# Fixed Point Results for Generalized Chatterjea Type Contractive Conditions in Partially OrderedG-Metric Spaces

Authors: Safeer Hussain Khan; Mujahid Abbas; Talat Nazir
Journal: The Scientific World Journal
 (2014)

---

## Abstract

In the framework of orderedG-metric spaces, fixed points of maps that satisfy the generalized (
ψ
,
φ
)-Chatterjea type contractive conditions are obtained. The results presented in the paper generalize and extend several well known comparable results in the literature.


---

## Body

## 1. Introduction and Preliminaries

The study of fixed points of mappings satisfying certain contractive conditions has been at the center of rigorous research activity. Mustafa and Sims [1] generalized the concept of a metric space. Based on the notion of generalized metric spaces, Mustafa et al. [1–5] obtained some fixed point theorems for mappings satisfying different contractive conditions. Abbas and Rhoades [6] initiated the study of a common fixed point theory in generalized metric spaces. Abbas et al. [7] and Chugh et al. [8] obtained some fixed point results for maps satisfying property P in G-metric spaces. Recently, Shatanawi [9] proved some fixed point results for self mappings in a completeG-metric space under some contractive conditions related to a nondecreasing map ϕ
:
R
+
→
R
+ with lim
⁡
n
→
∞
⁡
ϕ
n
(
t
)
=
0 for all t
≥
0. Recently, Saadati et al. [10] proved some fixed point results for contractive mappings in partially ordered G-metric spaces.Ran and Reurings [11] extended Banach contraction principle in partially ordered metric spaces with some applications to linear and nonlinear matrix equations, while Nieto and Rodríguez-López [12] extended the result of Ran and Reurings and applied their main result to obtain a unique solution for a first order ordinary differential equation with periodic boundary conditions. Bhaskar and Lakshmikantham [13] introduced the concept of mixed monotone mappings and obtained some coupled fixed point results. Also, they applied their results to a first order differential equation with periodic boundary conditions.Alber and Guerre-Delabriere [14] introduced the concept of weakly contractive mappings and proved the existence of fixed points of such mappings in Hilbert spaces. Thereafter, in 2001, Rhoades [15] proved the fixed point theorem which is one of the generalizations of Banach’s contraction mapping principle. Weakly contractive mappings are closely related to the mappings of Boyd and Wong [16] and of Reich types [17]. Recently, Dorić [18] proved a common fixed point theorem for generalized (
ψ
,
ϕ
)-weakly contractive mappings. Fixed point problems involving weak contractions and mappings satisfying weak contractive type inequalities have been studied by many authors (see [8, 14, 15, 18–21] and references cited therein).In this paper, we generalize the Chatterjea type contraction mappings to generalized(
ψ
,
φ
)-Chatterjea type contraction mappings and derive some fixed point results for single-valued mappings in ordered generalized metric spaces.Consistent with Mustafa and Sims [1], the following definitions and results will be needed in the sequel.Definition 1.
LetXbe a nonempty set. Suppose that a mappingG
:
X
×
X
×
X
→
R
+satisfies(G1)
G
(
x
,
y
,
z
)
=
0 if x
=
y
=
z;
(G2)
0
<
G
(
x
,
y
,
z
) for all x
,
y
,
z
∈
X, with x
≠
y;
(G3)
G
(
x
,
x
,
y
)
≤
G
(
x
,
y
,
z
) for all x
,
y
,
z
∈
X, with y
≠
z;
(G4)
G
(
x
,
y
,
z
)
=
G
(
x
,
z
,
y
)
=
G
(
y
,
z
,
x
)
=
⋯ (symmetry in all three variables);
(G5)
G
(
x
,
y
,
z
)
≤
G
(
x
,
a
,
a
)
+
G
(
a
,
y
,
z
) for all x
,
y
,
z
,
a
∈
X.Then, G is called a G-metric on X and (
X
,
G
) is called a G-metric space.Definition 2.
A sequence{
x
n
}in aG-metric spaceXis(i)
aG-Cauchy sequence if, for every ε
>
0, there is a natural number n
0 such that for all n
,
m
,
l
≥
n
0
,
G
(
x
n
,
x
m
,
x
l
)
<
ε,
(ii)
aG-Convergent sequence if, for any ε
>
0, there is an x
∈
X and an n
0
∈
ℕ, such that for all n
,
m
≥
n
0
,
G
(
x
n
,
x
m
,
x
)
<
ε.A G-metric space onXis said to beG-complete if everyG-Cauchy sequence inXisG-convergent inX. It is known that {
x
n
}
G-converges to x
∈
X if and only if G
(
x
m
,
x
n
,
x
)
→
0 as n
,
m
→
∞.Proposition 3 (see [1]).
