# Some Inevitable Remarks on “Tripled Fixed Point Theorems for Mixed Monotone Kannan Type Contractive Mappings”

Authors: Hamed H. Alsulami; Antonio-Francisco Roldán-López-de-Hierro; Erdal Karapınar; Stojan Radenović
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

We advise that the proof of Theorem 12 given by Borcut et al. (2014) is not correct, and it cannot be corrected using the same technique. Furthermore, we present some similar results as an approximation to the opened question if that statement is valid.


---

## Body

## 1. Introduction

The definition ofcoupled fixed point was firstly given by Guo and Lakshmikantham in [1]. This concept, in the context of metric space, was reconsidered by Gnana Bhaskar and Lakshmikantham [2] in 2006 and by Lakshmikantham and Ciric [3] in the coincidence case. Later, Karapınar investigated this notion in the context of cone metric space. After that, Berinde and Borcut [4] presented the notion oftripled fixed point obtaining similar results, and the same authors extended their work to the coincidence case in [5] (see also, e.g., [6–9]).Acoupled fixed point of
F
:
X
×
X
→
X is a point (
x
,
y
)
∈
X
2 such that F
(
x
,
y
)
=
x and F
(
y
,
x
)
=
y. In order to ensure existence and uniqueness of coupled fixed points, Bhaskar and Lakshmikantham introduced the concept of mapping having the mixed monotone property. Henceforth, let ≼ be a partial order on X. The mapping F is said to have themixed monotone property (with respect to
≼
) if F
(
x
,
y
) is monotone nondecreasing in x and monotone nonincreasing in y; that is, for any x
,
y
∈
X,
(1)
x
1
,
x
2
∈
X
,
x
1
≼
x
2
⟹
F
(
x
1
,
y
)
≼
F
(
x
2
,
y
)
,
y
1
,
y
2
∈
X
,
y
1
≼
y
2
⟹
F
(
x
,
y
1
)
≽
F
(
x
,
y
2
)
.Inspired by the previous notions, Berinde and Borcut defined the concepts oftripled fixed point andmixed monotone property as follows. Atripled fixed point of
F
:
X
×
X
×
X
→
X is a point (
x
,
y
,
z
)
∈
X
3 such that F
(
x
,
y
,
z
)
=
x, F
(
y
,
x
,
y
)
=
y, and F
(
z
,
y
,
z
)
=
z. The mapping F is said to have themixed monotone property (with respect to
≼
) if F
(
x
,
y
,
z
) is monotone nondecreasing in x and z, and it is monotone nonincreasing in y; that is, for any x
,
y
,
z
∈
X,
(2)
x
1
,
x
2
∈
X
,
x
1
≼
x
2
⟹
F
(
x
1
,
y
,
z
)
≼
F
(
x
2
,
y
,
z
)
,
y
1
,
y
2
∈
X
,
y
1
≼
y
2
⟹
F
(
x
,
y
1
,
z
)
≽
F
(
x
,
y
2
,
z
)
,
y
1
,
y
2
∈
X
,
y
1
≼
y
2
⟹
F
(
x
,
y
,
z
1
)
≼
F
(
x
,
y
2
,
z
)
.
The second equation that defines a tripled fixed point, that is, F
(
y
,
x
,
y
)
=
y, uses the point y twice in the arguments of F. This fact is necessary to ensure the existence of tripled fixed points of a nonlinear contration because, in such a case, the mixed monotone property is applicable.Very recently, as a continuation of their pioneering works in the tripled case, Borcut et al. announced in [10] the following result.Theorem 1 (Borcut et al. [10], Theorem 12).
Let(
X
,
≼
) be a partially ordered set and suppose there is a metric d on X such that (
X
,
d
) is a complete metric space. Let F
:
X
3
→
X be a mapping having the mixed monotone property on X. Assume that there exists a k
∈
[
0,1
) such that
(3)
d
(
F
(
x
,
y
,
z
)
,
F
(
u
,
v
,
w
)
)
≤
k
8
[
d
(
x
,
F
(
x
,
y
,
z
)
)
+
d
(
y
,
F
(
y
,
x
,
y
)
)
+
d
(
z
,
F
(
z
,
y
,
x
)
)
+
d
(
u
,
F
(
u
,
v
,
w
)
)
+
d
(
v
,
F
(
v
,
u
,
v
)
)
+
d
(
w
,
F
(
w
,
v
,
u
)
)
]
for all x
≽
u, y
≼
v, z
≽
w. Also suppose that either(a)
F is continuous, or
(b)
X has the following properties:
(i)
if a nondecreasing sequence{
x
m
}
→
x, then x
m
≼
x for all m;
(ii)
if a nonincreasing sequence{
y
m
}
→
y, then y
m
≽
y for all m.
