Abstract

The aim of this paper is to establish some new common fixed point theorems for generalized contractive maps in fuzzy metric space by using property (E.A.), common property (E.A.), JCLR_ST property and CLR_S property. Our results improve and extend the results of Chauhan et al. [1] and Sedghi et al. [2] besides several known results. We also furnish an illustrative example in support of our results.

Keywords:FM-Space, Weakly Compatible Maps, CLR_g Property, Property (E.A.), The Common Property (E.A.), JCLR_ST Property and Common Fixed Point

1. Introduction

Aamri et al. [3] generalized the concept of non compatibility by defining the notion of property (E.A.) and proved common fixed point theorems under strict contractive conditions. Many authors have proved common fixed point theorems in different settings for different contractive conditions. For details, we refer to [4] -[13] .

In 2005, Liu et al. [14] further improved it by common property (E.A) while proving common fixed point theorems under strict contractive conditions. Recently, Sintunavarat et al. [13] , defined the notion of (CLR_g) property which is more general than (E.A) property.

Very recently, Manro et al. [15] introduced the notion of (CLR_S) property and Chauhan et al. [4] introduced the notion of (JCLR_ST) property.

The aim of this paper is to establish some new common fixed point theorems for generalized contractive maps in fuzzy metric space by using property (E.A.), common property (E.A.), JCLR_ST property and CLR_S property. Our results improve and extend the results of Chauhan et al. [1] and Sedghi et al. [2] besides several known results.

2. Preliminaries

Definition 2.1. [16] Let X be any set. A fuzzy set in X is a function with domain X and values in [0,1].

The concept of triangular norms (t-norms) is originally introduced by Menger [17] in study of statistical metric spaces.

Definition 2.2. [18] A binary operation*: is continuous t-norm if * satisfies the following conditions:

i) * is commutative and associative;

ii) * is continuous;

iii) a * 1 = a for all;

iv) a * b £ c * d whenever a £ c and b £ d for all.

Examples of t-norms are:

a * b = min{a,b}, a * b = ab and a * b = max{a+b–1, 0} for all.

Definition 2.3. [6] A 3-tuple is a fuzzy metric space if X is an arbitrary set, * is a continuous t-norm and M is a fuzzy set on X² × [0, ∞) satisfying the following conditions for all and s, t > 0 i) M(x, y, t) > 0;

ii) M(x, y, t) = 1 for all t > 0 if and only if x = y;

iii);

iv);

v) is continuous.

The function M(x, y, t) denote the degree of nearness between x and y with respect to t.

In all that follows is a fuzzy metric space with the following property:

vi) for all and t > 0.

Now we give some interesting examples of FM-spaces:

Example 2.1. Let (X, d) be a metric space. Define a * b = a + b, for all; and t > 0.

Define. Then (X, M, *) is a FM-space.

Moreover, fuzzy metric M induced by a metric d is often referred to as the Standard fuzzy metric.

Definition 2.4. [6] . A sequence {x_n} in fuzzy metric space is i) convergent to a point if

for all t > 0ii) Cauchy sequence if

for all t > 0 and p > 0.

Definition 2.5. A pair of self maps (S, T) of a fuzzy metric space is i) compatible [19] if

for all t > 0, whenever is a sequence in X such that for some.

ii) non-compatible if there exists at least one sequence in X such that for some but either or non-existent.

iii) weakly compatible [20] if S and T commute at coincidence points, that is, STx = TSx whenever Sx = Tx.

iv) satisfy the property (E.A) [3] if there exist a sequence in X such that for some.

v) satisfies the common limit in the range of T property (CLR_T) [13] if there exist a sequence {x_n} in X such that for some.

Definition 2.6. Two pairs of self maps (A, S) and (B, T) of fuzzy metric space is i) satisfy the common property (E.A) [14] if there exist two sequences and in X such that

for some.

ii) satisfy the (JCLR_ST) property (with respect to maps S and T) [4] if there exist two sequences {x_n} and {y_n} in X such that

for some

iii) satisfy the (CLR_S) property (with respect to maps S) [15] if there exist two sequences {x_n} and{y_n} in X such that

for some

Lemma 2.1. [7] If for all t > 0 and for a number, then x = y.

3. Main Results

Let be the set of all increasing and continuous functions such that for all .

