1. Introduction

Many researchers have been generalising the notion of metric space in different ways and Menger space is one of such generalisations introduced by the great mathematician Karl Menger ^[1] in the year 1942 who used distribution functions instead of non-negative real numbers as the value of metric. Schweizer and Sklar ^[4] studied this concept and gave some fundamental results on this space. In 1972, Sehgal and Bharucha-Reid ^[2] obtained a generalization of Banach Contractive Principle on a complete Menger space which is a milestone in developing fixed point theory in Menger space.

In 1982, Sessa ^[3] improved the definition of commutativity in fixed point theorems by introducing the notion of weakly commuting maps. Then in 1986, Jungck ^[5] introduced the concept of compatible maps and this notion of compatible mappings in Menger space was introduced by Mishra ^[6]. Further this condition has been weakened by introducing the notion of weakly compatible mappings by Jungck and Rhoades ^[8]. Recently, Singh and Jain ^[10] introduced weakly compatible maps in Menger space to establish a common fixed point theorem.

Al. Thagafi and Shahzad ^[13] introduced the notion of occasionally weakly compatible mappings in metric space which is more general than weakly compatible mappings. Recently, Jungck and Rhoades ^[11] extensively studied the notion occasionally weakly compatible mappings in semi-metric space and Chauhan et.al. ^[13] extended the notion of occassionally weakly compatible mappings to PM-space.

Cho, Sharma and Sahu ^[7] introduced the concept of semi-compatibility in a d-complete topological space and using this concept of semi compatibility in Menger space, Singh et.al. ^[9] proved a fixed point theorem using implicit relation. Recently, Rohen and Chhatrajit ^[16] used the concept of semi compatible mappings in cone metric space to prove some common fixed point theorems.

In this paper, we prove a common fixed point theorem in Menger space using the concept of semi compatible and occasionally weakly compatible mappings. Some results are also given as corollaries. Our results generalise some similar results ^{[12, 15, 16]}.

2. Preliminaries

Definition 2.1 A triangular norm * (shortly t-norm) is a binary operation on the unit interval [0, 1] such that for all a, b, c, d [0, 1] the following conditions are satisfied:

i) a * 1 = a;

ii) a * b = b * a;

ii) a * b ≤ c * d whenever a ≤ c and b ≤ d;

iv) a * (b * c) = (a * b) * c.

Example: a * b = min {a, b}.

Definition 2.2 A distribution function is a function: which is left continuous on R, non-decreasing and, .

We will denote the family of all distribution functions on by ∆. is a special element of ∆ defined by

If is a non-empty set, ∆ is called a probabilistic distance on and is usually denoted by.

Definition 2.3 (Schweizer and Sklar ^[4]): The ordered pair is called a probabilistic metric space (shortly PM-space) if is a nonempty set and is a probabilistic distance satisfying the following conditions:

(i);

(ii);

(iii);

(iv), .

The ordered triplet is called Menger space if is a PM-space, * is a t-norm and the following condition is also satisfied i.e.

(v) *.

Proposition 2.4 (Sehgal and Bharucha-Reid ^[2]) Let be a metric space. Then the metric induces a distribution function defined by for all and > 0. If -norm * is a * b = min for all then is a Menger space. Further, is a complete Menger space if is complete.

Definition 2.5 (Mishra ^[6]) Let be a Menger space and * be a continuous norm.

i) A sequence in is said to converge to a point in (written as ) iff for every > 0 and λ , there exists an integer = (, λ ) such that > 1- λ for all ≥.

ii) A sequence in is said to be Cauchy if for every > 0 and λ , there exists an integer = (,λ ) such that () > 1-λ for all and > 0.

iii) A Menger space in which every Cauchy sequence is convergent is said to be complete.

Remark 2.6 If * is a continuous t- norm, it follows from definition 2.3 (v) that the limit of sequence in Menger space is uniquely determined.

Definition 2.7 (Mishra^[6]) Two self-maps S and T of a Menger space are said to be compatible if for all t > 0, whenever is a sequence in such that , for some in as .

Definition 2.8 (Singh and Jain ^[10]) Two self-maps and of a Menger space are said to be weakly compatible (or coincidentally commuting) if they commute at their coincident points i.e. if = for some then =.

Definition 2.9(Al Thagafi and Shahzad ^[12]) Two self-maps and of a Menger space are said to be occasionally weakly compatible (owc) if and only if and commute at their coincidence point.

