DOI: 10.2478/auom-2014-0058
An. Şt. Univ. Ovidius Constanţa
Vol. 22(3),2014, 179–203
Tripled and coincidence fixed point theorems
for contractive mappings satisfying Φ-maps in
partially ordered metric spaces
Wasfi Shatanawi, Mihai Postolache and Zead Mustafa
Abstract
In this paper, we study some tripled fixed and coincidence point theorems for two mappings F : X × X × X → X and g : X → X satisfying a
nonlinear contraction based on φ-maps. Our results extend and improve
many existing results in the literature. Also, we introduce an example
to support the validity of our results.
1
Introduction and Preliminaries
The notion of a coupled fixed point for a mapping F : X ×X → X was initiated
by Gnana Bhaskar and Lakshmikantham [1] and they proved some interesting
coupled fixed point theorems, while Ćirić and Lakshmikantham [2] introduced
the concept of the coupled fixed point for two mappings F : X × X → X
and g : X → X and established many existing theorems. Vasile Berinde and
Marin Borcut [3, 4] initiated the concept of a tripled fixed point of the mapping
F : X × X × X → X and the concept of a tripled coincidence point of the two
mappings F : X × X × X → X and g : X → X and extended the results
of Gnana Bhaskar and Lakshmikantham [1] and Ćirić and Lakshmikantham
[2] to the interesting tripled fixed and coincidence point theorems. For some
Key Words: Tripled fixed point, ordered sets, generalized metric spaces, mixed monotone property.
2010 Mathematics Subject Classification: Primary 54H25, 47H10; Secondary 54E50.
Received: June, 2013.
Revised: October, 2013.
Accepted: November, 2013.
179
TRIPLED AND COINCIDENCE FIXED POINT THEOREMS
180
coupled fixed point results in different metric spaces we refer the reader to
[5]-[25].
Consistent with Berinde and Borcut [3, 4], we give the following definitions
and preliminaries.
Definition 1.1 ([3]). Let X be a nonempty set and F : X × X × X → X be
a given mapping. An element (x, y, z) ∈ X × X × X is called a tripled fixed
point of F if
F (x, y, z) = x, F (y, x, y) = y
and F (z, y, x) = z.
Let (X, d) be a metric space. The mapping
d : X × X × X → R,
d((x, y, z), (u, v, w)) = d(x, u) + d(y, v) + d(z, w),
defines a metric on X × X × X, which will be denoted for convenience by d.
Definition 1.2 ([3]). Let (X, ≤) be a partially ordered set and F : X ×
X × X → X be a mapping. We say that F has the mixed monotone property
if F (x, y, z) is monotone non-decreasing in x and z and is monotone nonincreasing in y; that is, for any x, y, z ∈ X,
x1 , x2 ∈ X,
y1 , y2 ∈ X,
x1 ≤ x2
implies
F (x1 , y, z) ≤ F (x2 , y, z),
y1 ≤ y2
implies
F (x, y1 , z) ≥ F (x, y2 , z),
and
z1 , z2 ∈ X,
z1 ≤ z2
implies
F (x, y, z1 ) ≤ F (x, y, z2 ).
The main results of [3] are:
Theorem 1.1 ([3]). Let (X, ≤) be a partially ordered set and (X, d) be a
complete metric space. Let F : X × X × X → X be a continuous mapping such
that F has the mixed monotone property. Assume that there exist j, k, l ∈ [0, 1)
with j + k + l < 1 such that
d(F (x, y, z), F (u, v, w)) ≤ jd(x, u) + kd(y, v) + ld(z, w)
(1)
for all x, y, z, u, v, w ∈ X with x ≥ u, y ≤ v, and z ≥ w. If there exist
x0 , y0 , z0 ∈ X such that x0 ≤ F (x0 , y0 , z0 ), y0 ≥ F (y0 , x0 , y0 ), and z0 ≤
F (z0 , y0 , x0 ), then F has a tripled fixed point.
Theorem 1.2 ([3]). Let (X, ≤) be a partially ordered set and (X, d) be a
complete metric space. Let F : X ×X ×X → X be a mapping having the mixed
monotone property. Assume that there exist j, k, l ∈ [0, 1) with j + k + l < 1
such that
d(F (x, y, z), F (u, v, w)) ≤ jd(x, u) + kd(y, v) + ld(z, w)
TRIPLED AND COINCIDENCE FIXED POINT THEOREMS
181
for all x, y, z, u, v, w ∈ X with x ≥ u, y ≤ v, and z ≥ w. Assume that X has
the following properties:
i. if a non-decreasing sequence xn → x, then xn ≤ x for all n ∈ N,
ii. if a non-increasing sequence yn → y, then yn ≥ y for all n ∈ N.
If there exist x0 , y0 , z0 ∈ X such that x0 ≤ F (x0 , y0 , z0 ), y0 ≥ F (y0 , x0 , y0 ),
and z0 ≤ F (z0 , y0 , x0 ), then F has a tripled fixed point.
Also, Borcut and Berinde [4] introduced the concept of a tripled coincidence
point of mappings F : X × X × X → X and g : X → X.
Definition 1.3 ([4]). Let X be a nonempty set. Let F : X × X × X → X
and g : X → X be two mappings. An element (x, y, z) ∈ X × X × X is called
a tripled coincidence point of F and g if
F (x, y, z) = gx, F (y, x, y) = gy
and F (z, y, x) = gz.
Definition 1.4 ([4]). Let (X, ≤) be a partially ordered set and F : X × X ×
X → X be a mapping. We say that F has the mixed g-monotone property
if F (x, y, z) is monotone non-decreasing in x and z and is monotone nonincreasing in y; that is, for any x, y, z ∈ X,
x1 , x2 ∈ X,
y1 , y2 ∈ X,
gx1 ≤ gx2
implies
F (x1 , y, z) ≤ F (x2 , y, z),
gy1 ≤ gy2
implies
F (x, y1 , z) ≥ F (x, y2 , z),
and
z1 , z2 ∈ X,
gz1 ≤ gz2
implies
F (x, y, z1 ) ≤ F (x, y, z2 ).
The main results of [4] is:
Theorem 1.3 ([4]). Let (X, ≤) be a partially ordered set and (X, d) be a
complete metric space. Let F : X × X × X → X and g : X → X such that
F has the mixed g-monotone property. Assume that there exist j, k, l ∈ [0, 1)
with j + k + l < 1 such that
d(F (x, y, z), F (u, v, w)) ≤ jd(gx, gu) + kd(gy, gv) + ld(gz, gw)
(2)
for all x, y, z, u, v, w ∈ X with gx ≥ gu, gy ≤ gv, and gz ≥ gw. Syppose that
F (X × X × X ⊆ gX, g is continuous and commute with F and also suppose
either
(a) F is continuous or
(b) X has the following property:
TRIPLED AND COINCIDENCE FIXED POINT THEOREMS
182
(i) if a non-decreasing sequence xn → x, then xn ≤ x for all n,
(ii) if a non-increasing sequence yn → y, then yn ≥ y for all n.