If there existx
0
,
y
0
,
z
0
∈
X such that
(4)
x
0
≼
F
(
x
0
,
y
0
,
z
0
)
,
y
0
≽
F
(
y
0
,
x
0
,
y
0
)
,
z
0
≼
F
(
z
0
,
y
0
,
x
0
)
,
then F has a tripled fixed point in X; that is, there exist x
,
y
,
z
∈
X such that
(5)
F
(
x
,
y
,
z
)
=
x
,
F
(
y
,
x
,
y
)
=
y
,
F
(
z
,
y
,
x
)
=
z
.This note is to advise that the proof given by the authors of the previous result is not correct, and it cannot be corrected using the same technique. Furthermore, we present some similar results as an approximation to the opened question if the previous theorem is valid.


## 2. A Review of the Incorrect Proof

Let us review the lines of their proof. Based onx
0
,
y
0
,
z
0
∈
X, the authors defined, recursively, for all n
≥
0,
(6)
x
n
+
1
=
F
(
x
n
,
y
n
,
z
n
)
,
y
n
+
1
=
F
(
y
n
,
x
n
,
y
n
)
,
z
n
+
1
=
F
(
z
n
,
y
n
,
x
n
)
,
and they proved that {
x
n
} and {
z
n
} were monotone nondecreasing sequences and {
y
n
} was a monotone nonincreasing sequence in (
X
,
≼
). Then, they defined, for all n
∈
N,
(7)
D
x
n
+
1
=
d
(
x
n
+
1
,
x
n
)
,
D
y
n
+
1
=
d
(
y
n
+
1
,
y
n
)
,
D
z
n
+
1
=
d
(
z
n
+
1
,
z
n
)
,
D
n
+
1
=
D
x
n
+
1
+
D
y
n
+
1
+
D
z
n
+
1
.
Using the contractivity condition (3), they proved that, taking into account that x
n
≽
x
n
-
1, y
n
≼
y
n
-
1, and z
n
≽
z
n
-
1,
(8)
D
x
n
+
1
=
d
(
x
n
+
1
,
x
n
)
=
d
(
F
(
x
n
,
y
n
,
z
n
)
,
F
(
x
n
-
1
,
y
n
-
1
,
z
n
-
1
)
)
≤
k
8
[
d
(
x
n
,
F
(
x
n
,
y
n
,
z
n
)
)
+
d
(
y
n
,
F
(
y
n
,
x
n
,
y
n
)
)
+
d
(
z
n
,
F
(
z
n
,
y
n
,
x
n
)
)
+
d
(
x
n
-
1
,
F
(
x
n
-
1
,
y
n
-
1
,
z
n
-
1
)
)
+
d
(
y
n
-
1
,
F
(
y
n
-
1
,
x
n
-
1
,
y
n
-
1
)
)
+
d
(
z
n
-
1
,
F
(
z
n
-
1
,
y
n
-
1
,
x
n
-
1
)
)
]
=
k
8
[
d
(
x
n
,
x
n
+
1
)
+
d
(
y
n
,
y
n
+
1
)
+
d
(
z
n
,
z
n
+
1
)
+
d
(
x
n
-
1
,
x
n
)
+
d
(
y
n
-
1
,
y
n
)
+
d
(
z
n
-
1
,
z
n
)
]
=
k
8
[
D
x
n
+
D
y
n
+
D
z
n
+
D
x
n
+
1
+
D
y
n
+
1
+
D
z
n
+
1
]
.
