Common Fixed Point Results for Various Mappings in Fuzzy Metric Spaces with Application

In this paper, first we discuss the variants of the weakly commuting and compatible mappings in the context of coupled fixed point theory of fuzzy metric spaces. Secondly, we investigate the existence and uniqueness of the common fixed point for pairs of weakly compatible mappings satisfying a new contraction condition in the setup of fuzzy metric spaces with Hadz̆ić type t-norm . Further, we talk about some results for the variants of weakly commuting and compatible mappings. At the end, as an application, we obtain metrical version of the discussed results.


Introduction and Preliminaries
In 1965, Zadeh [28] introduced the notion of fuzzy sets that provides quick headways into different branches of mathematics and its areas of applications.In particular, the fuzzy version of the metric spaces has been given by various authors, resulting into different definitions of fuzzy metric spaces in numerous nonequivalent ways (see e.g., Deng [6], Erceg [8], George and Veeramani [9,10], Kaleva and Seikkala [19], Kramosil and Michalek [20]).
Grabiec [11] presented the fuzzy version of the famous Banach contraction principle, in the sense of Kramosil and Michalek (in short, KM) [20].George and Veeramani (in short, GV) [9,10] modified the concept of fuzzy metric spaces due to Kramosil and Michalek [20].Afterwards, various authors established several fixed point results in fuzzy metric spaces in the sense of GV.Some examples in this direction can be found in the works of Gregori and Sapena [12], Murthy et al. [22], Singh and Chauhan [24].The present work deals with the definition of the fuzzy metric space as discussed by George and Veeramani [9,10].
Over the years, in metric fixed point theory, authors are continuously making an effort to extend and generalize the Banach's contraction mapping principle in different directions in different spaces.In order to extend this famous contraction mapping principle for the pair of mappings, researchers have successively introduced the notions of commutative mappings, compatible mappings, weakly compatible mappings, variants of weakly commuting and compatible mappings.
During the same course of time, coupled fixed point results were receiving much attention in metric fixed point theory.Although the concept of coupled fixed point was introduced by Guo and Lakshmikantham [13] but the line of research in this direction developed rapidly after the worth mentioning work of Bhaskar and Lakshmikantham [2].In [2], the authors proved a contraction mapping theorem in partially ordered metric spaces in the context of coupled fixed point theory.Similar to ordinary fixed point theory, authors introduced the notions of commutative mappings, compatible mappings and weak compatible mappings in the context of coupled fixed theory.Various instances of such works can be found in [2,17,21].
In fuzzy metric spaces, coupled fixed point theorem for contraction mappings was first proved by Sedghi et al. [25].Unfortunately, Zhu and Xiao [27] proved the falsity of the work presented by Sedghi et al. [25] and thereby they presented a correct modification of the results proved by Sedghi et al. [25].On the other hand, Hu [14] presented a coupled common fixed point theorem for a pair of compatible mappings under a φ-contraction in fuzzy metric spaces, which was followed by the works of Choudhury et al. [4,5], Jain et al. [17,18], Hu et al. [15], etc. Subsequently, Abbas et al. [1] introduced the notion of w-compatible mappings as a generalization of compatible mappings.Recently, in order to obtain the existence and uniqueness of the coupled common fixed points for mappings in the setup of fuzzy metric spaces Jain et al. [18] introduced the notions of weakly commuting mappings and their variants, that is, R-commuting mappings, R-weakly commuting mappings of type (A F ), R-weakly commuting mappings of type (A g ), R-weakly commuting mappings of type (P ).On the other hand, Sumitra and Masmali [7] studied the notions of variants of compatible mappings that includes compatible mappings of type (A), compatible mappings of type (B), compatible mappings of type (C), compatible mappings of type (P ), compatible mappings of type (A F ), compatible mappings of type (A g ) in the context of coupled fixed point problems in fuzzy metric spaces.
The purpose of this paper is to present a discussion on the variants of weakly commuting and compatible mappings and to prove a common fixed point result for pairs of weakly compatible mappings satisfying a new contraction condition in the context of coupled fixed point theory of fuzzy metric spaces.Further, we give results for the variants of weakly commuting and compatible mappings.At the end, the metrical version of the notions and results discussed in different sections of the present manuscript are also established.
Here, we state some allied definitions and results which are required for the development of the present study.
Clearly, * 2 is an example of t-norm of H-type.

