ANALYSIS International Journal of Analysis 2314-4998 2314-498X Hindawi Publishing Corporation 934738 10.1155/2013/934738 934738 Research Article Common Fixed Point Theorems Using the E.A. and CLR Properties in 2-Menger Spaces Singh Balbir 1 Gupta Vishal 2 Kumar Sanjay 3 Qian Chuanxi 1 Department of Mathematics B.M. Institute of Engineering and Technology Sonipat-131001 Haryana India 2 Department of Mathematics M.M. University, Mullana Ambala-133203 Haryana India mmumullana.org 3 Department of Mathematics D.C.R. University of Science and Technology Murthal, Sonipat-131039 Haryana India dcrustm.ac.in 2013 27 5 2013 2013 31 01 2013 02 04 2013 04 04 2013 2013 Copyright © 2013 Balbir Singh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

First, we prove a common fixed point theorem using weakly compatible maps in 2-Menger space with t-norm of Hadzic type. Second, we prove a common fixed point theorem using the E.A. property along with weakly compatible maps. Further, we obtained a common fixed point theorem using the CLR property along with weakly compatible maps. At the end, we provide an application of our main theorem for four finite families of mappings.

1. Introduction

The theory of probabilistic metric spaces is an important part of stochastic analysis, and so it is of interest to develop the fixed point theory in such spaces. The first result from the fixed point theory in probabilistic metric spaces was obtained by Sehgal and Bharucha-Reid . Since then many fixed points theorems for single valued and multivalued mappings in probabilistic metric spaces have been proved in . The study of 2-metric spaces was initiated by Gähler  and some fixed point theorems in 2-metric spaces were proved in [2, 8, 1013]. In 1987, Zeng  gave the generalization of 2-metric to probabilistic 2-metric as follows.

A probabilistic 2-metric space is an ordered pair (X,F), where X is an arbitrary set and F is a mapping from X3 into the set of distribution functions. The distribution function Fx,y,z(t) will denote the value of Fx,y,z at the positive real number t. The function Fx,y,z is assumed to satisfy the following conditions:

Fx,y,z(0)=0 for all x,y,zX;

Fx,y,z  (t)=1 for all t>0 if and only if at least two of the three points x,y,z are equal;

for distinct points x,yX, there exists a point zX such that Fx,y,z  (t)1 for t>0;

Fx,y,z  (t)=Fy,z,x  (t)=Fz,x,y  (t)= for all x,y,zX and t>0;

if Fx,y,w  (t1)=1, Fx,w,z  (t2)=1, and Fw,y,z  (t3)=1, then Fx,y,z  (t1+t2+t3)=1 for all x,y,z,wX and t1,t2,t3>0.

In 2003, Shi et al.  gave the notion of nth order t-norm as follows.

Definition 1.

A mapping Δ:i=1n[0,1][0,1] is called an nth order t-norm if the following conditions are satisfied:

Δ(0,0,,0)=0, Δ(a,1,1,,1)=a for all a[0,1];

Δ(a1,a2,a3,,an)=Δ(a2,a1,a3,,an)=Δ(a2,a3,a1,,an)==Δ(a2,a3,a4,,an,a1);

aibi,i=1,2,3,n, implies Δ(a1,a2,a3,,an)Δ(b1,b2,b3,,bn);

Δ(Δ(a1,a2,a3,,an),b2,b3  ,,bn)

=Δ(a1,Δ(a2,a3,,an,b2),b3,bn)

=Δ(a1,a2,Δ(a3,a4,.,an,b2,b3),b4,,bn)

==Δ(a1,a2,a3,,an-1,Δ(an,b2,b3,,bn)).

For n=2, we have a binary t-norm, which is commonly known as t-norm.

Basic examples of t-norm are the Lukasiewicz t-norm ΔL,ΔL(a,b)=max(a+b-1,0),t-norm ΔP, ΔP(a,b)=ab, and t-norm ΔM  , ΔM(a,b)=min{a,b}.

