Using the Supremum Form of Auxiliary Functions to Study the Common Coupled Coincidence Points in Fuzzy Semi-Metric Spaces

: This paper investigates the common coupled coincidence points and common coupled ﬁxed points in fuzzy semi-metric spaces. The symmetric condition is not necessarily satisﬁed in fuzzy semi-metric space. Therefore, four kinds of triangle inequalities are taken into account in order to study the Cauchy sequences. Inspired by the intuitive observations, the concepts of rational condition and distance condition are proposed for the purpose of simplifying the discussions.


Introduction
The common coupled coincidence points and common coupled fixed points in conventional metric spaces and probabilistic metric spaces have been studied for a long time in which the symmetric condition is satisfied. In this paper, we shall consider the fuzzy semi-metric space in which the symmetric condition is not satisfied. In this case, the role of triangle inequality should be re-interpreted. Therefore, four kinds of triangle inequalities are considered, which can also refer to Wu [1].
Schweizer and Sklar [2][3][4] introduced probabilistic metric space, in which the (conventional) metric space is associated with the probability distribution functions. For more details on the theory of probabilistic metric space, we can refer to Hadžić and Pap [5] and Chang et al. [6]. An interesting special kind of probabilistic metric space is the so-called Menger space. Kramosil and Michalek [7] proposed the fuzzy metric space based on the idea of Menger space. The definition of fuzzy metric space is presented below. Let X be a nonempty universal set associated with a t-norm * . Given a mapping M from X × X × [0, ∞) into [0, 1], the 3-tuple (X, M, * ) is called a fuzzy metric space when the following conditions are satisfied: • for any x, y ∈ X, M(x, y, t) = 1 for all t > 0 if and only if x = y; • M(x, y, 0) = 0 for all x, y ∈ X; • M(x, y, t) = M(y, x, t) for all x, y ∈ X and t ≥ 0; and, • M(x, y, t) * M(y, z, s) ≤ M(x, z, t + s) for all x, y, z ∈ X and s, t ≥ 0.
The mapping M in the fuzzy metric space (X, M, * ) can be treated as a membership function of a fuzzy subset of the product space X × X × [0, ∞). According to the first and second conditions of fuzzy metric space, the function value M(x, y, t) means that the membership degree of the distance that is less than or equal to t between x and y.
In this paper, we are going to consider the semi-metric space that is completely different from the fuzzy metric space. The so-called fuzzy semi-metric space does not assume the symmetric condition M(x, y, t) = M(y, x, t). Without this condition, the concept of triangle inequalities should be carefully treated. In this paper, there are four kinds of different triangle inequalities considered. It will be realized that, when the symmetric condition is satisfied, these four different kinds of triangle inequalities will be equivalent to the classical one. Being inspired by the intuitive observations, the concepts of rational condition and distance condition are proposed for the purpose of simplifying the discussions regarding the common coupled coincidence points and common coupled fixed points in a fuzzy semi-metric space.
Rakić et al. [8,9] studied the fixed points in b-fuzzy metric spaces. Mecheraoui et al. [10] obtained the sufficient condition for a G-Cauchy sequence to be an M-Cauchy sequence in fuzzy metric space. On the other hand, Gu and Shatanawi [11] used the concept of w-compatible mappings for studying the common coupled fixed points of two hybrid pairs of mappings in partial metric spaces. Petruel [12,13] studied the fixed point for graphic contractions and fixed point for multi-valued locally contractive operators. Hu et al. [14], Mohiuddine and Alotaibi [15], Qiu and Hong [16], and the references therein studied the common coupled coincidence points and common coupled fixed points in fuzzy metric spaces. Wu [17] also studied the common coincidence points in fuzzy semi-metric spaces. In this paper, the common coupled coincidence points and common coupled fixed points in fuzzy semi-metric spaces will be studied by considering four kinds of triangle inequalities. Although the common coupled fixed points are the common coupled coincidence points, the sufficient conditions will be completely different when considering the uniqueness.
This paper is organized, as follows. In Section 2, the concept of fuzzy semi-metric spaces will be introduced. Because the symmetric condition is not satisfied, four different kinds of triangle inequalities will be taken into account to study the common coupled fixed points. In Section 3, in order to study the Cauchy sequence in fuzzy semi-metric space, the auxiliary functions that are based on the supremun are proposed. In Section 4, while using the auxiliary functions proposed in Section 3, the desired property regarding the Cauchy sequence in fuzzy semi-metric space will be presented. In Section 5, many kinds of common coupled coincidence points in fuzzy semi-metric spaces will be investigated by considering the four different kinds of triangle inequalities. Finally, in Section 6, the common coupled fixed points shown in fuzzy semi-metric spaces will also be studied based on the four different kinds of triangle inequalities.

Fuzzy Semi-Metric Spaces
The concept of fuzzy semi-metric space is based on the concept of t-norm (triangular norm), which will be introduced below. Let * : [0, 1] × [0, 1] → [0, 1] be a function that is defined on the product set [0, 1] × [0, 1]. We say that * is a t-norm when the following conditions are satisfied: The following properties regarding t-norm will be used in the further study.