If there exist x0 , y0 , z0 ∈ X such that gx0 ≤ F (x0 , y0 , z0 ), gy0 ≥ F (y0 , x0 , y0 ),
and gz0 ≤ F (z0 , y0 , x0 ), then F and g a tripled coincidence point.
By following Matkowski [26], we let Φ be the set of all non-decreasing
functions φ : [0, +∞) → [0, +∞) such that lim φn (t) = 0, for all t > 0,
n→+∞
where φn denotes the n-th iterate of φ. Then its an easy matter to show that:
(1) φ(t) < t for all t > 0,
(2) φ(0) = 0.
If φ ∈ Φ, then we call φ to be a φ-map.
In this paper, we utilize the notion of φ-map to prove a number of tripled
fixed and coincidence point results for two mapping F : X × X × X → X and
g : X → X in ordered metric spaces. Our results generalize Theorems 1.1 and
1.3. Also, we support our results by introducing a nontrivial example satisfying
our main results and doesn’t satisfy the conditions 1 and 2 of Theorems 1.1
and 1.3 respectively.
2
Main Results
Our first result is the following.
Theorem 2.1. Let (X, ≤) be a partially ordered set and (X, d) be a metric
space. Let F : X × X × X → X, g : X → X be two mappings. Suppose the
following
1. gX is a complete subspace of X,
2. F (X × X × X) ⊆ gX,
3. F has the mixed g-monotone property,
4. F is continuous and
5. g is continuous and commute with F .
Assume that there exists φ ∈ Φ such that
n
d(F (x, y, z), F (u, v, w)) ≤ φ max d(gx, gu), d(gy, gv), d(gz, gw),
o
d(F (x, y, z), gx), d(F (z, y, x), gz), d(F (u, v, w), gu), d(F (w, v, u), gw) ,
(3)
for all x, y, z, u, v, w ∈ X with x ≥ u, y ≤ v, and z ≥ w. If there exist
x0 , y0 , z0 ∈ X such that gx0 ≤ F (x0 , y0 , z0 ), gy0 ≥ F (y0 , x0 , y0 ), and gz0 ≤
F (z0 , y0 , x0 ), then F and g have a tripled coincidence point.
TRIPLED AND COINCIDENCE FIXED POINT THEOREMS
183
Proof. Suppose there exist x0 , y0 , z0 ∈ X such that gx0 ≤ F (x0 , y0 , z0 ), gy0 ≥
F (y0 , x0 , y0 ), and gz0 ≤ F (z0 , y0 , x0 ). Define gx1 = F (x0 , y0 , z0 ), gy1 =
F (y0 , x0 , y0 ) and gz1 = F (z0 , y0 , x0 ). Then gx0 ≤ gx1 , gy0 ≥ gy1 and
gz0 ≤ gz1 . Again, define gx2 = F (x1 , y1 , z1 ), gy2 = F (y1 , x1 , y1 ) and gz2 =
F (z1 , y1 , x1 ). Since F has the mixed g-monotone property, we have gx0 ≤
gx1 ≤ gx2 , gy2 ≤ gy1 ≤ gy0 and gz0 ≤ gz1 ≤ gz2 . Continuing this process, we
can construct three sequences (gxn ), (gyn ) and (gzn ) in X such that
gxn = F (xn−1 , yn−1 , zn−1 ) ≤ gxn+1 = F (xn , yn , zn ),
gyn+1 = F (yn , xn , yn ) ≤ gyn = F (yn−1 , xn−1 , yn−1 ),
and
gzn = F (zn−1 , yn−1 , xn−1 ) ≤ gzn+1 = F (zn , yn , xn ).
If, for some integer n, we have (gxn+1 , gyn+1 , gzn+1 ) = (gxn , gyn , gzn ), then
F (xn , yn , zn ) = gxn , F (yn , xn , yn ) = gyn , and F (zn , yn , xn ) = gzn ; that is,
(xn , yn , zn ) is a tripled coincidence point of F . Thus we shall assume that
(gxn+1 , gyn+1 , gzn+1 ) 6= (gxn , gyn , gzn ) for all n ∈ N; that is, we assume that
either gxn+1 6= gxn or gyn+1 6= gyn or gzn+1 6= gzn . For any n ∈ N, we have
d(gxn+1 , gxn )
= d(F (xn , yn , zn ), F (xn−1 , yn−1 , zn−1 )
n
≤ φ max d(gxn , gxn−1 ), d(gyn , gyn−1 ), d(gzn , gzn−1 ),
d(F (xn , yn , zn ), gxn ), d(F (zn , yn , xn ), gzn ),
d(F (xn−1 , yn−1 , zn−1 ), gxn−1 )), d(F (zn−1 , yn−1 , xn−1 ), gzn−1 ))
n
= φ max d(gxn , gxn−1 ), d(gyn , gyn−1 ), d(gzn , gzn−1 ),
o
d(gxn+1 , gxn ), d(gzn+1 , gzn )
n
≤ φ max d(gxn , gxn−1 ), d(gyn , gyn−1 ), d(gzn , gzn−1 ),
o
d(gxn+1 , gxn ), d(gyn+1 , gyn ), d(gzn+1 , gzn ) ,
o
(4)
TRIPLED AND COINCIDENCE FIXED POINT THEOREMS
184
d(gyn , gyn+1 )
= d(F (yn−1 , xn−1 , yn−1 ), F (yn , xn , yn )
n
≤ φ max d(gyn−1 , gyn ), d(gxn−1 , gxn ), d(F (yn−1 , xn−1 , yn−1 ), gyn−1 ),
o
d(F (yn , xn , yn ), gyn )
= φ(max{d(gyn−1 , gyn ), d(gxn−1 , gxn ), d(yn+1 , yn )})
n
≤ φ max d(gxn , gxn−1 ), d(gyn , gyn−1 ), d(gzn , gzn−1 ), d(gxn+1 , gxn ),
o
d(gyn+1 , gyn ), d(gzn+1 , gzn ) ,
(5)
and
d(gzn+1 , gzn )
=
≤
d(F (zn , yn , xn ), F (zn−1 , yn−1 , xn−1 )
n
φ max d(gzn , gzn−1 ), d(gyn , gyn−1 ), d(gxn , gxn−1 ),
d(F (zn , yn , xn ), gzn ), d(F (xn , yn , zn ), gxn ),
=
≤
d(F (zn−1 , yn−1 , xn−1 ), gzn−1 )), d(F (xn−1 , yn−1 , zn−1 ), gxn−1 ))
n
φ max d(gxn , gxn−1 ), d(gyn , gyn−1 ), d(gzn , gzn−1 ),
o
d(gxn+1 , gxn ), d(gzn+1 , gzn )
n
φ max d(gxn , gxn−1 ), d(gyn , gyn−1 ), d(gzn , gzn−1 ),
o
d(gxn+1 , gxn ), d(gyn+1 , gyn ), d(gzn+1 , gzn ) ,
o
(6)
From (4), (5), and (6), it follows that
max{d(gxn+1 , gxn ), d(gyn , gyn+1 ), d(gzn+1 , gzn )}
n
≤ φ max d(gxn , gxn−1 ), d(gyn , gyn−1 ), d(gzn , gzn−1 ),
o
d(gxn+1 , gxn ), d(gyn+1 , gyn ), d(gzn+1 , gzn ) .
