# Quasi-Triangular Spaces, Pompeiu-Hausdorff Quasi-Distances, and Periodic and Fixed Point Theorems of Banach and Nadler Types

Authors: Kazimierz Włodarczyk
Journal: Abstract and Applied Analysis
 (2015)

---

## Abstract

LetC
=
{
C
α
}
α
∈
A
∈
[
1
;
∞
)
A, A-index set. A quasi-triangular space (
X
,
P
C
;
A
) is a set X with family P
C
;
A
=
{
p
α
:
X
2
→
[
0
,
∞
)
,
α
∈
A
} satisfying ∀
α
∈
A
∀
u
,
v
,
w
∈
X
{
p
α
(
u
,
w
)
≤
C
α
[
p
α
(
u
,
v
)
+
p
α
(
v
,
w
)
]
}. For any P
C
;
A, a left (right) family J
C
;
A generated by P
C
;
A is defined to be J
C
;
A
=
{
J
α
:
X
2
→
[
0
,
∞
)
,
α
∈
A
}, where ∀
α
∈
A
∀
u
,
v
,
w
∈
X
{
J
α
(
u
,
w
)
≤
C
α
[
J
α
(
u
,
v
)
+
J
α
(
v
,
w
)
]
} and furthermore the property ∀
α
∈
A
{
lim
m
→
∞
p
α
(
w
m
,
u
m
)
=
0
}  (
∀
α
∈
A
{
lim
m
→
∞
p
α
(
u
m
,
w
m
)
=
0
}
) holds whenever two sequences (
u
m
:
m
∈
N
) and (
w
m
:
m
∈
N
) in X satisfy ∀
α
∈
A
{
lim
m
→
∞
sup
n
>
m
J
α
(
u
m
,
u
n
)
=
0 and lim
m
→
∞
J
α
(
w
m
,
u
m
)
=
0
}  (
∀
α
∈
A
{
lim
m
→
∞
sup
n
>
m
J
α
(
u
n
,
u
m
)
=
0 and lim
m
→
∞
J
α
(
u
m
,
w
m
)
=
0
}
). In (
X
,
P
C
;
A
), using the left (right) families J
C
;
A generated by P
C
;
A (P
C
;
A is a special case of J
C
;
A), we construct three types of Pompeiu-Hausdorff left (right) quasi-distances on 2
X; for each type we construct of left (right) set-valued quasi-contraction T
:
X
→
2
X, and we prove the convergence, existence, and periodic point theorem for such quasi-contractions. We also construct two types of left (right) single-valued quasi-contractions T
:
X
→
X and we prove the convergence, existence, approximation, uniqueness, periodic point, and fixed point theorem for such quasi-contractions. (X
,
P
C
;
A) generalize ultra quasi-triangular and partiall quasi-triangular spaces (in particular, generalize metric, ultra metric, quasi-metric, ultra quasi-metric, b-metric, partial metric, partial b-metric, pseudometric, quasi-pseudometric, ultra quasi-pseudometric, partial quasi-pseudometric, topological, uniform, quasi-uniform, gauge, ultra gauge, partial gauge, quasi-gauge, ultra quasi-gauge, and partial quasi-gauge spaces).


---

## Body

## 1. Introduction

Theset-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 allfixed points andperiodic 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
}. Adynamic process or atrajectory starting at
w
0
∈
X or amotion 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 thata 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 allfixed points andperiodic 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 aPicard iteration starting at
w
0 of the system (
X
,
T
).LetX be a (nonempty) set. Adistance on X is a map p
:
X
2
→
[
0
;
∞
). The set X, together with distances on X, is calleddistance spaces.The following distance spaces are important for several reasons.Definition 1.
LetX 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 calledultra 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 calledpartial 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 calledpartial
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 calledquasi-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 calledultra 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 distancep is calledpseudometric (or thegauge) 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 distancep is calledquasi-pseudometric (or thequasi-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 calledultra quasi-pseudometric (or theultra 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]).
LetX be a (nonempty) set, and let A be an index set.
(A) Each familyD
=
{
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 familyD
=
{
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 topologyinduced 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 agauge space and is denoted by (
X
,
D
).Definition 3 (see [13]).
LetX be a (nonempty) set, and let A be an index set.
(A) Each familyP
=
{
p
α
,
α
∈
A
} of quasi-pseudometrics p
α
:
X
2
→
0
,
∞
,
α
∈
A, is calledquasi-gauge on X.
(B) Let the familyP
=
{
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 topologyinduced 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 calledquasi-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.ThePompeiu-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]).
LetX
,
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 thequasi-triangular spaces
(
X
,
P
C
;
A
) (Section 2) and more generalquasi-triangular spaces
(
X
,
P
C
;
A
) withleft(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 areleft (right) J
C
;
A-admissible orleft (right) P
C
;
A-closed (Section 6). Examples are provided (Sections 10–12) and concluding remarks are given (Section 13).


