Some Common Fixed Point Theorems in Fuzzy Metric Spaces and Their Applications

The concept of Fuzzy sets was introduced initially by Zadeh [6]. Since then, to use this concept in topology and analysis many authors have expansively developed the theory of Fuzzy sets and applications. Especially, Erceg [8], Kramosil and Michalek [12], Kaleva and Seikkala [13], Deng [29] have introduced the concept of Fuzzy metric space in different ways. The study of common fixed points of mapping, satisfying some contractive type condition has been at the center of vigorous research activities; and a number of interesting results have been obtained by various authors. Most of these results deal either with commuting mappings or assume the notion of weak commutativity of mappings introduced by Seesa [19]. In 1986, Jungck [5] introduced the notion of compatible maps. This concept was frequently used to prove existence theorems in common fixed point theory. In 2002, Aamir and Moutawakil [9] studied a property for pair of maps namely the property E.A, which is the generalization of the concept of non compatible maps. Further, Pant and Pant [28] studied the common fixed points of a pair of non compatible maps and the property E.A in FM-space.


Introduction
The concept of Fuzzy sets was introduced initially by Zadeh [6].Since then, to use this concept in topology and analysis many authors have expansively developed the theory of Fuzzy sets and applications.Especially, Erceg [8], Kramosil and Michalek [12], Kaleva and Seikkala [13], Deng [29] have introduced the concept of Fuzzy metric space in different ways.
The study of common fixed points of mapping, satisfying some contractive type condition has been at the center of vigorous research activities; and a number of interesting results have been obtained by various authors.Most of these results deal either with commuting mappings or assume the notion of weak commutativity of mappings introduced by Seesa [19].In 1986, Jungck [5] introduced the notion of compatible maps.This concept was frequently used to prove existence theorems in common fixed point theory.In 2002, Aamir and Moutawakil [9] studied a property for pair of maps namely the property E.A, which is the generalization of the concept of non compatible maps.Further, Pant and Pant [28] studied the common fixed points of a pair of non compatible maps and the property E.A in FM-space.The result obtained by us gives generalization of many important fixed point theorems and open up a wider scope for the study of common fixed points under contractive type conditions.After these no fixed point theorems have been investigated to find the fixed point in fuzzy metric spaces.(See [7] , [15,16,17,18], [22,23,24,25,26,27]) In this paper, our objective is to prove some common fixed point theorems by removing the assumption of continuity and replacing the completeness of the space with a set of three conditions for self mappings in Fuzzy metric space.Our result generalizes the result of Tanmony Som [21].[20] The 3-tuple (X, M, * ) is said to be Fuzzy metric space if X is an arbitrary set, * is a continuous t-norm and M is a Fuzzy set in X 2 × [0, ∞) satisfying the following conditions:

Preliminaries Definition 2.1. [14] A binary operation
for all x, y, z ∈ X and s, t > 0.

Definition 2.3. [20]
A sequence {x n } in a Fuzzy metric space (X, M, * ) is said to be Cauchy sequence if and only if for each ǫ > 0, t > 0, there exist n 0 ∈ N such that M (x n , x m , t) > 1 − ǫ for all n, m ≥ n 0 .Definition 2.4.[20] A sequence {x n } in a Fuzzy metric space (X, M, * ) is said to be convergent sequence to a point x in X if and only if for each ǫ > 0, t > 0, there exist n 0 ∈ N such that M (x n , x, t) > 1 − ǫ for all n ≥ n 0 .Definition 2.5.[20] A Fuzzy metric space (X, M, * ) is said to be complete if every Cauchy sequence in it converges to a point in it.Definition 2.6.[2] Two self mappings A and S of a Fuzzy metric space (X, M, * ) are said to be compatible if and only if M (ASx n , SAx n , t) → 1 for all t > 0, whenever {x n } is a sequence in X such that Sx n , Ax n → p for some p in X, as n → ∞.Definition 2.7.[3] Two self mappings A and S of a Fuzzy metric space (X, M, * ) are said to be semi-compatible if and only if M (ASx n , Sp, t) → 1 for all t > 0, whenever {x n } is a sequence in X such that Sx n , Ax n → p for some pinX, as n → ∞.Definition 2.8.[4] Two self mappings A and S of a Fuzzy metric space (X, M, * ) are said to be weakly-compatible if they commute at their coincidence points, i.e Ax = Sx implies ASx=SAx.Definition 2.9.[10] Two self mappings A and S of a Fuzzy metric space (X, M, * ) are said to be occasionally-weakly compatible if and only if there is a point x in X which is coincidence point of A and S at which A and S commute.Definition 2.10.[11] Two self mappings A and S of a Fuzzy metric space (X, M, * ) are said to satisfy the property E.A if there exist sequence {x n } in X such that lim n→∞ Ax n = lim n→∞ Sx n = z for some z ∈ X.