Definition 2 (see [<xref ref-type="bibr" rid="B9">6</xref>]).

A special class of t-norms (called a Hadzic-type t-norm) is introduced as follows.

Let Δ be a t-norm and let Δn:[0,1][0,1](nN) be defined in the following way: (1)Δ1(x)=Δ(x,x),Δn+1(x)=Δ(Δn(x),x)(nN,x[0,1]).

We say that the t-norm Δ is of H type if Δ is continuous and the family {Δn(x),nN} is equicontinuous at x=1.

The family {Δn(x),nN} is equicontinuous at x=1 if for every λ(0,1) there exists δ(λ)(0,1) such that the following implication holds: (2)x>1-δ(λ)implies  Δn(x)>1-λnN.

A trivial example of t-norm of H type is Δ=ΔM.

Remark 3.

Every t-norm ΔM is of Hadzic type but the converse need not be true; see .

There is a nice characterization of continuous t-norm.

If there exists a strictly increasing sequence {bn}nN[0,1] such that limnbn=1  and Δ(bn,bn)=bn for all nN, then Δ is of Hadzic type.

If Δ is continuous and Δ is of Hadzic type, then there exists a sequence {bn}nN as in (i).

Definition 4 (see [<xref ref-type="bibr" rid="B10">7</xref>]).

If Δ is a t-norm and (x1,x2,x3,,xn)[0,1]n(nN), then Δi=1nxi is defined recurrely by 1, if n=1 and Δi=1nxi=Δ(Δi=1n-1xi,xn) for all n2. If {bn}nN is a sequence of numbers from [0,1], then Δi=1xi is defined as limnΔi=1nxi (this limit always exists) and Δi=1xi as Δi=1xn+i.

Definition 5.

Let X be any nonempty set and D the set of all left-continuous distribution functions. A triplet (X,F,Δ) is said to be a 2-Menger space if the probabilistic 2-metric space (X,F) satisfies the following condition:

Fx,y,z(t)Δ(Fx,y,w(t1), Fx,w,z(t2), Fw,y,z(t3)), where t1,t2,t3>0, t1+t2+t3=t; x,y,z,wX; and Δ is the 3rd order t-norm.

Definition 6.

A sequence {xn} in a 2-Menger space (X,F,Δ) is said to be

converge with limit x if limnFxn,x,a(t)=1 for all t>0 and for every aX,

Cauchy sequence in X, if given ϵ>0, λ>0, there exists a positive integer Nϵ,λ such that (3)Fxn,xm,a(ϵ)>1-λm,n>Nϵ,λ,foreveryaX,

complete if every Cauchy sequence in X is convergent in X.

In 1996, Jungck’s  introduced the notion of weakly compatible as follows.

Definition 7.

Two maps f and g are said to be weakly compatible if they commute at their coincidence points.

In 2002, Aamri and Moutawakil  generalized the notion of noncompatible mapping to the E.A. property. It was pointed out in  that the property E.A. buys containment of ranges without any continuity requirements besides minimizing the commutativity conditions of the maps to the commutativity at their points of coincidence. Moreover, the E.A. property allows replacing the completeness requirement of the space with a more natural condition of closeness of the range. Recently, some common fixed point theorems in probabilistic metric spaces/fuzzy metric spaces by the E.A. property under weak compatibility have been obtained in .

Definition 8 (see [<xref ref-type="bibr" rid="B1">17</xref>]).

Let f and g be two self-maps of a metric (X, d). The maps f and g are said to satisfy the E.A. property if there exists a sequence {xn} in X such that (4)limnfxn=limngxn=u  for  someuX.

Now in a similar mode, we can state the E.A. property in 2-Menger space as follows.

Definition 9.

A pair of self-mappings (f,g) of 2-Menger spaces (X,F,Δ) is said to hold the E.A. property if there exists a sequence {xn} in X such that (5)limnFfxn,gxn,p(t)=1t>0,pX.

