V. K. Gupta, Arihant Jain, Jaya Kushwah
Fixed point theorem in 2 non-archimedean Menger PM-Space using weakly L-compatible and weakly
M-compatible mappings
Abstract. In the present paper, we extend and generalize the result of Cho et. al.
[1]by introducing the notion of weakly L-compatible and weakly M-compatible maps in a 2 non-Archimedean Menger PM-space.
2000 Mathematics Subject Classification: Primary 47H10, Secondary 54H25.
Key words and phrases: Non-Archimedean Menger probabilistic metric space, Com- mon fixed points, compatible maps, weakly L-compatible and Weakly M-compatible maps.
1. Introduction. There have been a number of generalizations of metric space.
One such generalization is Menger space initiated by Menger [7]. It is a probabilistic generalization in which we assign to any two points x and y, a distribution function Fx,y . Schweizer and Sklar [9] studied this concept and gave some fundamental results on this space
The notion of compatible mapping in a Menger space has been introduced by Mishra [8]. Using the concept of compatible mappings of type (A), Jain et. al. [2, 3]
proved some interesting fixed point theorems in Menger space. Afterwards, Jain et.
al.[4] proved the fixed point theorem using the concept of weak compatible maps in Menger space.
The notion of non-Archimedean Menger space has been established by Istratescu and Crivat [6]. The existence of fixed point of mappings on non- Archimedean Menger space has been given by Istratescu [5]. This has been the extension of the results of Sehgal and Bharucha - Reid [10] on a Menger space. Cho. et. al. [1]
proved a common fixed point theorem for compatible mappings in non- Archimedean Menger PM-space. Recently, in 2009, Singh, Jain and Agarwal [11, 12] proved results in non-archimedean Menger PM-space using the concept of semi-compatibility and coincidentally commuting mappings.
In this paper, we extend and generalize the result of Cho et. al. [1] by introducing the notion of of weakly L-compatible maps and weakly M-compatible maps.
2. Preliminaries.
Definition 2.1 Let X be a non-empty set and D be the set of all left-continuous distribution functions. An ordered pair (X, F) is called a 2 non-Archimedean proba- bilistic metric space (briefly, a 2 N.A. PM-space) if Fis a mapping from X × X × X into D satisfying the following conditions (the distribution function F(u, v, w) is denoted by Fu,v,w for all u, v, w ∈ X):
Fu,v,w(x) = 1, for all x > 0, if and only if (PM-1)
at least two of the three points are equal;
(PM-2) Fu,v,w = Fu,w,v= Fw,v,u;
(PM-3) Fu,v,w(0) = 0;
If Fu,v,s(x1) = 1, Fu,s,w(x2) = 1 and Fs,v,w(x3) = 1 (PM-4)
then Fu,v,w(max{x1, x2, x3}) = 1, for all u, v, w, s ∈ X and x1, x2, x3 0.
Definition 2.2 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) = a for every a ∈ [0, 1].
Definition 2.3 A 2 N.A. Menger PM-space is an ordered triple (X, F, ∆), where (X, F) is a 2 non-Archimedean PM-space and ∆ is a t-norm satisfying the following condition:
Fu,v,w(max{x1, x2, x3}) ∆(Fu,v,s(x1), Fu,s,w(x2), Fs,v,w(x3)), (PM-5)
for all u, v, w, s ∈ X and x1, x2, x3 0.
Definition 2.4 A 2 N.A. PM-space (X, F) is said to be of type (C)gif there exists a g ∈ Ω such that
g(Fx,y,z(t)) ¬ g(Fx,y,a(t)) + g(Fx,a,z(t)) + g(Fa,y,z(t))
for all x, y, z, a ∈ X and t 0, where Ω = {g : g : [0, 1] → [0, ∞) is continuous, strictly decreasing, g(1) = 0 and g(0) < ∞}.
Definition 2.5 A 2 N.A. Menger PM-space (X, F, ∆) is said to be of type (D)g
if there exists a g ∈ Ω such that
g(∆(t1, t2, t3)) ¬ g(t1) + g(t2) + g(t3) for all t1, t2, t3∈ [0, 1].
Remark 2.6 (1) If a 2 N.A. Menger PM-space (X, F, ∆) is of type (D)g then (X, F, ∆) is of type (C)g.
(2) If a 2 N.A. Menger PM-space (X, F, ∆) is of type (D)g, then it is metrizable, where the metric d on X is defined by
(*) d(x, y) =
Z 1
0 g(Fx,y,a(t))d(t) for all x, y, a ∈ X.