Definition 2.10 (Singh B. and Jain S.^[10]) Two self-maps and of a Menger space are said to be semi compatible if for all > 0, whenever is a sequence in such that , , for some in , as .

Lemma 2.11(Singh B. and Jain S. ^[10]) Let be a sequence in a Menger space with continuous -norm * and. If there exists a constant such that for all and then is a Cauchy sequence in.

3. Main Results

Theorem 3.1 Let A, B, S, T, L and M be self-maps on a complete Menger space (X, F, *) with t*t ≥ t for all t [0,1], satisfying:

i) L(X) ST(X), M(X) AB(X);

ii) There exists a constant k (0,1) such that

for all x, y X and t > 0 where 0 < p, q < 1 such that p + q=1;

iii) AB = BA, ST = TS, LB = BL, MT = TM;

iv) Either AB or L is continuous;

v) The pair (L, AB) is semi compatible and (M, ST) is occasionally weakly compatible.

Then A, B, S, T, L and M have a unique common fixed point.

Proof: Let us choose an arbitrary point x₀ in X then by (i), there exist,X such that and . By induction we can construct sequences {x_n} and {y_n} in X such that and for n = 0, 1, 2, 3…

By (ii), we have

Hence, we have

Similarly, we also have

In general, for all n even or odd, we have

for k (0, 1) and t > 0. Thus, by Lemma 2.11, {} is a Cauchy sequence in X. Since (X, F, *) is complete, it converges to a point z in X. Also {} → z, {} →z, {} →z and {} →z.

First, let AB be continuous then we have, AB(AB)→ABz and (AB)→ABz. Since (L, AB) is semi compatible, we have L(AB)→ABz.

Again, by (ii), we have

Letting n→∞ we have

For k (0, 1) and all t > 0. Thus, we have z = ABz.

Now by (ii), we have

Letting n→∞ we have

Noting that ≤1 and using (iii) in definition 2.1, we have

Thus, we have z = Lz = ABz.

By (ii), we have

Since AB=BA and BL=LB, we have L(Bz)=B(Lz)=Bz and AB(Bz)=B(ABz)=Bz. Letting n→∞, we have

For k (0, 1) and all t > 0. Thus, we have z = Bz. Since z = ABz, we also have z = Az. Therefore, z = Az = Bz = Lz.

Since L(X) ST(X), there exists v X such that z = Lz = STv. By (ii), we have

Letting n→∞ we have

Noting that ≤1 and using (iii) in definition 2.1, we have

Thus, by Lemma 2.11, we have z = Mv and so z = Mv = STv. Since (M, ST) is occasionally weakly compatible, we have STMv = MSTv. Thus, STz = Mz.

By (ii), we have

Letting as n→∞ we have

Thus, we have z = Mz and therefore z = Az = Bz = Lz =Mz = STz.

By (ii), we have

Since MT = TM and ST = TS, we have MTz = TMz = Tz and ST(Tz) =T(STz) = Tz. Letting n→∞, we have

Thus, we have z = Tz. Since Tz = STz, we also have z = Sz. Therefore, z = Az = Bz = Lz =Mz = Sz = Tz, and hence z is the common fixed point of A, B, L, M, S and T.

Secondly, let L be continuous then we have, LL→Lz and L(AB) →Lz.

Since (L, AB) is semi compatible, we have L (AB) →ABz and ABz =Lz.

By (ii), we have

Letting n→∞ we have

Thus, we have z = Lz. Hence z = Lz = Mz = S z = Tz.

Since M(X) AB(X), there exists vϵ X such that z = Mz = ABv. By (ii), we have

Letting n→∞

Noting that and using (iii) in definition 2.1, we have

Thus, we have z = Lv = ABv. Since (L, AB) is occasionally weakly compatible, we have Lz = ABz and using z = Bz as shown above. Hence z = Az = Bz = Sz = Tz = Lz = Mz, that is, z is the common fixed point of the six mappings in this case also.

In order to prove the uniqueness of fixed point let w be another common fixed point of A, B, S, T, L and M. Then by (ii), we have

which implies that

Thus, we have z = w. This completes the proof of the theorem.

If we take B = T =( the identity map on X) in the main theorem, we have the following:

Corollary 3.2: Let A, S, L and M be self-maps on a complete Menger space (X, F, *) with t*t ≥ t for all t [0, 1], satisfying:

(i) L(X) S(X), M(X) A(X);

(ii)There exists a constant kϵ (0, 1) such that

for all x, y X and t > 0 where 0 < p, q < 1 such that p + q=1;

(iii)either A or L is continuous;

(iv)the pair (L, A) is semicompatible and (M, S) is occassionally weakly compatible.

