DEMONSTRATIO MATHEMATICA
Vol. XLII
No 4
2009
Servet Kutukcu, Sushil Sharma
A COMMON FIXED POINT THEOREM IN
NON-ARCHIMEDEAN MENGER PM-SPACES
Abstract. In the present work, we introduce two types of compatible maps in nonArchimedean Menger PM-spaces and obtain a common fixed point theorem for six maps.
1. Introduction and Preliminaries
In 1997, Cho et al. [2] introduced the concepts of compatible maps and
compatible maps of type (A) in non-Archimedean Menger probabilistic metric spaces and gave some fixed point theorems for these maps. In this paper,
we introduce the concept of compatible maps of type (A-1) and type (A-2),
show that they are equivalent to compatible maps under certain conditions
and illustrating with an example, prove a common fixed point theorem for
such maps in the spaces which generalizes, extends and fuzzifies several fixed
point theorems for contractive type maps on metric spaces and fuzzy metric
spaces.
Next, we recall some definitions and known results in Menger space. For
more details we refer the readers to [1-7].
Definition 1. A triangular norm ∗ (shorty t-norm) is a binary operation
on the unit interval [0, 1] which is associative, commutative, nondecreasing
in each coordinate and a ∗ 1 = a for all a ∈ [0, 1]. Some important examples
of t-norms are a ∗ b = max {a + b − 1, 0} and a ∗ b = min {a, b}.
Definition 2. A distribution function is a function F : [−∞, ∞] → [0, 1]
which is left continuous on R, non-decreasing and F (−∞) = 0, F (∞) = 1.
If X is a nonempty set, F : X × X → ∆ is called a probabilistic distance on
X and F (x, y) is usually detoned by Fxy .
2000 Mathematics Subject Classification: 47H10, 54H25, 54E70.
Key words and phrases: non-Archimedean Menger probabilistic metric space, compatible maps, mutually compatible maps, common fixed point.
838
S. Kutukcu, S Sharma
Definition 3. The ordered pair (X, F ) is called a non-Archimedean probabilistic metric space (shortly N. A. PM-space) if X is a nonempty set
and F is a probabilistic distance satisfying the following conditions: for
all x, y, z ∈ X and t, s ≥ 0,
(PM-1)
(PM-2)
(PM-3)
(PM-4)
Fxy (t) = 1, t > 0 ⇐⇒ x = y,
Fxy = Fyx ,
Fxy (0) = 0,
Fxy (t) = 1, Fyz (s) = 1 ⇒ Fxz (max{t, s}) = 1.
The ordered triple (X, F, ∗) is called a non-Archimedean Menger probabilistic metric space (shortly N. A. Menger space) if (X, F ) is a N. A.
PM-space, ∗ is a t-norm and the following condition is also satisfies: for all
x, y, z ∈ X and t, s > 0,
(PM-5) Fxz (max{t, s}) ≥ Fxy (t) ∗ Fyz (s).
The concept of neighbourhoods in Menger PM-spaces was introduced by
Schweizer and Sklar [6]. If x ∈ X, ε > 0 and λ ∈ (0, 1), then an (ε, λ)neighbourhood of x, Ux (ε, λ) is defined by
Ux (ε, λ) = {y ∈ X : Fxy (ε) > 1 − λ} .
If the t-norm ∗ is continuous and strictly increasing then (X, F, ∗) is a Hausdorff space in the topology induced by the family {Ux (ε, λ) : x ∈ X, ε >
0, λ ∈ (0, 1)} of neighbourhoods [6].
Definition 4. ([2]) A PM-space (X, F ) is said to be of type (C)g if there
exists a g ∈ Ω such that
g(Fxy (t)) ≤ g(Fxz (t)) + g(Fzy (t))
for all x, y, z ∈ X and t ≥ 0 where Ω = {g : g : [0, 1] → [0, ∞) is continuous,
strictly decreasing, g(1) = 0 and g(0) < ∞}.
Definition 5. ([2]) A N. A. Menger PM-space (X, F, ∗) is said to be of