Main results
Theorem 3.1.Let A, B, S and T be self mappings on a Fuzzy metric space (X, M, * ) satisfying the following condition: M (Sx, By, t) , M (T y, Ax, t) for all x, y ∈ X, where r : Also, suppose the pair (A, S) and (B, T ) share the common property (E.A), and S (X) and T (X) are closed subsets of X, then the pair (A, S) as well as (B, T ) have a coincidence point.Further A, B, S,T have a unique common fixed point provided the pair (A, S) is semi-compatible and (B, T ) is occasionally weakly compatible.
Proof: Since the pair (A, S) and (B, T ) share the common property (E.A) then there exist two sequences {x n } and {y n } in X such that lim n→∞ Ax n = lim n→∞ Sx n = lim n→∞ By n = lim n→∞ T y n = z, for some z ∈ X.Also, S (X) is closed subset of X, therefore lim n→∞ Sx n = z ∈ S(X) and there is a point u in X such that Su = z.Now, we claim that Au = z.If not, then by using (3.1), we have Taking n → ∞, we get this is a contradiction.Hence Au = z.
Thus we have Au = Su or u is a coincidence point of the pair (A, S).
Since T (X) is closed subset of X, therefore lim n→∞ T y n = z ∈ T (X) and there exists w ∈ X such that T w = z.Again using (3.1), we obtain On taking n → ∞, we get This implies Bw = z.Hence, we get T w = Bw = z.Thus w is a coincidence point of the pair (B, T ).Also, (A, S) is semi-compatible pair, so lim n→∞ ASx n = Sz and lim n→∞ ASx n = Az.Since the limit in Fuzzy metric space is unique so Sz = Az.Now, we claim that z is a common fixed point of the pair (A, S).Again, from (3.1), we obtain Taking n → ∞, we get Since w is a coincidence point of B and T and the pair (B, T ) is occasionally weakly compatible, so we have, BT w = T Bw ⇒ Bz = T z = z.Hence, z is the common fixed point of A, B, S and T .For Uniqueness, Let v be another common fixed point of A, B, S and T .Take x = z and y = v in (3.1), we get Taking n → ∞, we get n , then one can say the pairs (A, S) and (B, T ) share the common property (E.A) and the pair (A, S) is semi-compatible and (B, T ) is occasionally weakly compatible.Also, for different values of x, equation (3.1) is satisfied.We will discuss it in the following three cases: r min {M (Sx, T y, t) , M (Sx, Ax, t) , M (Sx, By, t) , M (T y, Ax, t)} = 1 therefore, equation ( for all x, y ∈ X, a, b, c ≥ 0, q > 0 and q < a + b + c, then each pair (A, S) and (B, T ) have a point of coincidence .Further, if the pair (A, S) is semi-compatible and (B, T ) is occasionally weakly compatible, then A, B, S and T have a unique common fixed point.
Proof: As the pair (A, S) and (B, T ) share the common property (E.A), then there exist two sequences {x n } and {y n } in X such that lim n→∞ Ax n = lim n→∞ Sx n = lim n→∞ By n = lim n→∞ T y n = z, for some z ∈ X.Since S (X) is a closed subset of X; therefore, there exists a point u ∈ X such that Su = z.Using the above condition (iii), we have this implies, for all t > 0, this implies, Au = z.Hence Au = Su, which shows that u is the coincidence point of (A, S).Again, T (X) is closed subset of X, therefore there is a point w in X such that T w = z.Now take x = x n and y = w in condition (iii), we get Taking n → ∞, we obtain qM (z, Bw, t) ≥ aM (z, z, t) + bM (z, Bw, t) + cM (z, Bw, t) + max {M (z, z, t) , M (Bw, z, t)} , this gives This implies Bw = z.Hence T w = Bw = z and thus w is the coincidence point of (B, T ).
Further, we assume that (A, S) is a semi-compatible pair, so lim n→∞ ASx n = Sz and lim n→∞ ASx n = Az.Since the limit in fuzzy metric space is unique, so Sz = Az.Now, we claim that z is a common fixed point of the pair A and S.
Taking n → ∞, we obtain qM (Az, z, t) ≥ aM (z, Az, t) + bM (Az, z, t) + cM (Az, z, t) Since w is a coincidence point B and T , and the pair (B, T ) is occasionally weak compatible.So, BT w = T Bw this implies Bz = T z = z.
Hence, z is the common fixed point of mappings A, S, B and T .The uniqueness of fixed point follows from taking x = z and y = v in condition (iii).✷ Taking A = B in the above theorem, we get the following corollary: Corollary 3.3.Let A, S and T be three self mappings of a Fuzzy metric space (X, M, * ), satisfying the following conditions: 1.The pairs (A, S) and (A, T ) share the common property (E.A) 2. S (X) and T (X) are closed subsets of X 3. qM (Ax, Ay, t) ≥ aM (T y, Sx, t) + bM (Sx, Ay, t) + cM (Ax, Ay, t) +max {M (Ax, Sx, t) , M (Ay, T y, t)} for all x, y ∈ X, a, b, c ≥ 0, q > 0 and q < a + b + c, then the pairs (A, S) and (A, T ) have a point of coincidence.Further, if the pair (A, S) is semi-compatible and (A, T ) is occasionally weakly compatible then A, S and T have a unique common fixed point.