Example 3.1: Let defined by for all. Clearly,.

Theorem 3.1: Let A, B, S and T be self mappings of a fuzzy metric space (X, M, *) satisfying the following:

(3.1)

for all, t > 0 and for some,;

(3.2) and;

(3.3) pair (A,S) or (B,T) satisfies the property (E.A);

(3.4) the range of one of the maps A, B, S or T is a closed subset of X.

Then pairs (A, S) and (B, T) have coincidence point. Further if (A, S) and (B, T) be weakly compatible pairs of self maps of fuzzy metric space (X, M, *) then A, B, S and T have a unique common fixed point in X.

Proof: If the pair (B,T) satisfies the property (E.A.), then there exist a sequence {x_n} in X such that

for some.

Since, , therefore, there exist a sequence {y_n} in X such that. Hence,. Also, since, there exist a sequence in X such that. Hence,.

Suppose that S(X) is a closed subset of X. Then z = Su for some. Therefore,

We first claim that Au = z.

If Au ≠ z, then there exist t₀ > 0 such that

. (3.5)

The inequality (3.5) is always true when Au ≠ z. To support our claim, we suppose on contrary that (3.5) is not true all t > 0, i.e.,

. (3.6)

Now, using equality (3.6) repeatedly, we get

as. This gives, for all t > 0. Hence, Au = z, which gives contradiction.

Therefore, inequality (3.5) is always true for some t₀ > 0.

Using (3.1), take x = u, y = y_n, we get

let then where and, we get

As, we get

which gives contradiction, hence Au = z.

Therefore, Au = z = Su which shows that u is a coincidence point of the pair (A, S). As A and S are weakly compatible. Therefore, ASu = SAu and then AAu = ASu = SAu = SSu.

On the other hand, since, there exist v in X such that Au = Tv.

Now, we show that Bv = z.

If Bv ≠ z, then again, as done above, there exist t₀ > 0 such that

. (3.7)

The inequality (3.7) is always true when Bv ≠ z.

Using (3.1), take x = u, y = v, we have

let then where, we get

As, we get

which gives contradiction, hence Bv = z.

Therefore, Bv = z = Au = Tv which shows that Bv = Tv, i.e., v is a coincidence point of the pair (B, T). As B and T are weakly compatible, therefore, BTv = TBv and hence, BTv = TBv = TTv = BBv.

Next, we show that AAu = Au, if not, then again as done above, there exist t₀ > 0 such that

. (3.8)

Using (3.1), take x = Au, y = v, we have

let then where, we get

As, we get which gives contradiction, hence AAu = Au.

Therefore, AAu = Au = SAu and Au are a common fixed point of A and S. Similarly, we can prove that Bv is a common fixed point of B and T. As Au = Bv, we conclude that Au is a common fixed point of A, B, S and T.

The proof is similar when T(X) is assumed to be a closed subset of X. The cases in which A(X) or B(X) is a closed subset of X are similar to the cases in which T(X) or S(X) respectively, is closed since

.

For uniqueness; let w be another fixed point of A, B, S and T. Then by (3.1), we have

let then where,

as, we get

a contradiction, hence, w = z. It implies that A, B, S and T have unique common fixed point in X.

Hence the result.

Now we attempt to drop containment of subspaces by replacing property (E.A.) by a weaker condition common property (E.A.) in Theorem 3.1.

Theorem 3.2: Let A, B, S and T be self mappings of a fuzzy metric space (X, M, *) satisfying condition (3.1) of Theorem 3.1 and the following:

(3.9) the pair (A, S) and (B, T) share the common (E.A.) property;

(3.10) S(X) and T(X) are closed subsets of X.

Then the pairs (A, S) and (B, T) have a point of coincidence each. Moreover, A, B, S and T have a unique common fixed point provided both the pairs (A, S) and (B, T) are weakly compatible.

Proof: In view of (3.2), there exist two sequences {x_n} and {y_n} in X such that

for some.

Since S(X) is a closed subset of X, therefore, there exists a point u in X such that z = Su.

We claim that Au = z. If Au ≠ z, then there exist t₀ > 0 such that

. (3.11)

The inequality (3.11) is always true when Au ≠ z. To support our claim, we suppose on contrary that (3.11) is not true all t > 0, i.e.,

. (3.12)

Now, using equality (3.12) repeatedly, we get

as. This gives, for all t > 0. Hence, Au = z, which gives contradiction.