If
max d(gxn , gxn−1 ), d(gyn , gyn−1 ), d(gzn , gzn−1
),
d(gx
,
gx
),
d(gy
,
gy
),
d(gz
,
gz
)
n
n+1
n
n+1
n
n+1
= max d(gxn+1 , gxn ), d(gyn , gyn+1 ), d(gzn+1 , gzn ) ,
(7)
TRIPLED AND COINCIDENCE FIXED POINT THEOREMS
185
then from (7) we have
max{d(gxn+1 , gxn ), d(gyn , gyn+1 ), d(gzn+1 , gzn )}
≤ φ max d(gxn , gxn−1 ), d(gyn , gyn−1 ), d(gzn , gzn−1 )
< max{d(gxn+1 , gxn ), d(gyn , gyn+1 ), d(gzn+1 , gzn )},
a contradiction. Thus (7) becomes
max{d(gxn+1 , gxn ), d(gyn , gyn+1 ), d(gzn+1 , gzn )}
≤ φ max d(gxn , gxn−1 ), d(gyn , gyn−1 ), d(gzn , gzn−1 ) .
(8)
By repeating (8) n-times, we get that
max{d(gxn+1 , gxn ), d(gyn , gyn+1 ), d(gzn+1 , gzn )}
≤ φ(max{d(gxn , gxn−1 ), d(gyn , gyn−1 ), d(gzn , gzn−1 )})
≤ φ2 (max{d(gxn−1 , gxn−2 ), d(gyn−1 , gyn−2 ), d(gzn−1 , gzn−2 )})
..
.
≤ φn (max{d(gx1 , gx0 ), d(gy1 , gy0 ), d(gz1 , gz0 )}).
(9)
Now, we shall show that (gxn ), (gyn ), and (gzn ) are Cauchy sequence in X.
Let > 0. Since
lim φn (max{d(gx1 , gx0 ), d(gy1 , gy0 ), d(gz1 , gz0 )}) = 0
n→+∞
and > φ(), there exist n0 ∈ N such that
φn (max{d(gx1 , gx0 ), d(gy1 , gy0 ), d(gz1 , gz0 )}) < − φ(),
for all n ≥ n0 .
This implies that,
max{d(gxn+1 , gxn ), d(gyn , gyn+1 ), d(gzn+1 , gzn )} < − φ(),
(10)
for all n ≥ n0 .
For m, n ∈ N, we will prove by induction on m that
max{d(gxn , gxm ), d(gyn , gym ), d(gzn , gzm )} < ,
(11)
for all m ≥ n ≥ n0 .
For m = n + 1, (11) is true by using (10) and noting that − φ() < .
Now suppose that (11) holds for m = k; that is,
max{d(gxn , gxk ), d(gyn , gyk ), d(gzn , gzk )} < .
(12)
TRIPLED AND COINCIDENCE FIXED POINT THEOREMS
186
For m = k + 1, we have
d(gxn , gxk+1 )
≤
d(gxn , gxn+1 ) + d(gxn+1 , gxk+1 )
≤
− φ() + d(F (xk , yk , zk ), F (xn , yn , zn ))
n
− φ() + φ max d(gxk , gxn ), d(gyk , gyn ), d(gzk , gzn ),
≤
d(F (xk , yk , zk ), gxk ), d(F (zk , yk , xk ), gzk ),
o
d(F (xn , yn , zn ), gxn ), d(F (zn , yn , xn ), gzn )
n
= − φ() + φ max d(gxn , gxk ), d(gyn , gyk ), d(gzn , gzk ), d(gxn+1 , gxn ),
o
d(gzn+1 , gzn ), d(gxk+1 , gxk ), d(gzk+1 , gzk )
n
≤ − φ() + φ max d(gxn , gxk ), d(gyn , gyk ), d(gzn , gzk ), d(gxn+1 , gxn ),
d(gyn+1 , gyn ), d(gzn+1 , gzn ), d(gxk+1 , gxk ),
o
d(gyk+1 , gyk ), d(gzk+1 , gzk ) .
(13)
Using (10) and (12), we have
n
max d(gxn , gxk ), d(gyn , gyk ), d(gzn , gzk ), d(gxn+1 , gxn ), d(gyn+1 , gyn ),
o
d(gzn+1 , gzn ), d(gxk+1 , gxk ), d(gyk+1 , gyk ), d(gzk+1 , gzk )
≤ max{, − φ()} = .
(14)
From (13), (14) and the properties of φ, we obtain
d(gxn , gxn+1 ) < − φ() + φ() = .
Similarly, we show that
d(gyn , gyk+1 ) < ,
and
d(gzn , gzk+1 ) < .
Hence, we have
max{d(gxn , gxk+1 ), d(gyn , gyk+1 ), d(gzn , gzk+1 )} < .
Thus (11) holds for all m ≥ n ≥ n0 . Hence (gxn ), (gyn ) and (gzn ) are Cauchy
sequences in gX.
TRIPLED AND COINCIDENCE FIXED POINT THEOREMS
187
Since gX is complete, there exist p, q, r ∈ gX such that (gxn ), (gyn ) and
(gzn ) converge to p, q, and r respectively. Choose x, y, z ∈ X such that
p = gx, q = gy and r = gz. Now, we show that (p, q, r) is a coincidence point
of F . Since F , g are commute and F is continuous, we have
ggxn+1 = g(F (xn , yn , zn )) = F (gxn , gyn , gzn ) → F (p, q, r).
Also, since g is continuous and gxn → p, we have ggxn → gp. By uniqueness
of limit, we get F (p, q, r) = gp. Similarly, we show that gq = F (q, p, q) and
gr = F (r, q, p). So (p, q, r) is a tripled coincidence point of F and g.