Example 10.

Let X=[0,) be the usual metric space. Define f,g:XX by fx=x/4 and gx=3x/4 for all xX. Consider the sequence {xn}=1/n. Since limnfxn=limngxn=0, then f and g satisfy the E.A. property.

Although E.A property is generalization of the concept of noncompatible maps, yet it requires either completeness of the whole space or any of the range spaces or continuity of maps. But on the contrary, the new notion of the CLR property (common limit range property) recently given by Sintunavarat and Kumam  does not impose such conditions. The importance of the CLR property ensures that one does not require the closeness of range subspaces.

Definition 11 (see [<xref ref-type="bibr" rid="B31">21</xref>]).

Two maps f and g on 2-Menger spaces X satisfy the common limit in the range of g (CLRg) property if limnfxn=limngxn=gx for some xX.

Example 12.

Let X=[0,) be the usual metric space. Define f,g:XX by fx=x+1 and gx=2x for all xX. Consider the sequence {xn}=1+(1/n). Since limnfxn=limngxn=2=g1, f and g satisfy the CLRg property.

Now we state a lemma which is useful in our study.

Lemma 13 (see [<xref ref-type="bibr" rid="B4">22</xref>]).

Let (X,F,Δ) be a 2-Menger space. If there exists q(0,1) such that F(x,y,z,qt)F(x,y,z,t) for all x,y,zX with zx,zy, and t>0, then x=y.

2. Weakly Compatible Maps Theorem 14.

Let (X,F,Δ) be a complete 2-Menger space with continuous t-norm Δ of H type. Let S and T be self-mappings of X. Then S and T have a unique common fixed point in X if and only if there exist two self-mappings A,B of X satisfying the following:

A(X)T(X) and B(X)S(X);

the pairs (A,S) and (B,T) are weakly compatible;

there exists q(0,1) such that for every x,y,pX and t>0, F(Ax,By,p,qt)min{F(Sx,Ty,p,t),F(Ax,Sx,p,t), F(By,Ty,p,t),F(Ax,Ty,p,t)};

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

Indeed, A,B,S, and T have a unique common fixed point in X.

Proof.

Suppose that S and T have a unique common fixed point, say zX.

Define A:XX by Ax=z for all xX and B:XX by Bx=z for all xX.

Then one can see that (2.1)–(2.4) are satisfied.

Conversely, assume that there exist two self-mappings A,B of X satisfying conditions (2.1)–(2.4). From condition (2.1) we can construct two sequences {xn} and {yn} of X such that (6)y2n-1=Tx2n-1=Ax2n-2,y2n=Sx2n=Bx2n-1forn=1,2,. Putting x=x2n and y=x2n+1 in (2.3), we have that for all pX and t>0(7)F(y2n+1,y2n+2,p,qt)=F(Ax2n,Bx2n+1,p,qt)min{F(Sx2n,Tx2n+1,p,t),F(Ax2n,Sx2n,p,t),F(Bx2n+1,Tx2n+1,p,t),F(Ax2n,Tx2n+1,p,t)}=min{F(y2n,y2n+1,p,qt),F(y2n+1,y2n+2,p,qt)} implies F(y2n+1,y2n+2,p,qt)F(y2n,y2n+1,p,t), because F is nondecreasing.

Also, letting x=x2n+2 and y=x2n+1 in (2.3), we have that (8)F(y2n+2,y2n+3,p,qt)F(y2n+1,y2n+2,p,t)pX,t>0. In general, we have (9)F(yn,yn+1,p,qt)F(yn-1,yn,p,t). Thus for all pX,t>0  and  n=1,2,3(10)F(yn,yn+1,p,t)F(y0,y1,p,tqn).

We now show that {yn} is a Cauchy sequence in X.