Definition 1.4 ( [9]
).The 3-tuple (X, M, * ) is called a fuzzy metric space (in the sense of GV), if X is an arbitrary non-empty set, * is a continuous t-norm and M is a fuzzy set on X × (0, ∞) satisfying the following conditions for each x, y, z ∈ X and t, s > 0: 2. Cauchy sequence if for each 0 < ε < 1 and t > 0, there exists a positive integer n 0 such that M (x n , x m , t) > 1 − ε for each n, m ≥ n 0 .

Discussion on Variants of Weakly Commuting and Compatible Mappings
In this section, we study the notions of the weakly commuting and compatible mappings, their variants and the weakly compatible mappings in the fuzzy metric spaces for problems concerning the computation of coupled coincidence and coupled fixed points.
Recently, Choudhury et al. [3] introduced the following notion of the compatible mappings to establish the existence of coupled coincidence points in ordered metric spaces: ).The mappings F : X × X → X and g : X → X are said to be compatible if Hu [14] defined the following notion as the fuzzy counterpart of the definition of compatibility, which was introduced in Choudhury et al. [3] for coupled fixed point problems in ordered metric spaces: Definition 2.2 ( [14]).The mappings F : X × X → X and g : X → X are said to be compatible if for all t > 0 whenever {x n } and {y n } are sequences in X, such that lim n→∞ The following notions were given by Jain et al. [18], which extend the definitions of variants of weakly commuting mappings from ordinary fixed point theory to coupled fixed point theory in the setup of fuzzy metric spaces: 18]).The mappings F : X × X → X and g : X → X are said to be weakly commuting if M (F (gx, gy), gF (x, y), t) ≥ M (F (x, y), gx, t), M (F (gy, gx), gF (y, x), t) ≥ M (F (y, x), gy, t) for all x, y in X and t > 0.
4. R-weakly commuting maps of type (P) if there exists some R > 0 such that M (F (F (x, y), F (y, x)), ggx, t) ≥ M (F (x, y), gx, t/R), M (F (F (y, x), F (x, y)), ggy, t) ≥ M (F (y, x), gy, t/R) for all x, y in X and t > 0. Now we present some illustrations and discuss the relations between these variants.
Example 2.1.Let X = (0, ∞).Define a * b = ab and M (x, y, t) = t t+|x−y| , for all x, y ∈ X and t > 0. Then (X, M, * ) is a FM-space.Define F : X × X → X as F (x, y) = x+y 2 for all x, y in X and g : X → X as g(x) = x 2 for all x in X.Then, clearly for all x, y in X and t > 0, we have which shows that the pair (F, g) is weakly commuting.Moreover, for all x, y in X and t > 0, we have which shows that the pair (F, g) is R-weakly commuting for each R > 0.
Further, we note the followings: Fixed Point Results for Various Mappings in Fuzzy Metric Spaces

39
For R ≥ 1 2 , the pair (F, g) are R-weakly commuting of type (A F ), since for all x, y in X and t > 0, we have For R ≥ 1 and x = y, the pair (F, g) satisfies the property of R-weakly commuting of type (A g ), since Finally, we proceed towards R-weakly commutativity of type (P).For R ≥ 3 2 and x = y, the pair (F, g) satisfies the property of R-weakly commuting of type (P) since, M (F (F (x, y), F (y, x)), ggx, t) = 4t 4t + (x + 2y) Clearly, the pair (F, g) is R-weakly commuting for each R > 0 but R-weakly commuting of type (A F ) for R ≥ 1 2 and R-weakly commuting of type (A g ) for R ≥ 1.
Remark 2.5.Example 2.1 shows that R-weakly commuting pair of mappings of type (A F ) need not be R-weakly commuting of type (A g ) nor it can be R-weakly commuting of type (P).
The following example illustrates that if the pair of mappings is R-weakly commuting for some value of R > 0, then that pair of mappings need not be weakly commuting, nor R weakly commuting of type (A F ), nor R-weakly commuting of type (A g ), nor weakly commuting of type (P) for the same value of R.
Also, for all x, y in X and t > 0, we have which shows that the pair (F, g) is R-weakly commuting of type (A F ) for R ≥ 7.
Further, we note that the pair (F, g) is R-weakly commuting for each R ≥ 5 but neither weakly commuting, nor R-weakly commuting of type (A g ), nor weakly commuting of type (P) for any R > 0.
