International Journal of Pure and Applied Mathematics
Volume 109 No. 4 2016, 919-939
ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)
url: http://www.ijpam.eu
doi: 10.12732/ijpam.v109i4.15
AP
ijpam.eu
COMMON FIXED POINTS FOR MINIMAL COMMUTING
MAPPINGS SATISFYING (ψ-φ)-CONTRACTIVE
CONDITIONS IN MULTIPLICATIVE METRIC SPACES
Young Chel Kwun1 , Poonam Nagpal2 , Sanjay Kumar3 ,
Sudhir Kumar Garg4 , Shin Min Kang5 §
1 Department
of Mathematics
Dong-A University
Busan, 49315, KOREA
2,3,4 Departement of Mathematics
Deenbandhu Chhotu Ram University of Science and Technology
Murthal, Sonepat 131039, Haryana, INDIA
5 Department of Mathematics and RINS
Gyeongsang National University
Jinju, 52828, KOREA
Abstract:
In this paper, we prove common fixed point theorems for minimal commuting
mappings satisfying (ψ-φ)-contractive conditions in multiplicative metric spaces.
AMS Subject Classification: 47H10, 54H25
Key Words: multiplicative metric spaces, minimal commuting mappings, altering distances
1. Introduction and Preliminaries
The fixed point theory has applications not only in many areas of mathematics
Received:
Revised:
Published:
August 18, 2016
September 10, 2016
October 7, 2016
§ Correspondence author
c 2016 Academic Publications, Ltd.
url: www.acadpubl.eu
920
Y.C. Kwun, P. Nagpal, S. Kumar, S.K. Garg, S.M. Kang
but also in many branches of quantitative sciences such as economics and computer sciences. The most famous result in this field is known as the Banach
contraction principle ([4]). Many authors generalized the Banach contraction
principle in various spaces such as quasi-metric spaces, fuzzy metric spaces,
cone metric spaces, partial metric spaces and generalized metric spaces.
In 2008, Bashirov et al. [5] introduced the notion of multiplicative metric spaces, and studied the concept of multiplicative calculus and proved the
fundamental theorem of multiplicative calculus.
In 2012, Florack and Assen [10] displayed the use of the concept of multiplicative calculus in biomedical image analysis. In 2011, Bashirov et al. [6]
exploit the efficiency of multiplicative calculus over the Newtonian calculus.
They demonstrated that the multiplicative differential equations are more suitable than the ordinary differential equations in investigating some problems in
various fields. They defined the multiplicative distance between two nonnegative real numbers as well as between two positive square matrices by using
multiplicative absolute value function. This provides us basic tool in proving
fixed points results in multiplicative metric spaces.
In 2012, Özavsar and Çevikel [17] gave the multiplicative metric spaces by
remarking its topological properties, and introduced concept of multiplicative
contraction mapping and proved some fixed point theorems of multiplicative
contraction mappings on multiplicative spaces.
It is well known that the set of positive real numbers R+ is not complete
according to the usual metric. To overcome this problem, in 2008, Bashirov et
al. [5] introduced the concept of multiplicative metric spaces as follows:
Definition 1.1. Let X be a nonempty set. A multiplicative metric is a
mapping d : X × X → R+ satisfying the following conditions:
(i) d(x, y) ≥ 1 for all x, y ∈ X and d(x, y) = 1 if and only if x = y;
(ii) d(x, y) = d(y, x) for all x, y ∈ X;
(iii) d(x, y) ≤ d(x, z) · d(z, y) for all x, y, z ∈ X (multiplicative triangle
inequality).
Then the mapping d together with X, that is, (X, d) is a multiplicative
metric space.
Example 1.2. ([17]) Let Rn+ be the collection of all n-tuples of positive
real numbers. Let d∗ : Rn+ × Rn+ → R be defined as follows:
∗
∗ ∗
xn x1 x2 ∗
d (x, y) = · · · · ,
y1
y2
yn
where x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ Rn+ and | · |∗ : R+ → R+ is defined
COMMON FIXED POINTS FOR MINIMAL COMMUTING...
921
by
∗
|a| =
(
a
1
a
if a ≥ 1,
if a < 1.
Then it is obvious that all conditions of a multiplicative metric are satisfied.
Therefore (Rn+ , d∗ ) is a multiplicative metric space.
Example 1.3. ([23]) Let d : R × R → [1, ∞) be defined as d(x, y) = a|x−y| ,
where x, y ∈ R and a > 1. Then d is a multiplicative metric and (R, d) is a
multiplicative metric space. We may call it usual multiplicative metric spaces.
Remark 1.4. We note that the Example 1.2 is valid for positive real
numbers and Example 1.3 is valid for all real numbers.
Example 1.5. ([23]) Let (X, d) be a metric space. Define a mapping da
on X by
(
1 if x = y,
d(x,y)
da (x, y) = a
=
a if x 6= y,
where x, y ∈ X and a > 1. Then da is a multiplicative metric and (X, da ) is
known as the discrete multiplicative metric space.