Proposition 1.
We have the following properties.
(i) Suppose that the t-norm * is left-continuous at 1 with respect to the first or second component. For any a, b ∈ (0, 1) with a > b, there exists r ∈ (0, 1) that satisfies a * r ≥ b. (ii) Suppose that the t-norm * is left-continuous at 1 with respect to the first or second component.
(iii) Given any fixed a, b ∈ [0, 1], suppose that the t-norm * is continuous at a and b with respect the first or second component, and that {a n } ∞ n=1 and {b n } ∞ n=1 are two sequences in [0, 1] satisfying a n → a and b n → b as n → ∞. Subsequently, we have a n * b n → a * b as n → ∞. (iv) Given any fixed a, b ∈ (0, 1], suppose that the t-norm * is left-continuous at a and b with respect to the first or second component, and that {a n } ∞ n=1 and {b n } ∞ n=1 are two sequences in [0, 1] satisfying a n → a− and b n → b− as n → ∞. Afterwards, we have a n * b n → a * b as n → ∞. (v) Given any fixed a, b ∈ [0, 1), suppose that the t-norm * is right-continuous at a and b with respect to the first or second component, and that {a n } ∞ n=1 and {b n } ∞ n=1 are two sequences in [0, 1] satisfying a n → a+ and b n → b+ as n → ∞. Subsequently, we have a n * b n → a * b as n → ∞.
Wu [1,17,18] proposed the concept of fuzzy semi-metric space. The formal definition is given below. Definition 1. Let X be a nonempty set and let M be a mapping from We say that (X, M) is fuzzy semi-metric space when the following conditions are satisfied: • for any x, y ∈ X, M(x, y, t) = 1 for all t ≥ 0 if and only if x = y; • M(x, y, 0) = 0 for all x, y ∈ X with x = y; The mapping M is said to satisfy the symmetric condition when M(x, y, t) = M(y, x, t) for any x, y ∈ X and t ≥ 0.

Definition 2.
Let (X, M) be a fuzzy semi-metric space. We say that M satisfies the distance condition when, for any x, y ∈ X with x = y, there exists t 0 > 0, such that M(x, y, t 0 ) = 0.
Because the symmetric condition is not necessarily be satisfied in fuzzy semi-metric space (X, M), by referring to Wu [1,17,18], four kinds of triangle inequalities are proposed below.
Definition 3. Let X be a nonempty set, let * be a t-norm, and let M be a mapping that is defined

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We say that M satisfies the -triangle inequality when the following inequality is satisfied: M(x, y, t) * M(y, z, s) ≤ M(x, z, t + s) for all x, y, z ∈ X and s, t > 0.
• We say that M satisfies the -triangle inequality when the following inequality is satisfied: M(x, y, t) * M(z, y, s) ≤ M(x, z, t + s) for all x, y, z ∈ X and s, t > 0.
• We say that M satisfies the -triangle inequality when the following inequality is satisfied: M(y, x, t) * M(y, z, s) ≤ M(x, z, t + s) for all x, y, z ∈ X and s, t > 0.
• We say that M satisfies the -triangle inequality when the following inequality is satisfied: for all x, y, z ∈ X and s, t > 0.

Remark 1.
Suppose that the mapping M satisfies the -triangle inequality. Subsequently, we have and In general, we have For the case of satisfying the -triangle inequality, -triangle inequality and -triangle inequality, we can refer to Wu [17].
Proposition 2 (Wu [1]). Let (X, M) be a fuzzy semi-metric space. Then we have the following properties.
(i) Suppose that the mapping M satisfies the -triangle inequality. Subsequently, M is nondecreasing in the sense of M(x, y, t 1 ) ≥ M(x, y, t 2 ) for any fixed x, y ∈ X and t 1 > t 2 . (ii) Suppose that the mapping M satisfies the -triangle inequality. Subsequently, M is symmetrically non-decreasing in the sense of M(x, y, t 1 ) ≥ M(y, x, t 2 ) for any fixed x, y ∈ X and t 1 > t 2 . (iii) Suppose that the mapping M satisfies the -triangle inequality or the -triangle inequality.
Afterwards, M is both non-decreasing and symmetrically non-decreasing.
Let {x n } ∞ n=1 be a sequence in the fuzzy semi-metric space (X, M).
Proposition 3 (Wu [17]). Let (X, M) be a fuzzy semi-metric space, and let {x n } ∞ n=1 be a sequence in X. Suppose that the t-norm * is left-continuous at 1 with respect to the first or second component. Afterwards, we have the following results.
(i) Assume that the mapping M satisfies the -triangle inequality or -triangle inequality.
Subsequently, we have the following properties.  Proposition 4 (Wu [18]). Let (X, M) be a fuzzy semi-metric space, and let {(x n , y n , t n )} ∞ n=1 be a sequence in X × X × (0, ∞). Assume that the t-norm * is left-continuous with respect to the first or second component. For any sequences {a n } ∞ n=1 and {b n } ∞ n=1 in [0, 1], we also assume that the following inequality is satisfied sup n (a n * b n ) ≥ sup n a n * sup n b n .
(i) Suppose that M satisfies the -triangle inequality, and that t n → t • , x n M −→ x • and y n M −→ y • as n → ∞. Subsequently, the following statements hold true.

