is impossible. This gives that
Fix
(
T
)
is a singleton.
Thus (57) and (58) hold.
10. Examples of Spaces
(
X
,
P
C
;
A
)
Example 1.
Let
X
=
[
0
;
6
]
,
γ
≥
81
and let
p
:
X
2
→
[
0
;
∞
)
be of the form
(65)
p
u
,
w
=
0
if
u
≥
w
,
u
,
w
∩
0
;
6
=
u
,
w
,
w
-
u
4
if
u
<
w
,
u
,
w
∩
0
;
6
=
u
,
w
,
γ
if
u
,
w
∩
0
;
6
≠
u
,
w
.
(
1
)
(
X
,
P
{
8
}
;
{
1
}
)
,
P
{
8
}
;
{
1
}
=
{
p
}
, is the quasi-triangular space. In fact,
(66)
∀
u
,
v
,
w
∈
X
p
u
,
w
≤
8
p
u
,
v
+
p
v
,
w
.
Inequality (66) is a consequence of Cases 1–6.
Case 1. If
u
,
v
,
w
∈
(
0
;
6
)
and
v
≤
u
<
w
, then
p
(
u
,
v
)
=
0
and
w
-
u
≤
w
-
v
. This gives
p
(
u
,
w
)
=
w
-
u
4
≤
w
-
v
4
<
8
w
-
v
4
=
8
[
p
(
u
,
v
)
+
p
(
v
,
w
)
]
.
Case 2. If
u
,
v
,
w
∈
(
0
;
6
)
,
u
<
w
and
u
≤
v
≤
w
, then
p
(
u
,
w
)
=
w
-
u
4
and
f
(
v
0
)
=
min
u
≤
v
≤
w
f
(
v
)
=
w
-
u
4
where, for
u
≤
v
≤
w
,
f
(
v
)
=
8
[
p
(
u
,
v
)
+
p
(
v
,
w
)
]
=
8
[
v
-
u
4
+
w
-
v
4
]
and
v
0
=
(
u
+
w
)
/
2
.
Case 3.
s
u
p
u
,
w
∈
(
0
;
6
)
;
u
<
w
p
(
u
,
w
)
=
s
u
p
u
,
w
∈
(
0
;
6
)
;
u
<
w
w
-
u
4
=
6
4
=
1296
and
s
u
p
u
,
w
∈
(
0
;
6
)
;
u
<
w
m
i
n
u
≤
v
≤
w
8
[
p
(
u
,
v
)
+
p
(
v
,
w
)
]
=
s
u
p
u
,
w
∈
(
0
;
6
)
;
u
<
w
m
i
n
u
≤
v
≤
w
8
[
v
-
u
4
+
w
-
v
4
]
=
8
[
3
-
0
4
+
6
-
3
4
]
=
8
[
81
+
81
]
=
1296
.
Case 4. If
u
,
v
,
w
∈
(
0
;
6
)
and
u
<
w
≤
v
, then
p
(
v
,
w
)
=
0
and
w
-
u
≤
v
-
u
. This gives
p
(
u
,
w
)
=
w
-
u
4
≤
v
-
u
4
<
8
v
-
u
4
=
8
[
p
(
u
,
v
)
+
p
(
v
,
w
)
]
.
Case 5. If
u
,
w
∈
(
0
;
6
)
,
u
<
w
and
v
∈
{
0,6
}
, then
p
(
u
,
w
)
≤
1296
≤
8
[
p
(
u
,
v
)
+
p
(
v
,
w
)
]
=
8
[
γ
+
γ
]
.
Case 6. If
{
u
,
w
}
∩
(
0
;
6
)
≠
{
u
,
w
}
, then
∀
v
∈
X
{
p
(
u
,
w
)
=
γ
<
8
γ
≤
8
[
p
(
u
,
v
)
+
p
(
v
,
w
)
]
}
.
(
2
)
P
{
8
}
;
{
1
}
=
{
p
}
is asymmetric. Indeed, we have that
0
=
p
(
5,1
)
≠
p
(
1,5
)
=
256
. Therefore, condition
∀
u
,
w
∈
X
{
p
(
u
,
w
)
=
p
(
w
,
u
)
}
does not hold.
(
3
)
P
{
8
}
;
{
1
}
=
{
p
}
does not vanish on the diagonal. Indeed, if
u
∈
{
0,6
}
, then
p
(
u
,
u
)
=
γ
≠
0
. Therefore, the condition
∀
u
∈
X
{
p
(
u
,
u
)
=
0
}
does not hold.
(
4
) For the constant sequence of the form
(
u
m
=
3
:
m
∈
N
)
⊂
X
the sets
L
I
M
(
u
m
:
m
∈
N
)
L
-
P
{
8
}
;
{
1
}
and
L
I
M
(
u
m
:
m
∈
N
)
R
-
P
{
8
}
;
{
1
}
are not singletons. Indeed, by (65), Remark 12, and Definition 16(B), we have that
L
I
M
(
u
m
:
m
∈
N
)
L
-
P
{
8
}
;
{
1
}
=
[
3
;
6
]
and
L
I
M
(
u
m
:
m
∈
N
)
R
-
P
{
8
}
;
{
1
}
=
[
0
;
3
]
.
Example 2.
Let
X
be a set (nonempty),
A
⊂
X
,
A
≠
⌀
,
A
≠
X
,
γ
>
0
, and let
p
:
X
2
→
[
0
;
∞
)
be of the form
(67)
p
u
,
w
=
0
if
A
∩
u
,
w
=
u
,
w
,
γ
if
A
∩
u
,
w
≠
u
,
w
.