Example 1.6. ([1]) Let X = C ∗ [a, b] be the collection of all real-valued
multiplicative continuous functions on [a, b] ⊂ R+ . Then (X, d) is a multi (x) for arbitrary
plicative metric space with d defined by d(f, g) = supx∈[a,b] fg(x)
f, g ∈ X.
Remark 1.7. ([23]) We note that multiplicative metrics and metric spaces
are independent.
Indeed, the mapping d∗ defined in Example 1.2 is multiplicative metric but
not metric as it does not satisfy triangular inequality. Consider
3
∗ 1
∗ 1 1
∗ 1
,
, 3 = + 6 = 7.5 < 9 = d
,3 .
d
+d
3 2
2
2
3
On the other hand the usual metric on R is not multiplicative metric as it
doesnt satisfy multiplicative triangular inequality, since
d(2, 3) · d(3, 6) = 3 < 4 = d(2, 6).
One can refer to [17] for detailed multiplicative metric topology.
Definition 1.8. Let (X, d) be a multiplicative metric space. Then a
sequence {xn } in X is said to be
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Y.C. Kwun, P. Nagpal, S. Kumar, S.K. Garg, S.M. Kang
(1) a multiplicative convergent to x if for every multiplicative open ball
Bǫ (x) = {y | d(x, y) < ǫ}, ǫ > 1, there exists N ∈ N such that xn ∈ Bǫ (x) for
all n ≥ N, that is, d(xn , x) → 1 as n → ∞.
(2) a multiplicative Cauchy sequence if for all ǫ > 1, there exists N ∈ N such
that d(xn , xm ) < ǫ for all m, n ≥ N , that is, d(xn , xm ) → 1 as n, m → ∞.
(3) We call a multiplicative metric space complete if every multiplicative
Cauchy sequence in it is multiplicative convergent to x ∈ X.
Remark 1.9. The set of positive real numbers R+ is not complete according to the usual metric. Let X = R+ and the sequence {xn } = { n1 }. It
is obvious {xn } is a Cauchy sequence in X with respect to usual metric and
X is not a complete metric space, since 0 ∈
/ R+ . In case of a multiplicative
1
metric space, we take a sequence {xn } = {a n }, where a > 1. Then {xn } is a
multiplicative Cauchy sequence since for n ≥ m,
1 xn a n 1 − 1 =
= a n m d(xn , xm ) = xm a m1 1
1
1
log a
≤ a m − n < a m < ǫ if m >
,
log ǫ
(
a if a ≥ 1,
Also, {xn } → 1 as n → ∞ and 1 ∈ R+ . Hence (X, d)
where |a| = 1
if a < 1.
a
is a complete multiplicative metric space.
In 2012, Özavsar and Çevikel [17] gave the concept of multiplicative contraction mappings and proved some fixed point theorem of such mappings in a
multiplicative metric space.
Definition 1.10. Let f be a mapping of a multiplicative metric space
(X, d) into itself. Then f is said to be a multiplicative contraction if there
exists a real number λ ∈ [0, 1) such that
d(f x, f y) ≤ dλ (x, y)
for all x, y ∈ X.
Altering distances is introduced by Khan et al. [14] in a metric space. These
are control functions which alter the distance between two points in a metric
space. Later on various author ([2, 3, 8, 9, 22, 24, 25] using altering distances
proved common fixed point theorems.
In the similar mode, now we use the altering distances in multiplicative
metric spaces. For this we define control functions as follows:
Ψ = {ψ | ψ : [1, ∞) → [1, ∞) is continuous and non-decreasing satisfying
ψ(t) = 1 if and only if t = 1}.
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COMMON FIXED POINTS FOR MINIMAL COMMUTING...
Φ = {φ | φ : [1, ∞) → [1, ∞) is lower semi-continuous and discontinuous at
t = 1, φ(t) > 1 for all t ≥ 1}.
2. Relations between Various Minimal Commuting Mappings
In 2015, Kumar et al. [15] introduced the notions of subcompatible and occasionally weakly compatible mappings in a multiplicative metric space.
Definition 2.1. Let f and g be mappings of a multiplicative (X, d) into
itself. Then f and g are called
(1) subcompatible if there exists a sequence {xn } in X such that lim f xn =
n→∞
lim gxn = t for some t ∈ X and which satisfy lim d(f gxn , gf xn ) = 1;
n→∞
n→∞
(2) occasionally weakly compatible if there exists x ∈ X, which is a coincidence point of f and g at which f and g commute, that is, there exists x ∈ X
such that f x = gx implies f gx = gf x.
Remark 2.2. Every occasionally weakly compatible mappings are subcompatible. However, the converse is not true, in general.
In 2015, Kang et al. [13] and Jung et al. [11] introduced the notions of
compatible and weakly compatible mappings in a multiplicative metric space,
respectively.
