COMMENTATIONES MATHEMATICAE Vol. 51, No. 2 (2011), 177-187
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, Common 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.
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Fixed point theorem in 2 non-archimedean Menger PM-Space
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 probabilistic 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 F u,v,w for all u, v, w ∈ X):
(PM-1)
Fu,v,w (x) = 1, for all x > 0, if and only if
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;
(PM-4)
If Fu,v,s (x1 ) = 1, Fu,s,w (x2 ) = 1 and Fs,v,w (x3 ) = 1
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:
(PM-5)
Fu,v,w (max{x1 , x2 , x3 }) ­ ∆(Fu,v,s (x1 ), Fu,s,w (x2 ), Fs,v,w (x3 )),
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)g if 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) < ∞}.
V. K. Gupta, A. Jain, J. Kushwah
179
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
Z 1
g(Fx,y,a (t))d(t) for all x, y, a ∈ X.
(*)
d(x, y) =
0
Throughout this paper, suppose (X, F, ∆) be a complete 2 N.A. Menger PM-space
of type (D)g with 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.
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Fixed point theorem in 2 non-archimedean Menger PM-Space
Proposition 2.10 Let B, T : X → X be mappings. If B and T are weakly Bcompatible 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 Bcompatible, 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 implies 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 Menger 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))) })
V. K. Gupta, A. Jain, J. Kushwah
181
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 Menger 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)
(3.5)
(L, AB) is weakly L-compatible and (M, ST ) is weakly M-compatible;
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 M x0 = ABx1 = y0 . Inductively, we can construct sequences {xn } and {yn } in
X such that
(3.6)
Lx2n = ST x2n+1 = y2n and M x2n+1 = ABx2n+2 = y2n+1
for n = 0, 1, 2, . . . .
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Fixed point theorem in 2 non-archimedean Menger PM-Space
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)
{M x2n+1 }→z and {ST x2n+1 }→z,
(3.8)
{Lx2n }→z and {ABx2n }→z.
Case I. AB is continuous.
As AB is continuous, (AB)2 x2n →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))) }).
V. K. Gupta, A. Jain, J. Kushwah
183
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. Therefore, 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))) }).
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Fixed point theorem in 2 non-archimedean Menger PM-Space
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 = M v. Hence,
ST v = z = M v. As (M, ST ) is weakly M-compatible, we have ST M v = M ST v.
Thus, ST z = M z.
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 = M z.
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 M T = T M and ST = T S we have M T z = T M z = 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 = M z = 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, L2 x2n →Lz and L(AB)x2n →Lz.
V. K. Gupta, A. Jain, J. Kushwah
185
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 M z = ST z = Sz = T z = z.
Step 9. As M (X) ⊂ AB(X), there exists w ∈ X such that z = M z = 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 = M u = 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))) }).
186
Fixed point theorem in 2 non-archimedean Menger PM-Space
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.
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187
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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: [email protected]
Jaya Kushwah
Department of Applied Mathematics, Prashanti Institute of Technology and Science
Ujjain (M.P.)
E-mail: [email protected]
(Received: 7.05.2011)