Throughout this paper, suppose (X, F, ∆) be a complete 2 N.A. Menger PM-space of type (D)gwith a continuous strictly increasing t-norm ∆. Let φ : [0, ∞) → [0, ∞) be a function satisfied the condition (Φ) :
(Φ) φ is upper-semicontinuous from the right and φ(t) < t for all t > 0.
Lemma 2.7 ([1]) If a function φ : [0, ∞) → [0, ∞) satisfies the condition (Φ), then we have
(1) For all t 0, limn→∞φn(t) = 0, where φn(t)is n-th iteration of φ(t).
(2) If {tn} is a non-decreasing sequence of real numbers and tn+1 ¬ φ(tn), n = 1, 2, . . . then limn→∞tn = 0. In particular, if t ¬ φ(t) for all t 0, then t = 0.
Definition 2.8 Let A, S : X → X be mappings. A and S are said to be compatible if limn→∞g(FSAxn,ASxn,a(t)) = 0 for all t > 0, whenever {xn} is a sequence in X such that limn→∞Axn= limn→∞Sxn = z for some z in X.
Definition 2.9 Let A, S : X → X be mappings. The ordered pair (A, S) is said to be weakly A-compatible at z if either limn→∞g(FSAxn,Az,a(t)) = 0 or limn→∞g(FSSxn,Az,a(t)) = 0 for all t > 0, whenever {xn} is a sequence in X such that limn→∞Axn = limn→∞Sxn = z and limn→∞ASxn = limn→∞AAxn = Az for some z in X.
Similarly, suppose B, T : X → X be mappings. The ordered pair (B, T ) is said to be weakly B-compatible at z if either limn→∞g(FT Bxn,Bz,a(t)) = 0 or limn→∞g(FT Txn,Bz,a(t)) = 0 for all t > 0, whenever {xn} is a sequence in X such that limn→∞Bxn= limn→∞T xn = z and limn→∞BT xn= limn→∞BBxn= Bz for some z in X.
Proposition 2.10 Let B, T : X → X be mappings. If B and T are weakly B- compatible and Bz = T z for some z in X, then T Bz = BBz = BT z = T T z.
Proof Suppose {xn} be a sequence in X defined by xn = z, n = 1, 2, 3, . . . and Bz = T z. Then we have Bxn, T xn→T z as n→∞. Since B and T are weakly B- compatible, by triangle inequality limn→∞g(FT B(xn),BB(xn),a(t)) ¬ limn→∞g(FT B(xn),Bz,a(t)) + limn→∞g(FBz,BB(xn),a(t)). Since T Bxn→Bz and BBxn→Bz as n→∞, then limn→∞g(FT B(xn),Bz,a(t)) = 0 and limn→∞g(FBz,BB(xn),a(t)) = 0 implies limn→∞g(FT B(xn),BB(xn),a(t)) = 0 im- plies limn→∞g(FT Bz,BBz,a(t)) = 0 i.e. T Bz = BBz. (1) Similarly, we can have BT z = T T z. (2) Hence, by (1) and (2), we have BT z = T Bz = BBz = T T z.
Proposition 2.11 Let B, T : X → X be weakly B-compatible mappings and let {xn} be a sequence in X such that limn→∞B(xn) = limn→∞T (xn) = z for some z in X. Then we have the following : (i) limn→∞BT (xn) = T z if T is continuous at z. (ii) T Bz = BT z and T z = Az if B and T are continuous at z.
Proof (i) Suppose T is continuous at z. Since limn→∞B(xn) = limn→∞T (xn) = z for some z in X and T T (xn)→T z as n→∞. Since (B, T ) is weakly B-compatible, hence either limn→∞g(FT B(xn),Bz,a(t)) = 0 or limn→∞g(FT T(xn),Bz,a(t)) = 0 for all t > 0, whenever {xn} is a sequence in X such that limn→∞B(xn) = limn→∞T (xn) = z and limn→∞BT (xn) = limn→∞BB(xn) = Bz for some z in X.
Now, by triangle inequality, limn→∞g(FBT(xn),T z,a(t))
¬ limn→∞g(FBT(xn),T T (xn),a(t))+ limn→∞g(FT T(xn),T z,a(t)).