By taking φ(t) = kt, k ∈ [0, 1) in Theorem 2.1, we have the following:
Corollary 2.1. Let (X, ≤) be a partially ordered set and (X, d) be a metric
space. Let F : X × X × X → X, g : X → X be two mappings. Suppose the
following:
1. gX is a complete subspace of X,
2. F (X × X × X) ⊆ gX,
3. F has the mixed g-monotone property,
4. F is continuous and
5. g is continuous and commute with F .
Assume that there exists k ∈ [0, 1) such that
d(F (x, y, z), F (u, v, w)) ≤ k max d(gx, gu), d(gy, gv), d(gz, gw), d(F (x, y, z), gx), d(F (z, y, x), gz), d(F (u, v, w), gu), d(F (w, v, u), gw) ,
for all x, y, z, u, v, w ∈ X with x ≥ u, y ≤ v, and z ≥ w. If there exist
x0 , y0 , z0 ∈ X such that gx0 ≤ F (x0 , y0 , z0 ), gy0 ≥ F (y0 , x0 , y0 ), and gz0 ≤
F (z0 , y0 , x0 ), then F and g have a tripled coincidence point.
As a consequence of Corollary 2.1, we have the following:
Corollary 2.2. Let (X, ≤) be a partially ordered set and (X, d) be a metric
space. Let F : X × X × X → X, g : X → X be two mappings. Suppose the
following
1. gX is a complete subspace of X,
2. F (X × X × X) ⊆ gX,
3. F has the mixed g-monotone property,
4. F is continuous and
5. g is continuous and commute with F .
P7
Assume that there exist a1 , a2 , a3 , a4 , a5 , a6 , a7 ∈ [0, 1) with i=1 ai < 1
such that
d(F (x, y, z), F (u, v, w)) ≤ a1 d(gx, gu) + a2 d(gy, gv) + a3 d(gz, gw)
+a4 d(F (x, y, z), gx) + a5 d(F (z, y, x), gz)
+a6 d(F (u, v, w), gu) + a7 d(F (w, v, u), gw),
TRIPLED AND COINCIDENCE FIXED POINT THEOREMS
188
for all x, y, z, u, v, w ∈ X with x ≥ u, y ≤ v, and z ≥ w. If there exist
x0 , y0 , z0 ∈ X such that gx0 ≤ F (x0 , y0 , z0 ), gy0 ≥ F (y0 , x0 , y0 ), and gz0 ≤
F (z0 , y0 , x0 ), then F and g have a tripled coincidence point.
By taking g = iX (the identity mapping on X) in Theorem 2.1, Corollary
2.1 and Corollary 2.2, we have the following:
Corollary 2.3. Let (X, ≤) be a partially ordered set and (X, d) be a complete
metric space. Let F : X × X × X → X be a continuous mapping having the
mixed monotone property. Assume that there exists φ ∈ Φ such that
d(F (x, y, z), F (u, v, w)) ≤ φ max d(x, u), d(y, v), d(z, w), d(F
(x, y, z), x),
d(F (z, y, x), z), d(F (u, v, w), u), d(F (w, v, u), w) ,
for all x, y, z, u, v, w ∈ X with x ≥ u, y ≤ v, and z ≤ w. If there exist
x0 , y0 , z0 ∈ X such that x0 ≤ F (x0 , y0 , z0 ), y0 ≥ F (y0 , x0 , y0 ), and z0 ≤
F (z0 , y0 , x0 ), then F has a tripled fixed point.
Corollary 2.4. Let (X, ≤) be a partially ordered set and (X, d) be a complete
metric space. Let F : X × X × X → X be a continuous mapping having the
mixed monotone property. Assume that there exists k ∈ [0, 1) such that
d(F (x, y, z), F (u, v, w)) ≤ k max d(x, u), d(y, v), d(z, w), d(F
(x, y, z), x),
d(F (z, y, x), z), d(F (u, v, w), u), d(F (w, v, u), w) ,
for all x, y, z, u, v, w ∈ X with x ≥ u, y ≤ v, and z ≤ w. If there exist
x0 , y0 , z0 ∈ X such that x0 ≤ F (x0 , y0 , z0 ), y0 ≥ F (y0 , x0 , y0 ), and z0 ≤
F (z0 , y0 , x0 ), then F has a tripled fixed point.
Corollary 2.5. Let (X, ≤) be a partiallyy ordered set and (X, d) be a complete
metric space. Let F : X × X × X → X be a continuous mapping having the
mixed monotone
P7 property. Assume that there exists a1 , a2 , a3 , a4 , a5 , a6 , a7 in
[0, 1) with i=1 ai < 1 such that
d(F (x, y, z), F (u, v, w)) ≤ a1 d(x, u) + a2 d(y, v) + a3 d(z, w)+
a4 d(F (x, y, z), x) + a5 d(F (z, y, x), z) + a6 d(F (u, v, w), u) + a7 d(F (w, v, u), w),
for all x, y, z, u, v, w ∈ X with x ≥ u, y ≤ v, and z ≤ w. If there exist
x0 , y0 , z0 ∈ X such that x0 ≤ F (x0 , y0 , z0 ), y0 ≥ F (y0 , x0 , y0 ), and z0 ≤
F (z0 , y0 , x0 ), then F has a tripled fixed point.
By adding an additional hypotheses, the continuity of F and g in Theorem
2.1 can be dropped.
TRIPLED AND COINCIDENCE FIXED POINT THEOREMS
189
Theorem 2.2. Let (X, ≤) be a partially ordered set and (X, d) be a complete
metric space. Let F : X × X × X → X and g : X → X be two mappings.
Suppose that there exists φ ∈ Φ such that
d(F (x, y, z), F (u, v, w)) ≤ φ max d(gx, gu), d(gy, gv), d(gz, gw),
d(F (x, y, z), gx), d(F (z, y, x), gz), d(F (u, v, w), gu), d(F (w, v, u), gw) ,
(15)
for all x, y, z, u, v, w ∈ X with gx ≥ gu, gy ≤ gv, and gz ≤ gw. Suppose the
following:
1. gX is a complete subspace of X,
2. F (X × X × X) ⊆ gX,
3. F has the mixed g-monotone property,
4. F and g are commute.
Also, assume that X has the following properties:
i. if a non-decreasing sequence xn → x, then xn ≤ x for all n ∈ N,
ii. if a non-increasing sequence yn → y, then yn ≥ y for all n ∈ N.
If there exist x0 , y0 , z0 ∈ X such that gx0 ≤ F (x0 , y0 , z0 ),
gy0 ≥ F (y0 , x0 , y0 ), and gz0 ≤ F (z0 , y0 , x0 ), then F and g have a tripled
coincidence point.
