1. Introduction
The concept of fuzzy metric space proposed by Kramosil and Michalek [
1] was inspired by the Menger space that is a special kind of probabilistic metric space by referring to Schweizer and Sklar [
2,
3,
4], Hadžić and Pap [
5], and Chang et al. [
6]. Kaleva and Seikkala [
7] proposed another concept of fuzzy metric space by considering the membership degree of the distance between any two different points. George and Veeramani [
8,
9] studied some properties of fuzzy metric spaces in the sense of Kramosil and Michalek [
1]. Gregori and Romaguera [
10,
11,
12] also extended the study of the properties of fuzzy metric spaces and fuzzy quasi-metric spaces in which the symmetric condition was not assumed.
The Hausdorff topology induced by the fuzzy metric space was studied in Wu [
13]. In this paper, we shall propose the concept of double fuzzy-semi metric in fuzzy semi-metric space and study its convergent properties. The potential application for using the convergence of dual double fuzzy semi-metric is to study the new type of fixed point theorems in fuzzy semi-metric space by considering the Cauchy sequences, which will be the future research and may refer to the previous work of Wu [
14] for studying the common coincidence points and common fixed points in fuzzy semi-metric spaces. Wu [
15] studied the so-called fuzzy semi-metric space without assuming the symmetric condition. In the fuzzy semi-metric space
, the symmetric condition
for all
and
is not assumed to be true. Therefore, four kinds of triangle inequalities should be considered.
In order to obtain the new type of fixed point theorems in fuzzy semi-metric space, we need to study the convergence using dual double fuzzy semi-metric. Based on the concept of t-norm ∗, we shall firstly define the double fuzzy semi-metric by considering the mapping
that is defined by:
where
is called a double fuzzy semi-metric.
The convergence using fuzzy semi-metric has been studied in Wu [
16], where the infimum type of dual fuzzy semi-metric is the function
defined by:
and the supremum type of dual fuzzy semi-metric is the function
defined by:
In this paper, we shall consider the double fuzzy semi-metric
to define the infimum and supremum types of dual double fuzzy semi-metric. The infimum type of dual double fuzzy semi-metric is the function
defined by:
and the supremum type of dual double fuzzy semi-metric is the function
defined by:
Using the infimum and supremum types of dual fuzzy semi-metric
and
, the convergence of sequences in
and the concept of Cauchy sequence in
have been studied in Wu [
16]. In this paper, we study the extended convergence of sequences in
and the concept of joint Cauchy sequence in
using the infimum and supremum types of dual double fuzzy semi-metric
and
. As we mentioned above, these convergences will be used in the near future to establish the new types of fixed point theorems in fuzzy semi-metric space
.
In
Section 2, we review some basic properties of fuzzy semi-metric space that will be used for further discussion. In
Section 3, we introduce the concept of double fuzzy semi-metric and derive the related triangle inequalities. In
Section 4 and
Section 5, the concepts of infimum and supremum types of dual double fuzzy semi-metric are proposed, and their convergent properties and triangle inequalities are studied.
3. Double Fuzzy Semi-Metric
Let
be a fuzzy semi-metric space along with a t-norm ∗. Given any four elements
, recall that the value
means the membership degree of the distance that is less than
t between
x and
y, and the value
means the membership degree of the distance that is less than
t between
u and
v. In this case, we can define a value:
which means the membership degree of the distance that is simultaneously less than
t between
x and
y and between
u and
v. In general, instead of considering the min function, we shall use the t-norm. The formal definition is given below.
Definition 1. Let be a fuzzy semi-metric space along with a t-norm ∗. We define the mapping by: Then ζ is called a double fuzzy semi-metric.
Example 2. Continued from Example 1, we consider: If we take t-norm as , then the double fuzzy semi-metric can be obtained as: The potential application for considering the double fuzzy semi-metric is to study the new type of fixed point theorems in fuzzy semi-metric space.
Proposition 2. (Triangle Inequalities for Dual Fuzzy Semi-Metric) Let be a fuzzy semi-metric space along with a t-norm ∗. Given any , we have the following properties:
- (i)
Suppose that M satisfies the ⋈-triangle inequality. Then we have the inequality:for . - (ii)
Suppose that M satisfies the ▹-triangle inequality. Then we have the inequality:for . - (iii)
Suppose that M satisfies the ◃-triangle inequality. Then we have the inequality:for . - (iv)
Suppose that M satisfies the ⋄-triangle inequality. Then we have the inequality:for .
Proof. It suffices to prove part (i); we have:
This completes the proof. □
Definition 2. Let be a fuzzy semi-metric space along with a t-norm ∗, and let ζ be a double fuzzy semi-metric given by: Given any fixed , we define the following concepts of monotonicity:
The mapping is said to be nondecreasing if and only if for . The mapping is said to be increasing if and only if for .
The mapping is is said to be symmetrically nondecreasing if and only if for . The mapping is said to be symmetrically increasing if and only if for .
The mapping is said to be ◃-semisymmetrically nondecreasing if and only if for . The mapping is said to be ◃-semisymmetrically increasing if and only if for .
The mapping is said to be ▹-semisymmetrically nondecreasing if and only if for . The mapping is said to be ▹-semisymmetrically increasing if and only if for .