type (D)g if there exists a g ∈ Ω such that
g(t ∗ s) ≤ g(t) + g(s)
for all s, t ∈ [0, 1].
Remark 1. If a N. A. Menger PM-space (X, F, ∗) is said to be of type
(D)g then (X, F, ∗) is of type (C)g . On the other hand, (X, F, ∗) is a N.
A. PM-space such that a ∗ b ≥ max {a + b − 1, 0} for all a, b ∈ [0, 1], then
(X, F, ∗) is of type (D)g for g ∈ Ω defined by g(t) = 1 − t, t ≥ 0.
Throughout this paper, let (X, F, ∗) be a complete N. A. Menger PMspace of type (D)g with a continuous strictly increasing t-norm ∗ and φ :
A common fixed point theorem
839
[0, ∞) → [0, ∞) be a function satisfying the condition (Φ): φ is uppersemicontinuous from the right and φ(t) < t for all t > 0.
Lemma 1. ([1]) If a function φ : [0, ∞) → [0, ∞) satisfies the condition
(Φ), then we have
(a) for all t ≥ 0, limn→∞ φn (t) = 0 where φn (t) is the n-th iteration of φ(t),
(b) 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.
Lemma 2. ([2]) Let {yn } be a sequence in X such that limn→∞ Fyn ,yn+1 (t) =
1 for all t > 0. If {yn } is not a Cauchy sequence in X, then there exist
ε0 > 0, t0 > 0 and two sequences {mi }, {ni } of positive integers such that
(a) mi > ni + 1 and ni → ∞ as i → ∞,
(b) Fymi ,yni (t0 ) < 1 − ε0 and Fymi −1 ,yni (t0 ) ≥ 1 − ε0 , i = 1, 2, . . . .
Definition 6. ([2]) Self maps A and B of a N. A. Menger PM-space
(X, F, ∗) are said to be compatible if g(FABxn BAxn (t)) → 0 for all t > 0,
whenever {xn } is a sequence in X such that Axn , Bxn → z for some z in X
as n → ∞.
Definition 7. ([2]) Self maps A and B of a N. A. Menger PM-space
(X, F, ∗) are said to be compatible of type (A) if g(FABxn BBxn (t)) → 0 and
g(FBAxn AAxn (t)) → 0 for all t > 0, whenever {xn } is a sequence in X such
that Axn , Bxn → z for some z in X as n → ∞.
Now we introduce the concept of compatible mappings of type (A-1) and
type (A-2) in N. A. Menger PM-spaces and show that they are equivalent
to compatible mappings under certain conditions.
Definition 8. Self maps A and B of a N. A. Menger PM-space (X, F, ∗)
are said to be compatible of type (A-1) if g(FABxn BBxn (t)) → 0 for all t > 0,
whenever {xn } is a sequence in X such that Axn , Bxn → z for some z in X
as n → ∞.
Definition 9. Self maps A and B of a N. A. Menger PM-space (X, F, ∗)
are said to be compatible of type (A-2) if g(FBAxn AAxn (t)) → 0 for all t > 0,
whenever {xn } is a sequence in X such that Axn , Bxn → z for some z in X
as n → ∞.
Remark 2. Clearly, if a pair of mappings (A, B) is compatible of type
(A-1) then the pair (B, A) is compatible of type (A-2). Such maps are called
mutually compatible of type (A). Further, if A and B compatible maps of
type (A) then the pair (A, B) is compatible of type (A-1) as well as type
(A-2).
840
S. Kutukcu, S Sharma
The following is an example of pair of self maps in a N. A. Menger
PM-space which are mutually compatible of type (A) but not compatible.
Example 1. Let (X, d) be a metric space with the usual metric d where
X = [0, 2] and (X, F, ∗) be the induced N. A. Menger PM-space with g(t) =
1 − t and Fxy (t) = H(t − d(x, y)) for all x, y ∈ X,t > 0. Define self maps A
and B as follows:
(
(
x, if 0 ≤ x < 1,
2 − x, if 0 ≤ x < 1,
and Bx =
Ax =
2, if 1 ≤ x ≤ 2.
2,
if 1 ≤ x ≤ 2,
Take xn = 1−1/n. Then FAxn 1 (t) = H(t−(1/n)) and limn→∞ g(FAxn 1 (t)) =