Since, T T (xn)→T z as n→∞ therefore, limn→∞g(FT T(xn),T z,a(t))→0 as n→∞, hence limn→∞g(FBT(xn),T z,a(t)) ¬ limn→∞g(FBT(xn),T T (xn),a(t)) limn→∞¬ g(FBT(xn),Bz,a(t)) + limn→∞g(FBz,T T(xn),a(t)). Since BT (xn)→Bz as n→∞ therefore, limn→∞g(FBT(xn),T z,a(t))→ 0 as n→∞, hence g(FBT(xn),T z,a(t))→0 as n→∞ which implies that limn→∞BT (xn) = T z.
ii) Suppose B and T are continuous at z. Since B(xn)→z as n→∞ and T is continuous at z if BT (xn)→T z as n→∞. On the other hand, since T (xn)→z as n→∞ and B is also continuous at z, BT (xn)→Bz as n→∞. Thus Bz = T z by the uniqueness of limit. Since (B, T ) is weakly B-compatible and Bz = T z for some z
in X, then T Bz = BBz = BT z = T T z.
Lemma 2.12 ([1]) Let A, B, S and T be self mappings of a non-Archimedean Men- ger PM-space (X, F, ∆) satisfying the conditions (1) and (2) as follows :
(1) A(X) ⊂ T (X) and B(X) ⊂ S(X).
(2)
g(FAx,By(t)) ¬ φ(max{ g(FSx,T y(t)), g(FSx,Ax(t)), g(FT y,By(t)), 0.5(g(FSx,By(t)) + g(FT y,Ax(t))) })
for all t > 0, where a function φ : [0, +∞)→[0, +∞) satisfies the condition (Φ).
Then the sequence {yn} in X, defined by Ax2n = T x2n+1 = y2n and Bx2n+1 = Sx2n+2 = y2n+1 for n = 0, 1, 2, . . . , such that limn→∞g(Fyn,yn+1(t)) = 0 for all t > 0 is a Cauchy sequence in X.
Cho, Ha and Kang [1] established the following result :
Theorem 2.13 ([1]) Let A, B, S, T : X→X be mappings satisfying the conditions (1), (2)
(3) S and T are continuous,
(4) the pairs (A, S) and (B, T ) are compatible maps.
Then A, B, S and T have a unique common fixed point in X.
3. Main Result. In the following, we extend the above result to six self maps and generalize it in other respects too.
Theorem 3.1 Let A, B, S, T, L and M be self maps of a 2 non-Archimedean Men- ger PM-space (X, F, ∆) satisfying the conditions
(3.1) L(X)⊂ ST (X), M(X) ⊂ AB(X);
(3.2) AB = BA, ST = T S, LB = BL, M T = T M ;
(3.3) either AB or L is continuous;
(3.4) (L, AB)is weakly L-compatible and (M, ST ) is weakly M-compatible;
(3.5) g(FLx,M y,a(t)) ¬ φ(max {g(FABx,ST y,a(t)), g(FABx,Lx,a(t)), g(FST y,M y,a(t)), 0.5(g(FABx,M y,a(t)) + g(FST y,Lx,a(t))) })
for all t > 0, where a function φ : [0, +∞)→[0, +∞) satisfies the condition (Φ).
Then A, B, S, T, L and M have a unique common fixed point in X.
Proof Let x0∈ X. From condition (3.1) ∃x1, x2∈ X such that Lx1= ST x2= y1
and Mx0= ABx1= y0. Inductively, we can construct sequences {xn} and {yn} in X such that
(3.6) Lx2n= ST x2n+1= y2n and Mx2n+1 = ABx2n+2 = y2n+1
for n = 0, 1, 2, . . . .
Step 1. We prove that limn→∞g(Fyn,yn+1,a(t)) = 0 for all t > 0. From (3.5) and (3.6), we have
g(Fy2n,y2n+1,a(t)) = g(FLx2n,M x2n+1,a(t)
¬ φ(max{g(FABx2n,ST x2n+1,a(t)), g(FABx2n,Lx2n,a(t)), g(FST x2n+1,M x2n+1,a(t)),
0.5(g(FABx2n,M x2n+1,a(t)) + g(FST x2n+1,Lx2n,a(t))) })
= φ(max{g(Fy2n−1,y2n,a(t)), g(Fy2n−1,y2n,a(t)), g(Fy2n,y2n+1,a(t)), 0.5(g(Fy2n−1,y2n+1,a(t)) + g(1)) })
¬ φ(max{g(Fy2n−1,y2n,a(t)), g(Fy2n,y2n+1,a(t)), 0.5(g(Fy2n−1,y2n,a(t)) + g(Fy2n,y2n+1,a(t))) }) If g(Fy2n−1,y2n,a(t)) ¬ g(Fy2n,y2n+1,a(t)) for all t > 0, then by (3.5)
g(Fy2n,y2n+1,a(t)) ¬ φ(g(Fy2n,y2n+1,a(t)))
on applying Lemma 2.7, we have g(Fy2n,y2n+1,a(t)) = 0 for all t > 0. Similarly, we have g(Fy2n+1,y2n+2,a(t)) = 0 for all t > 0.