(
1
) A pair
(
X
,
P
{
1
}
;
{
1
}
)
,
P
{
1
}
;
{
1
}
=
{
p
}
, is the quasi-triangular space. Indeed, formula (67) yields
∀
u
,
v
,
w
∈
X
{
q
(
u
,
w
)
≤
q
(
u
,
v
)
+
q
(
v
,
w
)
}
. Otherwise,
∃
u
0
,
v
0
,
w
0
∈
X
{
q
(
u
0
,
w
0
)
>
q
(
u
0
,
v
0
)
+
q
(
v
0
,
w
0
)
}
. It is clear that then
q
(
u
0
,
w
0
)
=
γ
,
q
(
u
0
,
v
0
)
=
0
, and
q
(
v
0
,
w
0
)
=
0
. From this we see that
A
∩
{
u
0
,
w
0
}
≠
{
u
0
,
w
0
}
,
A
∩
{
u
0
,
v
0
}
=
{
u
0
,
v
0
}
, and
A
∩
{
v
0
,
w
0
}
=
{
v
0
,
w
0
}
. This is impossible.
(
2
)
P
{
1
}
;
{
1
}
=
{
p
}
does not vanish on the diagonal. Indeed, if
u
∈
X
∖
A
, then
p
(
u
,
u
)
=
γ
≠
0
. Therefore, the condition
∀
u
∈
X
{
p
(
u
,
u
)
=
0
}
does not hold.
(
3
)
P
{
1
}
;
{
1
}
=
{
p
}
is symmetric. This follows from (67).
(
4
) We observe that
L
I
M
(
u
m
:
m
∈
N
)
L
-
P
{
1
}
;
{
1
}
=
L
I
M
(
u
m
:
m
∈
N
)
R
-
P
{
1
}
;
{
1
}
=
A
for each sequence
(
u
m
:
m
∈
N
)
⊂
A
. We conclude this from (67).
Example 3.
Let
X
=
[
0
;
6
]
and let
p
:
X
2
→
[
0
;
∞
)
be of the form
(68)
p
u
,
w
=
0
if
u
≥
w
,
w
-
u
3
if
u
<
w
.
(
1
)
(
X
,
P
{
4
}
;
{
1
}
)
,
P
{
4
}
;
{
1
}
=
{
p
}
, is the quasi-triangular space. In fact,
∀
u
,
v
,
w
∈
X
{
q
(
u
,
w
)
≤
4
[
q
(
u
,
v
)
+
q
(
v
,
w
)
]
}
holds. This is a consequence of Cases 1–3.
Case 1. If
v
≤
u
<
w
, then
p
(
u
,
v
)
=
0
,
w
-
u
≤
w
-
v
, and, consequently,
p
(
u
,
w
)
=
w
-
u
3
≤
w
-
v
3
<
4
w
-
v
3
=
4
p
(
v
,
w
)
=
4
[
p
(
u
,
v
)
+
p
(
v
,
w
)
]
.
Case 2. If
u
<
w
and
u
≤
v
≤
w
, then
q
(
u
,
w
)
=
(
w
-
u
)
3
and
f
(
v
0
)
=
min
u
⩽
v
⩽
w
f
(
v
)
=
w
-
u
3
where
v
0
=
(
u
+
w
)
/
2
is a minimum point of the map
f
(
v
)
=
4
[
p
(
u
,
v
)
+
p
(
v
,
w
)
]
=
4
(
w
-
u
)
[
w
2
+
w
u
+
u
2
+
3
v
2
-
3
v
(
w
+
u
)
]
.
Case 3. If
u
<
w
≤
v
, then
p
(
v
,
w
)
=
0
and, consequently,
p
(
u
,
w
)
=
w
-
u
3
≤
v
-
u
3
<
4
v
-
u
3
=
4
p
(
u
,
v
)
=
4
[
p
(
u
,
v
)
+
p
(
v
,
w
)
]
.
(
2
)
P
{
4
}
;
{
1
}
=
{
p
}
is asymmetric. Indeed, we have that
0
=
p
(
6,0
)
≠
p
(
0,6
)
=
216
. Therefore, condition
∀
u
,
w
∈
X
{
p
(
u
,
w
)
=
p
(
w
,
u
)
}
does not hold.
(
3
)
P
{
4
}
;
{
1
}
=
{
p
}
vanishes on the diagonal. In fact, by (68), it is clear that
∀
u
∈
X
{
p
(
u
,
u
)
=
0
}
.
(
4
) We observe that
L
I
M
(
u
m
:
m
∈
N
)
L
-
P
{
4
}
;
{
1
}
=
[
2
;
6
]
and
L
I
M
(
u
m
:
m
∈
N
)
R
-
P
{
4
}
;
{
1
}
=
[
1
;
2
]
for sequence
(
u
m
=
2
:
m
∈
N
)
. We conclude this from (68).
Example 4.
Let
X
=
[
0
;
6
]
and let
P
{
2
}
;
{
1
}
=
{
p
}
where
p
:
X
2
→
[
0
;
∞
)
is of the form
(69)
p
u
,
w
=
0
if
u
≥
w
,
u
-
w
2
if
u
<
w
.
Let
(70)
E
=
0
;
3
∪
3
;
6
and let
μ
≥
36
/
4
and
J
{
2
}
;
{
1
}
=
{
J
}
where
J
:
X
2
→
[
0
;
∞
)
is of the form
(71)
J
u
,
w
=
p
u
,
w
if
E
∩
u
,
w
=
u
,
w
,
μ
if
E
∩
u
,
w
≠
u
,
w
.
(
1
)
J
{
2
}
;
{
1
}
is not symmetric. In fact, by (69)–(71),
J
(
0,6
)
=
36
and
J
(
6,0
)
=
0
.
(
2
)
J
{
2
}
;
{
1
}
=
{
J
}
∈
J
(
X
,
P
{
2
}
;
{
1
}
)
L
∩
J
(
X
,
P
{
2
}
;
{
1
}
)
R
. See Theorem 14.
Remark 33.