g(H(t)) = 0. Hence Axn → 1 as n → ∞. Similarly, Bxn → 1 as n → ∞.
Also FABxn BAxn (t) = H(t − (1 − 1/n)) and limn→∞ g(FABxn BAxn (t)) =
g(H(t − 1)) 6= 0 for all t > 0. Hence the pair (A, B) is not compatible. But
FABxn BBxn (t) = H(t − (2/n)) and limn→∞ g(FABxn BBxn (t)) = g(H(t)) = 0
for all t > 0. Hence the pair (A, B) is compatible of type (A-1). Similarly,
the pair (A, B) is compatible of type (A-2). Therefore A and B are mutually
compatible but not compatible maps.
Next, we cite the following propositions which gives the condition under
which the Definitions 6, 8 and 9 becomes equivalent.
Proposition 1. Let A and B be self maps of a N. A. Menger PM-space
(X, F, ∗).
(a) If B is continuous then the pair (A, B) is compatible of type (A-1) iff A
and B are compatible.
(b) If A is continuous then the pair (A, B) is compatible of type (A-2) iff A
and B are compatible.
Proof. (a) Let {xn } be a sequence in X such that Axn , Bxn → z for some
z in X as n → ∞ and let the pair (A, B) be compatible of type (A-1). Since
B is continuous, we have BAxn → Bz and BBxn → Bz and so
g(FABxn BAxn (t)) ≤ g(FABxn BBxn (t)) + g(FBBxn BAxn (t)) → 0
as n → ∞. Hence the mappings A and B are compatible.
Now, let A and B be compatible. Therefore, using the continuity of B,
we have
g(FABxn BBxn (t)) ≤ g(FABxn BAxn (t)) + g(FBAxn BBxn (t)) → 0
as n → ∞. Hence the mappings A and B are compatible of type (A-1).
(b) The proof is similar with (a).
Next, we give some properties of compatible mappings of type (A-1) and
type (A-2) which will be used in our main theorem.
A common fixed point theorem
841
Proposition 2. Let A and B be self maps of a N. A. Menger PM-space
(X, F, ∗). If the pair (A, B) is compatible of type (A-1) and Az = Bz for
some z in X then ABz = BBz.
Proof. Let {xn } be a sequence in X defined by xn = x for n ∈ N and let
Az = Bz. Then we have Axn → Az and Bxn → Bz. Since the pair (A, B)
is compatible of type (A-1), we have g(FABzBBz (t)) = g(FABxn BBxn (t)) → 0
as n → ∞. Hence ABz = BBz.
Proposition 3. Let A and B be self maps of a N. A. Menger PM-space
(X, F, ∗). If the pair (A, B) is compatible of type (A-2) and Az = Bz for
some z in X then BAz = AAz.
Proof. The proof is similar with the proof of Proposition 2.
Proposition 4. Let A and B be self maps of a N. A. Menger PM-space
(X, F, ∗). If the pair (A, B) is compatible of type (A-1) and {xn } is a sequence
in X such that Axn , Bxn → z for some z in X as n → ∞ then BBxn → Az
if A is continuous at z.
Proof. Since A is continuous at z and the pair (A, B) is compatible of type
(A-1), we have ABxn → Az and g(FABxn BBxn (t)) → 0 as n → ∞. Therefore
g(FAzBBxn (t)) ≤ g(FAzABxn (t)) + g(FABxn BBxn (t)) → 0
as n → ∞. Hence BBxn → Az as n → ∞.
Proposition 5. Let A and B be self maps of a N. A. Menger PM-space
(X, F, ∗). If the pair (A, B) is compatible of type (A-2) and {xn } is a sequence
in X such that Axn , Bxn → z for some z in X as n → ∞ then AAxn → Bz
if B is continuous at z.
Proof. The proof is similar with the proof of Proposition 4.
2. Main results
Theorem 1. Let A, B, P, Q, S and T be self maps on a complete N. A.
Menger PM-space (X, F, ∗) satisfying:
(a) P (X) ⊆ ST (X), Q(X) ⊆ AB(X),
(b) g(FP x,Qy (t)) ≤ φ(g(FABx,ST y (t))),
(c)
g(FP x,Qy (t))