Proof. By following the same process in Theorem 2.1, we construct three
Cauchy sequences (gxn ), (gyn ) and (gzn ) in gX with
gx1 ≤ gx2 ≤ . . . ≤ gxn ≤ . . . ,
gy1 ≥ gy2 ≥ . . . ≥ gyn ≥ . . . ,
and
gz1 ≤ gz2 ≤ . . . ≤ gzn ≤ . . .
such that gxn → p = gx ∈ gX, gyn → q = gy ∈ gX, and gzn → r = gz ∈ gX,
where x, y, z ∈ X. By the hypotheses on X, we have gxn ≤ gx, gyn ≥ gy and
gzn ≤ gz for all n ∈ N. From (15), we have
d(F (x, y, z), xn+1 )
= d(F (x, y, z), F (xn , yn , zn ))
n
≤ φ max d(x, xn ), d(y, yn ), d(z, zn ), d(F (x, y, z), gx), d(F (z, y, x), gz),
o
d(F (xn , yn , zn ), gxn ), d(F (zn , yn , xn ), gzn ) ,
TRIPLED AND COINCIDENCE FIXED POINT THEOREMS
=
≤
190
n
φ max d(gx, gxn ), d(gy, gyn ), d(gz, gzn ), d(F (x, y, z), gx),
o
d(F (z, y, x), gz), d(gxn+1 , gxn ), d(gzn+1 , gzn ) ,
n
φ max d(gx, gxn ), d(gy, gyn ), d(gz, gzn ), d(F (x, y, z), gx),
d(F (z, y, x), gz), d(F (y, x, y), gy), d(gxn+1 , gxn ),
o
d(gyn+1 , gyn ), d(gzn+1 , gzn ) ,
(16)
d(yn+1 , F (y, x, y))
= d(F (yn , xn , yn ), F (y, x, y))
n
o
≤ φ max d(yn , y), d(xn , x), d(F (yn , xn , yn ), gyn ), d(F (y, x, y), gy)
n
o
= φ max d(gyn , gy), d(gxn , gx), d(gyn+1 , gyn ), d(F (y, x, y), gy)
n
≤ φ max d(gx, gxn ), d(gy, gyn ), d(gz, gzn ),
d(F (x, y, z), gx), d(F (z, y, x), gz), d(F (y, x, y), gy),
o
d(gxn+1 , gxn ), d(gyn+1 , gyn ), d(gzn+1 , gzn )
(17)
and
d(F (z, y, x), zn+1 )
= d(F (z, y, x), F (zn , yn , xn ))
n
≤ φ max d(z, zn ), d(y, yn ), d(x, xn ), d(F (z, y, x), gz), d(F (x, y, z), gx),
o
d(F (zn , yn , xn ), gzn ), d(F (xn , yn , zn ), gxn ) ,
n
≤ φ max d(gz, gzn ), d(gy, gyn ), d(gx, gxn ), d(F (z, y, x), gz),
o
d(F (x, y, z), gx), d(gzn+1 , gzn ), d(gxn+1 , gxn ) ,
n
≤ φ max d(gx, gxn ), d(gy, gyn ), d(gz, gzn ), d(F (x, y, z), gx),
d(F (z, y, x), gz), d(F (y, x, y), gy), d(gxn+1 , gxn ),
o
d(gyn+1 , gyn ), d(gzn+1 , gzn ) .
(18)
TRIPLED AND COINCIDENCE FIXED POINT THEOREMS
191
From (16)-(18), we have
≤
max{d(F (x, y, z), xn+1 ), d(yn+1 , F (y, x, y)), d(F (z, y, x), zn+1 )}
n
φ max d(gx, gxn ), d(gy, gyn ), d(gz, gzn ), d(F (x, y, z), gx),
d(F (z, y, x), gz), d(F (y, x, y), gy), d(gxn+1 , gxn ),
o
d(gyn+1 , gyn ), d(gzn+1 , gzn ) .
(19)
Our claim is:
max{d(F (x, y, z), gx), d(gy, F (y, x, y)), d(F (z, y, x), gz)} = 0.
To prove our claim, suppose that
max{d(F (x, y, z), gx), d(gy, F (y, x, y)), d(F (z, y, x), gz)} =
6 0.
Let
= max{d(F (x, y, z), gx), d(gy, F (y, x, y)), d(F (z, y, x), gz)}.
Since > 0, d(gx, gxn ) → 0, d(gy, gyn ) → 0, d(gz, gzn ) → 0, d(gxn , gxn+1 ) →
0, d(gyn , gyn+1 ) → 0, and d(gzn , gzn+1 ) → 0, we choose n0 ∈ N such that
for all n ≥ n0 ,
2
d(y, yn ) < for all n ≥ n0 ,
2
d(z, zn ) < for all n ≥ n0 ,
2
d(xn , xn+1 ) < for all n ≥ n0 ,
2
d(yn , yn+1 ) < e for all n ≥ n0 ,
2
d(x, xn ) <
and
d(zn , zn+1 ) <
for all n ≥ n0 .
2
Thus (19) becomes
max{d(F (x, y, z), xn+1 ), d(yn+1 , F (y, x, y)), d(F (z, y, x), zn+1 )}
n
o
, d(F (x, y, z), gx), d(F (z, y, x), gz), d(F (y, x, y), gy)
≤ φ max
n2
o
= φ max d(F (x, y, z), gx), d(F (z, y, x), gz), d(F (y, x, y), gy)
(20)
for all n ≥ n0 .
TRIPLED AND COINCIDENCE FIXED POINT THEOREMS
192
Letting n → +∞ in (20) it follows that
max{d(F (x, y, z), gx), d(gy, F (y, x, y)), d(F (z, y, x), gz)}
n
o
≤ φ max d(F (x, y, z), gx), d(F (z, y, x), gz), d(F (y, x, y), gy)
< max{d(F (x, y, z), gx), d(gy, F (y, x, y)), d(F (z, y, x), gz)},
a contradiction. Therefore
max{d(F (x, y, z), gx), d(gy, F (y, x, y)), d(F (z, y, x), gz)} = 0
and hence d(F (x, y, z), gx) = 0, d(gy, F (y, x, y)) = 0 and d(F (z, y, x), gz) = 0.
Thus F (x, y, z) = gx, F (y, x, y) = gy and F (z, y, x) = gz; that is (x, y, z) is a
tripled fixed point of F and g.
By taking φ(t) = kt, k ∈ [0, 1) in Theorem 2.2, we have the following:
Corollary 2.6. Let (X, ≤) be a partially ordered set and (X, d) be a complete
metric space. Let F : X × X × X → X and g : X → X be two mappings.