Proposition 3. Let be a fuzzy semi-metric space along with a t-norm ∗. Given any fixed , the double fuzzy semi-metric ζ satisfies the following properties:
- (i)
Suppose that M satisfies the ⋈-triangle inequality. Then the mapping from into is nondecreasing.
- (ii)
Suppose that M satisfies the ▹-triangle inequality or the ◃-triangle inequality. Then the mapping from into is simultaneously nondecreasing, symmetrically nondecreasing, ◃-semisymmetrically nondecreasing, and ▹-semisymmetrically nondecreasing.
- (iii)
Suppose that M satisfies the ⋄-triangle inequality. Then the mapping from into is symmetrically nondecreasing.
Proof. Part (i) of Proposition 1 says that the mappings and from into are nondecreasing. According to the increasing property of t-norm, we conclude that the mapping from into is nondecreasing, which proves part (i).
Part (ii) can be obtained from part (ii) of Proposition 1, and part (iii) can be obtained from part (iii) of Proposition 1. This completes the proof. □
By using the strictly increasing property of t-norm, the proof of Proposition 3 is still valid for obtaining the following results.
Proposition 4. Let be a fuzzy semi-metric space along with a t-norm ∗. Suppose that the t-norm satisfies the strictly increasing property. Given any fixed , the double fuzzy semi-metric ζ satisfies the following properties:
- (i)
Suppose that M satisfies the strict ⋈-triangle inequality. Then the mapping from into is increasing.
- (ii)
Suppose that M satisfies the strict ▹-triangle inequality or the strict ◃-triangle inequality. Then the mapping from into is simultaneously increasing, symmetrically increasing, ◃-semisymmetrically increasing, and ▹-semisymmetrically increasing.
- (iii)
Suppose that M satisfies the strict ⋄-triangle inequality. Then the mapping from into is symmetrically increasing.
Let
be a fuzzy semi-metric space. The motivation for considering the following two concepts can refer to Wu [
16].
M is said to satisfy the canonical condition if and only if:
M is said to satisfy the rational condition if and only if:
Proposition 5. Let be a fuzzy semi-metric space along with a t-norm ∗.
- (i)
Suppose that M satisfies the canonical condition. If the t-norm ∗ is left-continuous at 1 with respect to the first or second argument, then we have: - (ii)
Suppose that M satisfies the rational condition. If the t-norm ∗ is right-continuous at 0 with respect to the first or second argument, then we have:
Proof. To prove part (i), the canonical condition says that:
The left-continuity of t-norm ∗ at 1 also says that:
To prove part (ii), the rational condition says that:
The right-continuity of t-norm ∗ at 0 also says that:
This completes the proof. □
Example 3. Continued from Example 1, it is not hard to check that M satisfies both the canonical and rational conditions. Suppose that we take t-norm as . Then Proposition 5 says that: 4. Convergence Based on the Infimum
From Definition 1, we see that the double fuzzy semi-metric is a mapping from into . Here, we shall consider its dual sense by considering the mapping from into . The formal definition is given below.
Definition 3. Let be a fuzzy semi-metric space along with a t-norm ∗. We also assume that M satisfies the canonical condition, and that the t-norm ∗ is left-continuous at 1 with respect to the first or second argument. Given any fixed and any fixed , we consider the following set:which is used to define a mapping by: In this case, the mapping from into is called the infimum type of dual double fuzzy semi-metric.
Example 4. Continued from Example 2, we have:where: The potential application of dual double fuzzy semi-metric will be used to study the fixed point theorems in fuzzy semi-metric space. However, we first need to claim that the set
is nonempty. Suppose that
. The definition says that
for all
; that is:
which contradicts Equation (
8). Therefore, Definition 3 is well-defined and
.
Remark 2. The following observations will be useful for further discussion.
For any , we have:and: Given any fixed , if , then:
Proposition 6. Let be a fuzzy semi-metric space along with a t-norm ∗. We also assume that M satisfies the canonical condition, and that the t-norm ∗ is left-continuous at 1 with respect to the first or second argument. Given any fixed , suppose that the following conditions are satisfied:and: Moreover, the following limit exists: Proof. The assumption
says that we can consider
, i.e.,
for all
. Then we obtain
by taking
, which shows that
, i.e.,
On the other hand, suppose that
. Then
for all
. Therefore, we obtain
, which implies the desired equality (Equation (
12)). Further, the inequality (Equation (
11)) says that the limit (Equation (
13)) exists. This completes the proof. □
Proposition 7. Suppose that is a fuzzy semi-metric space along with a t-norm ∗. We also assume that M satisfies the canonical and rational conditions, and that the t-norm ∗ is left-continuous at 1 and right-continuous at 0 with respect to the first or second argument. If M satisfies the ○-triangle inequality for , then, for any fixed with or , we have for .
Proof. We first consider the case of
M, satisfying the ○-triangle inequality for
. From Equation (
10), we need to consider
or
. We want to assume
for
to obtain a contradiction. Using the concept of infimum from
, given any
, there exists
such that
and:
Parts (i) and (ii) of Proposition 3 say that the mapping
from
into
is nondecreasing. Therefore, we obtain:
Since
can be any positive real number, using Equation (
9), we must have:
which contradicts
.