 g(FABx,ST y (t)) + g(FP x,ABx (t)) + g(FQy,ST y (t)), 


≤ φ max
g(FP x,ABx (t)) + g(FQy,ABx (t)),





g(FP x,ST y (t)) + g(FQy,ST y (t))
842
S. Kutukcu, S Sharma
for all x, y ∈ X and t > 0, where a function φ : [0, ∞) → [0, ∞) satisfies
the condition (Φ),
(d) AB = BA, ST = T S, P B = BP, QT = T Q,
(e) either P or AB is continuous,
(f) the pairs (P, AB) and (Q, ST ) are mutually compatible of type (A).
Then A, B, P, Q, S and T have a unique common fixed point.
Proof. Let x0 be an arbitrary point of X. By (a), there exists x1 , x2 ∈ X
such that P x0 = ST x1 = y0 and Qx1 = ABx1 = y1 . Inductively, we can
construct sequences {xn } and {yn } in X such that P x2n = ST x2n+1 = y2n
and Qx2n+1 = ABx2n+2 = y2n+1 for n = 0, 1, 2, . . . .
Step1. We shall show that the sequence {yn } is a Cauchy sequence.
Since P x2n = ST x2n+1 , using (b), we have g(Fy2n y2n+1 (t)) =
g(FP x2n Qy2n+1 (t)) ≤ φ(g(Fy2n y2n+1 (t))) and since Qx2n+1 = ABx2n+2 , we
also have g(Fy2n, y2n−1 (t)) = g(FP x2n ,Qy2n−1 (t)) ≤ φ(g(Fy2n−1 ,y2n+2 (t))). Thus
g(Fyn ,yn+1 (t)) ≤ φ(g(Fyn−1, yn (t))) for n = 1, 2, . . . . Hence g(Fyn ,yn+1 (t)) ≤
φn (g(Fy0, y1 (t))) for n = 1, 2, . . . . Therefore, from Lemma 1,
(2.1)
g(Fyn ,yn+1 (t)) → 0 as n → ∞.
Suppose {yn } is a not Cauchy sequence. Since g is strictly decreasing, from
Lemma 2, there exist ε0 > 0, t0 > 0 and two sequences {mk }, {nk } of positive
integers such that
(a) mk > nk + 1 and nk → ∞ as k → ∞,
(b) g(Fymk ,ynk (t0 )) > g(1 − ε0 ) and g(Fymk −1 ,ynk (t0 )) ≤ g(1 − ε0 ) for
k = 1, 2, . . .
Therefore
g(1 − ε0 ) < g(Fymk ,ynk (t0 ))
≤ g(Fymk ,ymk −1 (t0 )) + g(Fymk −1 ,ynk (t0 ))
≤ g(Fymk ,ymk −1 (t0 )) + g(1 − ε0 )
and letting k → ∞, we have
(2.2)
lim g(Fymk ,ynk (t0 )) = g(1 − ε0 ).
n→∞
On the other hand, we have
(2.3)
g(1 − ε0 ) < g(Fymk ,ynk (t0 ))
≤ g(Fymk ,ynk +1 (t0 )) + g(Fynk +1 ,ynk (t0 )).
Without loss of generality assume that both mk and nk are even. Using (c),
A common fixed point theorem
843
we have
g(Fymk ,ynk +1 (t0 )) = g(FP xmk ,Qxmk +1 (t0 ))





g(F
(t
))
+
g(F
(t
))
y
,y
0
y
,y
0


mk mk −1
mk −1 nk









+g(F
(t
)),
ynk +1 ,ynk 0


≤ φ max



(t
))
+
g(F
(t
)),
g(F

ymk ,ymk −1 0
ymk −1 ,ynk +1 0








 g(F
(t
))
+
g(F
(t
))
ymk ,ynk 0
ynk +1 ,ynk 0



g(1 − ε0 ) + g(Fymk ,ymk −1 (t0 )) 