Suppose that there exists k ∈ [0, 1) such that
≤
d(F (x, y, z), F (u, v, w))
k max d(gx, gu), d(gy, gv), d(gz, gw), d(F (x, y, z), gx),
d(F (z, y, x), gz), d(F (u, v, w), gu), d(F (w, v, u), gw) ,
for all x, y, z, u, v, w ∈ X with gx ≥ gu, gy ≤ gv, and gz ≤ gw. Suppose the
following:
1. gX is a complete subspace of X,
2. F (X × X × X) ⊆ gX,
3. F has the mixed g-monotone property,
4. F and g are commute.
Also, assume that X has the following properties:
i. if a non-decreasing sequence xn → x, then xn ≤ x for all n ∈ N,
ii. if a non-increasing sequence yn → y, then yn ≥ y for all n ∈ N.
If there exist x0 , y0 , z0 ∈ X such that gx0 ≤ F (x0 , y0 , z0 ),
gy0 ≥ F (y0 , x0 , y0 ), and gz0 ≤ F (z0 , y0 , x0 ), then F and g have a tripled
coincidence point.
As a consequence result of Corollary 2.6, we have
Corollary 2.7. Let (X, ≤) be a partially ordered set and (X, d) be a complete
metric space. Let F : X × X × X → X and g : X → X be two mappings.
193
TRIPLED AND COINCIDENCE FIXED POINT THEOREMS
Suppose that there exist a1 , a2 , a3 , a4 , a5 , a6 , a7 ∈ [0, 1) with
that
P7
i=1
ai < 1 such
d(F (x, y, z), F (u, v, w))
≤ a1 d(gx, gu) + a2 d(gy, gv) + a3 d(gz, gw) + a4 d(F (x, y, z), gx) +
a5 d(F (z, y, x), gz) + da6 (F (u, v, w), gu) + a7 d(F (w, v, u), gw),
for all x, y, z, u, v, w ∈ X with gx ≥ gu, gy ≤ gv, and gz ≤ gw. Suppose the
following:
1. gX is a complete subspace of X,
2. F (X × X × X) ⊆ gX,
3. F has the mixed g-monotone property,
4. F and g are commute.
Also, assume that X has the following properties:
i. if a non-decreasing sequence xn → x, then xn ≤ x for all n ∈ N,
ii. if a non-increasing sequence yn → y, then yn ≥ y for all n ∈ N.
If there exist x0 , y0 , z0 ∈ X such that gx0 ≤ F (x0 , y0 , z0 ),
gy0 ≥ F (y0 , x0 , y0 ), and gz0 ≤ F (z0 , y0 , x0 ), then F and g have a tripled
coincidence point.
By taking g = iX (the identity mapping on X) in Theorem 2.2, Corollary
2.6 and Corollary 2.7, we have the following:
Corollary 2.8. Let (X, ≤) be a partially ordered set and (X, d) be a complete
metric space. Let F : X×X×X → X be a mapping having the mixed monotone
property. Assume that there exists φ ∈ Φ such that
n
d(F (x, y, z), F (u, v, w)) ≤ φ max d(x, u), d(y, v), d(z, w), d(F (x, y, z), x),
o
d(F (z, y, x), z), d(F (u, v, w), u), d(F (w, v, u), w)
for all x, y, z, u, v, w ∈ X with x ≥ u, y ≤ v, and z ≤ w. Assume also that X
has the following properties:
i. if a non-decreasing sequence xn → x, then xn ≤ x for all n ∈ N,
ii. if a non-increasing sequence yn → y, then yn ≥ y for all n ∈ N.
If there exist x0 , y0 , z0 ∈ X such that x0 ≤ F (x0 , y0 , z0 ), y0 ≥ F (y0 , x0 , y0 ),
and z0 ≤ F (z0 , y0 , x0 ), then F has a tripled fixed point.
Corollary 2.9. Let (X, ≤) be a partially ordered set and (X, d) be a complete
metric space. Let F : X×X×X → X be a mapping having the mixed monotone
property. Assume that there exists k ∈ [0, 1) such that
n
d(F (x, y, z), F (u, v, w)) ≤ k max d(x, u), d(y, v), d(z, w), d(F (x, y, z), x),
o
d(F (z, y, x), z), d(F (u, v, w), u), d(F (w, v, u), w)
TRIPLED AND COINCIDENCE FIXED POINT THEOREMS
194
for all x, y, z, u, v, w ∈ X with x ≥ u, y ≤ v, and z ≤ w. Assume also that X
has the following properties:
i. if a non-decreasing sequence xn → x, then xn ≤ x for all n ∈ N,
ii. if a non-increasing sequence yn → y, then yn ≥ y for all n ∈ N.
If there exist x0 , y0 , z0 ∈ X such that x0 ≤ F (x0 , y0 , z0 ), y0 ≥ F (y0 , x0 , y0 ),
and z0 ≤ F (z0 , y0 , x0 ), then F has a tripled fixed point.
Corollary 2.10 ([3]). Let (X, ≤) be a partially ordered set and (X, d) be a
complete metric space. Let F : X ×X ×X → X be a mapping having the mixed
monotone property. Assume that there exist a1 , a2 , a3 , a4 , a5 , a6 , a7 ∈ [0, 1)
P7
with i=1 ai < 1 such that
d(F (x, y, z), F (u, v, w)) ≤ a1 d(x, u) + a2 d(y, v) + a3 d(z, w) +
a4 d(F (x, y, z), x) + a5 d(F (z, y, x), z) + a6 d(F (u, v, w), u) + a7 d(F (w, v, u), w),
for all x, y, z, u, v, w ∈ X with x ≥ u, y ≤ v, and z ≥ w. Assume that X has
the following properties:
i. if a non-decreasing sequence xn → x, then xn ≤ x for all n ∈ N,
ii. if a non-increasing sequence yn → y, then yn ≥ y for all n ∈ N.
If there exist x0 , y0 , z0 ∈ X such that x0 ≤ F (x0 , y0 , z0 ), y0 ≥ F (y0 , x0 , y0 ),
and z0 ≤ F (z0 , y0 , x0 ), then F has a tripled fixed point.
Now, we prove a uniqueness theorem for a tripled fixed point.
Theorem 2.3. In addition to the hypotheses of Theorem 2.1 (respectfully
Theorem 2.2) suppose that
[(gx0 ≤ gy0 ) ∧ (gz0 ≤ gy0 )] ∨ [(gy0 ≤ gx0 ) ∧ (gy0 ≤ gz0 )].
Then gx = gy = gz.
Proof. Suppose to the contrary, that is gx 6= gy or gy 6= gz or gx 6= gz. Let
= max{d(gx, gy), d(gx, gz), d(gy, gz)}.