Let
X
be a
G
-metric space. Then, the following are equivalent.
(1)
The sequence
{
x
n
}
is
G
-convergent to
x
.
(2)
G
(
x
n
,
x
n
,
x
)
→
0
as
n
→
∞.
(3)
G
(
x
n
,
x
,
x
)
→
0
as
n
→
∞.
(4)
G
(
x
n
,
x
m
,
x
)
→
0
as
n
,
m
→
∞
.Proposition 4 (see [1]).
Let
X
be a
G
-metric space. Then, the following are equivalent.
(1)
The sequence
{
x
n
}
is
G
-Cauchy.
(2)
For every
ε
>
0
, there exists
n
0
∈
ℕ
, such that for all
n
,
m
≥
n
0
,
G
(
x
n
,
x
m
,
x
m
)
<
ε
; that is, if
G
(
x
n
,
x
m
,
x
m
)
→
0
as
n
,
m
→
∞
.Definition 5.
AG-metric onXis said to be symmetric if G
(
x
,
y
,
y
)
=
G
(
y
,
x
,
x
) for all x
,
y
∈
X.Proposition 6.
EveryG-metric on X defines a metricd
GonXby
(1)
d
G
(
x
,
y
)
=
G
(
x
,
y
,
y
)
+
G
(
y
,
x
,
x
)
,
∀
x
,
y
∈
X
.
For a symmetric G-metric space, one obtains
(2)
d
G
(
x
,
y
)
=
2
G
(
x
,
y
,
y
)
,
∀
x
,
y
∈
X
.
However, ifGis not symmetric, then the following inequality holds:
(3)
3
2
G
(
x
,
y
,
y
)
≤
d
G
(
x
,
y
)
≤
3
G
(
x
,
y
,
y
)
,
∀
x
,
y
∈
X
.
First, we recall some basic definitions and notations.Let(
X
,
d
) be a metric space. A map T
:
X
→
X is said to be(a)
Kannan type (see [22]) if there exists a k
∈
(
0
,
1
/
2
] such thatd
(
T
x
,
T
y
)
≤
k
[
d
(
x
,
T
x
)
+
d
(
y
,
T
y
)
]for allx
,
y
∈
X;
(b)
Chatterjea type [20] if there exists a k
∈
(
0
,
1
/
2
] such that d
(
T
x
,
T
y
)
≤
k
[
d
(
x
,
T
y
)
+
d
(
y
,
T
x
)
] for all x
,
y
∈
X.Definition 7.
We define two classes of mappings as follows:Ψ
=
{
ψ
∣
ψ
:
[
0
,
∞
)
→
[
0
,
∞
) is continuous and nondecreasing with ψ
(
t
)
=
0if and only if t
=
0
} and
Φ
=
{
φ
∣
φ
:
[
0
,
∞
)
5
→
[
0
,
∞
) is lower semi-continuous with φ
(
t
1
,
t
2
,
t
3
,
t
4
,
t
5
)
=
0 if and only if t
1
=
t
2
=
t
3
=
t
4
=
t
5
=
0
}.Definition 8.
An ordered partialG-metric space is said to have a sequential limit comparison property if for every nondecreasing sequence (nonincreasing sequence) {
x
n
} inXsuch that G
(
x
n
,
x
,
x
)
→
0 as n
→
∞ implies that x
n
⪯
x
(
x
⪯
x
n
), respectively.


## 2. Fixed Point Results

In this section, we obtain fixed point results for mappings satisfying generalized(
φ
,
ψ
)-Chatterjea type contractive conditions on partially ordered complete generalized metric space. We start with the following result.Theorem 9.
Let(
X
,
⪯
) be a partially ordered set and f be a nondecreasing self mapping on a complete G-metric space X satisfying
(4)
ψ
(
G
(
f
x
,
f
y
,
f
y
)
)
≤
ψ
(
M
(
x
,
y
,
y
)
)
-
φ
(
N
(
x
,
y
,
y
)
)
,
where ψ
∈
Ψ, φ
∈
Φ with
(5)
M
(
x
,
y
,
y
)
=
max
⁡
{
[
G
(
x
,
f
y
,
f
y
)
+
G
(
y
,
f
x
,
f
x
)
]
2
G
(
x
,
y
,
y
)
,
G
(
x
,
f
x
,
f
x
)
,
c
c
c
c
c
c
c
c
c
c
G
(
y
,
f
y
,
f
y
)
,
c
c
c
c
c
c
c
c
c
[
G
(
x
,
f
y
,
f
y
)
+
G
(
y
,
f
x
,
f
x
)
]
2
}
,
N
(
x
,
y
,
y
)
=
(
G
(
x
,
y
,
y
)
,
G
(
x
,
f
x
,
f
x
)
,
x
x
x
G
(
y
,
f
y
,
f
y
)
,
G
(
x
,
f
y
,
f
y
)
,
G
(
y
,
f
x
,
f
x
)
)