Based on this inequality, the authors immediately announced that
(9)
D
y
n
+
1
≤
k
8
[
D
x
n
+
2
D
y
n
+
D
x
n
+
1
+
2
D
y
n
+
1
]
,
(10)
D
z
n
+
1
≤
k
8
[
D
x
n
+
D
y
n
+
D
z
n
+
D
x
n
+
1
+
D
y
n
+
1
+
D
z
n
+
1
]
(see [10, page 4, inequalities (
2.4
) and (
2.5
)]). However, these last two inequalities are false. In fact, we can only prove that
(11)
D
y
n
+
1
=
d
(
y
n
+
1
,
y
n
)
=
d
(
F
(
y
n
,
x
n
,
y
n
)
,
F
(
y
n
-
1
,
x
n
-
1
,
y
n
-
1
)
)
≤
k
8
[
d
(
y
n
,
F
(
y
n
,
x
n
,
y
n
)
)
+
d
(
x
n
,
F
(
x
n
,
y
n
,
x
n
)
)
+
d
(
y
n
,
F
(
y
n
,
x
n
,
y
n
)
)
+
d
(
y
n
-
1
,
F
(
y
n
-
1
,
x
n
-
1
,
y
n
-
1
)
)
+
d
(
x
n
-
1
,
F
(
x
n
-
1
,
y
n
-
1
,
x
n
-
1
)
)
+
d
(
y
n
-
1
,
F
(
y
n
-
1
,
x
n
-
1
,
y
n
-
1
)
)
]
≤
k
8
[
2
d
(
y
n
,
y
n
+
1
)
+
d
(
x
n
,
F
(
x
n
,
y
n
,
x
n
)
)
+
2
d
(
y
n
-
1
,
y
n
)
+
d
(
x
n
-
1
,
F
(
x
n
-
1
,
y
n
-
1
,
x
n
-
1
)
)
]
≤
k
8
[
2
D
y
n
+
1
+
d
(
x
n
,
F
(
x
n
,
y
n
,
x
n
)
)
+
2
D
y
n
+
d
(
x
n
-
1
,
F
(
x
n
-
1
,
y
n
-
1
,
x
n
-
1
)
)
]
.
However, comparing (9) with (11), we notice that D
x
n
+
1
=
d
(
x
n
,
x
n
+
1
)
=
d
(
x
n
,
F
(
x
n
,
y
n
,
z
n
)
) does not necessarily coincide with d
(
x
n
,
F
(
x
n
,
y
n
,
x
n
)
) and, similarly, D
x
n
=
d
(
x
n
-
1
,
x
n
)
=
d
(
x
n
-
1
,
F
(
x
n
-
1
,
y
n
-
1
,
z
n
-
1
)
) is not necessarily equal to d
(
x
n
-
1
,
F
(
x
n
-
1
,
y
n
-
1
,
x
n
-
1
)
). Therefore, inequality (9) cannot be ensured.Exactly in the same way, it can be possible to see that (10) cannot be proved using the contractivity condition (3). In such a case, the proof given by the authors, which decisively used inequalities (9) and (10), is false.


## 3. Some Berinde and Borcut’s Type Tripled Fixed Point Theorems

For the moment, the question about whether Theorem1 is valid is opened. The following results are some approximations to this problem, using contractivity conditions that are inspired in (3). The main aim of this section is to show some results in this line of research using a well-known result by Ćirić [11]. Our technique is based on some very recent works which showed that most of coupled/tripled/quadrupled fixed point results can be reduced to their corresponding unidimensional theorems in different frameworks (see, for instance, [12–18]). Before that, let us introduce some notation and basic results.Given a binary relation≼ on X, let us define
(12)
(
x
,
y
,
z
)
⊑
(
u
,
v
,
w
)
⟺
[
x
≼
u
,
y
≽
v
,
z
≼
w
]
.
If ≼ is a partial order on X, then ⊑ is also a partial order on X
3.Given a metricd on X, let us define d
3
s
,
d
3
m
:
X
3
×
X
3
→
[
0
,
∞
), for all (
x
,
y
,
z
)
,
(
u
,
v
,
w
)
∈
X
3, by
(13)
d
3
s
(
(
x
,
y
,
z
)
,
(
u
,
v
,
w
)
)
=
d
(
x
,
u
)
+
d
(
y
,
v
)
+
d
(
z
,
w
)
,
d
3
m
(
(
x
,
y
,
z
)
,
(
u
,
v
,
w
)
)
=
max
⁡
⁡
{
d
(
x
,
u
)
,
d
(
y
,
v
)
,
d
(
z
,
w
)
}
.