By Examples 1–4 it follows that the distances
p
defined by (65) and (67)–(69) and
J
defined by (70) and (71) are not metrics, ultra metrics, quasi-metrics, ultra quasi-metrics,
b
-metrics, partial metrics, partial
b
-metrics, pseudometrics (gauges), quasi-pseudometrics (quasi-gauges), and ultra quasi-pseudometrics (ultra quasi-gauges).
11. Examples Illustrating Theorem 26
Example 1.
Let
X
=
[
0
;
6
]
, let
γ
>
2048
be arbitrary and fixed, and, for
u
,
w
∈
X
, let
(72)
p
u
,
w
=
0
if
u
≥
w
,
u
,
w
∩
0
;
6
=
u
,
w
,
w
-
u
4
if
u
<
w
,
u
,
w
∩
0
;
6
=
u
,
w
,
γ
if
u
,
w
∩
0
;
6
≠
u
,
w
.
Define the set-valued dynamic system
(
X
,
T
)
by
(73)
T
u
=
1
;
2
if
u
∈
0
;
3
∪
4
;
6
,
4
;
6
if
u
∈
3
;
4
.
Let
(74)
E
=
0
;
3
∪
4
;
6
and let
J
:
X
×
X
→
[
0
;
∞
)
be of the form
(75)
J
u
,
w
=
p
u
,
w
if
E
∩
u
,
w
=
u
,
w
,
γ
if
E
∩
u
,
w
≠
u
,
w
.
(
1
)
(
X
,
P
{
8
}
;
{
1
}
)
, where
P
{
8
}
;
{
1
}
=
{
p
}
, is the quasi-triangular space, and
J
{
8
}
;
{
1
}
=
{
J
}
is the left and right family generated by
P
{
8
}
;
{
1
}
. This is a consequence of Definitions 7 and 11, Example 1, and Theorem 14; we see that
γ
=
μ
>
81
.
(
2
)
(
X
,
T
)
is a
(
D
=
D
1
;
2
X
L
-
J
{
8
}
;
{
1
}
=
D
1
;
2
X
R
-
J
{
8
}
;
{
1
}
,
λ
∈
[
2048
/
γ
;
1
)
)
-quasi-contraction on
X
; that is,
∀
λ
∈
[
2048
/
γ
;
1
)
∀
x
,
y
∈
X
{
8
·
D
(
T
(
x
)
,
T
(
y
)
)
≤
λ
J
(
x
,
y
)
}
where
(76)
D
U
,
W
=
max
s
u
p
u
∈
U
J
u
,
W
,
s
u
p
w
∈
W
J
U
,
w
,
U
,
W
∈
2
X
.
Indeed, we see that this follows from (73)–(76) and from Cases 1–4 below.
Case 1. If
x
,
y
∈
[
0
;
3
)
∪
(
4
;
6
]
, then
T
(
x
)
=
T
(
y
)
=
[
1
;
2
]
=
U
⊂
E
and
sup
u
∈
U
{
inf
w
∈
U
J
(
u
,
w
)
}
=
sup
u
∈
U
{
J
(
u
,
u
)
=
p
(
u
,
u
)
=
0
}
=
0
. Thus
4
D
(
T
(
x
)
,
T
(
y
)
)
=
0
≤
λ
J
(
x
,
y
)
.
Case 2. If
x
∈
[
0
;
3
)
∪
(
4
;
6
]
and
y
∈
[
3
;
4
]
, then
T
(
x
)
=
[
1
;
2
]
=
U
⊂
E
,
T
(
y
)
=
(
4
;
6
)
=
W
⊂
E
, and
sup
u
∈
U
{
i
n
f
w
∈
W
J
(
u
,
w
)
}
=
sup
u
∈
U
{
i
n
f
w
∈
W
w
-
u
4
}
=
sup
u
∈
U
4
-
u
4
=
81
and
sup
w
∈
W
{
inf
u
∈
U
J
(
u
,
w
)
}
=
sup
w
∈
W
{
i
n
f
u
∈
U
w
-
u
4
}
=
sup
w
∈
W
w
-
2
4
=
256
. Thus
8
D
(
T
(
x
)
,
T
(
y
)
)
=
2048
. On the other hand,
y
∉
E
which gives
J
(
x
,
y
)
=
γ
. Therefore,
8
D
(
T
(
x
)
,
T
(
y
)
)
≤
λ
J
(
x
,
y
)
whenever
2048
≤
λ
γ
. This gives
2048
/
γ
≤
λ
<
1
whenever
γ
>
max
{
2048
;
81
}
.
Case 3. If
x
∈
[
3
;
4
]
and
y
∈
[
0
;
3
)
∪
(
4
;
6
]
, then
T
(
x
)
=
(
4
;
6
)
=
U
⊂
E
and
T
(
y
)
=
[
1
;
2
]
=
W
⊂
E
. Hence we obtain
sup
u
∈
U
{
inf
w
∈
W
J
(
u
,
w
)
}
=
sup
u
∈
U
{
i
n
f
w
∈
W
p
(
u
,
w
)
}
=
sup
w
∈
W
{
i
n
f
u
∈
U
J
(
u
,
w
)
}
=
sup
w
∈
W
{
i
n
f
u
∈
U
p
(
u
,
w
)
}
=
0
. Therefore,
8
D
(
T
(
x
)
,
T
(
y
)
)
=
0
≤
λ
J
(
x
,
y
)
.
Case 4. If
x
,
y
∈
[
3
;
4
]
, then
T
(
x
)
=
T
(
y
)
=
(
4
;
6
)
=
U
⊂
E
. Therefore
4
D
1
(
T
(
x
)
,
T
(
y
)
)
=
0
≤
λ
J
(
x
,
y
)
.