Then A, S, L, and M have a unique common fixed point.

If we take A = S, L = M and B = T = in the main Theorem, we have the following:

Corollary 3.3: Let (X, F, *) be a complete Menger space with t*t ≥ t for all t [0, 1] and let A and L be compatible maps on X such that L(X) A(X). If A is continuous and there exists a constant k (0, 1) such that

for all x, y X and t > 0 where 0 < p, q < 1 such that p + q =1, then A and L have a unique fixed point.

Example 3.4: Let X = [0, 1] with the metric d defined by d(x, y) =│x-y│ and defined =H (t- d (x, y)) for all x, y X, t > 0. Clearly (X, F, *) is a complete Menger space where t-norm * is defined by a*b = min{a, b} for all a, b [0, 1]. Let A, B, S, T, L and M be maps from X into itself defined as

for all x X. Then

Clearly AB = BA, ST = TS, LB = BL, MT = TM and AB, L are continuous. If we take k = and t=1, we see that the condition (ii) of the main Theorem is also satisfied. Moreover, the maps L and AB are semi compatible if , where {} is a sequence in X such that == 0 for 0 X. The maps M and ST are occasionally weakly compatible at 0. Thus, all conditions of the main Theorem are satisfied and 0 is the unique common fixed point of A, B, S, T, L, and M.

Acknowledgement

The first author is supported by UGC, New Delhi vide MRP F. No. 42-12/2013 (SR).

References

[1]	K. Menger, Statistical Metric, Proc. Nat. Acad. Sci. 28 (1942), 535-537.
	In article		CrossRef PubMed
[2]	Sehgal V. M. and Bharucha-Reid A.T., Fixed points of contraction maps on probabilistic metric spaces, Math. System Theory 6(1972), 97-102.
	In article		CrossRef
[3]	S. Sessa, On a weak commutative condition in fixed point consideration, Publ. Inst. Math. (Beograd) 32 (1982) 149-153.
	In article
[4]	Schweizer B., Sklar A., Probanilistic Metric Spaces, Elsevier, North- Holland, New York (1983).
	In article		PubMed
[5]	G. Jungck, Compatible mappings and common fixed points, Internat. J. Math. Math. Sci. (1986) 771-779.
	In article		CrossRef
[6]	S. N. Mishra, Common fixed points of compatible mappings in PM- spaces, Math. Japon. 36(1991) 283-289.
	In article
[7]	Cho. Y. J., Sharma B. K. and Sahu R. D., Semi-compatibility and fixed points, Math. Japon. 42(1), 1995, 91-98.
	In article
[8]	Jungck G., Rhoades B.E., Fixed points for set valued functions without continuity, Indian J. Pure Appl. Math., 29(1998), 227-238.
	In article
[9]	Singh B., Jain S., Semi-compatibility and fixed point theorems in Menger space using implicit relation, East Asian Math. J. 21(1), 2005, 65-76.
	In article
[10]	Singh B., Jain S., A fixed point theorem in Menger space through weak compatibility, J. Math. Anal. Appl., 301(2005), 439-448.
	In article		CrossRef
[11]	Jungck G., Rhoades B.E., Fixed point theorems for occasionally weakly compatible mappings, Fixed point theory 7, 2(2006) 286-296.
	In article
[12]	Servet Kutukcu, A fixed point theorem in Menger spaces, International Mathematical Forum, 1(32), 2006, 1543-1554.
	In article
[13]	Al-Thagafi M. A., Shahzad N., Generalized I-non expansive selfmaps and invariant approximations, Acta. Math. Sinica (English series) 24, 5(2008) 867-876.
	In article		CrossRef
[14]	Chauhan S., Kumar S., Pant B. D., Common fixed point theorems for occasionally weakly compatible mappings in Menger spaces, J. Adv. Research Pure Math. 3, 4(2011) 17-23.
	In article		CrossRef
[15]	Arihant Jain and Basant Chaudhary, On common fixed point theorems for semi-compatible and occasionally weakly compatible mappings in Menger space, IJRAS, 14(3), 2013, 662-670.
	In article
[16]	Yumnam Rohen and Th. Chhatrajit, Semicompatible mappings and fixed point theorems in cone metris space, Wulfenia Journal, 21(4), 2014, 216-224.
	In article