Then d
3
s and d
3
m are metrics on X
3. In addition to this, if d is complete, then d
3
s and d
3
m are also complete.Given a mappingF
:
X
3
→
X, let us denote by T
F
3
:
X
3
→
X
3 the mapping
(14)
T
F
3
(
x
,
y
,
z
)
=
(
F
(
x
,
y
,
z
)
,
F
(
y
,
x
,
y
)
,
F
(
z
,
y
,
x
)
)
∀
(
x
,
y
,
z
)
∈
X
3
.
Notice that a tripled fixed point of F is nothing but a fixed point of T
F
3. If F is d-continuous, then T
F
3 is d
3
s-continuous. Furthermore, if F has the mixed monotone property with respect to ≼, then T
F
3 is nondecreasing with respect to ⊑ (see [12]). We also recall the following result.Definition 2.
Letd be a metric on X and let ≼ be a partial order on X. We will say that (
X
,
d
,
≼
) isregular if it verifies the following two properties:(i)
if a nondecreasing sequence{
x
n
}
→
x, then x
n
≼
x for all n;
(ii)
if a nonincreasing sequence{
y
n
}
→
y, then y
n
≽
y for all n.Lemma 3.
If(
X
,
d
,
≼
) is regular, then (
X
3
,
d
3
s
,
⊑
) is also regular.The first version of the following theorem was given by Ćirić in 1972 (see [11]) in the case of metric spaces that were not necessarily partially ordered. A partially ordered version can be found, for example, in [19]. Our main results will be consequences of the next result.Theorem 4 (see e.g., [19]).
Let(
X
,
≼
) be a partially ordered set and suppose that there is a metric d on X such that (
X
,
d
) is a complete metric space. Let T
:
X
→
X be a nondecreasing mapping and let k
∈
[
0,1
) be such that
(15)
d
(
T
x
,
T
y
)
≤
k
max
⁡
⁡
(
d
(
x
,
T
y
)
+
d
(
y
,
T
x
)
2
d
(
x
,
y
)
,
d
(
x
,
T
x
)
,
d
(
y
,
T
y
)
,
d
(
x
,
T
y
)
+
d
(
y
,
T
x
)
2
)
for all x
,
y
∈
X such that x
≼
y. Also assume that F is continuous or (
X
,
d
,
≼
) is regular. If there exist x
0
∈
X such that x
0
≼
T
x
0, then F has a fixed point.In the following result, we found some terms that play an important role in the contractivity condition (3).Theorem 5.
Let(
X
,
≼
) be a partially ordered set and suppose there is a metric d on X such that (
X
,
d
) is a complete metric space. Let F
:
X
3
→
X be a mapping having the mixed monotone property on X. Suppose that there exists k
∈
[
0,1
) such that
(16)
d
(
F
(
x
,
y
,
z
)
,
F
(
u
,
v
,
w
)
)
≤
k
4
max
⁡
(
(
2
)
-
1
d
(
x
,
u
)
+
d
(
y
,
v
)
+
d
(
z
,
w
)
,
d
(
x
,
F
(
x
,
y
,
z
)
)
+
d
(
z
,
F
(
z
,
y
,
x
)
)
,
d
(
u
,
F
(
u
,
v
,
w
)
)
+
d
(
v
,
F
(
w
,
v
,
u
)
)
,
(
d
(
u
,
F
(
x
,
y
,
z
)
)
+
d
(
w
,
F
(
z
,
y
,
x
)
)
+
d
(
x
,
F
(
u
,
v
,
w
)
)
+
d
(
z
,
F
(
w
,
v
,
u
)
)
)
×
(
2
)
-
1
)
for all x
,
y
,
z
,
u
,
v
,
w
∈
X such that x
≼
u, y
≽
v, and z
≼
w. Also assume that F is continuous or (
X
,
d
,
≼
) is regular. If there exist x
0
,
y
0
,
z
0
∈
X such that
(17)
x
0
≼
F
(
x
0
,
y
0
,
z
0
)
,
y
0
≽
F
(
y
0
,
x
0
,
y
0
)
,
z
0
≼
F
(
z
0
,
y
0
,
x
0
)
,
then F has a tripled fixed point in X; that is, there exist x
,
y
,
z
∈
X such that
(18)
F
(
x
,
y
,
z
)
=
x
,
F
(
y
,
x
,
y
)
=
y
,
F
(
z
,
y
,
x
)
=
z
.Proof.