Applications
Theorem 4.1.Let A, B, S and T be self mappings on a Fuzzy metric space (X, M, * ) satisfying the condition where φ : R + → R + is a Lebesgue-integrable mapping which is summable, nonnegative such that ǫ 0 φ (t) dt > 0 for each ǫ > 0,and This gives, Bw = z.Hence T w = Bw = z or w is a coincidence point of the pair (B, T ).Also, (A, S) is a semi-compatible pair, so lim n→∞ ASx n = Sz and lim n→∞ ASx n = Az.
Since the limit in Fuzzy metric space is unique, therefore Sz = Az.We claim that z is a common fixed point of the pair (A, S). from (4.1), we have 1.The pairs (A, S) and (B, T ) share the common property (E.A); 2. S (X) and T (X) are closed subsets of X; for all x, y ∈ X, a, b, c ≥ 0, q > 0 and q < a + b + c, then the pairs (A, S) and (B, T ) have a point of coincidence each.Further, if the pair (A, S) is semi-compatible and (B, T ) is occasionally weakly compatible, then A, B, S and T have a unique common fixed point.
Proof: The proof follows from Theorem 4.1.✷

Conclusion
In this paper, Theorem 3.1, Theorem 3.2 are specially constructed for pairwise semi-compatible mappings and occasionally weakly compatible mappings(owc) in fuzzy metric spaces.An example and some applications are is given in support of our result.

Example 3 . 1 .
Thus z is the unique common fixed point of the mappings A, B, S and T .✷Let X =[2,20] and d be the usual metric on X.For each t ∈ [0, ∞), define M (x, y, t) = t t+|x−y| .Clearly (X, M, * ) is a Fuzzy metric space, where * is defined as a * b = ab.Define the mapping A, B, S, T as follows:

φφ✷Theorem 4 . 2 .
where r [m (z, w, t)] =r [min {M (Az, z, t) , M (Az, Az, t) , M (Az, z, t) , M (z, Az, t)}] =r [M (Az, z, t)] >M (Az, z, t) , (t) dt.this implies Az = z and hence Az = z = Sz.Since w is a coincidence point of B and T , and the pair (B, T ) is occasionally weakly compatible, so we have BT w = T Bw ⇒ Bz = T z = z.Hence z is the common fixed point of A, S, B and T .For Uniqueness, let v be another common fixed point of A, B, S and T .Take x = z and y = v in (4.1), we getM(Az,Bv,t) 0 φ (t) dt ≥ r[m(z,v,t)] 0 φ (t) dt or M(z,v,t) 0 φ (t) dt ≥ r[m(z,v,t)] 0 φ (t) dt, where r [m (z, v, t)] = r [min {M (z, v, t) , M (z, z, t) , M (z, v, t) , M (v, z, t)}] = r [M (Az, z, t)]> M (Az, z, t) , (t) dt, Some Common Fixed Point Theorems in Fuzzy Metric Spaces 151 this implies z = v and thus z is the unique common fixed point of the mappings A, B, S and T .Let A, B, S and T be four self mappings on a Fuzzy metric space (X, M, * ) satisfying the following conditions: 3.1) is satisfied.Case II: when 2 < x ≤ 5 M (Ax, By, t) = M (6, 7, t) = t t+1 and r [min {M (Sx, T y, t) , M (Sx, Ax, t) f, M (Sx, By, t) , M (T y, Ax, t)}] Theorem 3.2.Let A, B, S and T be four self mappings on a Fuzzy metric space (X, M, * ) the following conditions:1.The pairs (A, S) and (B, T ) share the common property (E.A) 2. S (X) and T (X) are closed subsets of X 3. qM (Ax, By, t) ≥ aM (T y, Sx, t) + bM (Sx, By, t) + cM (Ax, By, t) +max {M (Ax, Sx, t) , M (By, T y, t)} for each t < 1 and r(t) = 1 for t = 1., Also suppose that the pairs (A, S) and (B, T ) share the common property (E.A), and S (X) and T (X) are closed subsets of X.Then the pair (A, S) as well as (B, T ) have a coincidence point.Further, if A, B, S and T have a unique common fixed point provided the pair (A, S) is semi-compatible and (B, T ) is occasionally weakly compatible.Proof: Since the pair (A, S) and (B, T ) share the common property (E.A) then there exist two sequences {x n } and {y n } in X such that lim n→∞ Ax n = lim n→∞ Sx n = lim n→∞ By n = lim n→∞ T y n = z for some z ∈ X.Since S (X) is closed subset of X, then lim n→∞ Sx n = z ∈ S(X), therefore there is a point u in X such that Su = z.Su or u is a coincidence point of the pair (A, S).But T (X) is closed subset of X, then lim n→∞ T y n = z ∈ T (X), therefore there exists w ∈ X such that T w = z.