Therefore, inequality (3.11) is always true for some t₀ > 0. Using (3.1), take x = u, y = y_n, we get

let then where and, we get

As, we get which gives contradiction, hence Au = z.

Therefore, Au = z = Su which shows that u is a coincidence point of the pair (A, S).

Since T(X) is also a closed subset of X, therefore in T(X) and hence there exists v in X such that Tv = z = Au = Su. Now, we show that Bv = z.

If Bv ≠ z, then again as done above, there exist t₀ > 0 such that

. (3.13)

The inequality (3.13) is always true when Bv ≠ z.

Using (3.1), take x = u, y = v, we have

let then where, we get

As, we get

which gives contradiction, hence Bv = z.

Therefore, Bv = z = Tv which shows that v is a coincidence point of the pair (B, T).

Since the pairs (A, S) and (B, T) are weakly compatible and Au = Su, Bv = Tv, therefore, Az = ASu = SAu = Sz, Bz = BTv = TBv = Tz.

If Az ≠ z, then again as done above, there exist t₀ > 0 such that

. (3.14)

Using (3.1), take x = z, y = v, we have

let then where, we get

As, we get

which gives contradiction, hence Az = z.

Therefore, Az = z = Sz.

Similarly, one can prove that Bz = Tz = z. Hence, Az = Bz = Sz = Tz, and z is common fixed point of A, B, S and T.

Uniqueness easily follows by the use of inequality (3.1).

Hence the result.

Now we attempt to drop containment of subspaces by using weaker condition JCLR_ST property in Theorem 3.2.

Theorem 3.3: Let A, B, S and T be four selfmaps in fuzzy metric space satisfying condition (3.1) of Theorem 3.1 and (3.15) (A, S) and (B, T) shares the JCLR_ST property.

Then pairs (A, S) and (B, T) have coincidence point. Further if (A, S) and (B, T) be weakly compatible pair of self maps of X then A, B, S and T have a unique common fixed point in X.

Proof: The pairs (A, S) and (B, T) satisfy the (JCLR_ST) property, then there exist two sequences {x_n} and {y_n}

in X such that for some.

Firstly, we claim that Tu = Bu. Suppose not, then there exist t₀ > 0 such that

. (3.16)

The inequality (3.16) is always true when Tu ≠ Bu. To support our claim, we suppose on contrary that (3.16) is not true all t > 0, i.e.,

(3.17)

Now, using equality (3.17) repeatedly, we get

as. This gives, for all t > 0. Hence, Tu = Bu, which gives contradiction.

Therefore, inequality (3.16) is always true for some t₀ > 0.

Using (3.1), take x = x_n, y = u, we get

let then where and, we get

As, we get

which gives contradiction, hence Tu = Bu.

Next, we show that Au = Tu. Suppose not, then again as done above, there exist t₀ > 0 such that

. (3.18)

Using (3.1), take x = u, y = y_n, we get

let then where and, we get

As, we get which gives contradiction, hence Au = Tu. Hence, Au

= Bu = Su = Tu = z (say). Since the pair (A, S) is weakly compatible, ASu = SAu and then Az = Sz. Similarly, as the pair (B, T) is weakly compatible, BTu = TBu and then Tz = Bz.

Next, we claim that Az = z, suppose not. Then by (3.1), take x = z, y = u, we get

let then where,

As, we get

a contradiction, hence, Az = Bz = z. Therefore, z is a common fixed point of A and B. Similarly, we prove that Sz = Tz = z by taking x = u, y = z in (3.1). Therefore, we conclude that z = Az = Bz = Sz = Tz this implies that A, B, S and T have common fixed point in X.

Uniqueness easily follows by the use of inequality (3.1).

Next we attempt to drop closedness of range of maps and relax containment of two subspaces to one subspace by replacing property (E.A.) by a weaker condition CLR_S property in Theorem 3.1.

Theorem 3.4: Let A, B, S and T be four selfmaps fuzzy metric space (X, M, *) satisfying condition (3.1) of Theorem 3.1 and

(3.19) (A, S) and (B, T) shares the CLR_S property (CLR_T property)

(3.20).

