WEAKLY COMPATIBLE MAPS IN 2 -NON-ARCHIMEDEAN MENGER PM-SPACES

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WEAKLY COMPATIBLE MAPS IN 2 -NON-ARCHIMEDEAN MENGER PM-SPACES

RENU CHUGH and SANJAY KUMAR Received 19 March 2001

The aim of this paper is to introduce the concept of weakly compatible maps in 2-non- Archimedean Menger probabilistic metric (PM) spaces and to prove a theorem for these mappings without appeal to continuity. We also provide an application.

2000 Mathematics Subject Classification: 47H10, 54H25.

1. Introduction. In 1999, Chugh and Sumitra [2] introduced the concept of 2-N.A.

Menger PM-space as follows.

Definition1.1. LetXbe any nonempty set and Lthe set of all left continuous distribution functions. An ordered pair(X,F)is said to be a 2-non-Archimedean prob- abilistic metric space (briefly 2-N.A. PM-space) ifF is a mapping fromX×X×X into Lsatisfying the following conditions (where the value ofF atx,y,z∈X×X×X is represented byF_x,y,zorF(x,y,z)for allx,y,z∈X):

(i) F_x,y,z(t)=1 for allt >0 if and only if at least two of the three points are equal, (ii) Fx,y,z=Fx,z,y=Fz,y,x,

(iii) F_x,y,z(0)=0,

(iv) ifF_x,y,s(t1)=F_x,s,z(t2)=F_s,y,z(t3)=1, thenF_x,y,z(max{t1,t2,t3})=1.

Definition 1.2. A t-norm is a function∆:[0,1]×[0,1]×[0,1]→[0,1]which is associative, commutative, nondecreasing in each coordinate and∆(a,1,1)=afor everya∈[0,1].

Definition1.3. A 2-N.A. Menger PM-space is an order triplet(X,F,∆)where∆is at-norm and(X,F)is 2-N.A. PM-space satisfying the following condition:

(v)Fx,y,z(max{t1,t2,t3})≥∆(Fx,y,s(t1),Fx,s,z(t2),Fs,y,z(t3))for allx,y,z,s∈Xand t1,t2,t3≥0.

Definition 1.4. Let (X,F,∆)be a 2-N.A. Menger PM-space and ∆a continuous t-norm, then(X,F,∆)is a Hausdorff in the topology induced by the family of neigh- bourhoods ofx

U_x,λ,a1,a2,...,a_n, x, a_i∈X, >0, i=1,2,...,n, n∈Z⁺, (1.1) whereZ⁺is the set of all positive integers and

U_x,λ,a1,a2,...,a_n

=y∈X;F_x,y,ai() >1−λ, 1≤i≤n

= n

y∈X; Fx,y,a_i() >1−λ, 1≤i≤n

. (1.2)

Definition1.5. A 2-N.A. Menger PM-space(X,F,∆)is said to be of type(C)g if there exists ag∈Ωsuch that

gF_x,y,z(t)

≤gF_x,y,a(t)

+gF_x,a,z(t)

+gF_a,y,z(t)

(1.3)

for allx,y,z,a∈Xandt≥0, whereΩ= {g;g:[0,1]→[0,∞)}is continuous, strictly decreasing,g(1)=0 andg(0) <∞.

Definition1.6. A 2-N.A. Menger PM-space(X,F,∆)is said to be of type(D)g if there exists ag∈Ωsuch that

g

∆t1,t2,t3

≤gt1 +gt2

+gt3

∀t1,t2,t3∈[0,1]. (1.4)

Definition1.7. Let(X,F,∆)be a 2-N.A. Menger PM-space where∆is a continuous t-norm andA,S:X→Xbe mappings. The mappingsAandS are said to be weakly compatible if they commute at the coincidence point, that is, the mappingsAandS are weakly compatible if and only ifAx=SximpliesASx=SAx.

Remark 1.8. (1) If 2-N.A. PM-space(X,F,∆)is of type (D)g, then (X,F,∆) is of type(C)g.

(2) If(X,F,∆)is a 2-N.A. PM-space and∆≥∆m, where∆m(r ,s,t)=max{r+s+t− 1,0,0}, then(X,F,∆)is of type(D)g forg∈Ωdefined byg(t)=1−t.

Throughout this paper, let(X,F,∆)be a complete 2-N.A. Menger PM-space of type (D)gwith a continuous strictly increasingt-norm∆.

Letφ:[0,∞)→[0,∞)be a function satisfying the condition(Φ):

(Φ) φis upper semi-continuous from right andφ(t) < tfor allt >0.