Remark 2.6.In Example 2.2, the pair (F, g) of the mappings is R-weakly commuting but not R-weakly commuting of type (A F ) for R = 5.
We observe in general that, every pair of commuting mappings is always weakly commuting but converse need not be true.Further, the pair of R-weakly commuting mappings of type (A g ) need not be R-weakly commuting nor R-weakly commuting of type (A F ), nor R-weakly commuting of type (P) as shown in the following illustration: 2 for all x, y in X and g : X → X as g(x) = x 2 for all x in X.The mappings F and g are not commuting, since Again, for all x, y in X and t > 0, we have = M (F (y, x), gy, t), which shows that the pair (F, g) is weakly commuting.
Moreover, for all x, y in X and t > 0, M (F (gx, gy), gF (x, y), t) which shows that the pair (F, g) is R-weakly commuting for each R ≥ 1 2 .Further, we note the followings: • The pair (F, g) is not R-weakly commuting of type (A F ) for any R > 0.
• The pair (F, g) is R-weakly commuting of type (A g ) for R ≥ 1  4 .• The pair (F, g) is not R-weakly commuting of type (P) for any R > 0.
Clearly, for R = 1  4 , the pair (F, g) is R-weakly commuting of type (A g ) but not R-weakly commuting.
Example 2.4.Let X = [1, ∞).Define a * b = ab and M (x, y, t) = t t+|x−y| , for all x, y ∈ X and t > 0. Then (X, M, * ) is a FM-space.Define F : X × X → X as F (x, y) = x+y+1 2 for all x, y in X and g : X → X as g(x) = x 2 for all x in X.The mappings F and g are not commuting, since F (gx, gy = gF (x, y) for x, y in X.Now, for all x, y in X and t > 0, we have which shows that the pair (F, g) is R-weakly commuting for R ≥ 1 4 .Also, since for R = 1, the R-weakly commuting property coincides with weakly commuting property of the mappings, therefore, the pair (F, g) is also weakly commuting.
Also, for all x, y in X and t > 0, which shows that the pair (F, g) is R-weakly commuting of type (A F ) for each R ≥ 3 4 .Further, we note that the pair (F, g) is neither R-weakly commuting of type (A g ), nor R weakly commuting of type (P) for any R > 0.
Then, the pair (F, g) is not commuting; R-weakly commuting for each R ≥ 1 (and hence weakly commuting); R-weakly commuting of type (A F ) for each R ≥ 2; R-weakly commuting of type (A g ) for each R ≥ 3; R-weakly commuting of type (P) for each R ≥ 4.
In the setup of fuzzy metric spaces, now, we study the following notions of variants of compatible mappings, which are due to Sumitra and Masmali [7]: and for some x, y ∈ X and t > 0.

Definition 2.8 ( [7]
).The mappings F : X × X → X and g : X → X are said to be compatible of type (B) if and whenever {x n } and {y n } are sequences in X such that for some x, y ∈ X and t > 0.
Definition 2.9 ( [7]).The mappings F : X × X → X and g : X → X are said to be compatible of type (P) if Definition 2.10 ( [7]).The mappings F : X × X → X and g : X → X are said to be compatible of type (C) if and x for some x, y ∈ X and t > 0.

Definition 2.11 ( [7]
).The mappings F : X × X → X and g : X → X are said to be compatible of type whenever {x n } and {y n } are sequences in X such that lim n→∞ for some x, y ∈ X and t > 0.
Definition 2.12 ( [7]).The mappings F : X × X → X and g : X → X are said to be compatible of type y for some x, y ∈ X and t > 0.
We now discuss the relationship between these variants as follows: In the following example, we show that compatible mappings need not be compatible of type (A), nor compatible of type (P), nor compatible of type (A F ), nor compatible of type (A g ).
Example 2.6.Let X = R. Define a * b = ab and M (x, y, t) = t t+|x−y| , for all x, y ∈ X and t > 0. Then (X, M, * ) is a FM-space.Define the mappings F : X × X → X and g : X → X by We claim that the pair (F, g) is compatible but not compatible of type (A), nor compatible of type (P), nor compatible of type (A F ), nor compatible of type (A g ).
For, let {x n = n 2 , n ≥ 1} and {y n = 2n 2 , n ≥ 1}.Then and Also, since we have Thus, the pair (F, g) is none of the following: 1. compatible of type (A), 2. compatible of type (P),

compatible of type (A g ).