(
3
)
Property (26) holds; that is,
∀
x
∈
X
∀
β
∈
(
0
;
∞
)
∃
y
∈
T
(
x
)
{
J
(
x
,
y
)
<
J
(
x
,
T
(
x
)
)
+
β
}
. Indeed, this follows from Cases 1–4 below.
Case 1. If
x
0
=
0
and
y
0
=
1
∈
T
(
x
0
)
=
[
1
;
2
]
, then
J
(
x
0
,
y
0
)
=
γ
,
J
(
x
0
,
T
(
x
0
)
)
=
i
n
f
w
∈
[
1
;
2
]
J
(
x
0
,
w
)
=
γ
, and
∀
β
∈
(
0
;
∞
)
{
J
(
x
0
,
y
0
)
<
J
(
x
0
,
T
(
x
0
)
)
+
β
}
.
Case 2. If
x
0
∈
(
0
;
1
]
and
y
0
=
1
∈
T
(
x
0
)
=
[
1
;
2
]
, then
J
(
x
0
,
y
0
)
=
1
-
x
0
,
J
(
x
0
,
T
(
x
0
)
)
=
inf
w
∈
1
;
2
J
(
x
0
,
w
)
=
1
-
x
0
, and
∀
β
∈
(
0
;
∞
)
{
J
(
x
0
,
y
0
)
<
J
(
x
0
,
T
(
x
0
)
)
+
β
}
.
Case 3. If
x
0
∈
(
1
;
3
)
∪
(
4
;
6
)
and
y
0
=
1
∈
T
(
x
0
)
=
[
1
;
2
]
, then
J
(
x
0
,
y
0
)
=
0
,
J
(
x
0
,
T
(
x
0
)
)
=
0
, and
∀
β
∈
(
0
;
∞
)
{
J
(
x
0
,
y
0
)
<
J
(
x
0
,
T
(
x
0
)
)
+
β
}
.
Case 4. If
x
0
∈
[
3
;
4
]
and
y
0
∈
T
(
x
0
)
=
(
4
;
6
)
, then
J
(
x
0
,
y
0
)
=
γ
,
p
(
x
0
,
T
(
x
0
)
)
=
γ
, and
∀
β
∈
(
0
;
∞
)
{
J
(
x
0
,
y
0
)
<
J
(
x
0
,
T
(
x
0
)
)
+
β
}
.
Case 5. If
x
0
=
6
and
y
0
∈
T
(
x
0
)
=
[
1
;
2
]
, then
J
(
x
0
,
y
0
)
=
γ
,
p
(
x
0
,
T
(
x
0
)
)
=
γ
, and
∀
β
∈
(
0
;
∞
)
{
J
(
x
0
,
y
0
)
<
J
(
x
0
,
T
(
x
0
)
)
+
β
}
.
(
4
)
(
X
,
T
)
is left and right
J
{
8
}
;
{
1
}
-admissible in each point
w
0
∈
X
. In fact, if
w
0
∈
X
and
(
w
m
:
m
∈
{
0
}
∪
N
)
are such that
∀
m
∈
{
0
}
∪
N
{
w
m
+
1
∈
T
(
w
m
)
}
and
l
i
m
m
→
∞
sup
n
>
m
J
(
w
m
,
w
n
)
=
0
(
l
i
m
m
→
∞
sup
n
>
m
J
(
w
n
,
w
m
)
=
0
)
, then
∀
m
≥
2
{
w
m
∈
[
1
;
2
]
}
and, consequently, by (72),
∀
w
∈
[
2
;
6
)
⊂
X
{
l
i
m
m
→
∞
p
(
w
,
w
m
)
=
0
}
(
∀
w
∈
(
0
;
1
]
⊂
X
{
l
i
m
m
→
∞
p
(
w
m
,
w
)
=
0
}
)
. Hence, by (75) and (76), we get
∀
w
∈
[
2
;
3
)
∪
(
4
;
6
)
⊂
X
{
l
i
m
m
→
∞
J
(
w
,
w
m
)
=
0
}
(
∀
w
∈
(
0
;
1
]
⊂
X
{
l
i
m
m
→
∞
J
(
w
m
,
w
)
=
0
}
)
.
(
5
)
(
X
,
T
)
is a left and right
P
{
8
}
;
{
1
}
-closed on
X
. Indeed, let
(
x
m
:
m
∈
N
)
⊂
T
(
X
)
be a left (right)
P
{
8
}
;
{
1
}
-converging sequence in
X
(thus
L
I
M
(
x
m
:
m
∈
N
)
L
-
P
{
8
}
;
{
1
}
≠
⌀
(
L
I
M
(
x
m
:
m
∈
N
)
R
-
P
{
8
}
;
{
1
}
≠
⌀
)
)
and having subsequences
(
v
m
:
m
∈
N
)
and
(
u
m
:
m
∈
N
)
satisfying
∀
m
∈
N
{
v
m
∈
T
(
u
m
)
}
. Then
∀
m
≥
2
{
x
m
∈
[
1
;
2
]
}
,
2
∈
T
(
2
)
and
2
∈
L
I
M
(
x
m
:
m
∈
N
)
L
-
P
{
8
}
;
{
1
}
(
1
∈
T
(
1
)
and
1
∈
L
I
M
(
x
m
:
m
∈
N
)
R
-
P
{
8
}
;
{
1
}
)
.
(
6
)
All assumptions of Theorem 26 are satisfied. This follows from (1)–(5) in Example 1.
We conclude that
Fix
(
T
)
=
[
1
;
2
]
and we have shown the following.
Claim A.
2
∈
T
(
2
)
and
2
∈
L
I
M
(
w
m
:
m
∈
{
0
}
∪
N
)
L
-
P
{
8
}
;
{
1
}
for each
w
0
∈
X
and for each dynamic process
(
w
m
:
m
∈
{
0
}
∪
N
)
of the system
(
X
,
T
)
.
Claim B.