Definition 2.3. Let f and g be mappings of a multiplicative metric space
(X, d) into itself. Then f and g are called
(1) compatible if lim d(f gxn , gf xn ) = 1, whenever {xn } is a sequence in X
n→∞
such that lim f xn = lim gxn = t for some t ∈ X;
n→∞
n→∞
(2) weakly compatible if they commute at coincidence points, that is, if
f t = gt for some t ∈ X implies that f gt = gf t.
Remark 2.4. Every compatible mappings are weakly compatible. However, the converse is not true, in general.
In metric spaces, in 2010, Pant and Pant [21], introduced the notion of
conditionally commuting mappings. In 2012, Pant and Bisht [20] introduced the
notion of conditionally compatible mappings. In 1998, Pant [18] introduced the
notion of non-compatible mappings. In 2013, Bisht and Shahzad [7] introduced
the notion of faintly compatible mappings.
In similar mode, we introduce the notions in multiplicative metric spaces.
Definition 2.5. Let f and g be mappings of a multiplicative metric space
(X, d) into itself. Then f and g are called
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Y.C. Kwun, P. Nagpal, S. Kumar, S.K. Garg, S.M. Kang
(1) conditionally commuting if the pair commutes on a nonempty subset of
the set of coincidence points, whenever the set of coincidences is nonempty;
(2) conditionally compatible if whenever the set of sequences {yn } satisfying
lim f yn = lim gyn is nonempty, there exists a sequence {zn } (distinct from
n→∞
n→∞
{yn } such that lim f zn = lim gzn = t and lim d(f gzn , gf zn ) = 1;
n→∞
n→∞
n→∞
(3) non-compatible if the pair is not compatible, that is, there exists a
sequence {xn } in X such that lim f xn = lim gxn = t for some t ∈ X, and
n→∞
n→∞
lim d(f gxn , gf xn ) either 6= 1 or non-existent;
n→∞
(4) faintly compatible if f and g are conditionally compatible and f and
g commute on a nonempty subset of coincidence points, whenever the set of
coincidences is nonempty.
Clearly every commuting mappings are conditionally commuting. However,
the converse is not true, in general.
Example 2.6. Let X = [0, ∞) with a multiplicative metric d : X × X →
[1, ∞) be defined by d(x, y) = a|x−y| for all x, y ∈ X and a > 1.
Define mappings f, g : X → X by
x2
x+2
and gx =
for all x ∈ X.
2
2
Here f and g have a coincidence point x = 2. Further, f and g are conditionally
commuting since they commute at a coincidence point x = 2.
Also we have
x + 22
x2 + 4
and gf x =
.
f gx =
4
8
Hence f and g are not commuting.
fx =
Every weakly compatible mappings are occasionally weakly compatible.
However, the converse is not true, in general.
Example 2.7. Let X = [0, ∞) with multiplicative metric d : X × X →
[1, ∞) be defined by d(x, y) = a|x−y| for all x, y ∈ X and a > 1.
Define mappings f, g : X → X by
x2
for all x ∈ X.
3
Here f and g have two coincidence points x = 0, 9. Further, f and g are ccasionally weakly compatible since they commute at a coincidence point x = 0.
But f and g are not weakly compatible as they are not commuting at x = 9 as
f x = 3x
and gx =
f g9 = 81 6= 9 = gf 9.
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COMMON FIXED POINTS FOR MINIMAL COMMUTING...
Every occasionally weakly compatible mappings are conditionally compatible. However, the converse is not true, in general.
Example 2.8. Let X = [0, ∞) with a multiplicative metric d : X × X →
[1, ∞) be defined by d(x, y) = a|x−y| for all x, y ∈ X and a > 1.
Define mappings f, g : X → X by
(
x+6
if x ∈ [0, 9] ∪ (16, ∞),
2
f x = x for all x ∈ X and gx =
7x + 18 if x ∈ (9, 16].
Let {yn } = {3 − n1 } be a sequence in X. Then lim f yn = 9 and lim gyn =
n→∞
n→∞
9. Hence the set of sequences {yn } satisfying lim f yn = lim gyn is nonempty.
n→∞
n→∞
Now let {zn } = {3 + n1 } be a sequence in X. Then lim f zn = 9 and
n→∞
lim gzn = 9 and hence
n→∞
lim f gzn = 81,
n→∞
lim gf zn = 81,
n→∞
lim d(f gzn , gf zn ) = 1.
n→∞
So, f and g are conditionally compatible. Here 3 is a coincidence point at which
f g3 = 81 6= 15 = gf 3
and hence f and g are not commuting at 3. Thus, f and g are not occasionally
weakly compatible.
Every faintly compatible and non-compatible are independent each other.
Example 2.9. Let X = [1, 5) with a multiplicative metric d : X × X →
[1, ∞) be defined by d(x, y) = a|x−y| for all x, y ∈ X and a > 1.
Define mappings f, g : X → X by
(
(
3
if 1 ≤ x < 4,
5 if 1 ≤ x < 4,
and gx =
fx =
x − 3 if x ≥ 4.