Lemma1.9(see [1]). If a functionφ:[0,∞)→[0,∞)satisfies the condition(Φ), then (1) for allt≥0,lim_n→∞φⁿ(t)=0whereφⁿ(t)is thenth iteration ofφ(t);

(2) if{t_n}is a nondecreasing sequence of real numbers and t_n+1≤φ(t_n), n= 1,2,...,thenlim_n→∞t_n=0. In particular, ift≤φ(t)for allt≥0, thent=0.

Lemma1.10(see [1]). Let{y_n}be a sequence inXsuch thatlim_n→∞F_yn,yn+1,a(t)=1 for allt >0. If the sequence{yn}is not Cauchy sequence inX, then there exist0>0, t0>0, and two sequences{m_i}and{n_i}of positive integers such that

(i) m_i> n_i+1andn_i→ ∞asi→ ∞,

(ii) Fy_mi,y_ni,a(t0) <1−0andFy_mi−1,y_ni,a(t0) >1−0,i=1,2,....

Chugh and Sumitra [2] proved the following theorem.

Theorem1.11. LetA,B,S,T :X→Xbe mappings satisfying the following condi- tions:

(i) A(X)⊂T (X)andB(X)⊂S(X);

(ii) the pairsA,S andB,Tare weak compatible of type(A);

(iii) SandT are continuous;

(iv) for alla∈Xandt >0, gF_Ax,By,a(t)

≤φ

max

gF_{Sx,T y,a}(t),gF_Sx,Ax,a(t),gF_{T y,By,a}(t), 1

2

gF_Sx,By,a(t)

+gF_{T y,Ax,a}(t) ,

(1.5)

where a functionφ:[0,∞)→[0,∞)satisfies the condition(Φ).

ThenA,B,S, andT have a unique common fixed points inX.

Now we prove the following theorem.

Theorem1.12. LetA,B,S,T:X→Xbe mappings satisfying

A(X)⊂T (X), B(X)⊂S(X), (1.6)

the pairsA, SandB, T are weakly compatible, (1.7) g

FAx,By,a(t)

≤φ

max

g

FSx,T y,a(t) ,g

FSx,Ax,a(t) ,g

FT y,By,a(t) , 1

2

gF_Sx,By,a(t)

+gF_{T y,Ax,a}(t)

(1.8)

for allt >0,a∈Xwhere a functionφ:[0,∞)→(0,∞)satisfies the condition(Φ). Then A,B,S, andT have a unique common fixed point inX.

Proof. By (1.6) sinceA(X)⊂T (X), for anyx0∈X, there exists a pointx1∈Xsuch thatAx0=T x1. SinceB(X)⊂S(X), for thisx1, we can choose a pointx2∈Xsuch thatBx1=Sx2and so on, inductively, we can define a sequence{yn}inXsuch that

y2n=Ax2n=T x_2n+1, y_2n+1=Bx_2n+1=Sx_2n+2, forn=0,1,2,.... (1.9)

First we prove the following lemma.

Lemma1.13. LetA,B,S,T:X→Xbe mappings satisfying conditions (1.6) and (1.8), then the sequence{yn}defined by (1.9), such thatlim_n→∞g(Fyn,y_n+1,a(t))=0for all t >0,a∈X, is a Cauchy sequence inX.

Proof. Sinceg∈Ω, it follows that lim_n→∞(Fyn,y_n+1,a(t))=0 for all a∈X and t >0 if and only if lim_n→∞g(F_yn,yn+1,a(t))=0 for alla∈Xandt >0. ByLemma 1.10, if{y_n}is not a Cauchy sequence inX, there exist0>0,t0>0, and two sequences {mi},{ni}of positive integers such that

(A) m_i> n_i+1 andn_i→ ∞asi→ ∞,

(B) g(F_ymi,yni,a(t0)) > g(1−0)andg(F_{ymi−1,yni,a}(t0))≤g(1−0),i=1,2,....

Thus we have g

1−0< gF_ymi,yni,at0

≤gF_{ymi,yni,ymi−}1t0 +gF_{ymi,ymi−1,a}t0

+gF_{ymi−1,yni,a}t0

≤g

F_{ymi,yni,ymi−}1

t0

+g

F_{ymi,ymi−1,a} t0

+g 1−0

.

(1.10)

Lettingi→ ∞in (1.10), we have

limgF_ymi,yni,at0

=g

1−0. (1.11)

On the other hand, we have g

1−0

< g

Fy_mi,y_ni,a t0

≤g

Fy_mi,y_ni,y_ni+1 t0

+g

Fy_mi,y_ni+1,a t0

+g

Fy_ni+1,y_ni,a t0

. (1.12)

Now, considerg(F_ymi,yni+1,a(t0))in (1.12), without loss of generality, assume that bothniandmiare even.