+g(Fynk +1 ,ynk (t0 )),







.
max
≤ φ
g(F
(t
))
+
g(1
−
ε
)
y
,y
0
0
m
m
−1


k
k










+g(F
(t
)),


y
,y
0
n


k nk +1




g(Fynk +1 ,ynk (t0 )) + g(Fymk ,ynk (t0 ))
Substituting this in (3.3), letting k → ∞ and using (3.1) and (3.2), we have
g(1 − ε0 ) ≤ φ(g(1 − ε0 )) < g(1 − ε0 )
which is a contradiction. Hence {yn } is a Cauchy sequence. Since (X, F, ∗) is
complete, it converges to a point z in X. Also its subsequences converge as
follows: {P x2n } → z, {ABx2n } → z, {Qx2n+1 } → z and {ST x2n+1 } → z.
Case I. AB is continuous, and (P, AB) and (Q, ST ) are compatible of
type (A-1).
Since AB is continuous, AB(AB)x2n → ABz and (AB)P x2n → ABz.
Since (P, AB) is compatible of type (A-1), P P x2n → ABz.
Step 2. By taking x = P x2n , y = x2n+1 in (c), we have
g(FP P x2n ,Qx2n+1 (t))





g(F
(t))
+
g(F
(t))


(AB)P
x
,ST
x
P
P
x
,(AB)P
x
2n
2n+1
2n
2n









+g(F
(t)),
Qx
,ST
x
2n+1
2n+1


≤ φ max



g(F
(t))
+
g(F
(t)),




P
P
x
,(AB)P
x
Qx
,(AB)P
x
2n
2n
2n+1
2n





 g(F
P P x2n ,ST x2n+1 (t)) + g(FQx2n+1 ,ST x2n+1 (t))
this implies that, as n → ∞





 g(FABz,z (t)) + g(FABz,ABz (t)) + g(Fz,z (t)), 


g(FABz,z (t)) ≤ φ max
g(FABz,ABz (t)) + g(Fz,ABz (t)),





g(Fz,z (t)) + g(Fz,z (t))
= φ (g(FABz,z (t)))
which means that, by Lemma 1, g(FABz,z (t)) = 0 for all t > 0 and it follows
that z = ABz.
844
S. Kutukcu, S Sharma
Step 3. By taking x = z, y = x2n+1 in (c), we have
g(FP z,Qx2n+1 (t))









+g(F
(t)),
Qx2n+1 ,ST x2n+1


≤ φ max



g(F
(t))
+
g(F
(t)),


P z,ABz
Qx2n+1 ,ABz





 g(F
(t)) + g(F
(t)) 







g(FABz,ST x2n+1 (t)) + g(FP z,ABz (t))
P z,ST x2n+1
Qx2n+1 ,ST x2n+1
this implies that, as n → ∞





 g(Fz,z (t)) + g(FP z,z (t)) + g(Fz,z (t)), 


g(FP z,z (t)) ≤ φ max
g(FP z,z (t)) + g(Fz,z (t)),





g(FP z,z (t)) + g(Fz,z (t))
= φ(g(FP z,z (t)))
which means that z = P z. Therefore, z = ABz = P z.
Step 4. By taking x = Bz, y = x2n+1 in (c) and using (d), we have
g(FP (Bz),Qx2n+1 (t))





g(F
(t))
+
g(F
(t))


AB(Bz),ST
x
P
(Bz),AB(Bz)
2n+1









+g(F
(t)),
Qx
,ST
x
2n+1
2n+1


≤ φ max



g(F
(t))
+
g(F
(t)),




P (Bz),AB(Bz)
Qx2n+1 ,AB(Bz)




 g(F
(t)) + g(F
(t)) 
P (Bz),ST x2n+1
Qx2n+1 ,ST x2n+1
this implies that, as n → ∞