Let ϵ(0,1) be given. Since the t-norm Δ is of H type, there exists λ(0,1) such that for all m, nN with m>n(11)Δ2m-n(1-  λ)>1-ϵ. Since limnF(y0,y1,p,t/qn)=1, there exists n0N   such that for all pX and t>0,F(y0,y1,p,t/qn)>1-λ for all nn0.

From (10) we have for all pX and t>0, F(yn,yn+1,p,t)>1-λ for all nn0.

Let m>nn0. Then for all pX and t>0, we have (12)F(ym,yn,p,t)=F(ym,yn,p,t3+t3+t3)Δ(F(ym,yn,yn+1,t3),F(ym,yn+1,p,t3),F(yn+1,yn,p,t3))Δ(F(yn+1,yn,p,t3),Δ(F(yn+1,yn,ym,t3),F(yn+1,ym,p,t3)))Δ(Δ2((1-  λ),F(yn+1,ym,p,t3))).

Since (13)F(yn+1,ym,p,t3)Δ(F(yn+2,yn+1,p,t32),  Δ(F(yn+2,yn+1,ym,t32),  F(yn+2,ym,p,t32))).From (13), we get (14)F(ym,yn,p,t)Δ(Δ22(1-λ),F(yn+2,ym,p,t32)). Inductively, we obtain (15)F(ym,yn,p,t)Δ(Δ2m-n(1-λ),F(ym,ym,p,t  3m-n))=Δ2m-n(1-λ).

From (11) and (13) we get, for all pX and t>0,F(ym,yn,p,t)>1-ϵ for all m>nn0. Thus {yn} is a Cauchy sequence in X. Since X is complete, there exists a point z in X such that limnyn=z and this gives (16)limnSx2n=limnTx2n-1=limnAx2n-2= limnBx2n-1=znN. Without loss of generality, we assume that S(X) is a complete subspace of X; therefore, z=Su for some uX.

Subsequently, we have (17)limnSxn=limnTxn=limnAxn=limnBxn=z=Su. Next, we claim that Au=Su.

For this purpose, we put x=u and y=xn in (2.3); we have (18)F(Au,Bxn,p,qt)min{F(Su,Txn,p,t),F(Au,Su,p,t),F(Bxn,Txn,p,t),F(Au,Txn,p,t)}.

Taking limit as n(19)F(Au,z,p,qt)min{F(z,z,p,t),F(Au,z,p,t),F(z,z,p,t),F(Au,z,p,t)}F(Au,z,p,t). By Lemma 13, we have Au=z.

Hence Au=Su=z.

Since A(X)T(X), there exists a point vX such that Au=z=Tv.

Next we claim that Tv=Bv.

Putting x=u and y=v in (2.3), we have (20)F(Au,Bv,p,qt  )min{F(Su,Tv,p,t),F(Au,Su,p,t),F(Bv,Tv,p,t),F(Au,Tv,p,t)}min{F(z,z,p,t),F(z,z,p,t),F(Bv,Tv,p,t),F(z,z,p,t)}F(Bv,Tv,p,t).

That is, F(Tv,Bv,p,qt)F(Bv,Tv,p,t).

By Lemma 13, we have Tv=Bv.

Thus Au=Su=Tv=Bv=z.

Since the pairs (A,S) and (B,T) are weakly compatible and u and v are their points of coincidence, respectively, then (21)Az=A(Su)=S(Au)=Sz,Bz=B(Tv)=T(Bv)=Tz.

Now we prove that z is a common fixed point of A,B,S, and T.

For this purpose, puting x=z and y=v in (2.3), we get (22)F(Az,Bv,p,qt)min{F(Sz,Tv,p,t),F(Az,Sz,p,t),F(Bv,Tv,p,t),F(Az,Tv,p,t)}min{F(Az,Bv,p,t),F(Az,Az,p,t),F(Bv,Bv,p,t),F(Az,Bv,p,t)}F(Az,Bv,p,t).

By Lemma 13, we have Az=Bv=z.