Then pairs (A, S) and (B, T) have coincidence point. Further if (A, S) and (B, T) be weakly compatible pair of self maps of X then A, B, S and T have a unique common fixed point in X.

Proof: Proof of this theorem easily follows on same lines of Theorem 3.2.

On taking A = B and S = T in Theorem 3.1 then we get the following interesting result which is improved version of Theorem 1 of Sedghi et al. [2] .

Corollary 3.1: Let A and S be self mappings of a fuzzy metric space (X, M, *) satisfying the following:

(3.21)

for all, t > 0 and for some,;

(3.22);

(3.23) pair (A, S) satisfies the property (E.A)

(3.24) A(X) or S(X) is a closed subset of X.

Then pair (A, S) has a coincidence point. Further, if pair A and S be weakly compatible self maps of fuzzy metric space (X, M, *), then A and S have a unique common fixed point in X.

On taking A = B and S = T in Theorem 3.4 then we get the following interesting result which is improved version of Theorem 3.3 of Chauhan et al. [3] .

Corollary 3.2: Let A and S be self mappings of a fuzzy metric space (X, M, *) satisfying the following:

(3.25) (A, S) satisfies the CLR_S property.

Then pair (A, S) has a coincidence point. Further if pair A and S be weakly compatible self maps of X then A and S has a unique common fixed point in X.

Finally, we conclude this paper by furnishing example to demonstrate Theorem 3.3 besides exhibiting its superiority over earlier relevant results.

Example 3.2. Let be a fuzzy metric space where a * b = a. b for all and X = [3, 19). Let be defined as for all, Clearly,. Define A, B, S and T by

, ,

and

.

Take, clearly

for some.

Thus, (A, S) and (B,T) satisfies JCLR_ST property.

Also, , , , TX = (1, 13) and condition (3.1) is satisfied by maps A, B, S and T. Thus, the maps A, B, S and T satisfy all conditions of Theorem 3.3. Hence, A, B, S and T have a unique common fixed point x = 1.Moreover it should be noted that AX, BX, SX and TX are not closed subsets of X. Also, and. Also, A, B, S and T are all discontinuous maps at fixed point x = 1.

Definition 3.1 [21] Two families of self mappings and are said to be pairwise commuting if i)ii)iii)

As an application of Theorem 3.2, we prove a common fixed point theorem for four finite families of mappings on fuzzy metric spaces. While proving our result, we utilize Definition 3.1 which is a natural extension of commutativity condition to two finite families.

Theorem 3.4: Let, , and be four finite families of self mappings of a fuzzy metric space (X, M, *) such that, , and satisfying the conditions (3.1), (3.9), (3.10) and (3.26) the pairs of families and commute pairwise.

Then the pairs and have a point of coincidence each. Moreover, , and have a unique common fixed point.

Proof: By using (3.26), we first show that AS = SA as

.

Similarly one can prove that BT = TB. And hence, obviously the pair (A, S) is compatible and (B, T) is weakly compatible. Now using Theorem 3.1, we conclude that A, S, B and T have a unique common fixed point in X, say z.

Now, one needs to prove that z remains the fixed point of all the component mappings.

For this consider

.

Similarly, one can prove that

,

,

,

,

,

and

which show that (for all i, r, k and t) A_iz and S_kz are other fixed point of the pair (A, S) whereas B_rz and T_tz are other fixed points of the pair (B, T). As A, B, S and T have a unique common fixed point, so, we get

for all . which shows that z is a unique common fixed point of, and.

Remark 3.2: Theorem 3.4 is a slight but partial generalization of Theorem 3.2 as the commutativity requirements in this theorem are slightly stronger as compared to Theorem 3.1.

Remark 3.3. From the above results, it is asserted that for the existence of common fixed point of two pairs of self maps in fuzzy metric spaces satisfying JCLR_ST property the following conditions are never required:

a) the containment of ranges amongst the involved maps;

b) the completeness of the whole space/subspace;

c) the closedness of space/subspaces;

d) continuity requirement amongst the involved maps.

Acknowledgements

The first author wishes to thank Dr. B. E. Rhoades, Dr. Sanjay Kumar and Dr. S. S. Bhatia and the referees for their very helpful suggestions and many kind comments. The second author wishes to thank DSR, Jizan University, Saudi Arabia for financial support for the publication of this research.

References

NOTES

^*Corresponding author.