1 if x ≥ 4
Let {yn } = {4+ n1 } be a sequence in X. Then lim f yn = 1 and lim gyn = 1
n→∞
n→∞
and hence
lim f gyn = 5,
n→∞
lim gf yn = 3,
n→∞
lim d(f gyn , gf yn ) = a2 6= 1.
n→∞
So, f and g are non-compatible. Here 4 is a coincidence point of f and g. But
f g4 = 6 6= 3 = gf 2.
Hence f and g are not faintly compatible.
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Y.C. Kwun, P. Nagpal, S. Kumar, S.K. Garg, S.M. Kang
Example 2.10. Let X = [0, ∞) with a multiplicative metric d : X × X →
[1, ∞) be defined by d(x, y) = a|x−y| for all x, y ∈ X and a > 1.
Define mappings f, g : X → X by
(
1 if x ∈ (0, 2],
f x = x2 for all x ∈ X and gx =
0 if x ∈ {0} ∪ (2, ∞].
1
Let {yn } = {1 + n+1
} be a sequence in X. Then lim f yn = 1 and
n→∞
lim gyn = 1 and hence
n→∞
lim f gyn = 1,
n→∞
lim gf yn = 1,
n→∞
lim d(f gyn , gf yn ) = 1.
n→∞
So, f and g are not non-compatible.
Now let {zn } = {0} be a constant sequence. Then lim f zn = 0 and
n→∞
lim gzn = 0 and hence
n→∞
lim f gzn = 0,
n→∞
lim gf zn = 0,
n→∞
lim d(f gzn , gf zn ) = 1.
n→∞
Further 0 is the only one coincidence point in which f and g commute. So, f
and g are faintly compatible.
3. Main Results
In 1999, Pant [19] introduced a new notion of continuity, known as reciprocal
continuity. In similar mode, we introduced the notions in a multiplicative metric
space.
Definition 3.1. [12] Let f and g be mappings of a multiplicative (X, d)
into itself. Then f and g are called reciprocally continuous if lim f gxn = f t
n→∞
and lim gf xn = gt, whenever {xn } is a sequence in X such that lim f xn =
n→∞
n→∞
lim gxn = t for some t ∈ X.
n→∞
Now, we prove the following theorem for subcompatible mapping.
Theorem 3.2. Let A, B, S and T be mappings of a complete multiplicative
metric space (X, d) into itself satisfying a (ψ-φ)-contractive condition, that is,
(C1 )
ψ(d(Ax, By)) ≤
ψ(M (x, y))
φ(N (x, y))
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COMMON FIXED POINTS FOR MINIMAL COMMUTING...
for all x, y ∈ X with x 6= y, where ψ ∈ Ψ, φ ∈ Φ,
M (x, y) = max{d(Sx, T y), (d(Sx, Ax) · d(T y, By))1/2 ,
(d(Sx, By) · d(T y, Ax))1/2 }
and
N (x, y) = min{d(Sx, T y), d(Sx, Ax), d(T y, By)},
(C2 ) the pairs A, S and B, T are reciprocally continuous.
Assume that the pairs A, S and B, T are subcompatible. Then A, B, S and
T have a unique common fixed point in X.
Proof. Since the pairs A, S and B, T are subcompatible and reciprocally
continuous, there exists sequences {xn } and {yn } in X such that lim Axn =
n→∞
lim Sxn = w for some w ∈ X and which satisfy lim d(ASxn , SAxn ) =
n→∞
n→∞
d(Aw, Sw) = 1 and lim Byn = lim T yn = z for some z ∈ X and which
n→∞
n→∞
satisfy lim d(BT yn , T Byn ) = d(Bz, T z) = 1.
n→∞
Therefore, Aw = Sw and Bz = T z, that is w is a coincidence point of A
and S, and z is a coincidence point of B and T.
Now we claim that w = z. Putting x = xn and y = yn in (C1 ), we get
ψ(d(Axn , Byn )) ≤
ψ(M (xn , yn ))
,
φ(N (xn , yn ))
where
M (xn , yn ) = max{d(Sxn , T yn ), (d(Sxn , Ax2n ) · d(T yn , Byn ))1/2 ,
(d(Sxn , Byn ) · d(T yn , Axn ))1/2 }
and
N (xn , yn ) = min{d(Sxn , T yn ), d(Sxn , Ax2n ), d(T yn , Byn )}.
Letting n → ∞, we have
lim M (xn , yn ) = d(w, z)
n→∞
and
lim N (xn , yn ) = 1.
n→∞
Since φ is discontinuous at t = 1. Therefore, we have
ψ(d(t, z)) ≤
ψ(w, z)
< ψ(w, z),
lim φ(N (x2n , yn ))
n→∞
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Y.C. Kwun, P. Nagpal, S. Kumar, S.K. Garg, S.M. Kang
which is a contradiction. Hence w = z.
Next we show that Az = z. Putting x = z and y = yn in (C1 ), we have
ψ(d(Az, Byn )) ≤
ψ(M (z, yn ))
,
φ(N (z, yn ))
where
M (z, yn ) = max{d(Sz, T yn ), (d(Sz, Az) · d(T yn , Byn ))1/2 ,
(d(Sz, Byn ) · d(T yn , Az))1/2 }
and
N (z, yn ) = min{d(Sz, T yn ), d(Sz, Az), d(T yn , Byn )}.