## 2. Quasi-Triangular Spaces(
X
,
P
C
;
A
)

It is worth noticing that the distance spaces(
X
,
P
C
;
A
), introduced and described below, are not necessarily topological or Hausdorff or sequentially complete.Definition 7.
LetX be a (nonempty) set, let A be an index set, and let C
=
C
α
α
∈
A
∈
[
1
;
∞
)
A.
(A) One says that a familyP
C
;
A
=
{
p
α
:
X
2
→
[
0
,
∞
)
,
α
∈
A
} of distances is aquasi-triangular family on X if(4)
∀
α
∈
A
∀
u
,
v
,
w
∈
X
p
α
u
,
w
≤
C
α
p
α
u
,
v
+
p
α
v
,
w
.Aquasi-triangular space
(
X
,
P
C
;
A
) is a set X together with the quasi-triangular family P
C
;
A on X.
(B) Let(
X
,
P
C
;
A
) be the quasi-triangular space. One says that P
C
;
A isseparating if (5)
∀
u
,
w
∈
X
u
≠
w
⟹
∃
α
∈
A
p
α
u
,
w
>
0
∨
p
α
w
,
u
>
0
.
(C) If(
X
,
P
C
;
A
) is an quasi-triangular space and ∀
α
∈
A
∀
u
,
w
∈
X
{
p
α
-
1
(
u
,
w
)
=
p
α
(
w
,
u
)
}, then ∀
α
∈
A
∀
u
,
v
,
w
∈
X
{
p
α
-
1
(
u
,
w
)
≤
C
α
[
p
α
-
1
(
u
,
v
)
+
p
α
-
1
(
v
,
w
)
]
}. One says that the quasi-triangular space (
X
,
P
C
;
A
-
1
), P
C
;
A
-
1
=
{
p
α
-
1
:
X
2
→
[
0
,
∞
)
,
α
∈
A
}, is theconjugation of (
X
,
P
C
;
A
).Remark 8.
In the spaces(
X
,
P
C
;
A
), in general, the distances p
α
:
X
2
→
0
,
∞
,
α
∈
A, do not vanish on the diagonal; they are asymmetric and do not satisfy triangle inequality (i.e., the properties ∀
α
∈
A
∀
u
∈
X
{
p
α
(
u
,
u
)
=
0
} or ∀
α
∈
A
∀
u
,
w
∈
X
{
p
α
(
u
,
w
)
=
p
α
(
w
,
u
)
} or ∀
α
∈
A
∀
u
,
v
,
w
∈
X
{
p
α
(
u
,
w
)
≤
p
α
(
u
,
v
)
+
p
α
(
v
,
w
)
} do not necessarily hold); see Section 10.Definition 9.
LetX be a (nonempty) set, let A be an index set, and let C
=
{
C
α
}
α
∈
A
∈
[
1
;
∞
)
A.
(A) One says that a familyL
C
;
A
=
{
l
α
:
X
2
→
[
0
,
∞
)
,
α
∈
A
} of distances on X is aultra quasi-triangular family if(6)
∀
α
∈
A
∀
u
,
v
,
w
∈
X
l
α
u
,
w
≤
C
α
max
⁡
l
α
u
,
v
,
l
α
v
,
w
.Anultra quasi-triangular space
(
X
,
L
C
;
A
) is a set X together with the ultra quasi-triangular family L
C
;
A on X.
(B) One says that a familyS
C
;
A
=
{
s
α
:
X
2
→
[
0
,
∞
)
,
α
∈
A
} of distances on X is apartial quasi-triangular family if(7)
∀
α
∈
A
∀
u
,
v
,
w
∈
X
s
α
u
,
w
≤
C
α
s
α
u
,
v
+
s
α
v
,
w
-
s
α
v
,
v
.Apartial quasi-triangular space
(
X
,
S
C
;
A
) is a set X together with the partial quasi-triangular family S
C
;
A on X.Remark 10.
It is worth noticing that quasi-triangular spaces generalize ultra quasi-triangular and partial quasi-triangular spaces (in particular, generalize metric, ultra metric, quasi-metric, ultra quasi-metric,b-metric, partial metric, partial b-metric, pseudometric, quasi-pseudometric, ultra quasi-pseudometric, partial quasi-pseudometric, topological, uniform, quasi-uniform, gauge, ultra gauge, partial gauge, quasi-gauge, ultra quasi-gauge, and partial quasi-gauge spaces).