Hence z=Az=Sz, and z is a common fixed point of A and S. One can prove that Bv=z is also a common fixed point of B and T.

Uniqueness. Suppose zw is another fixed point of A,B,S, and T.

Then, for all pX with pz and pw and t>0, (23)F(z,w,p,qt)=F(Az,Bw,p,qt)min{F(Sz,Tw,p,t),F(Az,Sz,p,t),F(Bw,Tw,p,t),F(Az,Tw,p,t)}min{F(z,w,p,t),F(z,z,p,t),F(w,w,p,t),F(z,w,p,t)}F(z,w,p,t), which implies that F(z,w,p,qt)F(z,w,p,t).

Hence z=w is a unique common fixed point of A,B,S, and T.

This completes the proof of the theorem.

Corollary 15.

Let (X,F,Δ) be a 2-Menger space with continuous t-norm Δ of H type. Let S and T be self-mappings of X. Then S and T have a unique common fixed point in X if and only if there exist two self-mappings A,B of X satisfying the following:

A(X)S(X);

pair (A,S) is weakly compatible;

there exists q(0,1) such that for every x,y,pX and t>0, (24)F(Ax,Ay,p,qt)min{F(Sx,Sy,p,t),F(Ax,Sx,p,t),F(Ay,Sy,p,t),F(Ax,Sy,p,t)};

one of the subspaces A(X) or S(X) is a closed subspace of X.

Indeed, A and S have a unique common fixed point in X.

Proof .

We can easily prove the theorem by setting B=A and T=S, in the proof of the Theorem 14.

Corollary 16.

Let (X,F,Δ) be a 2-Menger space with continuous t-norm Δ of H type. Let S and T be self-mappings of X. Then S and T have a unique common fixed point in X if and only if there exist two self-mappings A,B of X satisfying (2.1), (2.2), (2.4), and the following:

there exists q(0,1) such that for every x,y,pX and t>0, F(Ax,By,p,qt)F(Sx,Ty,p,t).

Indeed, A,B,S, and T have a unique common fixed point in X.

Proof.

Proof easily follows by setting min{F(Sx,Ty,p,t), F(Ax,Sx,p,t), F(By,Ty,p,t), F(Ax,Ty,p,t)}=F(Sx,Ty,p,t) in the proof of Theorem 14.

3. E.A. Property and Weakly Compatible Maps

Now we prove our main result for weakly compatible maps along with the E.A. property as follows.

Theorem 17.

Let (X,F,Δ) be a 2-Menger space with continuous t-norm Δ of H type. Let A, B, S, and T be self-mappings of X satisfying (2.1), (2.2), (2.3), and (2.4) and the following:

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

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 exists a sequence {xn} in X such that limnBxn=limnTxn=z for some zX.

Since B(X)S(X), there exists a sequence {yn} in X such that Bxn=Syn. Hence limnSyn=z. Also A(X)T(X); there exists a sequence {yn} in X such that Ayn=Txn. Hence limnAyn=z.

Suppose that S(X) is a complete subspace of X. Then z=Su for some uX; subsequently; we have (25)limnBxn=limnTxn=z=limnAyn=limnSyn=z=Sufor  someuX.

Next, we claim that Au=Su.

For this purpose, we put x=u and y=xn in (2.3); this gives (26)F(Au,Bxn,p,qt)min{F(Su,Txn,p,t),F(Au,Su,p,t),F(Bxn,Txn,p,t),F(Au,Txn,p,t)}.

Taking limit as n(27)F(Au,z,p,qt)min{F(z,z,p,t),F(Au,z,p,t),F(z,z,p,t),F(Au,z,p,t)}F(Au,z,p,t).

By Lemma 13, we have Au=z.

Hence Au=Su=z.

Since A(X)T(X), there exists a point vX such that Au=z=Tv.

Next we claim that Tv=Bv.