Also, for the sequences {x n } and {y n }, with lim n→∞ Similarly, lim n→∞ M (gF (y n , x n ), F (gy n , gx n ), t) = 1, so that the mappings F and g are compatible.
Next we illustrate that mappings of compatible of type (A) need not be compatible.
Example 2.7.Let X = [0, 6].Define a * b = ab and M (x, y, t) = t t+|x−y| , for all x, y ∈ X and t > 0. Then (X, M, * ) is a FM-space.Define the mappings F : X × X → X and g : X → X by Let , n ≥ 1} be two sequences.Then, we obtain that By routine calculation, it is easy to notice that the pair (F, g) is compatible of type (A).Lemma 2.13.Let (X, M, * ) be a fuzzy metric space.Let F : X × X → X and g : X → X be two mappings such that the pair (F, g) is compatible of type (A) and one of the mappings F and g is continuous, then the pair (F, g) is compatible.
Proof.Without loss of generality, we assume that the mapping g is continuous.Let {x n } and {y n } be two sequences in X such that lim n→∞ since the mappings F and g are compatible of type (A) and by continuity of g, on letting n → ∞, it follows that lim Therefore, the mappings F and g are compatible.Analogously, it can be proved that if the mapping F is continuous and the pair (F, g) is compatible of type (A), then the pair (F, g) is also compatible.✷ Lemma 2.14.If the pair of mappings F : X × X → X and g : X → X is compatible and both the mappings F and g are continuous, then the pair (F, g) is compatible of type (A).
Lemma 2.15.Let (X, M, * ) be a fuzzy metric space.Let F : X × X → X and g : X → X be two mappings.If the mapping g is continuous, then the pair (F, g) is compatible of type (A F ) iff the pair (F, g) is compatible.
Proof.Let g be the continuous mapping.Let {x n } and {y n } be two sequences in for some x, y ∈ X.
Let the pair of mappings (F, g) be compatible of type (A F ), then on letting n → ∞ and by the continuity of the mapping g, it follows that Similarly, Hence, the pair (F, g) is compatible.We conclude the proof by showing that the pair (F, g) is compatible of type (A F ), if the pair (F, g) is compatible. For, then, by continuity of g, on letting n → ∞, it follows that lim Similarly, lim n→∞ M (F (g(y n ), g(x n )), g 2 y n , t) = 1.Thus, the pair (F, g) is compatible of type (A F ).This completes the proof.✷ The following lemma establishes the relationship between the pair of compatible mappings and the pair of compatible mappings of type (A g ): Lemma 2.16.Let (X, M, * ) be a fuzzy metric space.Let F : X × X → X and g : X → X be two mappings.If the mapping F is continuous, then the pair (F, g) is compatible of type (A g ) iff the pair (F, g) is compatible.
Proof.The result can be proved analogously as Lemma 2.3.✷ Lemma 2.17.Let (X, M, * ) be a fuzzy metric space.Let F : X × X → X and g : X → X be two mappings.If the pair (F, g) is compatible of type (A), then the pair (F, g) is 1. compatible of type (B), 2. compatible of type (P), 3. compatible of type (A F ),

compatible of type (A g ).
Proof.By using the definitions of variants of compatible mappings, the proof holds trivially.✷ Remark 2.18.Using Lemma 2.5, the Example 2.7 illustrates the fact that "the pair of the mappings that are compatible of type (B) or compatible of type (P) or compatible of type (A F ) or compatible of type (A g ) need not be compatible".
Lemma 2.19.Let (X, M, * ) be a fuzzy metric space.Let F : X × X → X and g : X → X be two continuous mappings.Then the pair (F, g) is compatible of type (B) (or compatible of type (C) or compatible of type (P)) iff the pair (F, g) is compatible.
Proof.First, assume that the pair (F, g) of the mappings is compatible of type (B).We shall show that the pair (F, g) of the mappings is compatible.For, let {x n } and {y n } are sequences in X, such that lim n→∞ g(y n ) = y for some x, y ∈ X.Then on using the continuity hypotheses of the mappings F and g in the definition of the compatible mappings of type (B), by the condition we have that M (F (x, y), gx, t) ≥ 1, that is F (x, y) = gx.Similarly, it can be obtained that F (y, x) = g(y).Now, for t > 0 then, on letting n → ∞, and using the continuity conditions of the mappings F and g in the last inequality, we obtain that Hence the pair (F, g) of the mappings is compatible.Interestingly, since the mappings F and g are continuous, so the conditions implies that gx = F (x, y) and gy = F (y, x), respectively, which has already been noted.