## 3. Left (Right) FamiliesJ
C
;
A Generated by P
C
;
A in Quasi-Triangular Spaces (
X
,
P
C
;
A
)

In the metric spaces(
X
,
d
) there are several types of distances (determined by d) which generalize metrics d. First these distances were introduced by Tataru [22]. More general concepts of distances in metric spaces (
X
,
d
) which generalize d, of this sort, are given by Kada et al. [23] (w-distances), Lin and Du [24] (τ-functions), Suzuki [25] (τ-distances), and Ume [26] (u-distance). Distances in uniform spaces were given by Vályi [27]. In the appearing literature, these distances and their generalizations in other spaces provide efficient tools to study various problems of fixed point theory; see, for example, [28–30] and references therein. In this paper we also generalize these ideas.LetP
C
;
A be the quasi-triangular family on X. It is natural to define the notions ofleft (right)families
J
C
;
A
generated by
P
C
;
A which provide new structures on X.Definition 11.
Let(
X
,
P
C
;
A
) be the quasi-triangular space.
(A) The familyJ
C
;
A
=
{
J
α
:
α
∈
A
} of distances J
α
:
X
2
→
0
,
∞
,
α
∈
A, is said to be aleft(right)family generated by
P
C
;
A if(
J
1
)
∀
α
∈
A
∀
u
,
v
,
w
∈
X
{
J
α
(
u
,
w
)
≤
C
α
[
J
α
(
u
,
v
)
+
J
α
(
v
,
w
)
]
}; and furthermore.
(
J
2
)
For any sequences(
u
m
:
m
∈
N
) and (
w
m
:
m
∈
N
) in X satisfying(8)
∀
α
∈
A
l
i
m
m
→
∞
s
u
p
n
>
m
J
α
u
m
,
u
n
=
0
,
(9)
∀
α
∈
A
l
i
m
m
→
∞
s
u
p
n
>
m
J
α
u
n
,
u
m
=
0
,
(10)
∀
α
∈
A
l
i
m
m
→
∞
J
α
w
m
,
u
m
=
0
,
(11)
∀
α
∈
A
l
i
m
m
→
∞
J
α
u
m
,
w
m
=
0
,
the following holds(12)
∀
α
∈
A
l
i
m
m
→
∞
p
α
w
m
,
u
m
=
0
,
(13)
∀
α
∈
A
l
i
m
m
→
∞
p
α
u
m
,
w
m
=
0
.
(B)J
(
X
,
P
C
;
A
)
L  (
J
(
X
,
P
C
;
A
)
R
) is the set of all left (right) families J
C
;
A on X generated by P
C
;
A.Remark 12.
From Definition11 if follows that P
C
;
A
∈
J
(
X
,
P
C
;
A
)
L
∩
J
(
X
,
P
C
;
A
)
R. Moreover, there are families J
C
;
A
∈
J
(
X
,
P
C
;
A
)
L and J
C
;
A
∈
J
(
X
,
P
C
;
A
)
R such that the distances J
α
,
α
∈
A, do not vanish on the diagonal, are asymmetric, and are quasi-triangular and thus are not metric, ultra metric, quasi-metric, ultra quasi-metric, b-metric, partial metric, partial b-metric, pseudometric (gauge), quasi-pseudometric (quasi-gauge), and ultra quasi-pseudometric (ultra quasi-gauge).