AsF has the mixed monotone property on X with respect to ≼, it follows that T
F
3 is ⊑-nondecreasing. Let us define, for all x
,
y
,
z
,
u
,
v
,
w
∈
X,
(19)
M
(
(
x
,
y
,
z
)
,
(
u
,
v
,
w
)
)
=
max
⁡
(
(
2
)
-
1
d
3
s
(
(
x
,
y
,
z
)
,
(
u
,
v
,
w
)
)
,
d
3
s
(
(
x
,
y
,
z
)
,
T
F
3
(
x
,
y
,
z
)
)
,
d
3
s
(
(
u
,
v
,
w
)
,
T
F
3
(
u
,
v
,
w
)
)
,
(
(
(
u
,
v
,
w
)
,
T
F
3
(
x
,
y
,
z
)
)
d
3
s
(
(
x
,
y
,
z
)
,
T
F
3
(
u
,
v
,
w
)
)
+
d
3
s
(
(
u
,
v
,
w
)
,
T
F
3
(
x
,
y
,
z
)
)
)
×
(
2
)
-
1
)
=
max
⁡
(
(
2
)
-
1
d
(
x
,
u
)
+
d
(
y
,
v
)
+
d
(
z
,
w
)
,
d
(
x
,
F
(
x
,
y
,
z
)
)
+
d
(
y
,
F
(
y
,
x
,
y
)
)
+
d
(
z
,
F
(
z
,
y
,
x
)
)
,
d
(
u
,
F
(
u
,
v
,
w
)
)
+
d
(
v
,
F
(
v
,
u
,
v
)
)
+
d
(
w
,
F
(
w
,
v
,
u
)
)
,
(
d
(
x
,
F
(
u
,
v
,
w
)
)
+
d
(
y
,
F
(
v
,
u
,
v
)
)
+
d
(
z
,
F
(
w
,
v
,
u
)
)
+
d
(
u
,
F
(
x
,
y
,
z
)
)
+
d
(
v
,
F
(
y
,
x
,
y
)
)
+
d
(
w
,
F
(
z
,
y
,
x
)
)
)
×
(
2
)
-
1
)
.
Assume that x
≼
u, y
≽
v, and z
≼
w; that is, (
x
,
y
,
z
)
⊑
(
u
,
v
,
w
). In this case, the contractivity condition (16) guarantees that
(20)
d
(
F
(
x
,
y
,
z
)
,
F
(
u
,
v
,
w
)
)
≤
k
4
max
⁡
(
(
2
)
-
1
d
(
x
,
u
)
+
d
(
y
,
v
)
+
d
(
z
,
w
)
,
d
(
x
,
F
(
x
,
y
,
z
)
)
+
d
(
z
,
F
(
z
,
y
,
x
)
)
,
d
(
u
,
F
(
u
,
v
,
w
)
)
+
d
(
v
,
F
(
w
,
v
,
u
)
)
,
(
d
(
u
,
F
(
x
,
y
,
z
)
)
+
d
(
w
,
F
(
z
,
y
,
x
)
)
+
d
(
x
,
F
(
u
,
v
,
w
)
)
+
d
(
z
,
F
(
w
,
v
,
u
)
)
(
d
(
u
,
F
(
x
,
y
,
z
)
)
+
d
(
w
,
F
(
z
,
y
,
x
)
)
)
×
(
2
)
-
1
)
≤
k
4
M
(
(
x
,
y
,
z
)
,
(
u
,
v
,
w
)
)
.