1
∈
T
(
1
)
and
1
∈
L
I
M
(
w
m
:
m
∈
{
0
}
∪
N
)
R
-
P
{
8
}
;
{
1
}
for each
w
0
∈
X
and for each dynamic process
(
w
m
:
m
∈
{
0
}
∪
N
)
of the system
(
X
,
T
)
.
Example 2.
Let
X
,
P
{
8
}
;
{
1
}
=
{
p
}
, and
(
X
,
T
)
be such as in Example 1.
(
1
)
For each
λ
∈
[
0
;
1
)
, condition
∀
x
,
y
∈
X
{
8
D
(
T
(
x
)
,
T
(
y
)
)
≤
λ
p
(
x
,
y
)
}
, where
D
(
U
,
W
)
=
max
{
sup
u
∈
U
p
u
,
W
,
sup
w
∈
W
p
(
U
,
w
)
}
,
U
,
W
∈
2
X
, does not hold. Suppose that
∃
λ
0
∈
[
0
;
1
)
∀
x
,
y
∈
X
{
8
D
(
T
(
x
)
,
T
(
y
)
)
≤
λ
0
p
(
x
,
y
)
}
. Letting
x
0
=
2
and
y
0
=
3
, it can be shown that
p
(
x
0
,
y
0
)
=
1
,
T
(
x
0
)
=
[
1
;
2
]
=
U
,
T
(
y
0
)
=
(
4
;
6
)
=
W
,
sup
u
∈
1
;
2
p
(
u
,
(
4
;
6
)
)
=
sup
u
∈
1
;
2
4
-
u
4
=
3
4
=
81
, and
sup
w
∈
4
;
6
p
(
[
1
;
2
]
,
w
)
=
sup
w
∈
4
;
6
w
-
2
4
=
4
4
=
256
. Therefore
2048
=
8
D
(
T
(
x
0
)
,
T
(
y
0
)
)
=
8
max
{
81
;
256
}
≤
λ
0
p
(
x
0
,
y
0
)
=
λ
0
, which is absurd.
Remark 34.
We make the following remarks about Examples 1 and 2 and Theorem 26: (a) By Example 1, we observe that we may apply Theorem 26 for set-valued dynamic systems
(
X
,
T
)
in the left and right quasi-triangular space
(
X
,
P
C
;
A
)
with left and right family
J
C
;
A
generated by
P
C
;
A
where
J
C
;
A
≠
P
C
;
A
. (b) By Example 2, we note, however, that we do not apply Theorem 26 in the quasi-triangular space
(
X
,
P
C
;
A
)
when
J
C
;
A
=
P
C
;
A
. (c) From (a) and (b) it follows that, in Theorem 26, the existence of left (right) families
J
C
;
A
generated by
P
C
;
A
and such that
J
C
;
A
≠
P
C
;
A
are essential.
Example 3.
Let
X
=
(
0
;
6
)
,
γ
>
0
, and
(77)
A
=
A
1
∪
A
2
,
A
1
=
0
;
2
,
A
2
=
4
;
6
.
Let
p
:
X
2
→
[
0
;
∞
)
be of the form
(78)
p
u
,
w
=
0
if
A
∩
u
,
w
=
u
,
w
,
γ
if
A
∩
u
,
w
≠
u
,
w
,
and let
J
{
1
}
;
{
1
}
=
P
{
1
}
;
{
1
}
=
{
p
}
. Define the set-valued dynamic system
(
X
,
T
)
by
(79)
T
u
=
A
2
for
u
∈
0
;
3
,
A
for
u
=
3
,
A
1
for
u
∈
3
;
6
.
(
1
)
(
X
,
P
{
1
}
;
{
1
}
)
is quasi-triangular space. See Example 2, Section 11.
(
2
)
(
X
,
T
)
is a
(
D
1
;
2
X
L
-
P
{
1
}
;
{
1
}
,
λ
∈
[
0
;
1
)
)
-quasi-contraction on
X
; that is,
∀
λ
∈
[
0
;
1
)
∀
x
,
y
∈
X
{
D
1
;
2
X
L
-
P
{
1
}
;
{
1
}
(
T
(
x
)
,
T
(
y
)
)
≤
λ
p
(
x
,
y
)
}
. Indeed, if
x
,
y
∈
X
, then, by (77)–(79),
T
(
x
)
,
T
(
y
)
⊂
A
and
max
{
sup
u
∈
T
x
p
u
,
T
y
,
sup
w
∈
T
y
p
(
T
(
x
)
,
w
)
}
=
0
.
(
3
)
Property (16) holds; that is,
∀
x
∈
X
∀
β
∈
(
0
;
∞
)
∃
y
∈
T
(
x
)
{
p
(
x
,
y
)
<
p
(
x
,
T
(
x
)
)
+
β
}
. Indeed, this follows from Cases 1–3 below.
Case 1. Let
x
0
∈
(
0
;
3
)
and
β
∈
(
0
;
∞
)
be arbitrary and fixed. If
y
0
∈
T
(
x
0
)
=
A
2
, then, by (78),
(80)
p
x
0
,
y
0
=
p
x
0
,
T
x
0
=
0
if
x
0
∈
A
1
,
γ
for
x
0
∈
0
;
3
∖
A
1
.
Therefore,
p
(
x
0
,
y
0
)
<
p
(
x
0
,
T
(
x
0
)
)
+
β
.
Case 2. Let
x
0
=
3
and let
β
∈
(
0
;
∞
)
be arbitrary and fixed. If
y
0
∈
T
(
x
0
)
=
A
, then, by (78),
p
(
x
0
,
y
0
)
=
p
(
x
0
,
T
(
x
0
)
)
=
γ
. Therefore,
p
(
x
0
,
y
0
)
<
p
(
x
0
,
T
(
x
0
)
)
+
β
.