Now we assume that
M satisfies the ⋄-triangle inequality. Suppose that
for
. Part (iii) of Proposition 3 says that the mapping
is symmetrically nondecreasing. Therefore, we can similarly obtain:
This completes the proof. □
Proposition 8. Let be a fuzzy semi-metric space along with a t-norm ∗. We also assume that M satisfies the canonical condition, and that the t-norm ∗ is left-continuous at 1 with respect to the first or second argument. Given any fixed and , we have the following properties:
- (i)
If is sufficiently small satisfying , then we have: - (ii)
Suppose that M satisfies the ○-triangle inequality for . For any , we have: - (iii)
Suppose that M satisfies the ○-triangle inequality for . For any , we have:and: - (iv)
Suppose that M satisfies the ○-triangle inequality for . For any , we have:
Proof. To prove part (i), we assume that:
The definition of says that . This contradiction implies .
To prove part (ii), using the concept of infimum from
, given any
, there exists
such that
and
. Parts (i) and (ii) of Proposition 3 says that the mapping
is nondecreasing. Therefore, we obtain:
To prove part (iii), using the concept of infimum from
, given any
, there exists
such that
and
. Part (ii) of Proposition 3 says that the mapping
is ◃-semisymmetrically nondecreasing. Therefore, we obtain:
Since the mapping
is also ▹-semisymmetrically nondecreasing, we can similarly obtain another inequality.
Since the mapping is semisymmetrically nondecreasing, using parts (ii) and (iii) of Proposition 3, we can similarly obtain part (iv). This completes the proof. □
Proposition 9. Let be a fuzzy semi-metric space along with a t-norm ∗. We also assume that M satisfies the canonical condition, and that the t-norm ∗ is left-continuous at 1 with respect to the first or second argument. Given any fixed and , the following statements hold true:
- (i)
Suppose that M satisfies the ○-triangle inequality for . If , then we have .
- (ii)
If , then we have the following properties:
Suppose that M satisfies the ○-triangle inequality for . Then we have .
Suppose that M satisfies the ○-triangle inequality for . Then we have and .
Suppose that M satisfies the ○-triangle inequality for . Then we have .
Proof. To prove part (i), the inequality says that there exists , satisfying . Therefore, we consider the following cases:
Suppose that
M satisfies the ○-triangle inequality for
. Parts (i) and (ii) of Proposition 3 say that the mapping
is nondecreasing. Therefore, using Equation (
15), we obtain:
Suppose that
M satisfies the ⋄-triangle inequality. Part (iii) of Proposition 3 says that the mapping
is symmetrically nondecreasing. Therefore, using Equation (
18), we obtain:
To prove part (ii), the inequality says that there exists , satisfying . Therefore, we consider the following cases:
Suppose that
M satisfies the ○-triangle inequality for
. Parts (i) and (ii) of Proposition 3 say that the mapping
is nondecreasing. Therefore, using Equation (
14), we obtain:
Suppose that
M satisfies the ○-triangle inequality for
. Part (ii) of Proposition 3 says that the mapping
is ◃-semisymmetrically nondecreasing. Therefore, using Equation (
14), we obtain:
We can similarly obtain another inequality using the fact that the mapping is also ▹-semisymmetrically nondecreasing.
Suppose that
M satisfies the ○-triangle inequality for
. Parts (ii) and (iii) of Proposition 3 say that the mapping
is symmetrically nondecreasing. Therefore, using Equation (
14), we obtain:
This completes the proof. □
Proposition 10. Let be a fuzzy semi-metric space along with a t-norm ∗. We also assume that M satisfies the canonical condition, and that the t-norm ∗ is left-continuous at 1 with respect to the first or second argument. Given any fixed and , the following statements hold true:
- (i)
If , then we have the following properties:
Suppose that M satisfies the ○-triangle inequality for . Then we have .
Suppose that M satisfies the ○-triangle inequality for . Then we have and .
Suppose that M satisfies the ○-triangle inequality for . Then we have .
- (ii)
Suppose that the t-norm ∗ satisfies the strictly increasing property. If for , then we have the following properties.
Suppose that M satisfies the strict ○-triangle inequality for . If , then we have .
Suppose that M satisfies the strict ○-triangle inequality for . If , then we have , and if , then we have .
Suppose that M satisfies the strict ○-triangle inequality for . If , then we have .
- (iii)
If , then we have the following properties:
Suppose that M satisfies the ○-triangle inequality for . Then we have .
Suppose that M satisfies the ○-triangle inequality for . Then we have and .
Suppose that M satisfies the ○-triangle inequality for . Then we have .
Proof. To prove part (i), three cases are separately considered below:
Suppose that M satisfies the ○-triangle inequality for . Using the contrapositive statement of part (i) of Proposition 9, we can obtain the desired result.
Suppose that
M satisfies the ○-triangle inequality for
. According to the concept of infimum, given any
, there exists
, satisfying
and
. Part (ii) of Proposition 3 says that the mapping
is ◃-semisymmetrically nondecreasing. Therefore, if
then
, which contradicts
. It says that:
Since can be any positive real number, we must have . We can similarly obtain another inequality using the fact of the mapping being ▹-semisymmetrically nondecreasing.