Conversely, assume that the pair (F, g) of the mappings be compatible.To show that it is compatible of type (B).For, let {x n } and {y n } are sequences in X, such that lim on letting n → ∞, and using the compatibility of the mappings F and g along with the continuity of the mapping g, we obtain that lim n→∞ M (F (gx n , gy n ), g 2 x n , t) ≥ 1, that is, lim n→∞ M (F (gx n , gy n ), g 2 x n , t) = 1.Also, on using the continuity hypothesis of the mapping F , we obtain that Hence, we can conclude that Similarly, we can show that if the pair (F, g) of the mappings is compatible and the mappings F and g are continuous, then all the other conditions for the mappings F and g to be the compatible of type (B) holds.Analogously, it can be easily proved that if the mappings F and g are both continuous, then the pair (F, g) is compatible of type (C) (or, compatible of type (P)) iff the pair (F, g) is compatible. ✷ Next example illustrates that compatible mappings of type (B) need not be compatible, nor compatible of type (A), nor compatible of type (C), nor compatible of type (P).
The pair (F, g) is not compatible but compatible of type (B).For, let x n = 1 n , n ≥ 3 and y n = 1 2n , n ≥ 3 .Then Since, hence, the pair (F, g) is not compatible.But the pair is compatible of type (B), since Also, Hence, it follows that Similarly, it can be easily checked that Thus, the pair (F, g) is compatible of type (B).Also, we note that the pair (F, g) is not compatible of type (A), since Further, the pair (F, g) is not compatible of type (P), since Further, simple calculation shows that the pair (F, g) is not compatible of type (C).Lemma 2.20.Let (X, M, * ) be a fuzzy metric space.Let F : X × X → X and g : X → X be two mappings.If the pair (F, g) is compatible of type (B) (or compatible of type (C)) and both the mappings F , g are continuous, then the pair (F, g) is compatible of type (A).
Proof.First, let us assume that the pair (F, g) of the mappings is compatible of type (B) and both the mappings F , g are continuous, then the pair (F, g) is compatible of type (A).For, let {x n } and {y n } are sequences in X, such that lim x, y ∈ X.Since the pair (F, g) is compatible of type (B), we have then, on using the continuity hypothesis of the mapping F on the right side of the above inequality, we obtain that lim Similarly, we can obtain that lim n→∞ M (F (gy n , gx n ), g 2 y n , t) = 1.
We now show that Since the pair (F, g) is compatible of type (B), we have then, on using the continuity hypothesis of the mapping g on the right side of the above inequality, we obtain that Similarly, we can obtain that lim Hence, the pair (F, g) is compatible of type (A).Analogously, it can be easily proved that if both the mappings F , g are continuous and the pair (F, g) is compatible of type (C), then it is compatible of type (A).✷ Remark 2.21.In view of the above discussion, various relations between the variants of compatible mappings could be easily established under certain conditions.For example, we can easily observe that "If the mappings F and g are both continuous, then the pair (F, g) is compatible of type (B) iff the pair (F, g) is compatible of type (C)".
Recently, Abbas et al. [1], introduced the concept of w-compatible mappings, following which, some authors established coupled common fixed point results for the similar notion of weakly compatible mappings.Works noted in [15,17,18] are some examples in this direction.
Interestingly, the concepts of w-compatible mappings and weakly compatible mappings are equivalent.Lemma 2.24.Let (X, M, * ) be a fuzzy metric space.Let F : X × X → X and g : X → X be two mappings.If F and g are compatible, or compatible of type (A), or compatible of type (P), or compatible of type (B), or compatible of type (C), or compatible of type (A F ), or compatible of type (A g ), then they are weakly compatible (or, w-compatible).
Proof.First, we shall show that if the pair (F, g) of the mappings be compatible, then it is also weakly compatible.For, if the pair (F, g) of the mappings be compatible, then by definition of compatible mappings, we have for all t > 0 whenever {x n } and {y n } are sequences in X, such that for some x, y ∈ X. Taking x n = a and y n = b, we obtain that ga = F (a, b) and gb = F (b, a) implies that gF (a, b) = F (ga, gb) and gF (b, a) = F (gb, ga).Hence every pair of compatible mappings is always weakly compatible (or, we can say w-compatible).