Thus, we have g(Fyn,yn+1,a(t)) = 0 for all t > 0.
On the other hand, if g(Fy2n−1,y2n,a(t)) g(Fy2n,y2n+1,a(t)) then by (3.5), we have g(Fy2n,y2n+1,a(t)) ¬ φ(g(Fy2n−1,y2n,a(t))) for all t > 0. Similarly,
g(Fy2n+1,y2n+2,a(t)) ¬ φ(g(Fy2n,y2n+1,a(t))) for all t > 0.
Thus, we have g(Fyn,yn+1,a(t)) ¬ φ(g(Fyn−1,yn,a(t))) for all t > 0 and n = 1, 2, . . . .
Therefore, by Lemma 2.7, g(Fyn,yn+1,a(t)) = 0 for all t > 0, which implies that {yn} is a Cauchy sequence in X by Lemma 2.12.
Since (X, F, ∆) is complete, the sequence {yn} converges to a point z ∈ X.
Also its subsequences converges as follows :
(3.7) {Mx2n+1}→z and {ST x2n+1}→z,
(3.8) {Lx2n}→z and {ABx2n}→z.
Case I. AB is continuous.
As AB is continuous, (AB)2x2n→ABz and (AB)Lx2n→ABz. As (L, AB) is weakly L-compatible, so by Proposition 2.11, L(AB)x2n→ABz.
Step 2. Putting x = ABx2n and y = x2n+1 for t > 0 in (3.5), we get g(FLABx2n,M x2n+1,a(t)) ¬ φ(max{g(FABABx2n,ST x2n+1,a(t)),
g(FST x2n+1,M x2n+1,a(t)), g(FABABx2n,LABx2n,a(t)), 0.5(g(FABABx2n,M x2n+1,a(t))
+g(FST x2n+1,LABx2n,a(t))) }).
Letting n → ∞, we get
g(FABz,z,a(t)) ¬ φ(max{g(FABz,z,a(t))), g(FABz,ABz,a(t)), g(Fz,z,a(t)), 0.5(g(FABz,z,a(t)) + g(Fz,ABz,a(t))) }) = φ(g(FABz,z,a(t)))
which implies that g(FABz,z,a(t)) = 0 by Lemma 2.7 and so we have ABz = z.
Step 3. Putting x = z and y = x2n+1 for t > 0 in (3.5), we get g(FLz,M x2n+1,a(t)) ¬ φ(max{g(FABz,ST x2n+1,a(t)), g(FABz,Lz,a(t)),
g(FST x2n+1,M x2n+1,a(t)),
0.5(g(FABz,M x2n+1,a(t)) + g(FST x2n+1,Lz,a(t))) }).
Letting n → ∞, we get
g(FLz,z,a(t)) ¬ φ(max{g(Fz,z,a(t)), g(Fz,Lz,a(t)), g(Fz,z,a(t)), 0.5(g(Fz,z,a(t)) + g(Fz,Lz,a(t))) }) = φ(g(FLz,z,a(t)))
which implies that g(FLz,z,a(t)) = 0 by Lemma 2.7 and so we have Lz = z. There- fore, ABz = Lz = z.
Step 4. Putting x = Bz and y = x2n+1 for t > 0 in (3.5), we get
g(FLBz,M x2n+1,a(t)) ¬ φ(max{g(FABBz,ST x2n+1,a(t)), g(FABBz,LBz,a(t)), g(FST x2n+1,M x2n+1,a(t)), 0.5(g(FABBz,M x2n+1,a(t)) +g(FST x2n+1,LBz,a(t))) }).