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If M is continuous with respect to the distance at t • , then M(x n , y n , t n ) → M(x • , y • , t • ) as n → ∞.
• If M is symmetrically continuous with respect to the distance at t • , then M(x n , y n , t n ) → M(y • , x • , t • ) as n → ∞.
(ii) Suppose that M satisfies the •-triangle inequality for • ∈ { , }, and that t n → t • , x n M −→ x • and y n M −→ y • as n → ∞. If M is continuous or symmetrically continuous with respect to the distance at t • , then M(x n , y n , (iii) Suppose that M satisfies the -triangle inequality, and that t n → t • as n → ∞, x n M −→ x • , and y n M −→ y • as n → ∞ simultaneously, or x n M −→ x • and y n M −→ y • as n → ∞ simultaneously. If M is continuous or symmetrically continuous with respect to the distance at t • , then M(x n , y n , n=1 be a sequence in the fuzzy semi-metric space (X, M).

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We say that {x n } ∞ n=1 is a >-Cauchy sequence when, given any pair (r, t) with t > 0 and 0 < r < 1, there exists n r,t ∈ N satisfying M(x m , x n , t) > 1 − r for all pairs (m, n) of integers m and n with m > n ≥ n r,t .

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We say that {x n } ∞ n=1 is a <-Cauchy sequence when, given any pair (r, t) with t > 0 and 0 < r < 1, there exists n r,t ∈ N satisfying M(x n , x m , t) > 1 − r for all pairs (m, n) of integers m and n with m > n ≥ n r,t .

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We say that {x n } ∞ n=1 is a Cauchy sequence when, given any pair (r, t) with t > 0 and 0 < r < 1, there exists n r,t ∈ N satisfying M(x m , x n , t) > 1 − r and M(x n , x m , t) > 1 − r for all pairs (m, n) of integers m and n with m, n ≥ n r,t and m = n.

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We say that (X, M) is (>, )-complete when each >-Cauchy sequence {x n } ∞ n=1 is convergent in the sense of x n Let (X, M) be a fuzzy semi-metric space along with a t-norm * . We define the mapping η : Subsequently, we have the following interesting result that will be used to define the auxiliary functions.
Proposition 5. Let (X, M) be a fuzzy semi-metric space, such that the mapping M satisfies the rational condition. Suppose that the t-norm * is right-continuous at 0 with respect to the first or second component. Subsequently, we have lim t→0+ η(x, y, u, v, t) = 0. (1) The following definition of auxiliary functions are based on X 4 . This new concept extends the auxiliary functions based on X 2 , as proposed by Wu [17].

Definition 7.
Let (X, M) be a fuzzy semi-metric space, such that M satisfies the rational condition in which the t-norm * is also right-continuous at 0 with respect to the first or second component. For any fixed x, y, u, v ∈ X and λ ∈ [0, 1) with x = y or u = v, we define a function Φ : By definition, we must have η(x, y, u, v, t) > 1 − λ for all t > 0. This says that lim t→0+ η(x, y, u, v, t) ≥ 1 − λ, which contradicts (1). Therefore, we indeed have {t > 0 : η(x, y, u, v, t) ≤ 1 − λ} = ∅, which says that the function Φ is well-defined. Proposition 6. Let (X, M) be a fuzzy semi-metric space such that the mapping M satisfies the rational condition in which the t-norm * is right-continuous at 0 with respect to the first or second component. Given any fixed x, y, u, v ∈ X and λ ∈ (0, 1), we have the following properties.
have the following properties.
• If the mapping M satisfies the -triangle inequality or the -triangle inequality or the -triangle inequality, then • If the mapping M satisfies the -triangle inequality or the -triangle inequality, then • If the mapping M satisfies the -triangle inequality or the -triangle inequality or the -triangle inequality, then Proof. The proof is similar to the argument in Wu [17] by considering X 4 instead of X 2 .
Proposition 7. Let (X, M) be a fuzzy semi-metric space, such that the mapping M satisfies the rational condition in which the t-norm * is right-continuous at 0 with respect to the first or second component. Given any fixed x, y, u, v ∈ X and λ ∈ (0, 1), we have the following properties.
• If the mapping M satisfies the -triangle inequality or the -triangle inequality or the -triangle inequality, then t ≤ Φ(λ, x, y, u, v).

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If the mapping M satisfies the -triangle inequality or the -triangle inequality or the -triangle inequality, then t ≤ Φ(λ, y, x, v, u).
(ii) We have the following results.
Proof. The proof is similar to the argument in Wu [17] by considering X 4 instead of X 2 .
Proposition 8. Let (X, M) be a fuzzy semi-metric space, such that M satisfies the rational condition, in which the t-norm * is right-continuous at 0 and left-continuous at 1 with respect to the first or second component.
• If p is even and Φ(µ, • If p is even and Φ(µ, x p , x 1 , y 1 , y p ) < ∞, then • If p is even and Φ(µ, • If p is odd and Φ(µ, Proof. The proof is similar to the argument put foward in Wu [17] by considering X 4 instead of X 2 .

Proposition 9.
Let (X, M) be a fuzzy semi-metric space, such that M satisfies the rational condition, in which the t-norm * is right-continuous at 0 with respect to the first or second component. Let {x n } ∞ n=1 and {y n } ∞ n=1 be two sequences in X. (i) Assume that M satisfies the -triangle inequality or the -triangle inequality or the -triangle inequality. Subsequently, we have the following results.
(ii) Assume that M satisfies the -triangle inequality. Then, we have the following results.
Proof. The proof is similar to the argument in Wu [17] by considering X 4 instead of X 2 .