Suppose that
M satisfies the ○-triangle inequality for
. According to the concept of infimum, given any
, there exists
, satisfying
and
. Parts (ii) and (iii) of Proposition 3 say that if
then
, which contradicts
. It says that:
Since can be any positive real number, we must have .
To prove part (ii), three cases are separately considered below:
Suppose that
M satisfies the ○-triangle inequality for
. According to the concept of infimum, given any
, there exists
, satisfying
and
. Regarding the strict property, parts (i) and (ii) of Proposition 4 say that if
then
, which contradicts
. It says that:
Since can be any positive real number, we must have . Now we assume that . The first case of part (ii) of Proposition 9 says that , which also contradicts . Therefore, we must have .
Suppose that M satisfies the ○-triangle inequality for . We can similarly obtain . Now we assume that . The second case of part (ii) of Proposition 9 says that , which also contradicts . Therefore, we must have . Another result can be similarly obtained.
Suppose that M satisfies the ○-triangle inequality for . We can similarly obtain . Now we assume that . The third case of part (ii) of Proposition 9 says that , which also contradicts . Therefore, we must have .
Part (iii) can be obtained from the contrapositive statement of part (ii) of Proposition 9. This completes the proof. □
Proposition 11. Let be a fuzzy semi-metric space along with a t-norm ∗. We also assume that M satisfies the canonical condition, and that the t-norm ∗ is left-continuous at 1 with respect to the first or second argument. Given any fixed and , the following statements hold true:
- (i)
Suppose that the mapping is left-continuous on . If , then we have: - (ii)
Suppose that the mapping is right-continuous on . Then the following statements hold true:
Suppose that M satisfies the ○-triangle inequality for . If , then we have: Suppose that M satisfies the ○-triangle inequality for . If , then we have:and if , then we have: Suppose that M satisfies the ○-triangle inequality for . If , then we have:
Proof. By applying
to the inequality Equation (
14), we obtain Equation (
19), which proves part (i). By applying
to the inequality (Equation (
15)), we obtain Equation (
20), which proves part (ii). The other inequalities can be similarly obtained by parts (iii) and (iv) of Proposition 8. This completes the proof. □
In order to establish the triangle inequalities for the infimum type of dual double fuzzy semi-metric, we provide a useful lemma.
Lemma 1. (Wu [
16])
Suppose that the t-norm ∗ is left-continuous at 1 with respect to the first or second argument. For any and any , there exists such that: Theorem 1. (Triangle Inequalities for Dual Double Fuzzy Semi-Metric) Let be a fuzzy semi-metric space along with a t-norm ∗. We also assume that M satisfies the canonical condition, and that the t-norm ∗ is left-continuous at 1 with respect to the first or second argument. Given any fixed and any fixed and distinct , we have the following inequalities:
- (i)
Suppose that M satisfies the ⋈-triangle inequality. Then, there exists , satisfying: - (ii)
Suppose that M satisfies the ▹-triangle inequality. Then, there exists , satisfying: - (iii)
Suppose that M satisfies the ◃-triangle inequality. Then, there exists , satisfying: - (iv)
Suppose that M satisfies the ⋄-triangle inequality. Then, there exists such that the following inequalities are satisfied:
Proof. To prove part (i), if
, then
. Therefore, the result is obvious. Now we assume
. Using Lemma 1, there exists
, satisfying:
Given any
, the first observation of Remark 1 says that:
and:
Now applying the increasing property and commutativity of t-norm to Equations (
35) and (
36), we obtain:
The definition of
says that:
By taking
, we obtain the desired inequality (Equation (
24)).
On the other hand, we also have:
and:
Now applying the increasing property and commutativity of t-norm to Equations (
37) and (
38), we also obtain:
The definition of
says that:
By taking
, we obtain the desired inequality (Equation (
25)). Since the other inequalities can be similarly obtained, we omit the details.
The above argument is still valid to obtain part (ii) by referring the second observation of Remark 1. Further, we can use the third observation of Remark 1 to obtain part (iii). Finally, part (iv) can be obtained by referring to the fourth observation of Remark 1. This completes the proof. □
Let
be a fuzzy semi-metric space, and let
be a sequence in
X. We write
as
if and only if:
We also write
as
if and only if:
The main convergence theorem is presented below. We first provide a useful lemma.
Lemma 2. Let ∗ be a t-norm. If then and .
Proof. Since
, the increasing property and boundary condition show that
. Suppose that
. Then we have
and:
A contradiction occurs. Therefore, we must have . We can similarly show that . This completes the proof. □
Theorem 2. Let be a fuzzy semi-metric space along with a t-norm ∗. We also assume that M satisfies the canonical condition, and that the t-norm ∗ is left-continuous at 1 with respect to the first or second argument. Let and be two sequences in X. Then we have the following properties:
and as if and only if as for all .
and as if and only if as for all .
and as if and only if as for all .
and as if and only if as for all .
Proof. For any fixed
, using Lemma 1, it follows that there exists
, satisfying:
We just prove the first case, since the other cases can be similarly obtained. Suppose that
and
as
for all
. Then, given any
and
, there exists
, satisfying
for
and
for
. Therefore, given any
, there exists
, satisfying:
for
. We also have:
for
. The increasing property of t-norm says that:
The definition of
says that:
for
. This shows that
as
.