Next, we shall show that if the pair (F, g) of the mappings be compatible of type (A), then it is also a weakly compatible pair.For, if the pair (F, g) of the mappings be compatible of type (A), then by definition of compatible mappings of type (A), we have y for some x, y ∈ X and t > 0. Taking x n = a and y n = b, we obtain that ga = F (a, b) = x and gb = F (b, a) = y.And the condition lim n→∞ M (F (gx n , gy n ), g 2 x n , t) = 1 becomes M (F (ga, gb), g 2 a, t) = 1, that is, M (F (ga, gb), gga, t) = 1, that is, M (F (ga, gb), gF a, b, t) = 1 which implies that F (ga, gb) = gF (a, b).Similarly, we can obtain that F (gb, ga) = gF (b, a).Therefore, ga = F (a, b) and gb = F (b, a) implies that F (ga, gb) = gF (a, b) and F (gb, ga) = gF (b, a).Hence, we can conclude that every pair of compatible mappings of type (A) is always weakly compatible (or, we can say w-compatible).Now, if the pair (F, g) of the mappings be compatible of type (B), then it is also a weakly compatible pair.For, if the pair (F, g) of the mappings be compatible of type (B), by taking x n = a and y n = b in the definition of compatible mappings of type (B), we obtain that ga = F (a, b) = x and gb = F (b, a) = y.Then, the condition in the definition of compatible mappings of type (B) becomes that is, that is, {M (F (ga, gb), F (ga, gb), t) + M (F (x, y), F (x, y), t)}, Fixed Point Results for Various Mappings in Fuzzy Metric Spaces 55 that is, M (F (ga, gb), gF (a, b), t) ≥ 1, that is, M (F (ga, gb), gF (a, b), t) = 1, hence, F (ga, gb) = F (ga, gb).Similarly, we can obtain that F (gb, ga) = gF (b, a).Therefore, ga = F (a, b) and gb = F (b, a) implies that F (ga, gb) = gF (a, b) and F (gb, ga) = gF (b, a).Hence, we can conclude that every pair of compatible mappings of type (B) is always weakly compatible (or, we can say w-compatible).
Similarly, we can prove that if the pair of the mappings (F, g) is compatible of type (P), or compatible of type (C), or compatible of type (A F ), or compatible of type (A g ), then it is weakly compatible (or, w-compatible).✷ The following example illustrates that weakly compatible mappings need not be compatible nor compatible of type (A), nor compatible of type (B), nor compatible of type (P), nor compatible of type (C), nor compatible of type (A F ).
Example 2.9.Let X = [1,20] and * being any continuous t-norm.Define M (x, y, t) = e −|x−y|/t , for all x, y ∈ X and t > 0. Then (X, M, * ) is a FM-space.Define the mappings F : X × X → X and g : X → X respectively by Then the only coupled coincidence point for the pair (F, g) is (1, 1).The mappings F and g are not compatible, since for the sequences {x n } and {y n } with x n := 4+ 1 2n and y n := 4 + 1 2n+1 for n ≥ 1, we have Also, for the above defined sequences {x n } and {y n }, we have as n → ∞, so that the functions F and g are not compatible of type (A) and not compatible of type (A F ).We next show that the mappings F and g are also not compatible of type (B).On the contrary, assume that the mappings F and g are compatible of type (B), then, we must have ) iff 2 ≥ e 7/t +e 3/t , which is not possible for t > 0. Hence, the mappings F and g are not compatible of type (B).In a similar way, we can easily show that the mappings F and g are neither compatible of type (C) nor compatible of type (P).But the mappings F and g are weakly compatible, since they commute at their coupled coincidence point (1, 1).
Similarly, we can obtain that ). Continuing in this way, for all n > 0, we can obtain that We are now ready to give our main result as follows: Theorem 3.3.Let (X, M, * ) be a Fuzzy Metric Space, * being continuous t-norm of H-type and M (x, y, t) → 1 as t → ∞, for all x, y ∈ X.Let A : X × X → X, B : X × X → X, S : X → X, T : X → X be four mappings satisfying (3.1), (3.2) and the following conditions: 8. the pairs (A, S) and (B, T ) are weakly compatible, 9. one of the subspaces A(X × X) or T (X) and one of B(X × X) or S(X) are complete.