Letting n → ∞, we get
lim M (z, yn ) = d(z, Az)
n→∞
and
lim N (z, yn ) = 1.
n→∞
Since φ is discontinuous at t = 1. Therefore, we have
ψ(d(Az, z)) ≤
ψ(z, Az)
< ψ(z, Az),
lim φ(N (z, yn ))
n→∞
which implies that Az = z.
Finally we claim that Bz = z. Putting x = z and y = z in (C1 ), we have
ψ(d(Az, Bz)) ≤
where
ψ(M (z, z))
,
φ(N (z, z))
M (z, z) = max{d(Sz, T z), (d(Sz, Az) · d(T z, Bz))1/2 ,
(d(Sz, Bz) · d(T z, Az))1/2 }
and
N (z, z) = min{d(Sz, T z), d(Sz, Az), d(T z, Bz)}.
Hence
M (z, z) = d(z, Bz)
and
N (z, z) = 1.
COMMON FIXED POINTS FOR MINIMAL COMMUTING...
929
Further we have
ψ(d(z, Bz)) = ψ(d(Az, Bz))
ψ(d(z, Bz))
≤
< ψ(d(z, Bz)),
φ(N (z, z))
which is a contradiction since φ(t) > 1 for t ≥ 1. Hence we have Bz = z. Hence
Bz = Az = Sz = T z = z, that is z is a common fixed point of A, B, S and T.
Uniqueness follows easily. Therefore A, B, S and T have a unique common
fixed point. This completes the proof.
Next, we prove the following theorem for occasionally weakly compatible
mappings without reciprocal continuity
Theorem 3.3. Let A, B, S and T be mappings of a complete multiplicative
metric space (X, d) into itself satisfying the condition (C1 ).
Assume that the pairs A, S and B, T are occasionally weakly compatible.
Then A, B, S and T have a unique common fixed point in X.
Proof. Since the pair A, S is occasionally weakly compatible, there exists
u ∈ X such that Au = Su = z (say) and ASu = SAu, that is, Az = Sz = z ′
(say). Since the pair B, T is occasionally weakly compatible, there exists v ∈ X
such that Bv = T v = w (say) and BT v = T Bv = w′ (say).
Now, we claim that z ′ = w′ . Putting x = z and y = w in (C1 ), we have
ψ(d(Az, Bw)) ≤
where
ψ(M (z, w))
,
φ(N (z, w))
M (z, w) = max{d(Sz, T w), (d(Sz, Az) · d(T w, Bw))1/2 ,
(d(Sz, Bw) · d(T w, Az))1/2 }
= d(z ′ , w′ )
and
N (u, w) = min{d(Su, T w), d(Su, Au), d(T w, Bw)}
= 1.
Hence we obtain
ψ(d(z ′ , w′ )) ≤
ψ(d(z ′ , w′ )
< ψ(d(z ′ , w′ ),
φ(N (z, w))
which is a contradiction since φ(t) > 1 for t ≥ 1. Thus z ′ = w′ . Hence we have
Az = Sz = z ′ = w′ = Bw = T w.
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Y.C. Kwun, P. Nagpal, S. Kumar, S.K. Garg, S.M. Kang
Next we prove that z = w′ . Putting x = u and y = w in (C1 ), we have
ψ(d(Au, Bw)) ≤
where
ψ(M (u, w))
,
φ(N (u, w))
M (u, w) = max{d(Su, T w), (d(Su, Au) · d(T w, Bw))1/2 ,
(d(Su, Bw) · d(T w, Au))1/2 }
= d(z, w′ )
and
N (z, w) = min{d(Sz, T w), d(Sz, Az), d(T w, Bw)}
= 1.
Hence we obtain
ψ(d(z, w′ )) ≤
ψ(d(z, w′ ))
< ψ(d(z, w′ )),
φ(N (z, w))
which is a contradiction. Thus z = w′ . Hence we have Az = Sz = z and
Bw = T w = z.
Again, we claim that z = w. Putting x = z and y = v in (C1 ), we have
ψ(d(Az, Bv)) ≤
where
ψ(M (z, v))
φ(N (z, v)),
M (z, v) = max{d(Sz, T v), (d(Sz, Az) · d(T v, Bv))1/2 ,
(d(Sz, Bv) · d(T v, Az))1/2 }
= d(z, w)
and
N (z, v) = min{d(Sz, T v), d(Sz, Az), d(T v, BV )}
= 1.
Hence we obtain
ψ(d(z, w)) ≤
ψ(d(z, w))
< ψ(d(z, w)),
φ(N (z, v))
which is a contradiction. So we have z = w. Thus we have Az = Sz = Bz =
T z = z. Hence z is a common fixed point of A, B, S and T.
Uniqueness follows easily. Therefore A, B, S and T have a unique common
fixed point. This completes the proof.