Taking into account that z
≼
w, y
≽
v, and x
≼
u, we also find the same upper bound:
(21)
d
(
F
(
z
,
y
,
x
)
,
F
(
w
,
v
,
u
)
)
≤
k
4
max
⁡
(
d
(
z
,
w
)
+
d
(
y
,
v
)
+
d
(
x
,
u
)
,
d
(
z
,
F
(
z
,
y
,
x
)
)
+
d
(
x
,
F
(
x
,
y
,
z
)
)
,
d
(
v
,
F
(
w
,
v
,
u
)
)
+
d
(
u
,
F
(
u
,
v
,
w
)
)
,
(
d
(
w
,
F
(
z
,
y
,
x
)
)
+
d
(
u
,
F
(
x
,
y
,
z
)
)
+
d
(
z
,
F
(
w
,
v
,
u
)
)
+
d
(
x
,
F
(
u
,
v
,
w
)
)
)
×
(
2
)
-
1
)
≤
k
4
M
(
(
x
,
y
,
z
)
,
(
u
,
v
,
w
)
)
.
Furthermore, as v
≼
y, u
≽
x, and v
≼
y,
(22)
d
(
F
(
y
,
x
,
y
)
,
F
(
v
,
u
,
v
)
)
≤
k
4
max
⁡
(
d
(
y
,
v
)
+
d
(
x
,
u
)
+
d
(
y
,
v
)
,
d
(
y
,
F
(
y
,
x
,
y
)
)
+
d
(
y
,
F
(
y
,
x
,
y
)
)
,
d
(
v
,
F
(
v
,
u
,
v
)
)
+
d
(
v
,
F
(
v
,
u
,
v
)
)
,
(
d
(
v
,
F
(
y
,
x
,
y
)
)
+
d
(
v
,
F
(
y
,
x
,
y
)
)
+
d
(
y
,
F
(
v
,
u
,
v
)
)
+
d
(
y
,
F
(
v
,
u
,
v
)
)
)
×
(
2
)
-
1
)
=
k
4
max
⁡
(
(
y
,
F
(
v
,
u
,
v
)
)
d
(
x
,
u
)
+
2
d
(
y
,
v
)
,
2
d
(
y
,
F
(
y
,
x
,
y
)
)
,
2
d
(
v
,
F
(
v
,
u
,
v
)
)
,
d
(
v
,
F
(
y
,
x
,
y
)
)
+
d
(
y
,
F
(
v
,
u
,
v
)
)
)
≤
k
4
2
M
(
(
x
,
y
,
z
)
,
(
u
,
v
,
w
)
)
=
k
2
M
(
(
x
,
y
,
z
)
,
(
u
,
v
,
w
)
)
.
Joining the last three inequalities, we deduce that, for all (
x
,
y
,
z
)
,
(
u
,
v
,
w
)
∈
X
3 such that (
x
,
y
,
z
)
⊑
(
u
,
v
,
w
),
(23)
d
3
s
(
T
F
3
(
x
,
y
,
z
)
,
T
F
3
(
u
,
v
,
w
)
)
=
d
3
s
(
(
F
(
x
,
y
,
z
)
,
F
(
y
,
x
,
y
)
,
F
(
z
,
y
,
x
)
)
,
(
F
(
u
,
v
,
w
)
,
F
(
v
,
u
,
v
)
,
F
(
w
,
v
,
u
)
)
)
=
d
(
F
(
x
,
y
,
z
)
,
F
(
u
,
v
,
w
)
)
+
d
(
F
(
y
,
x
,
y
)
,
F
(
v
,
u
,
v
)
)
+
d
(
F
(
z
,
y
,
x
)
,
F
(
w
,
v
,
u
)
)
=
(
k
4
+
k
4
+
k
2
)
M
(
(
x
,
y
,
z
)
,
(
u
,
v
,
w
)
)
=
k
M
(
(
x
,
y
,
z
)
,
(
u
,
v
,
w
)
)
.
As (
x
0
,
y
0
,
z
0
)
⊑
T
F
3
(
x
0
,
y
0
,
z
0
), Theorem 4 guarantees that T
F
3 has a fixed point; that is, F has a tripled fixed point.Theorem 6.