Putting x=u and y=v in (2.3), we have (28)F(Au,Bv,p,qt)min{F(Su,Tv,p,t),F(Au,Su,p,t),F(Bv,Tv,p,t),F(Au,Tv,p,t)}min{F(z,z,p,t),F(z,z,p,t),F(Bv,Tv,p,t),F(z,z,p,t)}  F(Bv,Tv,p,t). That is, F(Tv,Bv,p,qt)F(Bv,Tv,p,t).

By Lemma 13, we have Tv=Bv.

Thus Au=Su=Tv=Bv=z.

Since the pairs (A,S) and (B,T) are weakly compatible and u and v are their points of coincidence, respectively, then Az=A(Su)=S(Au)=Sz and Bz=B(Tv)=T(Bv)=Tz.

Now we prove that z is a common fixed point of A,B,S, and T.

For this purpose, we put x=z and y=v in (2.3); we get (29)F(Az,Bv,p,qt)min{F(Sz,Tv,p,t),F(Az,Sz,p,t),F(Bv,Tv,p,t),F(Az,Tv,p,t)}  min{F(Az,Bv,p,t),F(Az,Az,p,t),F(Bv,Bv,p,t),F(Az,Bv,p,t)}F(Az,Bv,p,t).

By Lemma 13, we have Az=Bv=z.

Hence z=Az=Sz and z is a common fixed point of A and S. One can prove that Bv=z is also a common fixed point of B and T.

Uniqueness. Suppose zw is another fixed point of A,B,S, and T.

Then, for all pX with pz and pw and t>0, (30)F(z,w,p,qt)=F(Az,Bw,p,qt)min{F(Sz,Tw,p,t),F(Az,Sz,p,t),F(Bw,Tw,p,t),F(Az,Tw,p,t)}min{F(z,w,p,t),F(z,z,p,t),F(w,w,p,t),F(z,w,p,t)}F(z,w,p,t), which implies that F(z,w,p,qt)F(z,w,p,t).

Hence z=w is a unique common fixed point of A,B,S, and T.

Corollary 18.

Let (X,F,Δ) be a 2-Menger space with continuous t-norm Δ of H type. Let A,B,S, and T be self-mappings of X satisfying (2.1), (2.2), (2.4), and (2.9).

Then A,B,S, and T have a unique common fixed point in X.

4. CLR Property and Weakly Compatible Maps

Now we prove our main result for weakly compatible maps along with the CLRS property as follows.

Theorem 19.

Let (X,F,Δ) be a 2-Menger space with continuous t-norm Δ of H type. Let A,B,S, and T be self mapping of X satisfying (2.1), (2.2), (2.3), and the following:

pair (A,S) or pair (B,T) satisfies CLRS property;

one of the subspaces A(X),B(X),S(X), or T(X) is a closed subspace of X.

Then A,B,S, and T have a unique common fixed point in X.

Proof.

If the pair (A,S) satisfies the CLRS property then there exists a sequence {xn} in X such that limnAxn=limnSxn=z, where zS(X). Therefore, there exists a point uX such that Su=z. Since T(X) is a closed subspace of X and A(X)T(X), for each {xn}X there corresponds a sequence {yn}X such that Axn=Tyn.

Therefore, limnTxn=limnAxn=z, where zS(X).

Thus in all, we have (31)limnAxn=limnSxn=limnTyn=z.

Now we are required to show that limnByn=z.

Putting x=xn and y=yn in (2.3), we get (32)F(Axn,Byn,p,qt)min{F(Sxn,Tyn,p,t),F(Axn,Sxn,p,t),  F(Byn,Tyn,p,t),F(Axn,Tyn,p,t)}.

Let limnByn=lz for t>0.Then taking limit as n, we have (33)F(z,l,p,qt)min{F(z,z,p,t),F(z,z,p,t),F(l,z,p,t),F(z,z,p,t)}F(l,z,p,t).

By Lemma 13, we have z=l, then limnByn=z.

Hence limnAxn = limnSxn=limnTyn=limnByn=z=Su, for some uX.