Then by (1.8), we have g

F_ymi,yni+1,a t0

=g

F_{Axmi,Bxni+1,a} t0

≤φ

max

g

F_{Sxmi,T xni+1,a} t0

, g

FSxm_i,Axm_i,a t0

,g

FT xn_i+1,Bxni+1,a t0

, 1

2 g

F_{Sxmi,Bxni+1,a} t0

+g

F_{T xni+1,Axmi+1,a} t0

=φ

max

gF_{ymi,−1,yni,a}t0, g

F_{ymi,−1,ymi,a} t0

,g

F_yni,yni+1,a t0

, 1

2

gF_{ymi,−1,yni+1,a}t0

+gF_yni,ymi,at0 . (1.13) By (1.11), (1.12), and (1.13), lettingi→ ∞in (1.13), we have

g 1−0

≤φ

maxg

1−0,0,0,g

1−0

=φg

1−0< g 1−0

(1.14) which is a contradiction. Therefore,{y_n}is a Cauchy sequence inX.

Now, we are ready to prove our main theorem.

If we prove lim_n→∞g(F_yn,yn+1,a(t))=0 for alla∈Xandt >0, then byLemma 1.13, the sequence {y_n}defined by (1.9) is a Cauchy sequence inX. First we prove that lim_n→∞g(Fyn,y_n+1,a(t))=0 for alla∈Xandt >0. In fact, by (1.8) and (1.9), we have

g

Fy2n,Y_2n+1,a(t)

=g

FAx2n,Bx_2n+1,a(t)

≤φ

max

g

FSx2n,T x_2n+1,a(t) , g

FSx2n,Ax2n,a(t) ,g

FT x_2n+1,Bx_2n+1,a(t) , 1

2 g

FSx2n,Bx_2n+1,a(t) +g

FT x_2n+1,Ax2n,a(t)

=φ

max

g

Fy_2n−1,y_2n,a(t) ,g

Fy_2n−1,y_2n,a(t) , g

F_y2n,y2n+1,a(t) ,1

2 g

F_{y2n−1,y2n+1,a}(t) +g(1)

≤φ

max

gF_{y2n−1,y2n,a}(t),gF_y2n,y2n+1,a(t), 1

2

gF_{y2n−1,y2n,a}(t)

+gF_y2n,y2n+1,a(t) .

(1.15)

Ifg(Fy_2n−1,y_2n,a(t))≤g(Fy_2n,y_2n+1,a(t))for allt >0, then by (1.8), g

Fy2n,y_2n+1,a(t)

≤φ g

Fy2n,y_2n+1,a(t)

(1.16) and thus, byLemma 1.9, g(F_y2n,y2n+1,a(t))=0 for alla∈Xandt >0. Similarly, we haveg(Fy_2n+1, y_2n+2, a(t))=0, thus we have lim_n→∞g(Fyn,y_n+1,a(t))=0 for alla∈X andt >0. On the other hand, ifg(F_{y2n−1,y2n,a}(t))≥g(F_y2n,y2n+1,a(t)), then by (1.8), we have

g

Fy2n,y_2n+1,a(t)

≤φ g

Fy_2n−1,y2n,a(t)

∀a∈X, t >0. (1.17) Similarly,g(F_{y2n+1,y2n+2,a}(t))≤φ(g(F_y2n,y2n+1,a(t)))for alla∈Xandt >0. Thus we haveg(F_yn,yn+1,a(t))≤φ(g(F_yn−1,yn,a(t)))for alla∈Xandt >0 andn=1,2,3,..., therefore byLemma 1.9, lim_n→∞g(Fyn,y_n+1,a(t))=0 for all a∈X and t >0, which implies that{y_n}is a Cauchy sequence inXbyLemma 1.13. Since(X,F,∆)is com- plete, the sequence{y_n}converges to a pointz∈Xand so the subsequences{Ax_2n}, {Bx_2n+1},{Sx2n},{T x_2n+1}of{yn}also converge to the limitz. SinceB(X)⊂S(X), there exists a pointu∈Xsuch thatz=Su.

Now g

FAu,z,a(t)

≤g

FAu,Bx_2n+1,Z(t) +g

FBx_2n+1,z,a(t) +g

FAu,Bx_2n+1,a(t)

. (1.18) From (1.8), we have

gF_Au,Bx2n+1,a(t)

≤φ max

gF_{Su,T x2n+1,a}(t),gF_Su,Au,a(t),gF_{T x2n+1,Bx2n+1,a}(t), 1

2 g

FSu,Bx_2n+1,a(t) +g

FT x_2n+1,Au,a(t) .