for all x
,
y
∈
X with x
⪯
y. Suppose that there exists x
0
∈
X with x
0
⪯
f
x
0. If f is continuous or X a sequential limit comparison property, then f has a fixed point in X.Proof.
Iff
x
0
=
x
0, there is nothing to prove. Suppose that f
x
0
≠
x
0. Since x
0
⪯
f
x
0 and f is nondecreasing, we have
(6)
x
0
⪯
f
x
0
⪯
f
2
x
0
⪯
⋯
⪯
f
n
x
0
⪯
⋯
.
Define a sequence{
x
n
}by x
n
=
f
n
x
0 so that x
n
+
1
=
f
x
n. We may assume that G
(
x
n
,
x
n
+
1
,
x
n
+
1
)
>
0 for everyn
∈
ℕ. If not, then x
n
=
x
n
+
1 for some n and x
n becomes a fixed point of f. Using (4), we obtain
(7)
ψ
(
G
(
x
n
+
1
,
x
n
+
2
,
x
n
+
2
)
)
=
ψ
(
G
(
f
x
n
,
f
x
n
+
1
,
f
x
n
+
1
)
)
≤
ψ
(
M
(
x
n
,
x
n
+
1
,
x
n
+
1
)
)
-
φ
(
N
(
x
n
,
x
n
+
1
,
x
n
+
1
)
)
,
where
(8)
M
(
x
n
,
x
n
+
1
,
x
n
+
1
)
=
max
⁡
{
[
G
(
x
n
,
f
x
n
+
1
,
f
x
n
+
1
)
+
G
(
x
n
+
1
,
f
x
n
,
f
x
n
)
]
2
G
(
x
n
,
x
n
+
1
,
x
n
+
1
)
,
G
(
x
n
,
f
x
n
,
f
x
n
)
,
c
c
c
c
c
c
c
c
c
c
c
c
c
G
(
x
n
+
1
,
f
x
n
+
1
,
f
x
n
+
1
)
,
c
c
c
c
c
c
c
c
c
c
c
c
[
G
(
x
n
,
f
x
n
+
1
,
f
x
n
+
1
)
+
G
(
x
n
+
1
,
f
x
n
,
f
x
n
)
]
2
}
=
max
⁡
{
[
G
(
x
n
,
x
n
+
2
,
x
n
+
2
)
+
G
(
x
n
+
1
,
x
n
+
1
,
x
n
+
1
)
]
2
G
(
x
n
,
x
n
+
1
,
x
n
+
1
)
,
G
(
x
n
,
x
n
+
1
,
x
n
+
1
)
,
c
c
c
c
c
c
c
c
c
c
c
c
c
c
G
(
x
n
+
1
,
x
n
+
2
,
x
n
+
2
)
,
c
c
c
c
c
c
c
c
c
c
c
c
c
[
G
(
x
n
,
x
n
+
2
,
x
n
+
2
)
+
G
(
x
n
+
1
,
x
n
+
1
,
x
n
+
1
)
]
2
}
=
max
⁡
{
G
(
x
n
,
x
n
+
1
,
x
n
+
1
)
,
G
(
x
n
+
1
,
x
n
+
2
,
x
n
+
2
)
}
,
N
(
x
n
,
x
n
+
1
,
x
n
+
1
)
=
(
G
(
x
n
,
x
n
+
1
,
x
n
+
1
)
,
G
(
x
n
,
f
x
n
,
f
x
n
)
,
c
c
c
c
l
c
c
c
c
G
(
x
n
+
1
,
f
x
n
+
1
,
f
x
n
+
1
)
,
G
(
x
n
,
f
x
n
+
1
,
f
x
n
+
1
)
,
c
c
c
c
c
c
c
c
G
(
x
n
+
1
,
f
x
n
,
f
x
n
)
)
=
(
G
(
x
n
+
1
,
x
n
+
1
,
x
n
+
1
)
G
(
x
n
+
1
,
x
n
+
1
,
x
n
+
1
)
G
(
x
n
,
x
n
+
1
,
x
n
+
1
)
,
G
(
x
n
,
x
n
+
1
,
x
n
+
1
)
,
c
c
c
c
c
c
l
c
G
(
x
n
+
1
,
x
n
+
2
,
x
n
+
2
)
,
G
(
x
n
,
x
n
+
2
,
x
n
+
2
)
,
c
c
c
c
c
c
c
G
(
x
n
+
1
,
x
n
+
1
,
x
n
+
1
)
)
=
(
G
(
x
n
,
x
n
+
1
,
x
n
+
1
)
,
G
(
x
n
,
x
n
+
1
,
x
n
+
1
)
,
c
c
c
c
c
c
c
G
(
x
n
+
1
,
x
n
+
2
,
x
n
+
2
)
,
G
(
x
n
,
x
n
+
2
,
x
n
+
2
)
,
0
)
.