 g(FBz,z (t)) + g(FBz,Bz (t)) + g(Fz,z (t)), 


g(FBz,z (t)) ≤ φ max
g(FBz,Bz (t)) + g(Fz,Bz (t)),





g(FBz,z (t)) + g(Fz,z (t))
= φ(g(FBz,z (t)))
which means that z = Bz. Since z = ABz, we have z = Az. Therefore,
z = Az = Bz = P z.
Step 5. Since P (X) ⊆ ST (X), there exists w ∈ X such that z = P z =
ST w. By taking x = x2n , y = w in (c), we have





g(F
(t))
+
g(F
(t))
ABx2n ,ST w
P x2n ,ABx2n











+g(F
(t)),
Qw,ST w


g(FP x2n ,Qw (t)) ≤ φ max



(t)),
(t))
+
g(F
g(F


Qw,ABx2n
P x2n ,ABx2n






 g(F
(t)) + g(F
(t)) 
P x2n ,ST w
Qw,ST w
A common fixed point theorem
845
this implies that, as n → ∞



g(F
(t))
+
g(F
(t))
+
g(F
(t)),


z,z
z,z
Qw,z




g(Fz,Qw (t)) ≤ φ max
g(Fz,z (t)) + g(FQw,z (t)),





g(Fz,z (t)) + g(FQw,z (t))
= φ(g(Fz,Qw (t)))
which means that z = Qw. Hence, ST w = z = Qw. Since (Q, ST ) is
compatible of type (A-1), we have Q(ST )w = ST (ST )w. Thus, ST z = Qz.
Step 6. By taking x = x2n , y = z in (c) and using Step 5, we have





g(F
(t))
+
g(F
(t))
ABx
,ST
z
P
x
,ABx


2n
2n
2n









+g(F
(t)),
Qz,ST
z


g(FP x2n ,Qz (t)) ≤ φ max



g(F
(t))
+
g(F
(t)),


P x2n ,ABx2n
Qz,ABx2n





 g(F
(t)) + g(F
(t)) 
P x2n ,ST z
Qz,ST z
this implies that, as n → ∞





 g(Fz,Qz (t)) + g(Fz,z (t)) + g(FQz,Qz (t)), 


g(Fz,Qz (t)) ≤ φ max
g(Fz,z (t)) + g(FQz,z (t)),





g(Fz,Qz (t)) + g(FQz,Qz (t))
= φ(g(Fz,Qz (t)))
which means that z = Qz. Since ST z = Qz, we have z = ST z. Therefore,
z = Az = Bz = P z = Qz = ST z.
Step 7. By taking x = x2n , y = T z in (c) and using (d), we have





g(F
(t))
+
g(F
(t))
P
x
,ABx


ABx
,ST
(T
z)
2n
2n
2n









+g(F
(t)),
Q(T
z),ST
(T
z)

g(FP x2n ,T z (t)) ≤ φ 

max  g(F

(t))
+
g(F
(t)),


P
x
,ABx


Q(T
z),ABx
2n
2n
2n





 g(F
(t)) + g(F
(t))
P x2n ,ST (T z)
Q(T z),ST (T z)
this implies that, as n → ∞





 g(Fz,T z (t)) + g(Fz,z (t)) + g(FT z,T z (t)), 


g(Fz,T z (t)) ≤ φ max
g(Fz,z (t)) + g(FT z,z (t)),





g(Fz,T z (t)) + g(FT z,T z (t))
= φ(g(Fz,T z (t)))
which means that z = T z. Since z = ST z, we have z = Sz. Therefore,
z = Az = Bz = P z = Qz = Sz = T z, that is, z is the common fixed point
of A, B, P, Q, S and T .
846
S. Kutukcu, S Sharma
Similarly, it is clear that z is also the common fixed point of A, B, P, Q, S
and T in the case AB is continuous, and (P, AB) and (Q, ST ) are compatible
of type (A-2).
Case II. P is continuous, and (P, AB) and (Q, ST ) are compatible of
type (A-1).
Since P is continuous, P P x2n → P z and P (AB)x2n → P z. Since
(P, AB) is compatible of type (A-1), AB(AB)x2n → P z.
Step 8. By taking x = ABx2n , y = x2n+1 in (c), we have
g(FP (AB)x2n ,Qx2n+1 (t))





g(F
(t))
+
g(F
(t))