Since > 0, d(gxn , gxn+1 ) → 0, d(gyn , gyn+1 ) → 0 and d(gzn , gzn+1 ) → 0
there exist n0 > 0 such that
for all n ≥ n0 ,
2
d(gyn , gyn+1 ) < for all n ≥ n0
2
d(gxn , gxn+1 ) <
and
d(gzn , gzn+1 ) <
for all n ≥ n0 .
2
TRIPLED AND COINCIDENCE FIXED POINT THEOREMS
195
Without loss of generality, we may assume that gx0 ≤ gy0 and gz0 ≤ gy0 .
By the mixed monotone property of F , we have gxn ≤ gyn and gzn ≤ gyn for
all n ∈ N. Thus by (3), we have
d(gyn+1 , gxn+1 )
= d(F (yn , xn , yn ), F (xn , yn , zn ))
n
≤ φ max d(gyn , gxn ), d(gyn , gzn ), d(F (yn , xn , yn ), gyn ),
o
d(F (xn , yn , zn ), gxn ), d(F (zn , yn , xn ), gzn )
n
≤ φ max d(gyn , gxn ), d(gyn , gzn ), d(gyn+1 ), gyn ),
o
d(gxn+1 , gxn ), d(gzn+1 , gzn )
(21)
and
d(gyn+1 , gzn+1 )
= d(F (yn , xn , yn ), F (zn , yn , xn ))
n
≤ φ max d(gyn , gxn ), d(gyn , gzn ), d(F (yn , xn , yn ), gyn ),
o
d(F (xn , yn , zn ), gxn ), d(F (zn , yn , xn ), gzn )
n
≤ φ max d(gyn , gxn ), d(gyn , gzn ), d(gyn+1 ), gyn ),
o
d(gxn+1 , gxn ), d(gzn+1 , gzn )
By (21) and (22), we have
max{d(gyn+1 , gxn+1 ), d(gyn+1 , gzn+1 )}
n
≤ φ max d(gyn , gxn ), d(gyn , gzn ), d(gyn+1 ), gyn ),
o
d(gxn+1 , gxn ), d(gzn+1 , gzn ) .
For n ≥ n0 , we have
max{d(gyn+1 , gxn+1 ), d(gyn+1 , gzn+1 )}
n
o
≤ φ max d(gyn , gxn ), d(gyn , gzn ),
2
n
o
2
≤ φ max d(gyn−1 , gxn−1 ), d(gyn−1 , gzn−1 ),
2
(22)
TRIPLED AND COINCIDENCE FIXED POINT THEOREMS
n
o
≤ φ3 max d(gyn−2 , gxn−2 ), d(gyn−2 , gzn−2 ),
2
..
.
n
o
.
≤ φn−n0 max d(gyn0 , gxn0 ), d(gyn0 , gzn0 ),
2
196
(23)
On letting n → +∞ in (23) and using the property of φ and the fact that
d is continuous on its variables, we get that max{d(gy, gx), d(gy, gz)} = 0.
Hence gy = gz = gx, a contradiction.
Corollary 2.11. In addition to the hypotheses of Corollary 2.1 (respectfully
Corollary 2.6) suppose that
[(gx0 ≤ gy0 ) ∧ (gz0 ≤ gy0 )] ∨ [(gy0 ≤ gx0 ) ∧ (gy0 ≤ gz0 )].
Then gx = gy = gz.
Now, we introduce an example to support the useability of our results:
Example 2.1. Let X = [0, 1] with usual ordered. Define d : X × X → X by
d(x, y) = |x − y|. Define g : X → X and F : X × X × X → X by gx = 43 x, and
0,
y ≥ min{x, z};
F (x, y, z) =
1
(min{x,
z}
−
y),
y < min{x, z}.
3
Then:
1. gX is a complete subspace of X.
2. F (X × X × X) ⊆ gX.
3. F and g are commute.
4. F has the mixed g-monotone property.
5. For x, y, z, u, v, w ∈ X we have
≤
d(F (x, y, z), F (u, v, w))
8
max d(gx, gu), d(gy, gv), d(gz, gw), d(F (x, y, z), gx),
9
d(F (z, y, x), gx), d(F (u, v, w), gu), d(F (w, v, u), gw)
holds, for all gx ≥ gu, gy ≤ gv and gz ≥ gw.
6. There are no j, k, l ∈ [0, 1) with j + k + l < 1 such that
d(F (x, y, z), F (u, v, w)) ≤ jd(x, u) + kd(y, v) + ld(z, w)
holds for all x ≥ u, y ≤ v and z ≥ w.
7. There are no j, k, l ∈ [0, 1) with j + k + l < 1 such that
d(F (x, y, z), F (u, v, w)) ≤ jd(gx, gu) + kd(gy, gv) + ld(gz, gw)
holds for all gx ≥ gu, gy ≤ gv and gz ≥ gw.
TRIPLED AND COINCIDENCE FIXED POINT THEOREMS
197
Proof. The proof of (1), (2) and (3) are clear. To prove (4), let x, y, z ∈ X. To
show that F (x, y, z) is monotone g-non-decreasing in x, let x1 , x2 ∈ X with
gx1 ≤ gx2 . Then 43 x1 ≤ 34 x2 and hence x1 ≤ x2 .
If y ≥ min{x1 , z}, then F (x1 , y, z) = 0 ≤ F (x2 , y, z).
If y < min{x1 , z}, then
F (x1 , y, z) =
1
1
(min{x1 , z} − y) ≤ (min{x2 , z} − y) = F (x2 , y, z).
3
3
Therefore, F (x, y, z) is monotone g-non-decreasing in x. Similarly, we may
show that F (x, y, z) is monotone g-non-decreasing in z and monotone g-nonincreasing in y. To prove (5), given x, y, z, u, v, w ∈ X with gx ≥ gu, gy ≤ gv
and gz ≥ gw. Then 43 x ≥ 34 u, 34 y ≤ 34 v and 34 z ≥ 34 w. Hence x ≥ u, y ≤ v
and z ≥ w. So, we have the following cases:
Case 1: y ≥ min{x, z} and v ≥ min{u, w}. Here, we have
≤
d(F (x, y, z), F (u, v, w)) = 0
1
max d(gx, gu), d(gy, gv), d(gz, gw), d(F (x, y, z), gx),
2
d(F (z, y, x), gx), d(F (u, v, w), gu), d(F (w, v, u), gw) .
Case 2: y ≥ min{x, z} and v < min{u, w}. This case is impossible since
y ≤ v < min{u, w} ≤ min{x, z}.
Case 3: y < min{x, z} and v ≥ min{u, w}.