Case 3. Let
x
0
∈
(
3
;
6
)
and
β
∈
(
0
;
∞
)
be arbitrary and fixed. If
y
0
∈
T
(
x
0
)
=
A
1
, then, by (78),
(81)
p
x
0
,
y
0
=
p
x
0
,
T
x
0
=
0
if
x
0
∈
A
2
,
γ
for
x
0
∈
3
;
6
∖
A
2
.
Therefore,
p
(
x
0
,
y
0
)
<
p
(
x
0
,
T
(
x
0
)
)
+
β
.
(
4
)
(
X
,
T
)
is left and right
P
{
1
}
;
{
1
}
-admissible in
X
. Assuming that
w
0
∈
X
is arbitrary and fixed we prove that if the dynamic process
(
w
m
:
m
∈
{
0
}
∪
N
)
of
(
X
,
T
)
starting at
w
0
is such that
l
i
m
m
→
∞
s
u
p
n
>
m
p
(
w
m
,
w
n
)
=
0
, then
∃
w
∈
X
{
l
i
m
m
→
∞
p
(
w
,
w
m
)
=
0
}
. Indeed, if
w
0
∈
X
, then, by (79),
∀
m
⩾
1
{
w
m
∈
T
(
w
m
-
1
)
⊂
A
}
and, by (78), we immediately get
A
=
L
I
M
(
w
m
:
m
∈
{
0
}
∪
N
)
L
-
P
{
1
}
;
{
1
}
=
L
I
M
(
w
m
:
m
∈
{
0
}
∪
N
)
R
-
P
{
1
}
;
{
1
}
.
(
5
)
Set-valued dynamic system
(
X
,
T
[
2
]
)
is a left and right
P
{
1
}
;
{
1
}
-closed on
X
. Indeed, if
(
x
m
:
m
∈
N
)
⊂
T
[
2
]
(
X
)
=
A
is a left or right
P
{
1
}
;
{
1
}
-converging sequence in
X
and having subsequences
(
v
m
:
m
∈
N
)
and
(
u
m
:
m
∈
N
)
satisfying
∀
m
∈
N
{
v
m
∈
T
(
u
m
)
}
, then, by (77)–(79), we have that
∃
m
0
∈
N
∀
m
⩾
m
0
{
x
m
∈
A
}
,
A
=
L
I
M
(
x
m
:
m
∈
N
)
L
-
P
{
1
}
;
{
1
}
=
L
I
M
(
x
m
:
m
∈
{
0
}
∪
N
)
R
-
P
{
1
}
;
{
1
}
, and
Fix
(
T
[
2
]
)
=
A
.
(
6
)
For
(
X
,
P
{
1
}
;
{
1
}
)
,
P
{
1
}
;
{
1
}
=
{
p
}
,
J
{
1
}
;
{
1
}
=
P
{
1
}
;
{
1
}
, and
(
X
,
T
)
defined by (77)–(79), all assumptions of Theorem 26 are satisfied. This follows from (1)–(5) in Example 3.
We conclude that
Fix
(
T
[
2
]
)
=
A
and we claim that if
w
0
∈
X
,
w
1
∈
T
(
w
0
)
, and
w
2
=
u
∈
T
(
w
1
)
are arbitrary and fixed, and
∀
m
⩾
3
{
w
m
=
u
}
, then sequence
(
w
m
:
m
∈
{
0
}
∪
N
)
is a dynamic process of
T
starting at
w
0
and left and right
P
{
1
}
;
{
1
}
-converging to each point of
A
. We observe also that
Fix
(
T
)
=
⌀
.
Example 4.
Let
X
=
[
0
;
6
]
and let
P
{
2
}
;
{
1
}
=
{
p
}
where
p
:
X
2
→
[
0
;
∞
)
is of the form
(82)
p
u
,
w
=
0
if
u
≥
w
,
u
-
w
2
if
u
<
w
.
Define the set-valued dynamic system
(
X
,
T
)
by
(83)
T
u
=
0
;
3
∪
3
;
6
∖
u
for
u
∈
0
;
6
.
Let
(84)
E
=
0
;
3
∪
3
;
6
and let
μ
≥
36
/
4
and
J
{
2
}
;
{
1
}
=
{
J
}
where
J
:
X
2
→
[
0
;
∞
)
is of the form
(85)
J
u
,
w
=
p
u
,
w
if
E
∩
u
,
w
=
u
,
w
,
μ
if
E
∩
u
,
w
≠
u
,
w
.
(
1
)
J
{
2
}
;
{
1
}
is not symmetric. In fact, by (82), (84), and (85),
J
(
0,6
)
=
36
and
J
(
6,0
)
=
0
.
(
2
)
J
{
2
}
;
{
1
}
=
{
J
}
∈
J
(
X
,
P
{
2
}
;
{
1
}
)
L
∩
J
(
X
,
P
{
2
}
;
{
1
}
)
R
. See Theorem 14.
(
3
)
(
X
,
T
)
is a
(
D
=
D
1
;
2
X
L
-
J
{
2
}
;
{
1
}
,
λ
∈
[
0
;
1
)
)
-contraction on
X
; that is,
∀
x
,
y
∈
X
{
2
·
D
(
T
(
x
)
,
T
(
y
)
)
≤
λ
J
(
x
,
y
)
}
where
λ
∈
[
0
;
1
)
and
(86)
D
U
,
W
=
max
sup
u
∈
U
J
u
,
W
,
sup
w
∈
W
J
U
,
w
,
U
,
W
∈
2
X
.
Indeed, we see that this follows from (1), (2) in Example 4, and from Cases 1–4 below.