If we take G
(
x
n
+
1
,
x
n
+
2
,
x
n
+
2
)
≥
G
(
x
n
,
x
n
+
1
,
x
n
+
1
) for some n
≥
0, it follows that φ
(
N
(
x
n
,
x
n
+
1
,
x
n
+
1
)
)
=
0, a contradiction. Therefore, for all n
≥
0,
(9)
G
(
x
n
+
1
,
x
n
+
2
,
x
n
+
2
)
<
G
(
x
n
,
x
n
+
1
,
x
n
+
1
)
so that M
(
x
n
,
x
n
+
1
,
x
n
+
1
)
=
G
(
x
n
,
x
n
+
1
,
x
n
+
1
). Now {
G
(
x
n
+
1
,
x
n
+
2
,
x
n
+
2
)
} is a decreasing sequence, so there exists L
≥
0 such that lim
⁡
n
→
∞
⁡
G
(
x
n
+
1
,
x
n
+
2
,
x
n
+
2
)
=
L. This gives lim
⁡
n
→
∞
⁡
G
(
x
n
,
x
n
+
1
,
x
n
+
1
) = lim
⁡
n
→
∞
⁡
M
(
x
n
,
x
n
+
1
,
x
n
+
1
)
=
L. By lower semicontinuity of φ,
(10)
φ
(
L
,
L
,
L
,
L
,
0
)
≤
lim
⁡
inf
⁡
n
→
∞
φ
(
N
(
x
n
,
x
n
+
1
,
x
n
+
1
)
)
.
We claim that L
=
0. Taking the upper limits as n
→
∞ on both sides of
(11)
ψ
(
G
(
x
n
+
1
,
x
n
+
2
,
x
n
+
2
)
)
≤
ψ
(
M
(
x
n
,
x
n
+
1
,
x
n
+
1
)
)
-
φ
(
N
(
x
n
,
x
n
+
1
,
x
n
+
1
)
)
,
we have
(12)
ψ
(
L
)
≤
ψ
(
L
)
-
lim
⁡
inf
⁡
n
→
∞
φ
(
N
(
x
n
,
x
n
+
1
,
x
n
+
1
)
)
≤
ψ
(
L
)
-
φ
(
L
,
L
,
L
,
L
,
0
)
.
This implies φ
(
L
,
L
,
L
,
L
,
0
)
=
0 and we conclude that
(13)
lim
⁡
n
→
∞
⁡
G
(
x
n
+
1
,
x
n
+
2
,
x
n
+
2
)
=
0
.Next, we show that{
x
n
} is a G-Cauchy sequence inX. If not, then there exist ε
>
0 and integers n
k and m
k with m
k
>
n
k
>
k such that
(14)
G
(
x
n
k
,
x
m
k
,
x
m
k
)
≥
ε
,
G
(
x
n
k
,
x
m
k
-
1
,
x
m
k
-
1
)
<
ε
.
A joint effect of (13) and (14) on
(15)
ε
≤
G
(
x
n
k
,
x
m
k
,
x
m
k
)
≤
G
(
x
n
k
,
x
m
k
-
1
,
x
m
k
-
1
)
+
G
(
x
m
k
-
1
,
x
m
k
,
x
m
k
)
yields
(16)
lim
⁡
k
→
∞
⁡
G
(
x
n
k
,
x
m
k
,
x
m
k
)
=
ε
.
Also,
(17)
G
(
x
n
k
,
x
m
k
,
x
m
k
)
≤
G
(
x
n
k
,
x
m
k
+
1
,
x
m
k
+
1
)
+
G
(
x
m
k
+
1
,
x
m
k
,
x
m
k
)
≤
G
(
x
n
k
,
x
m
k
+
1
,
x
m
k
+
1
)
+
G
(
x
m
k
,
x
m
k
+
1
,
x
m
k
+
1
)
+
G
(
x
m
k
,
x
m
k
+
1
,
x
m
k
+
1
)
implies that ε
≤
lim
⁡
k
→
∞
⁡
G
(
x
n
k
,
x
m
k
+
1
,
x
m
k
+
1
).On the other hand,(18)
G
(
x
n
k
,
x
m
k
+
1
,
x
m
k
+
1
)
≤
G
(
x
n
k
,
x
m
k
,
x
m
k
)
+
G
(
x
m
k
,
x
m
k
+
1
,
x
m
k
+
1
)
combined with (13) and (16) results in lim
⁡
k
→
∞
⁡
G
(
x
n
k
,
x
m
k
+
1
,
x
m
k
+
1
)
≤
ε so that
(19)
lim
⁡
k
→
∞
⁡
G
(
x
n
k
,
x
m
k
+
1
,
x
m
k
+
1
)
=
ε
.