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COMMON FIXED POINTS FOR MINIMAL COMMUTING...
Next, we prove the following theorem for conditionally commuting.
Theorem 3.4. Let A, B, S and T be mappings of a complete multiplicative
metric space (X, d) into itself satisfying the condition (C1 ). Suppos that
(C3 )
A(X) ⊂ T (X)
and B(X) ⊂ S(X),
(C4 ) One of the subspace AX or BX or SX or TX be closed.
Assume that the pairs A, S and B, T are conditionally commuting. Then
A, B, S and T have a unique common fixed point in X.
Proof. Let x0 ∈ X be an arbitrary point. Since A(X) ⊂ T (X) and B(X) ⊂
S(X), there exist a point x1 ∈ X such that Ax0 = T x1 and for this point
x1 ∈ X, there exists a point x2 ∈ X such that Bx1 = Sx2 . Continuing this
process, we can define a sequence {yn } in X such that
y2n+1 = Ax2n = T x2n+1 ,
y2n+2 = Bx2n+1 = Sx2n+2
for n ≥ 0.
From the proof of [16, Theorem 3.1], {yn } is a multiplicative metric space.
Since (X, d) is complete, there exists z ∈ X such that
Ax2n → z,
T x2n+1 → z,
Bx2n+1 → z,
Sx2n+2 → z.
Assume that SX is closed, there exists v ∈ X such that Sv = z.
Now we claim that Av = z. Putting x = v and y = x2n+1 in (C1 ), we have
ψ(d(Av, Bx2n+1 )) ≤
ψ(M (v, x2n+1 )
,
φ(N (v, x2n+1 ))
where
M (v, x2n+1 ) = max{d(Sv, T x2n+1 ), (d(Sv, Av) · d(T x2n+1 , Bx2n+1 ))1/2 ,
(d(Sv, Bx2n+1 ) · d(T x2n+1 , Av))1/2 }
and
N (v, x2n+1 ) = min{d(Sv, T x2n+1 ), d(Sv, Av), d(T x2n+1 , Bx2n+1 )}.
Letting n → ∞, we have
lim M (v, x2n+1 ) = max{d(Sv, z), (d(Sv, Av) · d(z, z))1/2 ,
n→∞
(d(Sv, z) · d(z, Av))1/2 }
= d1/2 (z, Av)
932
Y.C. Kwun, P. Nagpal, S. Kumar, S.K. Garg, S.M. Kang
and
lim N (v, x2n+1 ) = max{d(Sv, z), d(Sv, Av), d(z, z)}
n→∞
= 1.
Hence we get
ψ(d1/2 (z, Av))
< ψ(d1/2 (z, Av)),
lim φ(N (v, x2n+1 ))
ψ(d(Av, z)) ≤
n→∞
which is a contradiction since φ is discontinuous at t = 1. Thus we obtain
Av = z. Since AX ⊂ T X, there exists u ∈ X such that Av = T u = z.
Next, we claim that Bu = z. Putting x = x2n and y = u in (C1 ), we get
ψ(d(Ax2n , Bu)) ≤
ψ(M (x2n , u))
,
φ(N (x2n , u))
where
M (x2n , u) = max{d(Sx2n , T u), (d(Sx2n , Ax2n ) · d(T u, Bu))1/2 ,
(d(Sx2n , Bu) · d(T u, Ax2n ))1/2 }
and
N (x2n , u) = min{d(Sx2n , T u), d(Sx2n , Ax2n ), d(T u, Bu)}.
Letting n → ∞, we have
lim M (x2n , u) = max{d(z, T u), (d(z, z) · d(T u, Bu))1/2 ,
n→∞
(d(z, Bu) · d(T u, z))1/2 }
= d1/2 (z, Bu)
and
lim N (x2n , u) = min{d(z, T u), d(z, z), d(T u, Bu)}
n→∞
= 1.
Hence we get
ψ(d(z, Bu)) ≤
ψ(d1/2 (z, Bu))
< ψ(d1/2 (z, Bu)),
lim φ(N (x2n , u))
n→∞
which is a contradiction since φ is discontinuous at t = 1. So we have Bu = z.
Thus we have Av = Sv = T u = Bu = z. Hence we conclude that v is a
coincidence point of A and S and u is a coincidence point of B and T.
COMMON FIXED POINTS FOR MINIMAL COMMUTING...
933
Case 1. A and S commutes at v and B and T commutes at u.
Hence ASv = SAv, that is, Az = Sz and BT u = T Bu, that is, Bz = T z.
Now, we claim that Az = z. Putting x = z and y = x2n+1 in (C1 ), we have
ψ(d(Az, Bx2n+1 )) ≤
ψ(M (z, x2n+1 )
,
φ(N (z, x2n+1 ))
where
M (z, x2n+1 ) = max{d(Sz, T x2n+1 ), (d(Sz, Az) · d(T x2n+1 , Bx2n+1 ))1/2 ,
(d(Sz, Bx2n+1 ) · d(T x2n+1 , Az))1/2 }.
and
N (z, x2n+1 ) = min{d(Sz, T x2n+1 ), d(Sz, Az), d(T x2n+1 , Bx2n+1 )}.