Case 1. Let
x
,
y
∈
[
0
;
3
)
∪
(
3
;
6
]
. Then
x
,
y
∈
E
,
T
(
x
)
=
(
[
0
;
3
)
∪
(
3
;
6
]
)
∖
{
x
}
=
U
⊂
E
, and
T
(
y
)
=
(
[
0
;
3
)
∪
(
3
;
6
]
)
∖
{
y
}
=
W
⊂
E
. If
u
∈
U
, then we have
W
=
W
u
∪
W
u
and
(87)
inf
w
∈
W
J
u
,
w
≤
inf
w
∈
W
u
q
u
,
w
=
0
if
W
u
=
w
∈
W
:
u
≥
w
≠
⌀
,
inf
w
∈
W
u
u
-
w
2
=
0
if
W
u
=
w
∈
W
:
u
<
w
≠
⌀
and if
w
∈
W
, then we have
U
=
U
w
∪
U
w
and
(88)
inf
u
∈
U
J
u
,
w
≤
inf
u
∈
U
w
q
u
,
w
=
0
if
U
u
=
u
∈
U
:
u
≥
w
≠
⌀
,
inf
u
∈
U
w
u
-
w
2
=
0
if
U
u
=
u
∈
U
:
u
<
w
≠
⌀
.
By (86),
2
D
(
T
(
x
)
,
T
(
y
)
)
=
0
≤
λ
J
(
x
,
y
)
.
Case 2. If
x
=
y
=
3
, then
J
(
x
,
y
)
=
μ
and
T
(
x
)
=
T
(
y
)
=
[
0
;
3
)
∪
(
3
;
6
]
=
U
⊂
E
. Therefore,
2
D
(
T
(
x
)
,
T
(
y
)
)
=
2
D
(
U
,
U
)
=
0
≤
λ
J
(
x
,
y
)
.
Case 3. If
x
∈
[
0
;
3
)
∪
(
3
;
6
]
and
y
=
3
, then
x
∈
E
,
y
∉
E
,
J
(
x
,
y
)
=
μ
,
T
(
x
)
=
(
[
0
;
3
)
∪
(
3
;
6
]
)
∖
{
x
}
=
U
⊂
E
, and
T
(
y
)
=
[
0
;
3
)
∪
(
3
;
6
]
=
W
⊂
E
. We see that
s
u
p
u
∈
U
{
i
n
f
w
∈
W
J
(
u
,
w
)
}
=
0
since if
u
∈
U
, then also
w
=
u
∈
W
and
i
n
f
w
∈
W
J
(
u
,
w
)
=
q
(
u
,
u
)
=
0
. Next, we see that
s
u
p
w
∈
W
{
i
n
f
u
∈
U
J
(
u
,
w
)
}
=
0
since if
w
∈
W
, then
U
=
U
w
∪
U
w
and
(89)
inf
u
∈
U
J
u
,
w
≤
inf
u
∈
U
w
q
u
,
w
=
0
if
U
u
=
u
∈
U
:
u
≥
w
≠
⌀
,
inf
u
∈
U
w
u
-
w
2
=
0
if
U
u
=
u
∈
U
:
u
<
w
≠
⌀
.
Thus
2
D
(
T
(
x
)
,
T
(
y
)
)
=
0
⩽
λ
J
(
x
,
y
)
.
Case 4. If
x
=
3
and
y
∈
[
0
;
3
)
∪
(
3
;
6
]
, then
x
∉
E
,
y
∈
E
,
J
(
x
,
y
)
=
μ
,
T
(
x
)
=
[
0
;
3
)
∪
(
3
;
6
]
=
U
⊂
E
,
T
(
y
)
=
(
[
0
;
3
)
∪
(
3
;
6
]
)
∖
{
y
}
=
W
⊂
E
, and
s
u
p
u
∈
U
{
i
n
f
w
∈
W
J
(
u
,
w
)
}
=
0
since, for
u
∈
U
,
(90)
inf
w
∈
W
J
u
,
w
≤
inf
w
∈
W
u
q
u
,
w
=
0
if
W
u
=
w
∈
W
:
u
≥
w
≠
⌀
,
inf
w
∈
W
u
u
-
w
2
=
0
if
W
u
=
w
∈
W
:
u
<
w
≠
⌀
and
s
u
p
w
∈
W
{
i
n
f
u
∈
U
J
(
u
,
w
)
}
=
0
since
i
n
f
u
∈
U
J
(
u
,
w
)
=
J
(
w
,
w
)
=
0
for
w
∈
W
. Thus
2
D
(
T
(
x
)
,
T
(
y
)
)
=
0
≤
λ
J
(
x
,
y
)
.
(4) Property (26) holds; that is,
∀
x
∈
X
∀
γ
∈
(
0
;
∞
)
∃
y
∈
T
(
x
)
{
J
(
x
,
y
)
<
J
(
x
,
T
(
x
)
)
+
γ
}
. Indeed, this follows from Cases 1–3 below.
Case 1. Let
x
0
∈
[
0
;
3
)
and
γ
∈
(
0
;
∞
)
be arbitrary and fixed. If
y
0
∈
T
(
x
0
)
=
(
[
0
;
3
)
∪
(
3
;
6
]
)
∖
{
x
0
}
=
W
is such that
x
0
<
y
0
<
3
, then
J
(
x
0
,
y
0
)
=
(
x
0
-
y
0
)
2
and
J
(
x
0
,
T
(
x
0
)
)
=
i
n
f
w
∈
W
J
(
x
0
,
w
)
=
0
since
(91)
inf
w
∈
W
J
x
0
,
w
≤
inf
w
∈
W
x
0
q
x
0
,
w
=
0
if
W
x
0
=
w
∈
W
:
x
0
≥
w
≠
⌀
,
inf
w
∈
W
x
0
x
0
-
w
2
=
0
if
W
x
0
=
w
∈
W
:
x
0
<
w
≠
⌀
.