Now,
(20)
G
(
x
n
k
,
x
m
k
,
x
m
k
)
≤
G
(
x
n
k
,
x
m
k
+
2
,
x
m
k
+
2
)
+
G
(
x
m
k
+
2
,
x
m
k
,
x
m
k
)
≤
G
(
x
n
k
,
x
m
k
+
2
,
x
m
k
+
2
)
+
G
(
x
m
k
,
x
m
k
+
1
,
x
m
k
+
1
)
+
G
(
x
m
k
,
x
m
k
+
1
,
x
m
k
+
2
)
≤
G
(
x
n
k
,
x
m
k
+
2
,
x
m
k
+
2
)
+
G
(
x
m
k
,
x
m
k
+
1
,
x
m
k
+
1
)
+
G
(
x
m
k
,
x
m
k
+
1
,
x
m
k
+
1
)
+
G
(
x
m
k
+
1
,
x
m
k
+
1
,
x
m
k
+
2
)
≤
G
(
x
n
k
,
x
m
k
+
2
,
x
m
k
+
2
)
+
G
(
x
m
k
,
x
m
k
+
1
,
x
m
k
+
1
)
+
G
(
x
m
k
,
x
m
k
+
1
,
x
m
k
+
1
)
+
G
(
x
m
k
+
1
,
x
m
k
+
2
,
x
m
k
+
2
)
+
G
(
x
m
k
+
1
,
x
m
k
+
2
,
x
m
k
+
2
)
gives that ε
≤
lim
⁡
k
→
∞
⁡
G
(
x
n
k
,
x
m
k
+
2
,
x
m
k
+
2
), and
(21)
G
(
x
n
k
,
x
m
k
+
2
,
x
m
k
+
2
)
≤
G
(
x
n
k
,
x
m
k
,
x
m
k
)
+
G
(
x
m
k
,
x
m
k
+
2
,
x
m
k
+
2
)
≤
G
(
x
n
k
,
x
m
k
,
x
m
k
)
+
G
(
x
m
k
,
x
m
k
+
1
,
x
m
k
+
1
)
+
G
(
x
m
k
+
1
,
x
m
k
+
2
,
x
m
k
+
2
)
implies by (13) and (19) that
(22)
lim
⁡
k
→
∞
⁡
G
(
x
n
k
,
x
m
k
+
2
,
x
m
k
+
2
)
≤
ε
.
Hence,
(23)
lim
⁡
k
→
∞
⁡
G
(
x
n
k
,
x
m
k
+
2
,
x
m
k
+
2
)
=
ε
.
Also, from (16) and
(24)
G
(
x
n
k
,
x
m
k
,
x
m
k
)
≤
G
(
x
n
k
,
x
m
k
,
x
m
k
+
1
)
+
G
(
x
m
k
,
x
m
k
+
1
,
x
m
k
+
1
)
we obtain ε
≤
lim
⁡
k
→
∞
⁡
G
(
x
n
k
,
x
m
k
,
x
m
k
+
1
).But from(25)
G
(
x
n
k
,
x
m
k
,
x
m
k
+
1
)
≤
G
(
x
n
k
,
x
m
k
,
x
m
k
)
+
G
(
x
m
k
,
x
m
k
,
x
m
k
+
1
)
≤
G
(
x
n
k
,
x
m
k
,
x
m
k
)
+
G
(
x
m
k
,
x
m
k
+
1
,
x
m
k
+
1
)
+
G
(
x
m
k
,
x
m
k
+
1
,
x
m
k
+
1
)
together with (13) and (16), we get lim
⁡
k
→
∞
⁡
G
(
x
n
k
,
x
m
k
,
x
m
k
+
1
)
≤
ε. Thus,
(26)
lim
⁡
k
→
∞
⁡
G
(
x
n
k
,
x
m
k
,
x
m
k
+
1
)
=
ε
,
G
(
x
n
k
,
x
m
k
+
1
,
x
m
k
+
1
)
≤
M
(
x
n
k
,
x
m
k
+
1
,
x
m
k
+
1
)
=
max
⁡
{
G
(
x
n
k
,
x
m
k
+
1
,
x
m
k
+
1
)
,
G
(
x
n
k
,
f
x
n
k
,
f
x
n
k
)
,
c
c
c
c
c
c
c
c
c
c
c
c
c
G
(
x
m
k
+
1
,
f
x
m
k
+
1
,
f
x
m
k
+
1
)
,
c
c
c
c
c
c
c
c
c
c
c
c
c
[
G
(
x
n
k
,
f
x
m
k
+
1
,
f
x
m
k
+
1
)
vvvvvvvvvvv
+
G
(
x
m
k
+
1
,
f
x
n
k
,
f
x
n
k
)
]
×
(
2
)
-
1
}
=
max