Conversely, assume that
as
for all
. Now, given any
and
, there exists
, satisfying
for all
. Therefore, for any fixed
and given any
, there exists
, satisfying:
for
, which implies:
for
by part (i) of Proposition 9, i.e.,
for
. Lemma 2 says that:
for
. This shows that
and
as
, and the proof is complete. □
Example 5. From Example 1, we see that:and: From Example 4, we have:where: It is clear to see that and as if and only if as for all . The other convergence presented in Theorem 2 can be similarly verified.
Definition 4. Let be a fuzzy semi-metric space, and let be a sequence in X.
The sequence is said to be a >-Cauchy sequence in a metric sense if and only if, given any pair with and , there exists , satisfying for all pairs of integers m and n with .
The sequence is said to be a <-Cauchy sequence in a metric sense if and only if, given any pair with and , there exists , satisfying for all pairs of integers m and n with .
The sequence is said to be a Cauchy sequence in a metric sense if and only if, given any pair with and , there exists satisfying and for all pairs of integers m and n with and .
Definition 5. Let be a fuzzy semi-metric space such that M satisfies the canonical condition, and let and be two sequences in X.
Given any fixed , the sequences and are said to be the joint -Cauchy sequences with respect to if and only if, given any , there exists such that implies .
Given any fixed , the sequences and are said to be the joint -Cauchy sequences with respect to if and only if, given any , there exists such that implies .
Given any fixed , the sequences and are said to be the joint -Cauchy sequences with respect to if and only if, given any , there exists such that implies .
Given any fixed , the sequences and are said to be the joint -Cauchy sequences with respect to if and only if, given any , there exists such that implies .
Theorem 3. Let be a fuzzy semi-metric space along with a t-norm ∗. We also assume that M satisfies the canonical condition, and that the t-norm ∗ is left-continuous at 1 with respect to the first or second argument. Let and be two sequences in X. Then, we have the following properties:
- (i)
and are two >-Cauchy sequences in a metric sense if and only if and are the joint -Cauchy sequences with respect to for any .
- (ii)
is a >-Cauchy sequences and is a <-Cauchy sequences in a metric sense if and only if and are the joint -Cauchy sequences with respect to for any .
- (iii)
is a <-Cauchy sequences and is a >-Cauchy sequences in a metric sense if and only if and are the joint -Cauchy sequences with respect to for any .
- (iv)
and are two <-Cauchy sequences if and only if and are the joint -Cauchy sequences in a metric sense with respect to for any .
Proof. It suffices to just prove part (i), since the other cases can be similarly obtained. Suppose that
and
are >-Cauchy sequences. Then, given any
and
, there exists
such that
implies
and
. Now, given any
, there exists
such that
implies:
The increasing property of t-norm says that:
Further, by referring to the definition of
, we obtain:
for
.
Conversely, using the assumption, for any fixed
and given any
, there exists
such that
implies
. Using Proposition 9, we obtain:
for
, i.e.,
for
. Lemma 2 says that:
for
, which shows that
and
are >-Cauchy sequences. This completes the proof. □
5. Convergence Based on the Supremum
Using the infimum and assuming the canonical condition, the infimum type of dual double fuzzy semi-metric was proposed in the previous section. In this section, we shall consider the supremum to propose the so-called supremum type of dual double fuzzy semi-metric.
Recall that the purpose for considering the canonical condition is to guarantee the infimum type of dual fuzzy semi-metric space to be well-defined. Now, we shall consider the rational condition to guarantee the supremum type of dual fuzzy semi-metric space to be well-defined. The formal definition is given below.
Definition 6. Let be a fuzzy semi-metric space along with a t-norm ∗ such that M satisfies the rational condition, and that the t-norm ∗ is right-continuous at 0 with respect to the first or second argument. Given any fixed with or and any fixed , we consider the following set:which will be used to define a function by: The mapping from into is called the supremum type of dual double fuzzy semi-metric.
Example 6. Continued from Example 1, we have:where: For any
or
, we need to claim that the set
is nonempty. Suppose that
. The definition says that
for all
. Therefore, we obtain:
which contradicts Equation (
9). This says that Definition 6 is well-defined, which also says that
. We also have:
Moreover, if
, then:
Proposition 12. Let be a fuzzy semi-metric space along with a t-norm ∗. We also assume that M satisfies the rational condition, and that the t-norm ∗ is right-continuous at 0 with respect to the first or second argument. Given any fixed with or , suppose that . Then, the following statements hold true:
- (i)
Suppose that M satisfies the ○-triangle inequality for . Then we have for all .
- (ii)
Suppose that M satisfies the ○-triangle inequality for . Then we have and for all .
- (iii)
Suppose that M satisfies the ○-triangle inequality for . Then we have for all .
Proof. The fact
says that
for sufficiently large
in the sense of
. To prove part (i), we assume that there exists
, satisfying
. Parts (i) and (ii) of Proposition 3 say that the mapping
is nondecreasing. Therefore, if
, then:
which contradicts
for sufficiently large
.
To prove part (ii), we assume that there exists
, satisfying
. Part (ii) of Proposition 3 says that the mapping
is ◃-semisymmetrically nondecreasing. Therefore, if
, then:
which contradicts
for sufficiently large
. We can similarly obtain another inequality using the fact of the mapping
to be ▹-semisymmetrically nondecreasing.