## 4. Relations betweenJ
C
;
A and P
C
;
A

Remark 13.
The following result shows that Definition11 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 (J1) 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 (J1) 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
>
0holds.Proof.
We begin by supposing thatu
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 (J1) 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.


## 5. Left (Right)J
C
;
A-Convergences and Left (Right) J
C
;
A-Sequentially Completeness

Definition 16.
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.
(A) One says that a sequence(
u
m
:
m
∈
N
)
⊂
X isleft(right) J
C
;
A-Cauchy sequence if ∀
α
∈
A
{
l
i
m
m
→
∞
s
u
p
n
>
m
J
α
(
u
m
,
u
n
)
=
0
}  (
∀
α
∈
A
{
l
i
m
m
→
∞
s
u
p
n
>
m
J
α
(
u
n
,
u
m
)
=
0
}
).
(B) Letu
∈
X and let (
u
m
:
m
∈
N
)
⊂
X. One says that the sequence (
u
m
:
m
∈
N
) isleft (right) J
C
;
A-convergent to
u if u
∈
L
I
M
(
u
m
:
m
∈
N
)
L
-
J
C
;
A
≠
⌀  (
u
∈
L
I
M
(
u
m
:
m
∈
N
)
R
-
J
C
;
A
≠
⌀
) where(19)
L
I
M
u
m
:
m
∈
N
L
-
J
C
;
A
=
x
∈
X
:
∀
α
∈
A
l
i
m
m
→
∞
J
α
x
,
u
m
=
0
,
L
I
M
u
m
:
m
∈
N
R
-
J
C
;
A
=
x
∈
X
:
∀
α
∈
A
l
i
m
m
→
∞
J
α
u
m
,
x
=
0
.
(C) One says that a sequence(
u
m
:
m
∈
N
)
⊂
X isleft(right) J
C
;
A-convergent in
X if L
I
M
(
u
m
:
m
∈
N
)
L
-
J
C
;
A
≠
⌀  (
L
I
M
(
u
m
:
m
∈
N
)
R
-
J
C
;
A
≠
⌀
).
(D) If every left (right)J
C
;
A-Cauchy sequence (
u
m
:
m
∈
N
)
⊂
X is left (right) J
C
;
A-convergent in X (i.e., L
I
M
(
u
m
:
m
∈
N
)
L
-
J
C
;
A
≠
⌀  (
L
I
M
(
u
m
:
m
∈
N
)
R
-
J
C
;
A
≠
⌀
)
), then (
X
,
P
C
;
A
) is calledleft (right) J
C
;
A-sequentially complete.Remark 17.
The structures onX determined by left (right) families J
C
;
A generated by P
C
;
A are more general than the structure on X determined by P
C
;
A; see Remark 34.Remark 18.
Let(
X
,
P
C
;
A
) be the quasi-triangular space. It is clear that if (
u
m
:
m
∈
N
) is left (right) P
C
;
A-convergent in X, then L
I
M
(
u
m
:
m
∈
N
)
L
-
P
C
;
A
⊂
L
I
M
(
v
m
:
m
∈
N
)
L
-
P
C
;
A  (
L
I
M
(
u
m
:
m
∈
N
)
R
-
P
C
;
A
⊂
L
I
M
(
v
m
:
m
∈
N
)
R
-
P
C
;
A
) for each subsequence (
v
m
:
m
∈
N
) of (
u
m
:
m
∈
N
).Definition 19.
One says that(
X
,
P
C
;
A
) isleft (right)Hausdorff if for each left (right) P
C
;
A-convergent in X sequence (
u
m
:
m
∈
N
) the set L
I
M
(
u
m
:
m
∈
N
)
L
-
P
C
;
A  (
L
I
M
(
u
m
:
m
∈
N
)
R
-
P
C
;
A
) is a singleton.