AB(AB)x
,ST
x
P
(AB)x
,AB(AB)x
2n
2n+1
2n
2n









+g(F
(t)),
Qx
,ST
x
2n+1
2n+1


≤ φ max



g(F
(t))
+
g(F
(t)),




P (AB)x2n ,AB(AB)x2n
Qx2n+1 ,AB(AB)x2n




 g(F
(t)) + g(F
(t)) 
P (AB)x2n ,ST x2n+1
Qx2n+1 ,ST x2n+1
this implies that, as n → ∞





 g(FP z,z (t)) + g(FP z,P z (t)) + g(Fz,z (t)), 


g(FP z,z (t)) ≤ φ max
g(FP z,P z (t)) + g(Fz,P z (t)),





g(FP z,z (t)) + g(Fz,z (t))
= φ(g(FP z,z (t)))
which means that z = P z. Now using Step 5-7, we have z = Qz = ST z =
Sz = T z.
Step 9. Since Q(X) ⊆ AB(X), there exists w ∈ X such that z = Qz =
ABw. By taking x = w, y = x2n+1 in (c), we have
g(FP w,Qx2n+1 (t))










+g(F
(t)),
Qx
,ST
x
2n+1
2n+1


≤ φ max



g(F
(t))
+
g(F
(t)),


P w,ABw
Qx2n+1 ,ABw






 g(F
(t)) + g(F
(t)) 






g(FABw,ST x2n+1 (t)) + g(FP w,ABw (t))
P w,ST x2n+1
Qx2n+1 ,ST x2n+1
this implies that, as n → ∞





 g(Fz,z (t)) + g(FP w,z (t)) + g(Fz,z (t)), 


g(FP w,z (t)) ≤ φ max
g(FP w,z (t)) + g(Fz,z (t)),





g(FP w,z (t)) + g(Fz,z (t))
= φ(g(FP w,z (t)))
which means that z = P w. Since z = Qz = ABw, P w = ABw. Since
(P, AB) is compatible of type (A-1), we have P z = ABz. Also z = Bz
A common fixed point theorem
847
follows from Step 4. Thus, z = Az = Bz = P z. Hence, z is the common
fixed point of the six maps in this case also.
Similarly, it is clear that z is also the common fixed point of A, B, P, Q, S
and T in the case P is continuous, and (P, AB) and (Q, ST ) are compatible
of type (A-2).
Step 10. For uniqueness, let v (v 6= z) be another common fixed point
of A, B, P, Q, S and T . Taking x = z, y = v in (c), we have
g(FP z,Qv (t))





 g(FABz,ST v (t)) + g(FP z,ABz (t)) + g(FQv,ST v (t)), 


≤ φ max
g(FP z,ABz (t)) + g(FQv,ABz (t)),





g(FP z,ST v (t)) + g(FQv,ST v (t))
which implies that




 g(Fz,v (t)) + g(Fz,z (t)) + g(Fv,v (t)), 


g(Fz,v (t)) ≤ φ max
g(Fz,z (t)) + g(Fv,z (t)),





g(Fz,v (t)) + g(Fv,v (t))

= φ(g(Fz,v (t)))
so we have z = v. This completes the proof of the theorem.
If we take A = B = S = T = IX (the identity map on X) in Theorem 1,
we have the following:
Corollary 1. Let P and Q be self maps on a complete N. A. Menger
PM-space (X, F, ∗). If g(FP x,Qy (t)) ≤ φ(g(Fx,y (t))) and