To prove part (iii), we assume that there exists
, satisfying
. Parts (ii) and (iii) of Proposition 3 say that the mapping
is symmetrically nondecreasing. Therefore, if
, then:
which contradicts
for sufficiently large
. This completes the proof. □
Proposition 13. Let be a fuzzy semi-metric space along with a t-norm ∗. We also assume that M satisfies the rational and canonical conditions, and that the t-norm ∗ is right-continuous at 0 and left-continuous at 1 with respect to the first or second argument. Then, given any fixed with or , we have for .
Proof. We assume that
, which means that
for sufficiently large
t in the sense of
. Using Equation (
8), we obtain
which leads to a contradiction for
. This completes the proof. □
Proposition 14. Let be a fuzzy semi-metric space along with a t-norm ∗. Assume that M satisfies the canonical and rational conditions. We also assume that the t-norm ∗ is left-continuous at 1 and right-continuous at 0 with respect to the first or second argument, and that the t-norm ∗ also satisfies the strictly increasing property. For any fixed with or , the following statements hold true:
- (i)
Suppose that M satisfies the strict ○-triangle inequality for . Then we have:for each . - (ii)
Suppose that M satisfies the strict ○-triangle inequality for . Then we have:for each . - (iii)
Suppose that M satisfies the strict ○-triangle inequality for . Then we have:for each .
Proof. Proposition 13 says that for all . According to the concept of supremum, given any , there exists , satisfying and . To prove part (i), parts (i) and (ii) of Proposition 10 say that , which implies . Since can be any positive real number, we obtain the desired inequality.
To prove part (ii), parts (i) and (ii) of Proposition 10 say that , which implies . Since can be any positive real number, we obtain . Another inequality can be similarly obtained.
To prove part (iii), parts (ii) and (iii) of Proposition 10 say that , which implies . Since can be any positive real number, we obtain the desired inequality. This completes the proof. □
Proposition 15. Let be a fuzzy semi-metric space along with a t-norm ∗. We also assume that M satisfies the rational condition, and that the t-norm ∗ is right-continuous at 0 with respect to the first or second argument. For any fixed with or , and any fixed , we assume .
- (i)
For any , we have the following inequality: - (ii)
If is sufficiently small satisfying , then the following statements hold true:
Suppose that M satisfies the ○-triangle inequality for . Then we have: Suppose that M satisfies the ○-triangle inequality for . Then we have: Suppose that M satisfies the ○-triangle inequality for . Then we have:
Proof. To prove part (i), given any , we assume that . The definition of says that . This contradiction shows that .
To prove part (ii), according to the concept of supremum for , given any with , there exists , satisfying and . Therefore, we consider three cases below:
Suppose that
M satisfies the ○-triangle inequality for
. Parts (i) and (ii) of Proposition 3 say that the mapping
is nondecreasing. Therefore, we have:
Suppose that
M satisfies the ○-triangle inequality for
. Part (ii) of Proposition 3 says that the mapping
is ◃-semisymmetrically nondecreasing. Therefore, we have:
We can similarly obtain another inequality using the fact of the mapping to be ▹-semisymmetrically nondecreasing.
Suppose that
M satisfies the ○-triangle inequality for
. Parts (ii) and (iii) of Proposition 3 say that the mapping
is symmetrically nondecreasing. Therefore, we have:
This completes the proof. □
Proposition 16. Let be a fuzzy semi-metric space along with a t-norm ∗. We also assume that M satisfies the rational condition, and that the t-norm ∗ is right-continuous at 0 with respect to the first or second argument. Given any fixed with or , and any fixed , the following statements hold true:
- (i)
Suppose that . Then, we have the following properties:
If M satisfies the ○-triangle inequality for , then .
If M satisfies the ○-triangle inequality for , then and .
If M satisfies the ○-triangle inequality for , then .
- (ii)
We have the following properties:
Suppose that M satisfies the ○-triangle inequality for . If , then .
Suppose that M satisfies the ○-triangle inequality for . If or or , then .
Suppose that M satisfies the ○-triangle inequality for . If or , then .
Proof. To prove part (i), the fact says that there exists , satisfying . We consider three cases below:
Suppose that
M satisfies the ○-triangle inequality for
. Parts (i) and (ii) of Proposition 3 say that the mapping
is nondecreasing. Therefore, using Equation (
41), we obtain:
Suppose that
M satisfies the ○-triangle inequality for
. Part (ii) of Proposition 3 says that the mapping
is both ◃-semisymmetrically nondecreasing and ▹-semisymmetrically nondecreasing.Therefore, using Equation (
41), we obtain:
and:
Suppose that
M satisfies the ○-triangle inequality for
. Parts (ii) and (iii) of Proposition 3 say that the mapping
is symmetrically nondecreasing. Therefore, using Equation (
41), we obtain:
To prove part (ii), we consider three cases below:
Suppose that
M satisfies the ○-triangle inequality for
. Using part (i) of Proposition 12, if
, then it is done. Now, for
, the fact
says that there exists
, satisfying
. Using Equation (
42), we obtain:
Suppose that
M satisfies the ○-triangle inequality for
. Using part (ii) of Proposition 12, if
or
, then it is done. Now, for
, using Equation (
43), we obtain:
and:
Suppose that
M satisfies the ○-triangle inequality for
. Using part (iii) of Proposition 12, if
, then it is done. Now, for
, using Equation (
44), we obtain:
This completes the proof. □
Proposition 17. Let be a fuzzy semi-metric space along with a t-norm ∗. We also assume that M satisfies the rational condition, and that the t-norm ∗ is right-continuous at 0 with respect to the first or second argument. Given any fixed with or , and any fixed , the following statements hold true:
- (i)
Suppose that for . Then, we have the following properties:
If M satisfies the ○-triangle inequality for , then .