Proof.Since M has n-property on X 2 ×(0, ∞), we have that lim n→∞ [M (x, y, k n t)] n p = 1, whenever x, y ∈ X, k > 1 and p > 0. We claim that M (x, y, t) → 1 as t → ∞, for all x, y ∈ X.If not, then using the fact that M (x, y, •) ∈ [0, 1] and the nondecreasing property of M (x, y, •), we can conclude the existence of some a, b ∈ X such that lim t→+∞ M (a, b, t) = γ < 1, then for any t > 0 and k > 1, we have that k n t → +∞ as n → ∞ and hence we obtain that lim n→∞ [M (x, y, k n t)] n p = 0 for p > 0, which is a contradiction.Now, the proof follows immediately by applying Theorem 3.1.✷ On taking φ(t) = kt, for t > 0, where k ∈ (0, 1) and taking ω, γ to be the identity mapping on their respective domains, we obtain the following result: Theorem 3.5.Let (X, M, * ) be a Fuzzy Metric Space, * being continuous t-norm of H-type and M (x, y, t) → 1 as t → ∞, for all x, y ∈ X.Let A : X × X → X, B : X × X → X, S : X → X, T : X → X be four mappings satisfying (3.1), (3.8), (3.9) and the following condition: 1. there exists k ∈ (0, 1) such that M (A(x, y), B(u, v), kt) * M (A(y, x), B(v, u), kt) ≥ M (Sx, T u, t) * M (Sy, T v, t), for all x, y, u, v in X and t > 0. Then there exists a unique point α in X such that A(α, α) = S(α) = α = T (α) = B(α, α).
Taking A = B = F and S = T = g in Theorem 3.1, we have the following result: Corollary 3.6.Let (X, M, * ) be a Fuzzy Metric Space, * being continuous t-norm of H-type and M (x, y, t) → 1 as t → ∞, for all x, y ∈ X.Let F : X × X → X and g : X → X be two mappings and there exists φ ∈ Φ such that ≥ γM (gx, gu, t) * M (gy, gv, t), for all x, y, u, v in X and t > 0, where γ ∈ V and ω ∈ W such that γ(a) ≥ ω(a) for a ∈ [0, 1].Suppose that F (X × X) ⊆ g(X) and F and g are weakly compatible.If one of the range spaces of F or g is complete, then there exits a unique α in X such that α = g(α) = F (α, α).Theorem 4.10.Let (X, d) be a metric space and suppose that A : X × X → X, B : X × X → X, S : X → X, T : X → X be four mappings satisfying the condition that there exists some k ∈ (0, 1) such that 1. max{d(A(x, y), B(u, v)), d(A(y, x), B(v, u))} ≤ k 2 [d(Sx, T u) + d(Sy, T v)], for all x, y, u, v ∈ X.Also, suppose that A(X × X) ⊆ T (X), B(X × X) ⊆ S(X), the pairs (A, S) and (B, T ) are weakly compatible, one of the subspaces A(X × X) or T (X) and one of B(X × X) or S(X) are complete.Then there exists a unique point α in X such that A(α, α) = S(α) = α = T (α) = B(α, α).

Conclusion
In coupled fixed point theory of fuzzy metric spaces, we have discussed the relation among various variants of weakly commuting mappings and also among the variants of compatible mappings.The obtained main result for two pair of weakly compatible mappings has been extended for the variants of weakly commuting and compatible mappings.The corresponding notions of these variants have been discussed in the coupled fixed point theory of metric spaces.As application of the results proved in the setup of fuzzy metric spaces, the analogous results have been established in metric spaces.Further, due to the assumption of the new contraction condition, the proof of the main result of this paper is quite shorter and simpler than the proof of the results already present in the literature.

Definition 4 . 3 .Definition 4 . 4 .Definition 4 . 5 .Definition 4 . 6 .Definition 4 . 7 .Definition 4 . 8 .Remark 4 . 9 .