Cauchy Sequences
Given any a ∈ [0, 1], for convenience, we write ( * a) n = n times a * a * · · · * a and * η a, b, c, d, t k n The following results will be used for further discussion. (i) Suppose that M satisfies the -triangle inequality. Subsequently, we have the following results.
To prove part (ii), we consider two cases below.
• Suppose that the mapping M satisfies the -triangle inequality. While using part (ii) of Proposition 8, we have By referring to (19), we can similarly obtain By using (19), (22), (21) and referring to (20), we have Using the above argument, we can show that {x n } ∞ n=1 and {y n } ∞ n=1 are both >-Cauchy and <-Cauchy sequences in metric sense.

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Assume that the conditions (10), (11), (16) and (17) are satisfied. If p is even, then, using (2) and (5) in part (iv) of Proposition 8, we can similarly show that {x n } ∞ n=1 and {y n } ∞ n=1 are both >-Cauchy and <-Cauchy sequences in metric sense. If p is odd, then, using (6) and (9) in Proposition 8, we can similarly obtain the desired results. • Assume that the conditions (12), (13), (14) and (15) are satisfied. If p is even, then, using (3) and (4) in part (iv) of Proposition 8, we can similarly show that {x n } ∞ n=1 and {y n } ∞ n=1 are both >-Cauchy and <-Cauchy sequences in the metric sense. If p is odd, then, using (7) and (8) in Proposition 8, we can similarly obtain the desired results. This completes the proof.

Common Coupled Coincidence Points
In this section, we are going to investigate the common coupled coincidence points in fuzzy semi-metric space under some suitable conditions. We consider two mappings T : X × X → X and f : X → X.

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Recall that the mappings T and f commute when Recall that an element (x * , y * ) ∈ X × X is called a coupled coincidence point of mappings T and f when T(x * , y * ) = f (x * ) and T(y * , x * ) = f (y * ). In particular, if x * = f (x * ) = T(x * , y * ) and y * = f (y * ) = T(y * , x * ), then (x * , y * ) is called a common coupled fixed point of T and f .
Let {T n } ∞ n=1 be a sequence of mappings from the product space X × X into X, and let f be a mapping from X into itself satisfying T n (X, X) ⊆ f (X) for all n ∈ N. Given any two initial elements x 0 , y 0 ∈ X, since T n (X, X) ⊆ f (X), there exist x 1 , y 1 ∈ X satisfying f (x 1 ) = T 1 (x 0 , y 0 ) and f (y 1 ) = T 1 (y 0 , x 0 ).
Continuing this process, we can construct two sequences {x n } ∞ n=1 and {y n } ∞ n=1 , satisfying f (x n ) = T n (x n−1 , y n−1 ) and f (y n ) = T n (y n−1 , x n−1 ) for n ∈ N.
In the sequel, the common coupled coincidence points will be separately studied by considering the four different types of triangle inequalities.
Theorem 1 (Satisfying the -Triangle Inequality). Let (X, M) be a fuzzy semi-metric space, such that the mapping M satisfies the rational condition and the -triangle inequality. Suppose that the following conditions are satisfied.

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The t-norm * is left-continuous with respect to the first or second component.

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The mappings T n : X × X → X and f : X → X satisfy the inclusions T n (X, X) ⊆ f (X) for all n ∈ N.
• The mappings f and T n commute; that is, f (T n (x, y)) = T n ( f (x), f (y)) for all x, y ∈ X and all n ∈ N. • Given any x, y, u, v ∈ X, the following contractive inequality is satisfied: where k ij satisfies 0 < k ij ≤ k < 1 for all i, j ∈ N and for some constant k. Subsequently, we have the following results.
(i) Suppose that there exist x * , y * ∈ X satisfying sup λ∈(0,1) and that any one of the following conditions is satisfied: Afterwards, the mappings {T n } ∞ n=1 and f have a common coupled coincidence point (x • , y • ). We further assume that the following conditions are satisfied.

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The inequality (24) is replaced by the following inequality where the t-norm * is replaced by the product of real numbers.

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The mapping M satisfies the distance condition in Definition 2.

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For any fixed x, y ∈ X and t > 0, the following mapping is differentiable on (0, ∞). Afterwards, we have the following results.

(A)
Suppose that (x,ȳ) is another coupled coincidence point of mappings f and T n 0 for some n 0 ∈ N. Subsequently, f (x • ) = f (x) and f (y • ) = f (ȳ).

(B)
There exists (x • , y • ) ∈ X × X such that ( f (x • ), f (y • )) ∈ X × X is the common coupled fixed point of the mappings {T n } ∞ n=1 . Moreover, the point (x • , y • ) ∈ X × X can be obtained, as follows.