g(F
(t))
+
g(F
(t))
+
g(F
(t)),


x,y
P
x,x
Qy,y




g(FP x,Qy (t)) ≤ φ max
g(FP x,x (t)) + g(FQy,x (t)),





g(FP x,y (t)) + g(FQy,y (t))
for all x, y ∈ X and t > 0, where a function φ : [0, ∞) → [0, ∞) satisfies the
condition (Φ), then P and Q have a unique common fixed point.
In [7], Sehgal and Bharucha-Reid presented the probabilistic version of
the Banach contraction theorem. Next we prove such theorem for N. A.
Menger PM-spaces as follows:
Corollary 2. Let P be self maps on a complete N.A. Menger PM-space
(X, F, ∗). If
g(FP xP y (t)) ≤ φ(g(Fxy (t)))
for all x, y ∈ X and t > 0, then P has a unique common fixed point.
848
S. Kutukcu, S Sharma
Proof. The proof follows from Corollary 1 since P = Q and
g(Fxy (t)) = max{g(Fxy (t)), g(FxP x (t)), g(FyQy (t)), g(FyP x(t)), g(FxQy (t))}.
Example 2. Let (X, d) be a metric space with the usual metric d where
X = [0, 1] and (X, F, ∗) be the induced N. A. Menger PM-space with g(t) =
1 − t and Fxy (t) = H(t − d(x, y)) for all x, y ∈ X, t > 0. Let A, B, P, Q, S
and T be maps from X into itself defined as
Ax = x/5, Bx = x/3, P x = x/6, Qx = 0, Sx = x, T x = x/2
for all x ∈ X. Then
1
1
P (X) = 0,
⊂ 0,
= ST (X)
6
2
and
1
Q(X) = {0} ⊂ 0,
= AB(X).
15
If we take t = 1 and α = 1, we see that the condition (b) and (c) of the main
Theorem is satisfied. Clearly, conditions (d) and (e) of the main Theorem
are also satisfied. Moreover, the pairs (P, AB) and (Q, ST ) are mutually
compatible of type (A). In fact, if limn→∞ xn = 0, where {xn } is a sequence
in X such that limn→∞ P xn = limn→∞ ABxn = 0 and limn→∞ Qxn =
limn→∞ ST xn = 0 for some 0 ∈ X, then
lim g(FP (AB)xn ,AB(AB)xn (t)) = g(H(t)) = 0
n→∞
and
lim g(F(AB)P xn ,P P xn (t)) = g(H(t)) = 0
n→∞
hence (P, AB) and (Q, ST ) are compatible of type (A-1). Similarly, the pairs
(AB, P ) and (ST, Q) are also compatible of type (A-2). Thus, all conditions
of the main Theorem are satisfied and 0 is the unique common fixed point
of A, B, P, Q, S and T .
References
[1] S. S. Chang, Fixed point theorems for single-valued and multi-valued mappings in nonArchimedean Menger probabilistic metric spaces, Math. Japon. 35 (5) (1990), 875–885.
[2] Y. J. Cho, K. S. Ha, S. S. Chang, Common fixed point theorems for compatible maps of
type (A) in non-Archimedean Menger PM-spaces, Math. Japon. 46 (1) (1997), 169–179.
[3] Y. J. Cho, H. K. Pathak, S. M. Kang, J. S. Jung, Common fixed points of compatible
maps of type (β) on fuzzy metric spaces, Fuzzy Sets and Systems 93 (1998), 99–111.
[4] O. Hadzic, A note on Istratescu’s fixed point theorem in non-Archimedean Menger
spaces, Bull. Math. Soc. Sci. Math. R. S. Roumanie 24 (72) (1980), 277–280.
A common fixed point theorem
849
[5] S. Kutukcu, C. Yildiz, A common fixed point theorem of compatible and weak compatible maps on Menger spaces, Kochi Math. J. (2008), to appear in vol. 3.
[6] B. Schweizer, A. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960), 313–324.
[7] V. M. Sehgal, A. T. Bharucha-Reid, Fixed point of contraction mapping on PM spaces,
Math. Systems Theory 6 (1972), 97–100.
Servet Kutukcu:
DEPARTMENT OF MATHEMATICS,
FACULTY OF SCIENCE AND ARTS
ONDOKUZ MAYIS UNIVERSITY
55139 KURUPELIT, SAMSUN, TURKEY
E-mail: [email protected]
Sushil Sharma
DEPARTMENT OF MATHEMATICS
MADHAV SCIENCE COLLEGE
UJJAIN, M.P. 456 001, INDIA
E-mail: [email protected]
Received October 19, 2008.