The mappings F : X × X → X and g : X → X are said to be compatible of type (A) if lim n→∞ d(F (gx n , gy n ), g 2 x n ) = 0, lim n→∞ d(F (gy n , gx n ), g 2 y n ) = 0 and lim n→∞ d(gF (x n , y n ), F (F (x n , y n ), F (y n , x n ))) = 0, lim n→∞ d(gF (y n , x n ), F (F (y n , x n ), F (x n , y n ))) = 0, whenever {x n } and {y n } are sequences in X such that lim n→∞F (x n , y n ) = lim n→∞ g(x n ) = x, lim n→∞ F (y n , x n ) = lim n→∞ g(y n ) = y for some x, y ∈ X.The mappings F : X × X → X and g : X → X are said to be compatible of type (B) if lim n→∞ d(F (gx n , gy n ), g 2 x n ) (gx n , gy n ), F (x, y)) + lim n→∞ d(F (x, y), F (F (x n , y n ), F (y n , x n )))}, lim n→∞ d(F (gy n , gx n ), g 2 y n ) (gy n , gx n ), F (y, x)) + lim n→∞ d(F (y, x), F (F (y n , x n ), F (x n , y n )))} and lim n→∞ d(gF (x n , y n ), F (F (x n , y n ), F (y n , x n ))) (x n , y n ), gx) + lim n→∞ d(gx, g 2 x n )}, lim n→∞ d(gF (y n , x n ), F (F (y n , x n ), F (x n , y n )), t) (y n , x n ), gy) + lim n→∞ d(gy, g 2 y n )} whenever {x n } and {y n } are sequences in X such that lim n→∞ F (x n , y n ) = lim n→∞ g(x n ) = x, lim n→∞ F (y n , x n ) = lim n→∞ g(y n ) = y for some x, y ∈ X.The mappings F : X × X → X and g : X → X are said to be compatible of type (P) if lim n→∞ d(F (F (x n , y n ), F (y n , x n )), g 2 x n ) = 0, lim n→∞ d(F (F (y n , x n ), F (x n , y n )), g 2 y n ) = 0 whenever {x n } and {y n } are sequences in X such that lim n→∞ F (x n , y n ) = lim n→∞ g(x n ) = x, lim n→∞ F (y n , x n ) = lim n→∞ g(y n ) =y for some x, y ∈ X.The mappings F : X × X → X and g : X → X are said to be compatible of type (C) iflim n→∞ d(F (gx n , gy n ), g 2 x n ) (gx n , gy n ), F (x, y)) + lim n→∞ d(F (x, y), g 2 x n ) + lim n→∞ d(F (x, y), F (F (x n , y n ), F (y n , x n ))) }, lim n→∞ d(F (gy n , gx n ), g 2 y n ) (gy n , gx n ), F (y, x)) + lim n→∞ d(F (y, x), g 2 y n ) + lim n→∞ d(F (y, x), F (F (y n , x n ), F (x n , y n )))Fixed Point Results for Various Mappings in Fuzzy Metric Spaces 67 andlim n→∞ d(gF (x n , y n ), F (F (x n , y n ), F (y n , x n ))) (x n , y n ), gx) + lim n→∞ d(gx, F (F (x n , y n ), F (y n , x n ))) + lim n→∞ d(gx, g 2 x n ) , lim n→∞ d(gF (y n , x n ), F (F (y n , x n ), F (x n , y n ))) (y n , x n ), gy) + lim n→∞ d(gy, F (F (y n , x n ), F (x n , y n ))) + lim n→∞ d(gy, g 2 y n ) whenever {x n } and {y n } are sequences in X such that lim n→∞ F (x n , y n ) = lim n→∞ g(x n ) = x, lim n→∞ F (y n , x n ) = lim n→∞ g(y n ) = y for some x, y ∈ X.The mappings F : X × X → X and g : X → X are said to be compatible of type (A F ) if lim n→∞ d(F (gx n , gy n ), ggx n ) = 0, lim n→∞ d(F (gy n , gx n ), ggy n ) = 0 whenever {x n } and {y n } are sequences in X such that lim n→∞ F (x n , y n ) = lim n→∞ g(x n ) =x, lim n→∞ F (y n , x n ) = lim n→∞ g(y n ) = y for some x, y ∈ X.The mappings F : X × X → X and g : X → X are said to be compatible of type (A g ) iflim n→∞ d(gF (x n , y n ), F (F (x n , y n ), F (y n , x n ))) = 1, lim n→∞ d(gF (y n , x n ), F (F (y n , x n ), F (x n , y n ))) = 1 whenever {x n } and {y n } are sequences in X such that lim n→∞ F (x n , y n ) = lim n→∞ g(x n ) =x, lim n→∞ F (y n , x n ) = lim n→∞ g(y n ) = y for some x, y ∈ X.Interestingly, the comparison and relation between various mappings in the setup of fuzzy metric spaces established earlier in the Section 2 of the present manuscript also holds among the metrical versions of those mappings.