## 6. Left (Right)J
C
;
A-Admissible and Left (Right) P
C
;
A-Closed Set-Valued Maps

The following terminologies will be much used in the sequel.Definition 20.
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 set-valued dynamic system, T
:
X
→
2
X.
(A) Givenw
0
∈
X, One says that (
X
,
T
) isleft (right) J
C
;
A-admissible in
w
0 if, for each dynamic processes (
w
m
:
m
∈
{
0
}
∪
N
) starting at w
0, ∀
m
∈
{
0
}
∪
N
{
w
m
+
1
∈
T
(
w
m
)
}, 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(20)
∀
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) One says that(
X
,
T
) isleft (right) J
C
;
A-admissible on
X, if (
X
,
T
) isleft (right) J
C
;
A-admissible in each point w
0
∈
X.Remark 21.
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 set-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 but the converse not necessarily holds.We can define also the following generalization of continuity.Definition 22.
Let(
X
,
P
C
;
A
) be the quasi-triangular space. Let (
X
,
T
) be the set-valued dynamic system, T
:
X
→
2
X, and let k
∈
N. The set-valued dynamic system (
X
,
T
[
k
]
) is said to be aleft (right) P
C
;
A-closed on X if for every 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
)
).


## 7. Left (Right) Pompeiu-Hausdorff Quasi-Distances and Left (Right) Set-Valued Quasi-Contractions

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 define three types of left (right) Pompeiu-Hausdorff quasi-distances on 2
X, and for each type a left (right) set-valued quasi-contraction T
:
X
→
2
X is constructed.Definition 23.
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 λ
=
λ
α
α
∈
A
∈
[
0
;
1
)
A, let (
X
,
T
) be a set-valued dynamic system, T
:
X
→
2
X, and let η
∈
{
1,2
,
3
}. Let(21)
∀
α
∈
A
∀
x
∈
X
∀
V
∈
2
X
J
α
x
,
V
=
inf
⁡
J
α
x
,
v
:
v
∈
V
∧
J
α
V
,
x
=
inf
⁡
J
α
v
,
x
:
v
∈
V
.
(A) LetJ
C
;
A
∈
J
(
X
,
P
C
;
A
)
L. If(22)
∀
α
∈
A
∀
U
,
W
∈
2
X
D
1
;
2
X
;
α
L
-
J
C
;
A
U
,
W
=
max
⁡
sup
u
∈
U
⁡
J
α
u
,
W
,
sup
w
∈
W
⁡
J
α
U
,
w
,
∀
α
∈
A
∀
U
,
W
∈
2
X
D
2
;
2
X
;
α
L
-
J
C
;
A
U
,
W
=
max
⁡
sup
u
∈
U
⁡
J
α
u
,
W
,
sup
w
∈
W
⁡
J
α
w
,
U
,
∀
α
∈
A
∀
U
,
W
∈
2
X
D
3
;
2
X
;
α
L
-
J
C
;
A
U
,
W
=
sup
u
∈
U
⁡
J
α
u
,
W
,then a family D
η
;
2
X
L
-
J
C
;
A
=
{
D
η
;
2
X
;
α
L
-
J
C
;
A
,
α
∈
A
} is said to beleft
D
η
;
2
X
L
-
J
C
;
A-quasi-distance on
2
X.
If(23)
∀
α
∈
A
∀
x
,
y
∈
X
C
α
·
D
η
;
2
X
;
α
L
-
J
C
;
A
T
x
,
T
y
≤
λ
α
J
α
x
,
y
,then we say that (
X
,
T
) is aleft
(
D
η
;
2
X
L
-
J
C
;
A
,
λ
)-quasi-contraction on
X.
(B) LetJ
C
;
A
∈
J
(
X
,
P
C
;
A
)
R. If(24)
∀
α
∈
A
∀
U
,
W
∈
2
X
D
1
;
2
X
;
α
R
-
J
C
;
A
U
,
W
=
max
⁡
sup
u
∈
U
⁡
J
α
u
,
W
,
sup
w
∈
W
⁡
J
α
U
,
w
,
∀
α
∈
A
∀
U
,
W
∈
2
X
D
2
;
2
X
;
α
R
-
J
C
;
A
U
,
W
=
max
⁡
sup
u
∈
U
⁡
J
α
u
,
W
,
sup
w
∈
W
⁡
J
α
w
,
U
,
∀
α
∈
A
∀
U
,
W
∈
2
X
D
3
;
2
X
;
α
R
-
J
C
;
A
U
,
W
=
sup
u
∈
U
⁡
J
α
u
,
W
,then a family D
η
;
2
X
R
-
J
C
;
A
=
{
D
η
;
2
X
;
α
R
-
J
C
;
A
,
α
∈
A
} is said to beright
D
η
;
2
X
R
-
J
C
;
A-quasi-distance on
2
X.
If(25)
∀
α
∈
A
∀
x
,
y
∈
X
C
α
·
D
η
;
2
X
;
α
R
-
J
C
;
A
T
x
,
T
y
≤
λ
α
J
α
x
,
y
,then we say that (
X
,
T
) is aright
(
D
η
;
2
X
R
-
J
C
;
A
,
λ
)-quasi-contraction on
X.Remark 24.
Observe thatD
η
;
2
X
L
-
J
C
;
A and D
η
;
2
X
R
-
J
C
;
A extend (2). Quasi-contractions (23) and (25) extend (3).Remark 25.
Each(
D
η
;
2
X
L
-
J
C
;
A
,
λ
)-quasi-contraction (
(
D
η
;
2
X
R
-
J
C
;
A
,
λ
)-quasi-contraction), η
∈
{
1,2
}, is (
D
3
;
2
X
L
-
J
C
;
A
,
λ
)-quasi-contraction (
(
D
3
;
2
X
R
-
J
C
;
A
,
λ
)-quasi-contraction) but the converse does not necessarily hold.