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Suppose that condition (a) is satisfied. Afterwards, the point (x • , y • ) ∈ X × X can be obtained by taking the limit f (x n ) The sequences {x n } ∞ n=1 and {y n } ∞ n=1 are generated from the initial element (x 0 , y 0 ) = (x * , y * ) ∈ X × X according to (23).
(ii) Suppose that there exist x * , y * ∈ X satisfying sup λ∈(0,1) and that any one of the following conditions is satisfied: Afterwards, we have the same result as part (i).
Proof. We can generate two sequences {x n } ∞ n=1 and {y n } ∞ n=1 from the initial element x 0 = x * and y 0 = y * according to (23). Then we have To prove part (i), from (23) and (24), we obtain and By induction, we can obtain and Part (i) of Proposition 2 says that the mapping M(x, y, ·) is nondecreasing. Because k i,i+1 ≤ k for each i ∈ N, using the increasing property of t-norm to (27) and (28), we also have and Using the increasing property of t-norm to (29) and (30), we have From part (i) of Proposition 10, it follows that { f (x n )} ∞ n=1 and { f (y n )} ∞ n=1 are <-Cauchy sequences. We consider the following cases Since f is simultaneously ( , )-continuous and ( , )-continuous with respect to M, which say that, for all t > 0, Because f is simultaneously ( , )-continuous and ( , )-continuous with respect to M, we can similarly obtain (33)-(36).
Using Proposition 1 and applying (33) and (34) to (39), we obtain which says that Therefore, we obtain Using the -triangle inequality, we see that While using the left-continuity of t-norm * to (35) and (40), we obtain M( f (x • ), T n (x • , y • ), 2t) = 1 for all t > 0. Therefore we must have f (x • ) = T n (x • , y • ) for all n ∈ N. We can similarly show that f (y • ) = T n (y • , x • ) for all n ∈ N.
To prove property (A), let (x,ȳ) be another coupled coincidence point of f and T n 0 for some n 0 ∈ N, i.e., f (x) = T n 0 (x,ȳ) and f (ȳ) = T n 0 (ȳ,x). Because the mapping M(x, y, ·) is non-decreasing, by (25), we have and Therefore we obtain (41) and (42)) (by repeating to use (41) and (42)) (since M(x, y, t) ≤ 1 for any x, y ∈ X and t > 0), which can be rewritten as We are going to claim that there existst > 0, such that M( f (y • ), f (ȳ), t) = 0 for all t ≥t. We consider the following two cases.
, then the distance condition says that there exitst > 0 such that M( f (y • ), f (ȳ),t) = 0. Part (i) of Proposition 2 says that the mapping M(x, y, ·) is nondecreasing. It follows that M( f (y • ), f (ȳ), t) = 0 for all t ≥t. Therefore, from (44), for any fixed t > 0 with t ≥t, we have Because 0 < k < 1 and the mapping M(x, y, ·) is non-decreasing, the function defined in (26) is non-increasing, which says that (α) ≤ 0 on (0, ∞). Because M satisfies the rational condition, we have lim for any fixed x, y ∈ X with x = y. We consider (using the l'Hospital's rule) To prove property (B), using the commutativity of T n and f , we have and By regardingx as T n (x • , y • ) andȳ as T n (y • , x • ), the equalities (48) and (49) say that Therefore, using property (A), we must have which says that ( f (x • ), f (y • )) ∈ X × X is the common coupled fixed point of the mappings {T n } ∞ n=1 .
To prove part (ii), we can similarly obtain From part (i) of Proposition 10, it follows that { f (x n )} ∞ n=1 and { f (y n )} ∞ n=1 are >-Cauchy sequences. We consider two cases below.

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Suppose that condition (c) is satisfied.
The remaining proof follows from the similar argument of part (i). This completes the proof.
In Theorem 1, since the fuzzy semi-metric M is not necessarily symmetric, if the contractive inequalities (24) and (25) are not satisfied and, alternatively, the following converse-contractive inequalities and are satisfied, then we can also obtain the desired results by assuming the different conditions.
Theorem 2 (Satisfying the -Triangle Inequality: Converse-Contractive Inequality). Let (X, M) be a fuzzy semi-metric space, such that the mapping M satisfies the rational condition and the -triangle inequality. Suppose that the following conditions are satisfied.
• The first four conditions in Theorem 1 are satisfied.

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For any x, y, u, v ∈ X, the following converse-contractive inequality is satisfied: where k ij satisfies 0 < k ij ≤ k < 1 for all i, j ∈ N and for some constant k. Subsequently, we have the following results.
(i) Suppose that there exist x * , y * ∈ X satisfying sup λ∈(0,1) and that any one of the following conditions is satisfied: Subsequently, the mappings {T n } ∞ n=1 and f have a common coupled coincidence point (x • , y • ). We further assume that the following conditions are satisfied.

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The inequality (51) is replaced by the following inequality where the t-norm * is replaced by the product of real numbers; • The mapping M satisfies the distance condition in Definition 2.

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For any fixed x, y ∈ X and t > 0, the following mapping (α) = M x, y, k log 2 α · t is differentiable on (0, ∞). Afterwards, we have the following results.

(A)
Suppose that (x,ȳ) is another coupled coincidence point of f and T n 0 for some n 0 ∈ N.
) ∈ X × X is the common coupled fixed point of the mappings {T n } ∞ n=1 . Moreover, the point (x • , y • ) ∈ X × X can be obtained, as follows.