(1.19) From (1.18) and (1.19), lettingn→ ∞, we have

g

FAu,z,a(t)

≤φ

max

g

FSu,z,a(t) ,g

FSu,Au,a(t) ,g

Fz,z,a(t) , 1

2 g

FSu,z,a(t) +g

Fz,Au,a(t)

=φ g

Fz,Au,a(t)

∀a∈X, t >0,

(1.20)

which meansz=Au=Su. SinceA(X)⊂T (X), there exists a pointv∈Xsuch that z=T v. Then, again using (1.8), we have

g

Fz,Bv,a(t)

=g

FAu,Bv,a(t)

≤φ

max

g

FSu,T v,a(t) ,g

FSu,Au,a(t) ,g

FT v,Bv,a(t) , 1

2 g

FSu,Bv,a(t) +g

FT v,Au,a(t)

=φ g

Fz,Bv,a(t)

, ∀a∈X, t >0,

(1.21)

Bv=z=T v.

Since pairs of mapsAandSare weakly compatible, thenASu=SAu, that is,Az= Sz. Now we show thatzis a fixed point ofA. IfAz≠z, then by (1.8),

gF_Az,z,a(t)

=gF_Az,Bv,a(t)

≤φ max

gF_{Sz,T v,a}(t),gF_Sz,Az,a(t),gF_{T v,Bv,a}(t), 1

2 g

FSz,Bv,a(t) +g

FT v,Az,a(t)

=φ max

g

FAz,z,a(t)

, impliesAz=z.

(1.22)

Similarly, pairs of mapsBandTare weakly compatible, we haveBz=T z. Therefore, g

FAz,z,a(t)

=g

FAz,Bz,a(t)

≤φ

max

g

F_{Sz,T z,a}(t) ,g

F_Sz,Az,a(t) ,g

F_{T z,Bz,a}(t) , 1

2

gF_Sz,Bz,a(t)

+gF_{T z,Az,a}(t)

=φ max

g

F_{z,T z,a}(t) .

(1.23)

Thus we haveBz=T z=z.

Therefore,Az=Bz=Sz=T zandzis a common fixed point ofA,B,S, andT. The uniqueness follows from (1.8).

2. Application

Theorem2.1. Let(X,F,∆)be a complete 2-N.A. Menger PM-space andA,B,S, and T be the mappings from the productX×XtoXsuch that

AX×{y}

⊆TX×{y}, BX×{y}

⊆X×{y}, gFA(T (x,y),y),T (A(x,y),y),a(t)

≤gFA(x,y),T (x,y),a(t), g

FB(S(x,y),y),S(B(x,y),y),a(t)

≤g

FB(x,y),S(x,y),a(t) (2.1) for alla∈Xandt >0and

gF_{A(x,y),B(x,y),a}(t)

≤φ

max

g

FS(x,y),T (x,y),a(t) ,g

FS(x,y),A(x,y),a(t) ,g

FT (x,y),B(x,y),a(t) , 1

2 g

FS(x,y),B(x,y),a(t) +g

FT (x,y),A(x,y),a(t)

(2.2)

for alla∈X,t >0, andx,y,x,y inX, then there exists only one pointbinXsuch that A(b,y)=S(b,y)=B(b,y)=T (b,y) ∀yinX. (2.3) Proof. By (2.2),

g

FA(x,y),B(x,y)(t)

≤φ

max

g

FS(x,y),T (x,y),a(t) ,g

FS(x,y),A(x,y),a(t) ,g

FT (x,y),B(x,y),a(t) , 1

2 g

F_{S(x,y),B(x,y),a(t)} +g

F_{T (x,y),A(x,y),a}(t)

(2.4)

for alla∈Xandt >0; therefore byTheorem 1.12, for eachyinX, there exists only onex(y)inXsuch that

Ax(y),y

=Sx(y),y

=Bx(y),y

=Tx(y),y

=x(y) (2.5)

for everyy,y inX, gF_x(y),x(y),a(t)

=g

FA(x(y),y),A(x(y),y),a(t)

≤φ

max

g

F_{A(x,y),A(x,y),a}(t) ,g

FA(x,y),A(x,y),a(t) ,g

F_{T (x,y),A(x,y),a}(t) , 1

2

gF_{A(x,y),A(x,y),a}(t)

+gF_{A(x,y),A(x,y),a}(t)

=gF_x(y),x(y),a(t).

(2.6)

This impliesx(y)=x(y )and hencex(y)is some constantb∈Xso that

A(b,y)=b=T (b,y)=S(b,y)=B(b,y) ∀yinX. (2.7)

References

[1] Y. J. Cho, K. S. Ha, and S. S. Chang,Common fixed point theorems for compatible mappings of type (A) in non-Archimedean Menger PM-spaces, Math. Japon.46(1997), no. 1, 169–179.

[2] R. Chugh and Sumitra,Common fixed point theorems in 2 non-Archimedean Menger PM- space, Int. J. Math. Math. Sci.26(2001), no. 8, 475–483.

Renu Chugh: Department of Mathematics, Maharshi Dayanand University, Rohtak- 124001, India

Sanjay Kumar: Department of Mathematics, Maharshi Dayanand University, Rohtak-124001, India

E-mail address:[email protected]