## 8. Convergence, Existence, Approximation, and Periodic Point Theorem of Nadler Type for Left (Right) Set-Valued Quasi-Contractions

The following result extends Theorem6 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) familyJ
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 everyx
∈
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
) isleft (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
) isleft (right) P
C
;
A-convergent to
w.Proof.
We prove only the case whenJ
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
;
∞
Asuch 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 Definition23(A), we get that there exists a sequence (
w
m
:
m
∈
N
) in X satisfying(41)
∀
m
∈
N
w
m
+
1
∈
T
w
mand 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
mwe 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 nowm
<
n. Using (J1), 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
mand 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, definingu
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
=
0and 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 Definition27 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) IfJ
C
;
A
∈
J
(
X
,
P
C
;
A
)
L, then we define theleft
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
) isleft
(
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) IfJ
C
;
A
∈
J
(
X
,
P
C
;
A
)
R, then one defines theright
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
) isright
(
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) Givenw
0
∈
X, One says that (
X
,
T
) isleft (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
) isleft (right) J
C
;
A-admissible on
X, if (
X
,
T
) isleft (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 aleft (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 Theorem5 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) familyJ
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 pointw
∈
X such that the sequence (
w
m
=
T
[
m
]
(
w
0
)
:
m
∈
{
0
}
∪
N
) starting at w
0 isleft(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 isleft(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 familyP
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 isleft(right) P
C
;
A-convergent to
w, and(58)
∀
α
∈
A
J
α
w
,
w
=
0
.Proof.
By Theorem26, 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 thatu
,
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.
LetX
=
[
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.
LetX 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.
LetX
=
[
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.
LetX
=
[
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
;
6and 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
}
isnotsymmetric. 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 Examples1–4 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 Theorem26

Example 1.
LetX
=
[
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
;
6and 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 Theorem26 are satisfied. This follows from (1)–(5) in Example 1.
We conclude thatFix
⁡
(
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.
LetX, 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 Examples1 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.
LetX
=
(
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 thatFix
⁡
(
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.
LetX
=
[
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
;
6and 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 that6
∈
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 thatF
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
)
=
⌀.