Let(
X
,
≼
) be a partially ordered set and suppose that there is a metric d on X such that (
X
,
d
) is a complete metric space. Let F
:
X
3
→
X be a mapping having the mixed monotone property on X. Suppose that there exists k
∈
[
0,1
) such that
(24)
d
(
F
(
x
,
y
,
z
)
,
F
(
u
,
v
,
w
)
)
≤
k
max
⁡
⁡
{
d
(
x
,
F
(
x
,
y
,
z
)
)
,
d
(
z
,
F
(
z
,
y
,
x
)
)
,
d
(
u
,
F
(
u
,
v
,
w
)
)
,
d
(
w
,
F
(
w
,
v
,
u
)
)
}
for all x
,
y
,
z
,
u
,
v
,
w
∈
X such that x
≼
u, y
≽
v, and z
≼
w. Also assume that F is continuous or (
X
,
d
,
≼
) is regular. If there exist x
0
,
y
0
,
z
0
∈
X such that
(25)
x
0
≼
F
(
x
0
,
y
0
,
z
0
)
,
y
0
≽
F
(
y
0
,
x
0
,
y
0
)
,
z
0
≼
F
(
z
0
,
y
0
,
x
0
)
,
then F has a tripled fixed point in X; that is, there exist x
,
y
,
z
∈
X such that
(26)
F
(
x
,
y
,
z
)
=
x
,
F
(
y
,
x
,
y
)
=
y
,
F
(
z
,
y
,
x
)
=
z
.Proof.
Following the lines of the previous proof, considerX
3 provided with the metric d
3
m. Assume that x
≼
u, y
≽
v, and z
≼
w; that is, (
x
,
y
,
z
)
⊑
(
u
,
v
,
w
). In this case, the contractivity condition (24) guarantees that
(27)
d
(
F
(
x
,
y
,
z
)
,
F
(
u
,
v
,
w
)
)
≤
k
max
⁡
⁡
{
d
(
x
,
F
(
x
,
y
,
z
)
)
,
d
(
z
,
F
(
z
,
y
,
x
)
)
,
d
(
u
,
F
(
u
,
v
,
w
)
)
,
d
(
w
,
F
(
w
,
v
,
u
)
)
}
,
and the same upper bound is valid for d
(
F
(
z
,
y
,
x
)
,
F
(
w
,
v
,
u
)
). Moreover, as v
≼
y, u
≽
x, and v
≼
y,
(28)
d
(
F
(
v
,
u
,
v
)
,
F
(
y
,
x
,
y
)
)
≤
k
max
⁡
⁡
{
d
(
v
,
F
(
v
,
u
,
v
)
)
,
d
(
y
,
F
(
y
,
x
,
y
)
)
}
.
Therefore(29)
d
3
m
(
T
F
3
(
x
,
y
,
z
)
,
T
F
3
(
u
,
v
,
w
)
)
=
d
3
m
(
(
F
(
x
,
y
,
z
)
,
F
(
y
,
x
,
y
)
,
F
(
z
,
y
,
x
)
)
,
(
F
(
u
,
v
,
w
)
,
F
(
v
,
u
,
v
)
,
F
(
w
,
v
,
u
)
)
(
F
(
x
,
y
,
z
)
,
F
(
y
,
x
,
y
)
,
F
(
z
,
y
,
x
)
)
)
=
max
⁡
⁡
{
d
(
F
(
x
,
y
,
z
)
,
F
(
u
,
v
,
w
)
)
,
d
(
F
(
y
,
x
,
y
)
,
F
(
v
,
u
,
v
)
)
,
d
(
F
(
z
,
y
,
x
)
,
F
(
w
,
v
,
u
)
)
}
≤
k
max
⁡
⁡
{
d
(
x
,
F
(
x
,
y
,
z
)
)
,
d
(
y
,
F
(
y
,
x
,
y
)
)
,
d
(
z
,
F
(
z
,
y
,
x
)
)
,
d
(
u
,
F
(
u
,
v
,
w
)
)
,
d
(
v
,
F
(
v
,
u
,
v
)
)
,
d
(
w
,
F
(
w
,
v
,
u
)
)
}
=
k
max
⁡
⁡
{
d
3
m
(
(
x
,
y
,
z
)
,
T
F
3
(
x
,
y
,
z
)
)
,
d
(
(
u
,
v
,
w
)
,
T
F
3
(
u
,
v
,
w
)
)
}
.
Theorem 4 guarantees that T
F
3 has a fixed point; that is, F has a tripled fixed point.The following particularization is also inspired by some Berinde and Borcut’s results.Corollary 7.