Then we see that
J
(
x
0
,
y
0
)
=
(
x
0
-
y
0
)
2
<
γ
implies
y
0
<
x
0
+
γ
1
/
2
. From this we conclude that if
y
0
∈
(
x
0
;
m
i
n
{
3
,
x
0
+
γ
1
/
2
}
)
, then
J
(
x
0
,
y
0
)
<
J
(
x
0
,
T
(
x
0
)
)
+
γ
.
Case 2. Let
x
0
=
3
. Assume that
y
0
∈
T
(
x
0
)
=
[
0
;
3
)
∪
(
3
;
6
]
is arbitrary and fixed. Then
J
(
x
0
,
y
0
)
=
μ
,
J
(
x
0
,
T
(
x
0
)
)
=
i
n
f
w
∈
[
0
;
3
)
∪
(
3
;
6
]
J
(
x
0
,
w
)
=
μ
and, for each
γ
∈
(
0
;
∞
)
,
J
(
x
0
,
y
0
)
<
J
(
x
0
,
T
(
x
0
)
)
+
γ
.
Case 3. Let
x
0
∈
(
3
;
6
]
and
γ
∈
(
0
;
∞
)
be arbitrary and fixed. If
y
0
∈
T
(
x
0
)
=
(
[
0
;
3
)
∪
(
3
;
6
]
)
∖
{
x
0
}
=
W
is such that
3
<
y
0
<
x
0
, then
J
(
x
0
,
y
0
)
=
0
and, analogously as in Case 1, we get
J
(
x
0
,
T
(
x
0
)
)
=
i
n
f
w
∈
W
J
(
x
0
,
w
)
=
0
. Therefore,
J
(
x
0
,
y
0
)
<
J
(
x
0
,
T
(
x
0
)
)
+
γ
.
(
5
)
(
X
,
T
)
is left
J
{
2
}
;
{
1
}
-admissible in
X
. Assuming that
w
0
∈
X
is arbitrary and fixed we prove that if the dynamic process
(
w
m
:
m
∈
{
0
}
∪
N
)
of
(
X
,
T
)
starting at
w
0
is such that
l
i
m
m
→
∞
s
u
p
n
>
m
J
(
w
m
,
w
n
)
=
0
, then
∃
w
∈
X
{
l
i
m
m
→
∞
J
(
w
,
w
m
)
=
0
}
. We consider the following cases.
Case 1. If
w
0
∈
[
0
;
3
)
∪
(
3
;
6
]
, then
w
1
∈
T
(
w
0
)
=
(
[
0
;
3
)
∪
(
3
;
6
]
)
∖
{
w
0
}
and
∀
m
≥
2
{
w
m
∈
T
(
w
m
-
1
)
⊂
[
0
;
3
)
∪
(
3
;
6
]
}
and using (82) we immediately get
6
∈
LI
M
(
w
m
:
m
∈
{
0
}
∪
N
)
L
-
J
{
2
}
;
{
1
}
.
Case 2. If
w
0
=
3
, then
w
1
∈
T
(
w
0
)
=
[
0
;
3
)
∪
(
3
;
6
]
,
w
2
∈
T
(
w
1
)
=
(
[
0
;
3
)
∪
(
3
;
6
]
)
∖
{
w
1
}
, and
∀
m
⩾
3
{
w
m
∈
T
(
w
m
-
1
)
⊂
[
0
;
3
)
∪
(
3
;
6
]
}
and using (82) we also immediately get
6
∈
LI
M
(
w
m
:
m
∈
{
0
}
∪
N
)
L
-
J
{
2
}
;
{
1
}
.
This shows that
6
∈
LI
M
(
w
m
:
m
∈
{
0
}
∪
N
)
L
-
J
{
2
}
;
{
1
}
for each
w
0
∈
X
and for each dynamic process
(
w
m
:
m
∈
{
0
}
∪
N
)
of the system
(
X
,
T
)
; we see that here property
l
i
m
m
→
∞
s
u
p
n
>
m
J
(
w
m
,
w
n
)
=
0
of
(
w
m
:
m
∈
{
0
}
∪
N
)
is not required.
(
6
) Set-valued dynamic system
(
X
,
T
[
2
]
)
is a left
P
{
2
}
;
{
1
}
-quasi-closed on
X
. Indeed, if
(
x
m
:
m
∈
N
)
⊂
T
[
2
]
(
X
)
=
[
0
;
3
)
∪
(
3
;
6
]
is a left
P
{
2
}
;
{
1
}
-converging sequence in
X
and having subsequences
(
v
m
:
m
∈
N
)
and
(
u
m
:
m
∈
N
)
satisfying
∀
m
∈
N
{
v
m
∈
T
(
u
m
)
}
, then, by (83), we have that
∃
m
0
∈
N
∀
m
≥
m
0
{
x
m
∈
[
0
;
3
)
∪
(
3
;
6
]
}
. Therefore, in particular,
6
∈
LI
M
(
x
m
:
m
∈
N
)
L
-
P
{
2
}
;
{
1
}
and
6
∈
T
[
2
]
(
6
)
.
(
7
) For
P
{
2
}
;
{
1
}
=
{
p
}
,
J
{
2
}
;
{
1
}
=
{
J
}
, and
(
X
,
T
)
defined by (82)–(85), all assumptions of Theorem 26 in the case of “left” are satisfied. This follows from (1)–(6) in Example 4.
We conclude that
F
i
x
(
T
[
2
]
)
=
[
0
;
3
)
∪
(
3
;
6
]
and we claim that
6
∈
T
[
2
]
(
6
)
and that
6
∈
LI
M
(
w
m
:
m
∈
{
0
}
∪
N
)
L
-
P
{
2
}
;
{
1
}
for each
w
0
∈
X
and for each dynamic process
(
w
m
:
m
∈
{
0
}
∪
N
)
of the system
(
X
,
T
)
. We observe also that
F
i
x
(
T
)
=
⌀
.