Letting n → ∞, we have
lim M (z, x2n+1 ) = max{d(Sz, z), (d(Sz, Az) · d(z, z))1/2 ,
n→∞
(d(Sz, z) · d(z, Az))1/2 }
= d(z, Az)
and
lim N (z, x2n+1 ) = max{d(Sz, z), d(Sz, Az), d(z, z)}
n→∞
= 1.
Hence we get
ψ(d(Az, z)) ≤
ψ(d(z, Az))
< ψ(d(z, Az)),
lim φ(N (z, x2n+1 ))
n→∞
which is a contradiction since φ is discontinuous at t = 1. So we have Az = z.
Next we show that Az = Bz. Putting x = z and y = z in (C1 ), we have
ψ(d(Az, Bz)) ≤
where
ψ(M (z, z))
,
φ(N (z, z))
M (z, z) = max{d(Sz, T z), (d(Sz, Az) · d(T z, Bz))1/2 ,
(d(Sz, Bz) · d(T z, Az))1/2 }
= d(Az, Bz)
934
Y.C. Kwun, P. Nagpal, S. Kumar, S.K. Garg, S.M. Kang
and
N (z, z) = min{d(Sz, T z), d(Sz, Az), d(T z, Bz)}
= 1.
Hence we have
ψ(d(Az, Bz)) ≤
ψ(d(Az, Bz))
< ψ(d(Az, Bz)),
φ(N (z, z))
which is a contradiction since φ(t) > 1 for t ≥ 1. Thus, we have Bz = T z =
Az = Sz = z, that is, z is a common fixed point of A, B, S and T.
Case 2. A and S commute at v, but B and T do not commute at u.
Since A and S commute, ASv = SAv = z ′ (say), that is, Az = Sz = z ′ .
Since Av = Sv = z. by B and T is conditionally commuting, there exists r ∈ X
such that Br = T r = w (say) and also BT r = T Br = w′ (say), that is,
Bw = T w = w′ .
Now we shall prove that z ′ = w′ . Putting x = z and y = w in (C1 ), we have
ψ(d(Az, Bw)) ≤
where
ψ(M (z, w))
,
φ(N (z, w))
M (z, w) = max{d(Sz, T w), (d(Sz, Az) · d(T w, Bw))1/2 ,
(d(Sz, Bw) · d(T w, Az))1/2 }
= d(z ′ , w′ )
and
N (z, w) = min{d(Sz, T w), d(Sz, Az), d(T w, Bw)}
= 1.
Hence we obtain
ψ(d(z ′ , w′ )) ≤
ψ(d(z ′ , w′ ))
< ψ(d(z ′ , w′ )),
φ(N (z, w))
which is a contradiction since φ(t) > 1 for t ≥ 1. Hence z ′ = w′ . Thus, we have
Az = Sz = z ′ = Bw = T w.
Now we claim that z = z ′ . Putting x = v and y = w in (C1 ), we have
ψ(d(Av, Bw)) ≤
where
ψ(M (v, w))
,
φ(N (v, w))
M (v, w) = max{d(Sv, T w), (d(Sv, Av) · d(T w, Bw))1/2 ,
(d(Sv, Bw) · d(T w, Av))1/2 }
= d(z, z ′ )
COMMON FIXED POINTS FOR MINIMAL COMMUTING...
935
and
N (v, w) = min{d(Sv, T w), d(Sv, Av), d(T w, Bw)}
= 1.
Hence, we obtain
ψ(d(z, z ′ )) ≤
ψ(d(z, z ′ ))
< ψ(d(z, z ′ )),
φ(N (z, w))
which is a contradiction Thus z = z ′ . Hence Az = Sz = z = Bw = T w.
Next we claim that z = w. Putting x = z and y = r in (C1 ), we have
ψ(d(Az, Br)) ≤
where
ψ(M (z, r))
,
φ(N (z, r))
M (z, r) = max{d(Sz, T r), (d(Sz, Az) · d(T r, Br))1/2 ,
(d(Sz, Br) · d(T r, Az))1/2 }
= d(z, w)
and
N (z, r) = min{d(Sz, T r), d(Sz, Az), d(T r, Br)}
= 1.
Hence we obtain
ψ(d(z, w)) ≤
ψ(d(z, w)
< ψ(d(z, w)),
φ(N (z, r))
which is a contradiction. So we have z = w. Thus Az = Sz = z = Bz = T z,
that is, z is a common fixed point of A, B, S and T.
Case 3. B and T commute at u, but A and S do not commute at v.
This case is similar to Case 2.
Case 4. A and S do not commute at v, but B and T do not commute at u.
Since A and S be conditionally commuting, there exists p ∈ X such that
Ap = Sp = p′ (say) and ASp = SAp, that is, Ap′ = Sp′ . Also B and T is
conditionally commuting, there exists q ∈ X such that Bq = T q = q (say) and
BT q = T Bq, that is, Bq ′ = T q ′ . The rest proof is similar to the proof of Case
2.