Uniqueness follows from Theorem 14.

Corollary 20.

Let (X,F,Δ) be a 2-Menger space with continuous t-norm Δ of H type. Let A,B,S, and T be self-mappings of X satisfying (2.1), (2.2), (2.9), (4.1), and (4.2).

Then A,B,S, and T have a unique common fixed point in X.

5. Application

As an application of Theorem 14, we prove a common fixed point theorem for four finite families of mappings which runs as follows.

Theorem 21.

Let {A1,A2,,Am}, {B1,B2,,Bn}, {S1,S2,,Sp}, and {T1,T2,,Tq} be four finite families of self-mappings of a 2-Menger space (X,F,Δ) with continuous t-norm of Hadzic type such that A=A1A2Am, B=B1B2Bn, S=S1S2Sp, and T=T1T2Tq satisfy condition (2.1), (2.2), (2.3), and (2.4); then

A and S have a point of coincidence;

B and T have a point of coincidence.

Moreover, if AiAj=AjAi, BkBl=BlBk, SrSs=SsSr, TtTu=TuTt,   AiSr=SrAi, and BkTt=TtBk for all i,jI1={1,2,,m}, k, lI2={1,2,,n}, r, sI3={1,2,,p}, and t, uI4={1,2,,q}, then (for all iI1,kI2,rI3, and tI4)  Ai, Sr, Bk, and Tt have a common fixed point.

Proof.

The conclusions (i) and (ii) are immediate as A,S,B, and T satisfy all the conditions of Theorem 14. Now appealing to component wise commutativity of various pairs, one can immediately prove that AS=SA and BT=TB, and hence, obviously both pairs (A,S) and (B,T) are coincidently commuting. Note that all the conditions of Theorem 14 (for mappings A,S,B, and T) are satisfied ensuring the existence of a unique common fixed point, say z. Now one needs to show that z remains the fixed point of all the component maps. For this consider (34)A(Aiz)=((A1A2Am)Ai)z=(A1A2Am-1)((AmAi)z)=(A1A2Am-1)(AiAmz)=(A1A2Am-2)(Am-1Ai(Amz))=(A1A2Am-2)(AiAm-1(Amz))==  A1Ai(A2A3A4Amz)=AiA1(A2A3A4Amz)=Ai(Az)=Aiz. Similarly, one can show that (35)A(Srz)=Sr(Az)=Srz,S(Srz)=Sr(Sz)=Srz,S(Aiz)=Ai(Sz)=Aiz,B(Bkz)=Bk(Bz)=Bkz,B(Ttz)=Tt(Bz)=Ttz,T(Ttz)=Tt(Tz)=Ttz,T(Bkz)=Bk(Tz)=Bkz, which show that (for all i,r,k, and t) Aiz  and  Srz are other fixed points of the pair (A,S) whereas Bkz  and  Ttz are other fixed points of the pair (B,T). Now appealing to the uniqueness of common fixed points of both pairs separately, we get (36)z=Aiz=Srz=Bkz=Ttz, which shows that z is a common fixed point of Ai,  Sr,  Bk, and Tt for all i,r,k, and t.

By setting A=A1=A2==Am,B=B1=B2==Bn,S=S1=S2==Sp, and T=T1=T2==Tq, one deduces the following for certain iterates of maps, which run as follows.

Corollary 22.

Let A,B,S, and T be four self-mappings of a 2-Menger space (X,F,Δ) such that Am,  Bn,  Sp, and Tq satisfy the conditions (2.1) and (2.3). If one of Am(X),  Bn(X),  Sp(X), or Tq(X) is a closed subset of X, then A,B,S, and T have a unique common fixed point provided (A,S) and (B,T) commute.

Acknowledgments

One of the authors (S. Kumar) would like to acknowledge the UGC for providing financial grant through Major Research Project under Ref. 39-41/2010 (SR). I am also thankful to the referee for careful reading of the paper and giving useful comments to improve it.