⁡
{
G
(
x
n
k
,
x
m
k
+
1
,
x
m
k
+
1
)
,
G
(
x
n
k
,
x
n
k
+
1
,
x
n
k
+
1
)
,
c
c
c
c
c
c
c
c
c
c
c
c
c
G
(
x
m
k
+
1
,
x
m
k
+
2
,
x
m
k
+
2
)
,
c
c
c
c
c
c
c
c
c
c
c
c
c
[
G
(
x
n
k
,
x
m
k
+
2
,
x
m
k
+
2
)
vvvvvvvvvvv
+
G
(
x
m
k
+
1
,
x
n
k
+
1
,
x
n
k
+
1
)
]
×
(
2
)
-
1
}
≤
max
⁡
{
G
(
x
n
k
,
x
m
k
+
1
,
x
m
k
+
1
)
,
G
(
x
n
k
,
x
n
k
+
1
,
x
n
k
+
1
)
,
c
c
c
c
c
c
c
c
c
c
c
c
c
G
(
x
m
k
+
1
,
x
m
k
+
2
,
x
m
k
+
2
)
,
c
c
c
c
c
c
c
c
c
c
c
c
c
[
G
(
x
n
k
,
x
m
k
+
2
,
x
m
k
+
2
)
+
G
(
x
n
k
,
x
n
k
+
1
,
x
n
k
+
1
)
c
c
c
c
c
c
c
c
c
c
c
c
c
c
+
G
(
x
n
k
,
x
m
k
,
x
m
k
+
1
)
]
×
(
2
)
-
1
}
.
This gives
(27)
ε
≤
lim
⁡
k
→
∞
M
(
x
n
k
,
x
m
k
+
1
,
x
m
k
+
1
)
≤
max
⁡
{
ε
,
0,0
,
[
ε
+
ε
]
2
}
=
ε
and so
(28)
lim
⁡
k
→
∞
⁡
M
(
x
n
k
,
x
m
k
+
1
,
x
m
k
+
1
)
=
ε
.
Also,
(29)
lim
⁡
k
→
∞
⁡
N
(
x
n
k
,
x
m
k
+
1
,
x
m
k
+
1
)
=
(
ε
,
0,0
,
ε
,
ε
)
.
From (4), we obtain
(30)
ψ
(
G
(
x
n
k
+
1
,
x
m
k
+
2
,
x
m
k
+
2
)
)
=
ψ
(
G
(
f
x
n
k
,
f
x
m
k
+
1
,
f
x
m
k
+
1
)
)
≤
ψ
(
M
(
x
n
k
,
x
m
k
+
1
,
x
m
k
+
1
)
)
-
φ
(
N
(
x
n
k
,
x
m
k
+
1
,
x
m
k
+
1
)
)
,
which on taking the upper limit as k
→
∞ implies that
(31)
ψ
(
ε
)
≤
ψ
(
ε
)
-
φ
(
ε
,
0,0
,
ε
,
ε
)
,
a contradiction asε
>
0.It follows that{
x
n
} is a G-Cauchy sequence and by G-completeness of X, there exists u
∈
X such that {
x
n
}
G-converges to u as n
→
∞. If f is continuous, then it is clear that f
u
=
u. Next, if X has a sequential limit comparison property, then we have x
n
⪯
u for all n
∈
ℕ. From (4), we have
(32)
ψ
(
G
(
x
n
+
1
,
f
u
,
f
u
)
)
=
ψ
(
G
(
f
x
n
,
f
u
,
f
u
)
)
≤
ψ
(
M
(
x
n
,
u
,
u
)
)
-
φ
(
N
(
x
n
,
u
,
u
)
)
,
where
(33)
M
(
x
n
,
u
,
u
)
=
max
⁡
{
[
G
(
x
n
,
f
u
,
f
u
)
+
G
(
u
,
f
x
n
,
f
x
n
)
]
2
G
(
x
n
,
u
,
u
)
,
G
(
x
n
,
f
x
n
,
f
x
n
)
,
G
(
u
,
f
u
,
f
u
)
,
c
c
c
c
c
c
c
c
c
c
c
c
c
[
G
(
x
n
,
f
u
,
f
u
)
+
G
(
u
,
f
x
n
,
f
x
n
)
]
2
}
=
max
⁡
{
[
G
(
x
n
,
f
u
,
f
u
)
+
G
(
u
,
x
n
+
1
,
x
n
+
1