As BL = LB, AB = BA, so we have L(Bz) = B(Lz) = Bz and AB(Bz) = B(ABz) = Bz. Letting n→ ∞, we get
g(FBz,z,a(t)) ¬ φ(max{g(FBz,z,a(t)), g(FBz,Bz,a(t)), g(Fz,z,a(t)), 0.5(g(FBz,z,a(t)) + g(Fz,Bz,a(t))) }) = φ(g(FBz,z,a(t)))
which implies that g(FBz,z,a(t)) = 0 by Lemma 2.7 and so we have Bz = z. Also, ABz = z and so Az = z. Therefore,
(3.9) Az = Bz = Lz = z.
Step 5. As L(X) ⊂ ST (X), there exists v ∈ X such that z = Lz = ST v.
Putting x = x2n and y = v for t > 0 in (3.5), we get
g(FLx2n,M v,a(t)) ¬ φ(max{g(FABx2n,ST v,a(t)), g(FABx2n,Lx2n,a(t)), g(FST v,M v,a(t)),
0.5(g(FABx2n,M v,a(t)) + g(FST v,Lx2n,a(t))) }).
Letting n → ∞ and using equation (3.8), we get
g(Fz,M v,a(t)) ¬ φ(max{g(Fz,z,a(t)), g(Fz,z,a(t)), g(Fz,M v,a(t)),
0.5(g(Fz,M v,a(t)) + g(Fz,z,a(t))) }) = φ(g(Fz,M v,a(t)))
which implies that g(Fz,M v,a(t)) = 0 by Lemma 2.7 and so we have z = Mv. Hence, ST v = z = M v. As (M, ST ) is weakly M-compatible, we have ST M v = M ST v.
Thus, ST z = Mz.
Step 6. Putting x = x2n, y = z for t > 0 in (3.5), we get
g(FLx2n,M z,a(t)) ¬ φ(max{g(FABx2n,ST z,a(t)), g(FABx2n,Lx2n,a(t)), g(FST z,M z,a(t)),
0.5(g(FABx2n,M z,a(t)) + g(FST z,Lx2n,a(t))) }).
Letting n → ∞ and using equation (3.8) and Step 5 we get
g(Fz,M z,a(t)) ¬ φ(max{g(Fz,M z,a(t)), g(Fz,z,a(t)), g(FM z,M z,a(t)), 0.5(g(Fz,M z,a(t)) + g(FM z,z,a(t))) }) = φ(g(Fz,M z,a(t)))
which implies that g(Fz,M z,a(t)) = 0 by Lemma 2.7 and so we have z = Mz.
Step 7. Putting x = x2n and y= T z for t > 0 in (3.5), we get
g(FLx2n,M T z,a(t)) ¬ φ(max{g(FABx2n,ST T z,a(t)), g(FABx2n,Lx2n,a(t)), g(FST T z,M T z,a(t)),
0.5(g(FABx2n,M T z,a(t)) + g(FST T z,Lx2n,a(t))) }).
As MT = T M and ST = T S we have MT z = T Mz = T z and ST (T z) = T (ST z) = T z. Letting n→ ∞ we get
g(Fz,T z,a(t)) ¬ φ(max{g(Fz,T z,a(t)), g(Fz,z,a(t)), g(FT z,T z,a(t)), 0.5(g(Fz,T z,a(t)) + g(FT z,z,a(t))) }) = φ(g(Fz,T z,a(t)))
which implies that g(Fz,T z,a(t)) = 0 by Lemma 2.7 and so we have z = T z.
Now ST z = T z = z implies Sz = z. Hence
(3.10) Sz = T z = M z = z.
Combining (3.9) and (3.10), we get Az = Bz = Lz = Mz = T z = Sz = z. Hence, the six self maps have a common fixed point in this case.
Case II. L is continuous.
As L is continuous, L2x2n→Lz and L(AB)x2n→Lz.
As (L, AB) is weakly L-compatible, so by Proposition 2.11, (AB)Lx2n→Lz.
Step 8. Putting x = Lx2n and y = x2n+1 for t > 0 in (3.5), we get g(FLLx2n,M x2n+1,a(t)) ¬ φ(max{g(FABLx2n,ST x2n+1,a(t)),
g(FABLx2n,LLx2n,a(t)), g(FST x2n+1,M x2n+1,a(t)), 0.5(g(FABLx2n,M x2n+1,a(t))
+g(FST x2n+1,LLx2n,a(t))) }).