Let(
X
,
≼
) be a partially ordered set and suppose that there is a metric d on X such that (
X
,
d
) is a complete metric space. Let F
:
X
3
→
X be a mapping having the mixed monotone property on X. Suppose that there exists a
,
b
,
c
,
e
∈
[
0,1
) such that a
+
b
+
c
+
e
<
1 and
(30)
d
(
F
(
x
,
y
,
z
)
,
F
(
u
,
v
,
w
)
)
≤
a
d
(
x
,
F
(
x
,
y
,
z
)
)
+
b
d
(
z
,
F
(
z
,
y
,
x
)
)
+
c
d
(
u
,
F
(
u
,
v
,
w
)
)
+
e
d
(
w
,
F
(
w
,
v
,
u
)
)
for all x
,
y
,
z
,
u
,
v
,
w
∈
X such that x
≼
u, y
≽
v, and z
≼
w. Also assume that F is continuous or (
X
,
d
,
≼
) is regular. If there exist x
0
,
y
0
,
z
0
∈
X such that
(31)
x
0
≼
F
(
x
0
,
y
0
,
z
0
)
,
y
0
≽
F
(
y
0
,
x
0
,
y
0
)
,
z
0
≼
F
(
z
0
,
y
0
,
x
0
)
,
then F has a tripled fixed point in X; that is, there exist x
,
y
,
z
∈
X such that
(32)
F
(
x
,
y
,
z
)
=
x
,
F
(
y
,
x
,
y
)
=
y
,
F
(
z
,
y
,
x
)
=
z
.Proof.
Let us definek
=
a
+
b
+
c
+
e
∈
[
0,1
). If (
x
,
y
,
z
)
,
(
u
,
v
,
w
)
∈
X
3 are such that (
x
,
y
,
z
)
⊑
(
u
,
v
,
w
), then
(33)
d
(
F
(
x
,
y
,
z
)
,
F
(
u
,
v
,
w
)
)
≤
a
d
(
x
,
F
(
x
,
y
,
z
)
)
+
b
d
(
z
,
F
(
z
,
y
,
x
)
)
+
c
d
(
u
,
F
(
u
,
v
,
w
)
)
+
e
d
(
w
,
F
(
w
,
v
,
u
)
)
≤
(
a
+
b
+
c
+
e
)
max
⁡
⁡
{
d
(
x
,
F
(
x
,
y
,
z
)
)
,
d
(
z
,
F
(
z
,
y
,
x
)
)
,
d
(
u
,
F
(
u
,
v
,
w
)
)
,
d
(
w
,
F
(
w
,
v
,
u
)
)
}
,
so the previous theorem is applicable.The following result presents a contractivity condition more similar to (3) than (16). It follows from the previous result using a
=
b
=
c
=
e
=
k
/
4.Corollary 8.
Let(
X
,
≼
) be a partially ordered set and suppose that there is a metric d on X such that (
X
,
d
) is a complete metric space. Let F
:
X
3
→
X be a mapping having the mixed monotone property on X. Suppose that there exists k
∈
[
0,1
) such that
(34)
d
(
F
(
x
,
y
,
z
)
,
F
(
u
,
v
,
w
)
)
≤
k
4
[
d
(
x
,
F
(
x
,
y
,
z
)
)
+
d
(
z
,
F
(
z
,
y
,
x
)
)
+
d
(
u
,
F
(
u
,
v
,
w
)
)
+
d
(
w
,
F
(
w
,
v
,
u
)
)
]
for all x
,
y
,
z
,
u
,
v
,
w
∈
X such that x
≼
u, y
≽
v, and z
≼
w. Also assume that F is continuous or (
X
,
d
,
≼
) is regular. If there exist x
0
,
y
0
,
z
0
∈
X such that
(35)
x
0
≼
F
(
x
0
,
y
0
,
z
0
)
,
y
0
≽
F
(
y
0
,
x
0
,
y
0
)
,
z
0
≼
F
(
z
0
,
y
0
,
x
0
)
,
then F has a tripled fixed point in X; that is, there exist x
,
y
,
z
∈
X such that
(36)
F
(
x
,
y
,
z
)
=
x
,
F
(
y
,
x
,
y
)
=
y
,
F
(
z
,
y
,
x
)
=
z
.



---

*Source: 392301-2014-09-11.xml*