If M satisfies the ○-triangle inequality for , then and .
If M satisfies the ○-triangle inequality for , then .
- (ii)
We have the following properties:
Suppose that M satisfies the ○-triangle inequality for . If , then and .
Suppose that M satisfies the ○-triangle inequality for . If , then , and .
Suppose that M satisfies the ⋄-triangle inequality:
- -
If , then .
- -
If and , then .
Proof. To prove part (i), we consider three cases below:
Suppose that M satisfies the ○-triangle inequality for . It is clear to see that the fact implies . Now, for , using the contraposition of first property of part (i) of Proposition 16, we see that if , then .
Suppose that M satisfies the ○-triangle inequality for . It is clear to see that the fact implies . Now, for , using the contraposition of second property of part (i) of Proposition 16, we see that if , then . We can similarly show that if , then
Suppose that M satisfies the ○-triangle inequality for . It is clear to see that the fact implies . Now, for , using the contraposition of third property of part (i) of Proposition 16, we see that if , then .
To prove part (ii), we consider three cases below:
Suppose that M satisfies the ○-triangle inequality for . Using the contraposition of part (i) of Proposition 12 and the contraposition of first property of part (ii) of Proposition 16, we can obtain the desired result.
Suppose that M satisfies the ○-triangle inequality for . Using part (ii) of Proposition 12, if , then and . Using part (iii) of Proposition 12, if , then .
Suppose that M satisfies the ○-triangle inequality for . Using part (iii) of Proposition 12, if , then . Using the contraposition of third property of part (ii) of Proposition 16, if and then .
This completes the proof. □
Proposition 18. Let be a fuzzy semi-metric space along with a t-norm ∗. We also assume that M satisfies the rational condition, and that the t-norm ∗ is right-continuous at 0 with respect to the first or second argument. Given any fixed with or , and any fixed , the following statements hold true:
- (i)
Suppose that , and that the mapping is right-continuous on . Then we have: - (ii)
Suppose that the mapping is left-continuous on . Then, the following statements hold true:
Suppose that M satisfies the ○-triangle inequality for . Suppose that M satisfies the ○-triangle inequality for .and: Suppose that M satisfies the ○-triangle inequality for .
- (iii)
Suppose that M satisfies the ○-triangle inequality for , and that the mapping is continuous on .
Proof. To prove part (i), by taking the limit
to the inequality (Equation (
41)), we obtain Equation (
45). To prove part (ii), by taking the limit
to the inequalities (Equations (
42)–(
44)), we also obtain the desired results. Part (iii) follows from parts (i) and (ii) immediately. This completes the proof. □
Theorem 4. (Triangle Inequalities for Dual Double Fuzzy Semi-Metric). Let be a fuzzy semi-metric space along with a t-norm ∗. We also assume that M satisfies the rational condition, and that the t-norm ∗ is right-continuous at 0 and left-continuous at 1 with respect to the first or second argument. Given any distinct fixed and any fixed , we have the following properties:
- (i)
Suppose that M satisfies the ⋈-triangle inequality. There exists , satisfying: - (ii)
Suppose that M satisfies the ▹-triangle inequality. There exists , satisfying: - (iii)
Suppose that M satisfies the ◃-triangle inequality. There exists , satisfying: - (iv)
Suppose that M satisfies the ⋄-triangle inequality. There exists such that the following inequalities are satisfied:
If p is even and , then: If p is even and , then: If p is even and , then: If p is even and , then: If p is odd and , then: If p is odd and , then: If p is odd and , then: If p is odd and , then:
Proof. Lemma 1 says that there exists
, satisfying:
To prove part (i), we assume that
for all
. Given any
, the first observation of Remark 1 says that:
and:
Now, applying the increasing property and commutativity of t-norm to Equations (
57) and (
58), we obtain:
Therefore, we consider the following cases:
On the other hand, we also have:
and:
Now, applying the increasing property and commutativity of t-norm to Equations (
60) and (
61), we obtain:
The inequality (Equation (
47)) can be similarly obtained using the above argument. Further, the other inequalities can be similarly obtained.
The above argument is still valid by applying the second observation of Remark 1 to obtain part (ii). We can also apply the third observation of Remark 1 to obtain part (iii). Finally, part (iv) can be obtained using the fourth observation of Remark 1. This completes the proof. □
Theorem 5. Let be a fuzzy semi-metric space along with a t-norm ∗. We also assume that M satisfies the rational condition, and that the t-norm ∗ is right-continuous at 0 with respect to the first or second argument. Let and be two sequences in X. Then we have the following properties:
- (i)
Suppose that M satisfies the ○-triangle inequality for . Then the following statements hold true:
and as if and only if as for all .
and as if and only if as for all .
and as if and only if as for all .
and as if and only if as for all .