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Suppose that condition (a) is satisfied. Then the point (x • , y • ) ∈ X × X can be obtained by taking the limit f (x n ) The sequences {x n } ∞ n=1 and {y n } ∞ n=1 are generated from the initial element (x 0 , y 0 ) = (x * , y * ) ∈ X × X, according to (23).
(ii) Suppose that there exist x * , y * ∈ X satisfying sup λ∈(0,1) and that any one of the following conditions is satisfied: Subsequently, we have the same result as part (i).
Theorem 3 (Satisfying the -Triangle Inequality). Let (X, M) be a fuzzy semi-metric space, such that the mapping M satisfies the rational condition and the -triangle inequality. Let (x 0 , y 0 ) ∈ X × X be an initial element that generates the sequences {x n } ∞ n=1 and {y n } ∞ n=1 according to (23). Suppose that the following conditions are satisfied.

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The first four conditions in Theorem 1 are satisfied.

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The following contractive inequalities is satisfied or the following converse-contractive inequalities is satisfied where k ij satisfies 0 < k ij ≤ k < 1 for all i, j ∈ N and for some constant k.

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There exist x * , y * ∈ X satisfying sup λ∈(0,1) • Any one of the following conditions is satisfied: Subsequently, the mappings {T n } ∞ n=1 and f have a common coupled coincidence point (x • , y • ). We further assume that the following conditions are satisfied.

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The inequality (52) is replaced by the following inequality and the inequality (53) is replaced by the following inequality where the t-norm * is replaced by the product of real numbers, such that any one of the inequalities (54) and (55) is satisfied.

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The mapping M satisfies the distance condition in Definition 2.

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For any fixed x, y ∈ X and t > 0, the following mapping (α) = M x, y, k log 2 α · t is differentiable on (0, ∞). Afterwards, we have the following results.
(A) Suppose that (x,ȳ) is another coupled coincidence point of f and T n 0 for some n 0 ∈ N. Subsequently, f (x • ) = f (x) and f (y • ) = f (ȳ). (B) There exists (x • , y • ) ∈ X × X, such that ( f (x • ), f (y • )) ∈ X × X is the common coupled fixed point of the mappings {T n } ∞ n=1 . Moreover, the point (x • , y • ) ∈ X × X can be obtained as follows.

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Suppose that condition (a) is satisfied. Subsequently, the point (x • , y • ) ∈ X × X can be obtained by taking the limit f (x n ) The sequences {x n } ∞ n=1 and {y n } ∞ n=1 are generated from the initial element (x 0 , y 0 ) = (x * , y * ) ∈ X × X according to (23). Theorem 4 (Satisfying the -Triangle Inequality). Let (X, M) be a fuzzy semi-metric space such that the mapping M satisfies the rational condition and the -triangle inequality. Let (x 0 , y 0 ) ∈ X × X be an initial element that generates the sequences {x n } ∞ n=1 and {y n } ∞ n=1 according to (23). Suppose that the following conditions are satisfied.

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The first four conditions in Theorem 1 are satisfied.

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The following contractive inequalities is satisfied or the following converse-contractive inequalities is satisfied where k ij satisfies 0 < k ij ≤ k < 1 for all i, j ∈ N and for some constant k.

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There exist x * , y * ∈ X satisfying sup λ∈(0,1) • Any one of the following conditions is satisfied: Subsequently, the mappings {T n } ∞ n=1 and f have a common coupled coincidence point (x • , y • ). We further assume that the following conditions are satisfied.

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The inequality (56) is replaced by the following inequality and the inequality (57) is replaced by the following inequality where the t-norm * is replaced by the product of real numbers, such that any one of the inequalities (58) and (59) is satisfied.

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The mapping M satisfies the distance condition in Definition 2.

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For any fixed x, y ∈ X and t > 0, the following mapping (α) = M x, y, k log 2 α · t is differentiable on (0, ∞). Subsequently, we have the following results.
(A) Suppose that (x,ȳ) is another coupled coincidence point of f and T n 0 for some n 0 ∈ N.
) ∈ X × X is the common coupled fixed point of the mappings {T n } ∞ n=1 . Moreover, the point (x • , y • ) ∈ X × X can be obtained, as follows.

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Suppose that condition (a) is satisfied. Afterwards, the point (x • , y • ) ∈ X × X can be obtained by taking the limit f (x n ) The sequences {x n } ∞ n=1 and {y n } ∞ n=1 are generated from the initial element (x 0 , y 0 ) = (x * , y * ) ∈ X × X according to (23).
Theorem 5 (Satisfying the -Triangle Inequality). Let (X, M) be a fuzzy semi-metric space, such that the mapping M satisfies the rational condition and the -triangle inequality. Suppose that the following conditions are satisfied.

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All five conditions in Theorem 1 are satisfied.

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There exist x * , y * ∈ X satisfying sup λ∈(0,1) • Any one of the following conditions is satisfied: Subsequently, the mappings {T n } ∞ n=1 and f have a common coupled coincidence point (x • , y • ). Moreover, the point (x • , y • ) ∈ X × X can be obtained, as follows.

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Suppose that condition (a) is satisfied. Afterwards, the point (x • , y • ) ∈ X × X can be obtained by taking the limit f (x n ) The sequences {x n } ∞ n=1 and {y n } ∞ n=1 are generated from the initial element (x 0 , y 0 ) = (x * , y * ) ∈ X × X according to (23).
Theorem 6 (Satisfying the -Triangle Inequality: Converse-Contractive Inequality). Let (X, M) be a fuzzy semi-metric space, such that the mapping M satisfies the rational condition and the -triangle inequality. Let (x 0 , y 0 ) ∈ X × X be an initial element that generates the sequences {x n } ∞ n=1 and {y n } ∞ n=1 according to (23). Suppose that the following conditions are satisfied. • The first four conditions in Theorem 1 are satisfied.