Letting n → ∞ we get
g(FLz,z,a(t)) ¬ φ(max{g(FLz,z,a(t)), g(FLz,Lz,a(t)), g(Fz,z,a(t)), 0.5(g(FLz,z,a(t)) + g(Fz,Lz,a(t))) }) = φ(g(FLz,z,a(t))),
which implies that g(FLz,z,a(t)) = 0 by Lemma 2.7 and so we have Lz = z. Now, using steps 5-7 gives us Mz = ST z = Sz = T z = z.
Step 9. As M(X) ⊂ AB(X), there exists w ∈ X such that z = Mz = ABw.
Putting x = w and y = x2n+1 for t > 0 in (3.5), we get
g(FLw,M x2n+1,a(t)) ¬ φ(max{g(FABw,ST x2n+1,a(t)), g(FABw,Lw,a(t)), g(FST x2n+1,M x2n+1,a(t)),
0.5(g(FABw,M x2n+1,a(t)) + g(FST x2n+1,Lw,a(t))) }).
Letting n → ∞, we get
g(FLw,z,a(t)) ¬ φ(max{g(Fz,z,a(t)), g(Fz,Lw,a(t)), g(Fz,z,a(t)),
0.5(g(Fz,z,a(t)) + g(Fz,Lw,a(t))) }) = φ(g(FLw,z,a(t))),
which implies that g(FLw,z,a(t)) = 0 by Lemma 2.7 and so we have Lw = z.
Thus, we have Lw = z = ABw. Since (L, AB) is weakly L-compatible and so by Proposition 2.10, LABw = ABLw and hence, we have Lz = ABz. Also, Bz = z follows from Step 4. Thus, Az = Bz = Lz = z and we obtain that z is the common fixed point of the six maps in this case also.
Step 10. (Uniqueness) Let u be another common fixed point of A, B, S, T, L and M; then Au = Bu = Su = T u = Lu = Mu = u. Putting x = z and y = u for t > 0 in (3.5), we get
g(FLz,M u,a(t)) ¬ φ(max{g(FABz,ST u,a(t)), g(FABz,Lz,a(t)), g(FST u,M u,a(t)),
0.5(g(FABz,M u,a(t)) + g(FST u,Lz,a(t))) }).
Letting n → ∞ we get
g(Fz,u,a(t)) ¬ φ(max{g(Fz,u,a(t)), g(Fz,z,a(t)), g(Fu,u,a(t)), 0.5(g(Fz,u,a(t)) + g(Fu,z,a(t))) }) = φ(g(Fz,u,a(t))),
which implies that g(Fz,u,a(t)) = 0 by Lemma 2.7 and so we have z = u.
Therefore, z is a unique common fixed point of A, B, S, T, L and M. This completes
the proof.
Remark 3.2 If we take B = T = I, the identity map on X in Theorem 3.1, then the condition (3.2) is satisfied trivially and we get
Corollary 3.3 Let A, S, L, M : X→X be mappings satisfying the conditions :
(3.11) L(X)⊂ S(X), M(X) ⊂ A(X);
(3.12) Either A or L is continuous;
(3.13) (L, A)is weakly L-compatible and (M, S) is weakly M-compatible pairs;
(3.14) g(FLx,M y,a(t)) ¬ φ(max{g(FAx,Sy,a(t)), g(FAx,Lx,a(t)), g(FSy,M y,a(t)), 0.5(g(FAx,M y,a(t)) + g(FSy,Lx,a(t))) })
for all t > 0, where a function φ : [0, +∞)→[0, +∞) satisfies the condition (Φ).
Then A, S, L and M have a unique common fixed point in X.
Remark 3.4 In view of Remark 3.2, Corollary 3.3 is a generalization of the result of Cho et. al. [1] in the sense that condition of compatibility of the pairs of self maps in a non-Archimedean Menger PM-space has been restricted to weakly L-compatible and weakly M-compatible self maps in a 2 non-Archimedean Menger PM-space and only one of the mappings of the weakly L-compatible or weakly M-compatible pair is needed to be continuous.
References
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V. K. Gupta
Department of Mathematics, Govt. Madhav Science College Ujjain (M.P.) 456010
Arihant Jain
Department of Applied Mathematics, Shri Guru Sandipani Institute of Technology and Science Ujjain (M.P.) 456550
E-mail: arihant2412@gmail.com Jaya Kushwah
Department of Applied Mathematics, Prashanti Institute of Technology and Science Ujjain (M.P.)
E-mail: kushwahjaya@gmail.com
(Received: 7.05.2011)