- (ii)
Suppose that M satisfies the ⋄-triangle inequality. Then the following statements hold true:
If and as , given any fixed , we have for all imply as .
If and as , given any fixed , we have for all imply as .
If and as , given any fixed , we have for all imply as .
If and as , given any fixed , we have for all imply as .
If as for all , then and as .
If as for all , then and as .
If as for all , then and as .
If as for all , then and as .
Proof. For any fixed
, using Lemma 1, it follows that there exists
, satisfying:
To prove part (i), we just prove the first case, since the other cases can be similarly obtained. Suppose that
and
as
for all
. Then, given any
and
, there exists
, satisfying
for
and
for
. Given any
, there exists
, satisfying
for
. We also have:
for
. The increasing property of t-norm says that:
The first result of part (ii) of Proposition 17 says that:
for
. This shows that
as
.
To prove the converse, suppose that
as
for all
. Given any
and
, there exists
, satisfying
for all
. For any fixed
and given any
, there exists
, satisfying:
for
, which implies:
for
by the first result of part (i) of Proposition 16, i.e.,
for
. Lemma 2 says that:
for
. This shows that the sequences
and
in
X converge to
x and
y, respectively.
To prove part (ii), the first result to the fourth result can be similarly obtained by the third result of part (ii) of Proposition 17. For proving the fifth result, the fact
implies the inequality (Equation (
62)). The third result of part (i) of Proposition 16 says that
, which implies
and
. In other words, we have
and
as
. The remaining three results can be similarly obtained. This completes the proof. □
Example 7. From Example 1, we see that:and: From Example 6, we have:where: It is clear to see that and as if and only if as for all . The other convergence presented in Theorem 5 can be similarly verified.
According to Definition 5, we can similarly define the concepts of four types of the joint Cauchy sequences with respect to . We omit the details.
Theorem 6. Let be a fuzzy semi-metric space along with a t-norm ∗. We also assume that M satisfies the rational condition, and that the t-norm ∗ is right-continuous at 0 with respect to the first or second argument. Let and be two sequences in X. Then we have the following properties:
- (i)
Suppose that M satisfies the ○-triangle inequality for . Then the following statements hold true:
and are two >-Cauchy sequences in a metric sense if and only if and are the joint -Cauchy sequences with respect to for any .
is a >-Cauchy sequences in a metric sense and is a <-Cauchy sequences in a metric sense if and only if and are the joint -Cauchy sequences with respect to for any .
is a <-Cauchy sequences in a metric sense and is a >-Cauchy sequences in a metric sense if and only if and are the joint -Cauchy sequences with respect to for any .
and are two <-Cauchy sequences in a metric sense if and only if and are the joint -Cauchy sequences with respect to for any .
- (ii)
Suppose that M satisfies the ⋄-triangle inequality. Then the following statements hold true:
Let and be two >-Cauchy sequences in a metric sense. Given any fixed , if for all , then and are the joint -Cauchy sequences with respect to for any .
Let be a >-Cauchy sequence in a metric sense and let be a <-Cauchy sequence in a metric sense. Given any fixed , if for all , then and are the joint -Cauchy sequences with respect to for any .
Let be a <-Cauchy sequence in a metric sense and let be a >-Cauchy sequence in a metric sense. Given any fixed , if for all , then and are the joint -Cauchy sequences with respect to for any .
Let and be two <-Cauchy sequences in a metric sense. Given any fixed , if for all , then and are the joint -Cauchy sequences with respect to for any .
Suppose that and are the joint -Cauchy sequences with respect to for any Then and are two >-Cauchy sequences in a metric sense.
Suppose that and are the joint -Cauchy sequences with respect to for any . Then is a <-Cauchy sequences in a metric sense and is a <-Cauchy sequences in a metric sense.
Suppose that and are the joint -Cauchy sequences with respect to for any Then is a <-Cauchy sequences in a metric sense and is a >-Cauchy sequences in a metric sense.
Suppose that and are the joint -Cauchy sequences with respect to for any . Then and are two <-Cauchy sequences in a metric sense.
Proof. For any fixed
, using Lemma 1, it follows that there exists
, satisfying:
To prove part (i), we just prove the first case, since the other cases can be similarly obtained. Suppose that
and
are >-Cauchy sequences in a metric sense. Therefore, given any
and
, there exists
such that
implies
and
. Now, given any
, there exists
such that
implies:
The increasing property of t-norm says that:
Further, the first result of part (ii) of Proposition 17 says that:
for
.
To prove the converse, from the assumption, we see that for any fixed
and given any
, there exists
such that
implies
. Therefore, using the first result of part (i) of Proposition 16, we obtain
for
, i.e.,
for
. Lemma 2 says that:
for
. This shows that
and
are >-Cauchy sequences in a metric sense.
To prove part (ii), the first result to the fourth result can be similarly obtained by the third result of part (ii) of Proposition 17. For proving the fifth result, using the assumption, we see that for any fixed
and given any
, there exists
such that
implies
. The third result of part (i) of Proposition 16 says that
for
. Therefore, we obtain:
for
. This shows that
and
are <-Cauchy sequences in a metric sense. The remaining three results can be similarly obtained. This completes the proof. □