## 12. Example Illustrating Theorem32

Example 1.
LetX
=
(
0
;
6
), A, and J
{
1
}
;
{
1
}
=
P
{
1
}
;
{
1
}
=
{
p
} be as in Example 3. Define the single-valued dynamic system (
X
,
T
) by(92)
T
u
=
4
for
u
∈
0
;
3
,
2
for
u
∈
3
;
6
.
(
1
)  (
X
,
T
)
is a
(
D
1
;
X
L
-
P
{
1
}
;
{
1
}
,
λ
∈
[
0
;
1
)
)-quasi-contraction on
X; that is, ∀
λ
∈
[
0
;
1
)
∀
x
,
y
∈
X
{
D
1
;
X
L
-
P
1
;
1
(
T
(
x
)
,
T
(
y
)
)
=
max
⁡
{
p
(
T
(
x
)
,
T
(
y
)
)
,
p
(
T
(
y
)
,
T
(
x
)
)
}
≤
λ
p
(
x
,
y
)
}
and
Fix
⁡
(
T
)
=
⌀. Indeed, we see that if x
,
y
∈
X, then T
(
x
)
,
T
(
y
)
∈
A and, by (77) and (78), D
1
;
X
L
-
P
{
1
}
;
{
1
}
(
T
(
x
)
,
T
(
y
)
)
=
0
≤
λ
p
(
x
,
y
).
(
2
)  (
X
,
T
)
is left and right
P
{
1
}
;
{
1
}-admissible in
X. Assume that w
0
∈
X is arbitrary and fixed, (
w
m
:
m
∈
{
0
}
∪
N
) satisfies ∀
m
∈
{
0
}
∪
N
{
w
m
+
1
=
T
(
w
m
)
}, and l
i
m
m
→
∞
s
u
p
n
>
m
p
(
w
m
,
w
n
)
=
0. Then, by (92) and (78), we have ∀
m
∈
N
{
w
m
∈
A
}. This gives 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
}.
(
3
)
Single-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
)
=
{
2,4
} is a left 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
[
2
]
(
u
m
)
}, then, by (77), (78), and (92), we have that A
=
L
I
M
(
x
m
:
m
∈
N
)
L
-
P
{
1
}
;
{
1
}. In particular, 2
=
T
[
2
]
(
2
)
∈
L
I
M
(
x
m
:
m
∈
N
)
L
-
P
{
1
}
;
{
1
} and 4
=
T
[
2
]
(
4
)
∈
L
I
M
(
x
m
:
m
∈
N
)
L
-
P
{
1
}
;
{
1
}.
(
4
)
Property (56)holds. Indeed, we have ∀
v
∈
Fix
⁡
(
T
[
2
]
)
=
{
2,4
}
{
p
(
v
,
T
(
v
)
)
=
p
(
T
(
v
)
,
v
)
=
0
} since T
(
2
)
=
4, T
(
4
)
=
2, and T
(
{
2,4
}
)
=
{
2,4
}
⊂
A.
(
5
)  P
{
1
}
;
{
1
}
=
{
p
}
is not separating on
X. Indeed, if u
,
w
∈
X
/
A, then p
(
u
,
w
)
=
p
(
w
,
u
)
=
γ
>
0.
(
6
)
For
P
{
1
}
;
{
1
}
=
{
p
}, (
X
,
T
),and
J
{
1
}
;
{
1
}
=
P
{
1
}
;
{
1
}
defined by (77), (78), and (79) parts (B1) and (B2) of Theorem 32 hold but part (B3) of Theorem 32 does not hold. This follows from (1)–(5) in Example 1.


## 13. Concluding Remarks

Remark 2.
In Theorems5 and 6 the following play an important role: (i) Distances d and H
d, as metrics, satisfy conditions (A) of Definition 1 on X and C
B
(
X
), respectively. (ii) (
X
,
d
) and (
C
B
(
X
)
,
H
d
), as metric spaces, are topological and Hausdorff spaces and the completeness of (
X
,
d
) implies completeness of (
C
B
(
X
)
,
H
d
). (iii) The continuity of d and H
d on X
×
X and C
B
(
X
)
×
C
B
(
X
), respectively; (iv) The continuity of maps T
:
(
X
,
d
)
→
(
X
,
d
) and T
:
(
X
,
d
)
→
(
C
B
(
X
)
,
H
d
) (as consequences of contractive properties defined in (1) and (3), resp.); (v) In Theorem 6 the assumption that, for each x
∈
X, T
(
x
)
∈
C
B
(
X
).Remark 3.
Conclusions in Theorems5 and 6 concern only fixed points but not periodic points; this is a consequence of separability of spaces (
X
,
d
) and (
C
B
(
X
)
,
H
d
) and also continuity of T.Remark 4.
In Theorems26 and 32, properties concening the spaces and maps such as mentioned above generally need not hold, since spaces (
X
,
P
C
;
A
) with left (right) families J
C
;
A generated by P
C
;
A are very general, which is an obstruction to use Nadler’s and Banach’s reasoning. Theorems 26 and 32 show how to rectify this situation and are obtained without restrictively required assumptions and with conclusions more profound as in the well known results of this sort existing in the literature.



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*Source: 201236-2015-07-26.xml*