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For any x, y, u, v ∈ X, the following converse-contractive inequality is satisfied: where k ij satisfies 0 < k ij ≤ k < 1 for all i, j ∈ N and for some constant k.

Common Coupled Fixed Points
Consider the mappings T : X × X → X and f : X → X. Recall that an element (x * , y * ) ∈ X × X is called a common coupled fixed point when x * = f (x * ) = T(x * , y * ) and y * = f (y * ) = T(y * , x * ).
The common coupled fixed points are also the common coupled coincidence points. The uniqueness of common coupled coincidence points presented above was not guaranteed. In this section, we shall investigate the uniqueness of common coupled fixed points.
The contractive inequality and converse-contractive inequality should consider the product of real numbers instead of t-norm * in order to obtain the unique common coupled fixed point.
Theorem 7 (Satisfying the -Triangle Inequality). Let (X, M) be a fuzzy semi-metric space, such that the mapping M satisfies the rational condition and the -triangle inequality. Suppose that the following conditions are satisfied.

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For any sequences {a n } ∞ n=1 and {b n } ∞ n=1 in [0, 1], the following inequality is satisfied: sup n (a n * b n ) ≥ sup n a n * sup n b n .
• The t-norm * is left-continuous with respect to the first or second component.

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Given any fixed x, y ∈ X, the mapping M(x, y, ·) The mapping M satisfies the distance condition in Definition 2.

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The mappings T n : X → X and f : X → X satisfy the inclusion T n (X, X) ⊆ f (X) for all n ∈ N. • The mappings f and T n commute. • Any one of the following conditions is satisfied: the mapping f is simultaneously ( , )-continuous and ( , )-continuous with respect to M; -the mapping f is simultaneously ( , )-continuous and ( , )-continuous with respect to M.
• for any x, y, u, v ∈ X, the following contractive inequality is satisfied: where k ij satisfies 0 < k ij ≤ k < 1 for all i, j ∈ N and for some constant k. Subsequently, we have the following results.
(ii) Suppose that the space (X, M) is simultaneously (>, )-complete and (>, )-complete. We also assume that there exist x * , y * ∈ X satisfying sup λ∈(0,1) Then the mappings {T n } ∞ n=1 and f have a unique common coupled fixed point (x • , y • ). Moreover, the point (x • , y • ) ∈ X × X can be obtained as follows.

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The point x • can be obtained by taking the limit f (x n ) which is equivalent to We are going to claim that there existst > 0, such that M(y • , f (y • ), t) = 0 for all t ≥t. We consider the following cases.
If f (y • ) = y • , then the distance condition says that there exitst > 0, such that M(y • , f (y • ),t) = 0. Part (i) of Proposition 2 says that the mapping M(x, y, ·) is nondecreasing. Therefore, we have M(y • , f (y • ), t) = 0 for all t ≥t. From (63), for any fixed t > 0 with t ≥t, we have Because 0 < k < 1 and the mapping M(x, y, ·) is nondecreasing, the function that is defined in (26) is non-increasing, which says that (α) ≤ 0 on (0, ∞). Because M satisfies the rational condition, we have lim t→0+ M(x, y, t) = 0 (65) for any fixed x, y ∈ X with x = y. We consider Therefore, (65) says that (α) → 0+ as α → ∞. Subsequently, we obtain Applying (66) to (64), we obtain M(x • , f (x • ), t) = 0 for all t ≥t. Because f (x • ) = x • , the distance condition says that there exits t 0 > 0 such that M(x • , f (x • ), t 0 ) = 0, i.e., M(x • , f (x • ), t) = 0 for all t ≥ t 0 by the nondecreasing property of M(x, y, ·), which contradicts M(x • , f (x • ), t) = 0 for all t ≥t. Therefore we must have f (x • ) = x • . We can similarly obtain f (y • ) = y • ; that is, In order to prove the uniqueness, let (x,ȳ) be another common coupled fixed point of f and {T n } ∞ n=1 , i.e.,x = f (x) = T n (x,ȳ) andȳ = f (ȳ) = T n (ȳ,x) for all n ∈ N. The inequality (43) is equivalent to We can similarly show that there exists t > 0, such that M(y • ,ȳ, t) = 0 for all t ≥ t. Therefore, from (67), for any fixed t > 0 with t ≥ t, we have By referring to (68), it follows that M(x • ,x, t) = 0 for all t ≥ t. Becausex = x • , the distance condition says that there exits t 0 > 0, such that M(x • ,x, t 0 ) = 0, i.e., M(x • ,x, t) = 0 for all t ≥ t 0 by the non-decreasing property of M(x, y, ·), which contradicts M(x • ,x, t) = 0 for all t ≥ t. Therefore, we must havex = x • . We can similarly obtainȳ = y • . This proves the uniqueness. Finally, part (ii) can be obtained by applying part (ii) of Theorem 1 to the above argument. This completes the proof.
Theorem 8 (Satisfying the -Triangle Inequality: Converse-Contractive Inequality). Let (X, M) be a fuzzy semi-metric space such that the mapping M satisfies the rational condition and the -triangle inequality. Suppose that the following conditions are satisfied.