Similarly, we can also complete the proof when AX or BX or T X is closed.
Uniqueness follows easily. Therefore A, B, S and T have a unique common
fixed point. This completes the proof.
936
Y.C. Kwun, P. Nagpal, S. Kumar, S.K. Garg, S.M. Kang
Next we prove the following theorem for conditionally compatible and noncompatible.
Theorem 3.5. Let A, B, S and T be mappings of a complete multiplicative
metric space (X, d) into itself satisfying the conditions (C1 ) and (C2 ).
Assume that the pairs A, S and B, T are conditionally compatible and noncompatible. Then A, B, S and T have a unique common fixed point in X.
Proof. Since A and S is non-compatible, there exists a sequence {xn } in X
such that lim Axn = lim Sxn = t1 for some t1 ∈ X and lim d(ASxn , SAxn ) 6=
n→∞
n→∞
n→∞
1. Also since B and T is non-compatible, there exists a sequence {yn } in X such
that lim Byn = lim T yn = t2 for some t2 ∈ X and lim d(BT yn , T Byn ) 6= 1.
n→∞
n→∞
n→∞
Since the paies A, S and B, T are conditionally compatible, there exists {zn }
and {vn } in X such that lim Azn = lim Szn = z for some z ∈ X and
n→∞
n→∞
lim d(ASzn , SAzn ) = 1. and lim Bvn = lim T vn = v for some v ∈ X and
n→∞
n→∞
n→∞
lim d(BT vn , T Bvn ) = 1.
n→∞
Also since the pairs A, S and B, T are reciprocally continuous, lim ASzn =
n→∞
Az and lim SAzn = Sz and hence Az = Sz. Similarly lim BT vn = Bv and
n→∞
n→∞
lim T Bvn = T v and hence Bv = T v.
n→∞
Now we claim that z = v. Putting x = zn and y = vn in (C1 ), we have
ψ(d(Azn , Bvn )) ≤
ψ(M (zn , vn ))
φ(N (zn , vn )),
where
M (zn , vn ) = max{d(Szn , T vn ), (d(Szn , Azn ) · d(T vn , Bvn ))1/2 ,
(d(Szn , Bvn ) · d(T vn , Azn ))1/2 }
and
N (zn , vn ) = min{d(Szn , T vn ), d(Szn , Azn ), d(T vn , Bvn )}.
Letting n → ∞, we have
lim M (zn , vn ) = max{d(z, v), 1, (d(z, v) · d(v, z))1/2
n→∞
= d(u, v)
and
lim N (zn , vn )) = 1.
n→∞
COMMON FIXED POINTS FOR MINIMAL COMMUTING...
937
Since φ is discontinuous at t = 1, we obtain
ψ(d(z, v) ≤
ψ(d(z, v))
< ψ(d(z, v)),
lim φ(N (zn , vn ))
n→∞
which is a contradiction. Hence z = v.
Next, we show that Az = z. Putting x = z and y = vn in (C1 ), we have
ψ(d(Az, Bvn )) ≤
ψ(M (z, vn ))
,
φ(N (z, vn ))
where
M (z, vn ) = max{d(Sz, T vn ), (d(Sz, Az) · d(T vn , Bvn ))1/2 ,
(d(Sz, Bvn ) · d(T vn , Az))1/2 }
and
N (z, vn ) = min{d(Sz, T vn ), d(Sz, Az), d(T vn , Bvn )}.
Letting n → ∞, we have
lim M (z, vn ) = max{d(Az, z), 1, (d(Az, z) · d(v, Az))1/2 }
n→∞
= d(Az, z)
and
lim N (z, vn ) = min{d(Az, z, 1, 1} = 1.
n→∞
Since φ is discontinuous at t = 1, we obtain
ψ(d(Az, z) ≤
ψ(d(Az, z))
< ψ(d(Az, z)),
lim φ(N (z, vn ))
n→∞
which is a contradiction. Hence Az = z.
Similarly putting x = zn and y = z in (C1 ), we obtain Bz = z. Thus we
have Az = Bz = Sz = T z = z. Hence u is a common fixed point of A, B, S and
T.
Uniqueness follows easily. Therefore A, B, S and T have a unique common
fixed point. This completes the proof.
Next we prove the following theorem for faintly compatible and non-compatible.
938
Y.C. Kwun, P. Nagpal, S. Kumar, S.K. Garg, S.M. Kang
Theorem 3.6. Let A, B, S and T be mappings of a complete multiplicative
metric space (X, d) into itself satisfying the conditions (C1 ) and (C2 ).
Assume that the pairs A, S and B, T are faintly compatible and non-compatible. Then A, B, S and T have a unique common fixed point in X.
Proof. Since the pairs A, S and B, T are faintly compatible, by definition of
faintly compatible, the pairs A, S and B, T are also conditionally compatible.
The result follows immediately from Theorem 3.5.
Acknowledgment. This work was supported by the Dong-A University
research fund.
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