)
]
2
G
(
x
n
,
u
,
u
)
,
G
(
x
n
,
x
n
+
1
,
x
n
+
1
)
,
G
(
u
,
f
u
,
f
u
)
,
c
c
c
c
c
c
c
c
c
c
c
c
c
[
G
(
x
n
,
f
u
,
f
u
)
+
G
(
u
,
x
n
+
1
,
x
n
+
1
)
]
2
}
,
N
(
x
n
,
u
,
u
)
=
(
G
(
x
n
,
u
,
u
)
,
G
(
x
n
,
f
x
n
,
f
x
n
)
,
G
(
u
,
f
u
,
f
u
)
,
c
c
c
c
c
c
c
G
(
x
n
,
f
u
,
f
u
)
,
G
(
u
,
f
x
n
,
f
x
n
)
)
=
(
G
(
x
n
,
u
,
u
)
,
G
(
x
n
,
x
n
+
1
,
x
n
+
1
)
,
G
(
u
,
f
u
,
f
u
)
,
c
c
c
c
c
c
c
G
(
x
n
,
f
u
,
f
u
)
,
G
(
u
,
x
n
+
1
,
x
n
+
1
)
)
.
This implies that lim
⁡
n
→
∞
⁡
M
(
x
n
,
u
,
u
)
=
G
(
u
,
f
u
,
f
u
). Thus, from (32), we obtain
(34)
ψ
(
G
(
u
,
f
u
,
f
u
)
)
=
lim
⁡
sup
⁡
n
→
∞
ψ
(
G
(
f
x
n
,
f
u
,
f
u
)
)
≤
lim
⁡
sup
⁡
n
→
∞
[
ψ
(
M
(
x
n
,
u
,
u
)
)
cccccccccccc
-
φ
(
M
(
x
n
,
u
,
u
)
)
]
≤
ψ
(
G
(
u
,
f
u
,
f
u
)
)
-
φ
(
0,0
,
G
(
u
,
f
u
,
f
u
)
,
ccccccc
G
(
u
,
f
u
,
f
u
)
,
0
)
.
This givesφ
(
0,0
,
G
(
u
,
f
u
,
f
u
)
,
G
(
u
,
f
u
,
f
u
)
,
0
)
=
0so thatG
(
u
,
f
u
,
f
u
)
=
0and, hence, f
u
=
u.Corollary 10.
Let(
X
,
⪯
)be a partially ordered set andfbe a nondecreasing self mapping on a complete G-metric spaceXsatisfying
(35)
ψ
(
G
(
f
x
,
f
y
,
f
y
)
)
≤
ψ
(
M
(
x
,
y
,
y
)
)
-
φ
(
N
(
x
,
y
,
y
)
)
,
where ψ
∈
Ψ, φ
∈
Φ with
(36)
M
(
x
,
y
,
y
)
=
a
1
G
(
x
,
y
,
y
)
+
a
2
G
(
x
,
f
x
,
f
x
)
+
a
3
G
(
y
,
f
y
,
f
y
)
,
c
c
c
c
a
4
[
G
(
x
,
f
y
,
f
y
)
+
G
(
y
,
f
x
,
f
x
)
]
,
N
(
x
,
y
,
y
)
=
(
G
(
x
,
y
,
y
)
,
G
(
x
,
f
x
,
f
x
)
,
x
x
x
G
(
y
,
f
y
,
f
y
)
,
G
(
x
,
f
y
,
f
y
)
,
G
(
y
,
f
x
,
f
x
)
)
for all x
,
y
∈
X with x
⪯
y with a
i
≥
0 for all i
=
1,2
,
3,4 with a
1
+
a
2
+
a
3
+
2
a
4
≤
1. Suppose that there exists x
0
∈
X with x
0
⪯
f
x
0. If f is continuous or X has a sequential limit comparison property, then f has a fixed point in X.Now we give an example to illustrate above result.Example 11.
LetX
=
[
0,1
] and G
(
x
,
y
,
z
)
=
max
⁡
{
|
x
-
y
|
,
|
y
-
z
|
,
|
z
-
x
|
} be a G-metric on X. Define f
:
X
→
X by
(37)
f
(
x
)
=
x
12
∀
x
∈
X
.