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The first eight conditions of Theorem 7 are satisfied.

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For any x, y, u, v ∈ X, the following converse-contractive inequality is satisfied: where k ij satisfies 0 < k ij ≤ k < 1 for all i, j ∈ N and for some constant k. Subsequently, we have the following results.
Moreover, the point (x • , y • ) ∈ X × X can be obtained, as follows.

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The point x • can be obtained by taking the limit f (x n ) • The point y • can be obtained by taking the limit f (y n ) M −→ y • or the limit f (y n ) The sequences {x n } ∞ n=1 and {y n } ∞ n=1 are generated from the initial element (x 0 , y 0 ) = (x * , y * ) ∈ X × X, according to (23). Theorem 9 (Satisfying the -Triangle Inequality). Let (X, M) be a fuzzy semi-metric space, such that the mapping M satisfies the rational condition and the -triangle inequality. Suppose that the following conditions are satisfied.

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The first eight conditions of Theorem 7 are satisfied.

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The following contractive inequalities is satisfied or the following converse-contractive inequalities is satisfied where k ij satisfies 0 < k ij ≤ k < 1 for all i, j ∈ N and for some constant k.

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The point x • can be obtained by taking the limit f (x n ) • The point y • can be obtained by taking the limit f (y n ) The sequences {x n } ∞ n=1 and {y n } ∞ n=1 are generated from the initial element (x 0 , y 0 ) = (x * , y * ) ∈ X × X according to (23).
Theorem 10 (Satisfying the -Triangle Inequality). Let (X, M) be a fuzzy semi-metric space, such that the mapping M satisfies the rational condition and the -triangle inequality. Suppose that the following conditions are satisfied.

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The first eight conditions of Theorem 7 are satisfied.

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The following contractive inequalities is satisfied or the following converse-contractive inequalities are satisfied where k ij satisfies 0 < k ij ≤ k < 1 for all i, j ∈ N and for some constant k.
• The mapping f is ( , )-continuous or ( , )-continuous with respect to M. Subsequently, the mappings {T n } ∞ n=1 and f have a unique common coupled fixed point (x • , y • ). Moreover, the point (x • , y • ) ∈ X × X can be obtained, as follows.

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The point x • can be obtained by taking the limit f (x n ) • The point y • can be obtained by taking the limit f (y n ) M −→ y • or the limit f (y n ) M −→ y • .
Theorem 11 (Satisfying the -Triangle Inequality). Let (X, M) be a fuzzy semi-metric space such that the mapping M satisfies the rational condition and the -triangle inequality. Suppose that the following conditions are satisfied.

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All nine conditions of Theorem 7 are satisfied.
Theorem 12 (Satisfying the -Triangle Inequality: Converse-Contractive Inequality). Let (X, M) be a fuzzy semi-metric space, such that the mapping M satisfies the rational condition and the -triangle inequality. Suppose that the following conditions are satisfied.

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The first eight conditions of Theorem 7 are satisfied.

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The following converse-contractive inequalities are satisfied M(T i (x, y), T j (u, v), k ij · t) ≥ M( f (u), f (x), t) · M( f (v), f (y), t), where k ij satisfies 0 < k ij ≤ k < 1 for all i, j ∈ N and for some constant k.
• Any one of the following conditions is satisfied: Subsequently, the mappings T and f have a unique common coupled fixed point (x • , y • ) Moreover, the point (x • , y • ) ∈ X × X can be obtained, as follows.

Conclusions
Four different kinds of triangle inequalities play the important role of studying the common coupled coincidence points and common coupled fixed points in fuzzy semimetric spaces. We separately present the theorems of common coupled coincidence points that are based on the different kinds of triangle inequalities.

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Suppose that the fuzzy semi-metric space satisfies the -triangle inequality. Theorem 1 studies the common coupled coincidence points. Because the symmetric condition is not satisfied. Theorem 2 also studies the common coupled coincidence points by considering the so-called converse-contractive inequality. • Theorems 3 and 4 study the common coupled coincidence points when the fuzzy semimetric space satisfies the -triangle inequality and -triangle inequality, respectively. • Suppose that the fuzzy semi-metric space satisfies the -triangle inequality. Theorem 5 studies the common coupled coincidence points, and Theorem 6 studies the common coupled coincidence points by considering the so-called converse-contractive inequality.
Because the common coupled fixed points are the common coupled coincidence points, Theorems 1-6 can also be used to present the common coupled fixed points. However, the uniqueness cannot be realized from Theorems 1-6. Section 6 studies the uniqueness of common coupled fixed points.

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Suppose that the fuzzy semi-metric space satisfies the -triangle inequality. Theorem 7 studies the uniqueness of common coupled fixed points, and Theorem 8 also studies the uniqueness of common coupled fixed points by considering the so-called conversecontractive inequality. • Theorems 9 and 10 study the uniqueness of common coupled fixed points when the fuzzy semi-metric space satisfies the -triangle inequality and -triangle inequality, respectively.

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Suppose that the fuzzy semi-metric space satisfies the -triangle inequality. Theorem 11 studies the uniqueness of common coupled fixed points and Theorem 12 studies the uniqueness of common coupled fixed points by considering the so-called conversecontractive inequality.