Suppose w ≤ v, then w − y ≤ v − y and hence
min{x, z} − y
≤
z−y =z−w+w−y
≤
z−w+v−y
i
3
4h3
(z − w) + (v − y)
3 4
4
i
4h
(gz − gw) + (gv − gy)
3
4
[d(gz, gw) + d(gv, gy)]
3
8
max{d(gy, gv), d(gz, gw)}
3
8
max d(gx, gu), d(gy, gv), d(gz, gw), d(F (x, y, z), gx),
3
d(F (z, y, x), gx), d(F (u, v, w), gu), d(F (w, v, u), gw) .
=
=
=
≤
≤
TRIPLED AND COINCIDENCE FIXED POINT THEOREMS
Therefore,
=
=
≤
d(F (x, y, z), F (u, v, w))
1
d
(min{x, z} − y), 0
3
1
(min{x, z} − y)
3
8
max d(gx, gu), d(gy, gv), d(gz, gw), d(F (x, y, z), gx),
9
d(F (z, y, x), gx), d(F (u, v, w), gu), d(F (w, v, u), gw) .
Suppose u ≤ v, then u − y ≤ v − y and hence
min{x, z} − y
≤
x−y
=
x−u+u−y
≤
x−u+v−y
i
3
4h3
(x − u) + (u − y)
3 4
4
i
4h
(gx − gu) + (gu − gy)
3
i
4h
d(gx, gu) + d(gv, gy)
3
8
max{d(gx, gu), d(gy, gv))}
3
8
max d(gx, gu), d(gy, gv), d(gz, gw), d(F (x, y, z), gx),
3
d(F (z, y, x), gx), d(F (u, v, w), gu), d(F (w, v, u), gw) .
=
=
=
≤
≤
Therefore,
=
=
≤
d(F (x, y, z), F (u, v, w))
1
d
(min{x, z} − y), 0
3
1
(min{x, z} − y)
3
8
max d(gx, gu), d(gy, gv), d(gz, gw), d(F (x, y, z), gx),
9
d(F (z, y, x), gx), d(F (u, v, w), gu), d(F (w, v, u), gw) .
Case 4: y < min{x, z} and v < min{u, w}.
198
TRIPLED AND COINCIDENCE FIXED POINT THEOREMS
Since x ≥ u and z ≥ w, then min{x, z} ≥ min{u, w}. Thus
=
=
=
d(F (x, y, z), F (u, v, w))
1
1
(min{x, z} − y), (min{u, w} − v)
d
3
3
1 (min{x, z} − min{u, w}) + (v − y)
3
i
1h
(min{x, z} − min{u, w}) + (v − y)
3
If min{u, w} = u, then min{x, z} − min{u, w} ≤ x − u, and hence
≤
=
=
=
≤
≤
d(F (x, y, z), F (u, v, w))
i
1h
(x − u) + (v − y)
3
1 4 h 3
i
3
(x − u) + (v − y)
3 3 4
4
i
4h
(gx − gu) + (gv − gy)
9
i
4h
d(gx, gu) + d(gy, gv)
9
8
max d(gx, gu), d(gy, gv)
9
8
max d(gx, gu), d(gy, gv), d(gz, gw), d(F (x, y, z), gx),
9
d(F (z, y, x), gx), d(F (u, v, w), gu), d(F (w, v, u), gw) .
If min{u, w} = w, then min{x, z} − min{u, w} ≤ z − w, and hence
≤
=
=
=
≤
≤
d(F (x, y, z), F (u, v, w))
i
1h
(z − w) + (v − y)
3
i
1 4 h 3
3
(z − w) + (v − y)
3 3 4
4
i
4h
(gz − gw) + (gv − gy)
9
4
[d(gz, gw) + d(gy, gv)]
9
8
max{d(gy, gv), d(gz, gw)}
9
8
max d(gx, gu), d(gy, gv), d(gz, gw), d(F (x, y, z), gx),
9
d(F (z, y, x), gx), d(F (u, v, w), gu), d(F (w, v, u), gw) .
199
TRIPLED AND COINCIDENCE FIXED POINT THEOREMS
200
To prove (6), suppose there exist j, k, l ∈ [0, 1) with j + k + l < 1 such that
d(F (x, y, z), F (u, v, w)) ≤ jd(x, u) + kd(y, v) + ld(z, w)
holds for all x ≥ u, y ≤ v and z ≥ w. Then
1
1
d(F (1, 0, 1), F (0, 0, 1)) = d( , 0) = ≤ j,
3
3
(24)
1
1
d(F (1, 0, 1), F (1, 1, 1)) = d( , 0) = ≤ k,
3
3
(25)
and
1
1
d(F (1, 0, 1), F (1, 0, 0)) = d( , 0) = ≤ l.
3
3
From (24), (25), and (26), we have j + k + l ≥ 1, a contradiction.
(26)
To prove (7), suppose there exist j, k, l ∈ [0, 1) with j + k + l < 1 such that
d(F (x, y, z), F (u, v, w)) ≤ jd(gx, gu) + kd(gy, gv) + ld(gz, gw)
holds for all gx ≥ gu, gy ≤ gv and gz ≥ gw. Then
1
3
1
d(F (1, 0, 1), F (0, 0, 1)) = d( , 0) = ≤ j,
3
3
4
1
1
3
d(F (1, 0, 1), F (1, 1, 1)) = d( , 0) = ≤ k,
3
3
4
(27)
(28)
and
1
3
1
d(F (1, 0, 1), F (1, 0, 0)) = d( , 0) = ≤ l.
3
3
4
12
From (27), (28), and (29), we have j + k + l ≥ 9 , a contradiction.
(29)
Thus by Theorems 2.1 and 2.3, F and g have a tripled coincidence point.
Here, (0, 0, 0) is the tripled coincidence point of F and g.
Remarks:
1.
[3]).
2.
[4]).
3.
2.7.
4.
5.
Example 2.1 does not satisfy condition 1 of Theorem 1.1 (Theorem 7 of
Example 2.1 does not satisfy condition 2 of Theorem 1.3 (Theorem 4 of
Theorem 1.3 (Theorem 4 of [4]) is a special case of Corollaries 2.2 and
Theorem 1.1 (Theorem 7 of [3]) is a special case of Corollary 2.5.
Theorem 1.2 (Theorem 8 of [3]) is a special case of Corollary 2.10.
TRIPLED AND COINCIDENCE FIXED POINT THEOREMS
201
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Wasfi SHATANAWI,
Department of Mathematics,
Hashemite University, Zarqa-Jordan,
Email: swasfi@hu.edu.jo
Mihai POSTOLACHE,
Department of Mathematics and Informatics,
University Politehnica of Bucharest,
313 Splaiul Independenȩti, Romania.
Email: mihai@mathem.pub.ro
Zead MUSTAFA,
Department of Mathematics,
Hashemite University, Zarqa-Jordan,
Email: zmagablh@hu.edu.jo
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