We take ψ
(
t
)
=
(
3
/
4
)
t and φ
(
t
1
,
t
2
,
t
3
,
t
4
,
t
5
)
=
(
1
/
12
)
(
t
1
+
t
2
+
t
3
+
t
4
+
t
5
) for all t
,
t
1
,
t
2
,
t
3, t
4
,
t
5
∈
[
0
,
∞
).Now, for allx
,
y
∈
X with x
⪯
y, we have
(38)
G
(
x
,
f
x
,
f
x
)
=
11
x
12
,
G
(
y
,
f
y
,
f
y
)
=
11
y
12
,
[
G
(
x
,
f
y
,
f
y
)
+
G
(
y
,
f
x
,
f
x
)
]
2
=
|
12
x
-
y
|
+
12
y
-
x
24
.
So that(39)
G
(
f
x
,
f
y
,
f
y
)
=
1
12
(
y
-
x
)
≤
3
4
[
(
|
12
x
-
y
|
+
12
y
-
x
24
)
1
6
(
y
-
x
)
+
1
6
(
11
x
12
)
+
1
6
(
11
y
12
)
ccccc
+
1
6
(
|
12
x
-
y
|
+
12
y
-
x
24
)
]
-
1
12
[
|
12
x
-
y
|
+
12
y
-
x
24
)
(
y
-
x
)
+
11
x
12
+
11
y
12
cccccccc
+
|
12
x
-
y
|
+
12
y
-
x
24
]
=
3
4
M
(
x
,
y
,
y
)
-
1
12
N
(
x
,
y
,
y
)
=
ψ
(
M
(
x
,
y
,
y
)
)
-
φ
(
N
(
x
,
y
,
y
)
)
.
Thus, (35) is satisfied with a
1
=
a
2
=
a
3
=
a
4
=
1
/
6, where a
1
+
a
2
+
a
3
+
2
a
4
≤
1. Hence, the conditions of Corollary 10 are satisfied and 0 is the fixed point of f.Corollary 12.
Let(
X
,
⪯
) be a partially ordered set and f be a nondecreasing self mapping on a complete G-metric space X satisfying
(40)
G
(
f
x
,
f
y
,
f
y
)
≤
M
(
x
,
y
,
y
)
-
φ
(
M
(
x
,
y
,
y
)
)
,
whereφ
∈
Φ,
(41)
M
(
x
,
y
,
y
)
=
max
⁡
{
[
G
(
x
,
f
y
,
f
y
)
+
G
(
y
,
f
x
,
f
x
)
]
2
G
(
x
,
y
,
y
)
,
G
(
x
,
f
x
,
f
x
)
,
c
c
c
c
c
c
c
c
c
c
G
(
y
,
f
y
,
f
y
)
,
c
c
c
c
c
c
c
c
c
[
G
(
x
,
f
y
,
f
y
)
+
G
(
y
,
f
x
,
f
x
)
]
2
}
,
N
(
x
,
y
,
y
)
=
(
G
(
x
,
y
,
y
)
,
G
(
x
,
f
x
,
f
x
)
,
G
(
y
,
f
y
,
f
y
)
,
cccc
G
(
x
,
f
y
,
f
y
)
,
G
(
y
,
f
x
,
f
x
)
)
for all x
,
y
∈
X with x
⪯
y. Suppose that there exists x
0
∈
X with x
0
⪯
f
x
0. If f is continuous or X has a sequential limit comparison property, then f has a fixed point in X.Corollary 13.
Let(
X
,
⪯
) be a partially ordered set and f be a nondecreasing self mapping on a complete G-metric space X satisfying
(42)
G
(
f
x
,
f
y
,
f
y
)
≤
M
(
x
,
y
,
y
)
-
φ
(
N
(
x
,
y
,
y
)
)
,
where φ
∈
Φ,
(43)
M
(
x
,
y
,
y
)
=
a
1
G
(
x
,
y
,
y
)
+
a
2
G
(
x
,
f
x
,
f
x
)
+
a
3
G
(
y
,
f
y
,
f
y
)
+
a
4
[
G
(
x
,
f
y
,
f
y
)
ccccccccccccccccccccccccc
+
G
(
y
,
f
x
,
f
x
)
]
,
N
(
x
,
y
,
y
)
=
(
G
(
x
,
y
,
y
)
,
G
(
x
,
f
x
,
f
x
)
,
G
(
y
,
f
y
,
f
y
)
,
c
c
c
c
G
(
x
,
f
y
,
f
y
)
,
G
(
y
,
f
x
,
f
x
)
)
for all x
,
y
∈
X with x
⪯
y, where a
i
>
0 for i
=
1,2
,
3,4 with a
1
+
a
2
+
a
3
+
2
a
4
≤
1. Suppose that there existsx
0
∈
Xwithx
0
⪯
f
x
0. Iffis continuous orXhas a sequential limit comparison property, then f has a fixed point inX.



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*Source: 341751-2014-02-11.xml*