# Some Extensions of Fixed Point Results over Quasi-J

Authors: Maha Noorwali; Hamed H. Alsulami; Erdal Karapınar
Journal: Journal of Function Spaces
 (2016)

---

## Abstract

We introduce the notion of quasi-J
S-metric space. After defining the basic topological properties of quasi-J
S-metric space, we investigate fixed point of certain mapping in the frame of complete quasi-J
S-metric space. Our results unify and cover several existing fixed point theorems in distinct structures (such as standard quasi-metric spaces, quasi-b-metric spaces, dislocated quasi-metric spaces, and quasi-modular spaces) in the literature.


---

## Body

## 1. Introduction and Preliminaries

Jleli and Samet [1] combined a number of existing fixed point results, by introducing a new distance (that includes, as particular cases, standard metric spaces, b-metric spaces, dislocated metric spaces, and modular spaces). In this paper, our aim is to refine the new distance by omitting a symmetry condition. Hence, our new approaches cover and combine several more interesting existing fixed point results (that includes, as particular cases, standard quasi-metric spaces, quasi-b-metric spaces, dislocated quasi-metric spaces, and quasi-modular spaces) including the results of Jleli and Samet [1].For the sake of completeness, we collect some basic concepts and results from the literature. LetN
0 denote the set N
∪
0 where N represent the set of all positive integers. Let X be a nonempty set and let D
:
X
×
X
→
[
0
,
∞
] be a given mapping. For every x
∈
X, define the sets (1)
C
L
D
,
X
,
x
=
x
n
⊆
X
:
lim
n
→
∞
⁡
D
x
,
x
n
=
0
,
C
R
D
,
X
,
x
=
x
n
⊆
X
:
lim
n
→
∞
⁡
D
x
n
,
x
=
0
.Definition 1.
We say thatD
:
X
×
X
→
[
0
,
∞
] is a quasi-J
S-metric space on a nonempty set X if it fulfils the following conditions:(D1)
D
x
,
y
=
D
y
,
x
=
0
⇒
x
=
y, for every x
,
y
∈
X.
(D2)
There existsC
>
0 such that (2)
if
x
,
y
∈
X
,
x
n
∈
C
L
D
,
X
,
x
,
then
D
x
,
y
≤
C
limsup
n
→
∞
⁡
D
x
n
,
y
,
if
x
,
y
∈
X
,
x
n
∈
C
R
D
,
X
,
x
,
then
D
y
,
x
≤
C
limsup
n
→
∞
⁡
D
y
,
x
n
.In this case, the pair X
,
D is called a quasi-J
S-metric space.Remark 2.
If, in addition to the conditions in Definition1, the equality, (D3)
D
x
,
y
=
D
y
,
x,is satisfied for each x
,
y
∈
X, then, (
X
,
D
) is called J
S-metric space [1].In what follows we shall define some basic topological notions for quasi-J
S-metric space.Definition 3.
LetX
,
D be a quasi-J
S-metric space. Let x
n be a sequence in X and x
∈
X
. Then,(i)
x
n is said to be left D-convergent to x if x
n
∈
C
L
D
,
X
,
x; in this case x is said to be a left D-limit of x
n.
(ii)
x
n is said to be right D-convergent to x if x
n
∈
C
R
D
,
X
,
x; in this case x is said to be a right D-limit of x
n.
(iii)
x
n is said to be D-convergent to x if x
n is both left and right D-convergent to x; in this case x is said to be a D-limit of x
n (see [1]).Proposition 4.
TheD-limit of any sequence in a quasi-J
S-metric space is unique.Proof.
LetX
,
D be a quasi-J
S-metric space. Let x
n be a sequence in X. Assume that x
,
y are both D-limits of x
n. On account of (D2) and regarding the definition of D-convergence, we find that (3)
D
y
,
x
≤
C
limsup
n
→
∞
⁡
D
y
,
x
n
=
0
,
since
x
n
∈
C
R
D
,
X
,
x
,
x
n
∈
C
L
D
,
X
,
yand also (4)
D
x
,
y
≤
C
limsup
n
→
∞
⁡
D
x
,
x
n
=
0
,
since
x
n
∈
C
R
D
,
X
,
y
,
x
n
∈
C
L
D
,
X
,
x
.Thus, we have D
x
,
y
=
0
=
D
y
,
x. By (D1), we find x
=
y.Definition 5.
LetX
,
D be a quasi-J
S-metric space. Let x
n be a sequence in X.(i)
x
n is said to be left D-Cauchy sequence if (5)
lim
m
,
n
→
∞
⁡
D
x
m
+
n
,
x
n
=
0
.
(ii)
x
n is said to be right D-Cauchy sequence if(6)
lim
m
,
n
→
∞
⁡
D
x
n
,
x
m
+
n
=
0
.
(iii)
x
n is said to be D-Cauchy sequence if it is both left and right D-Cauchy sequence (see [1]).Definition 6.
LetX
,
D be a quasi-J
S-metric space. (i)
X is said to be left D-complete if every left D-Cauchy sequence in X is left D-convergent to some element in X
.
(ii)
X is said to be right D-complete if every right D-Cauchy sequence in X is right D-convergent to some element in X
.
(iii)
X is said to be D-complete if and only if it is left and right D-complete, so that every D-Cauchy sequence in X is D-convergent to some element in X (see [1]).Example 7.
LetX
=
R
∪
∞ and define D
:
X
×
X
→
[
0
,
∞
] by(7)
D
x
,
y
=
∞
if
x
o
r
y
=
∞
x
-
y
o
t
h
e
r
w
i
s
e
.
Then, clearly,D satisfies (D1). Let x
∈
X; we have two cases:Case 1: ifx
=
∞, then C
L
D
,
X
,
x
=
C
R
D
,
X
,
x
=
∅
.
Case 2: ifx
<
∞, let z
n
∈
C
L
D
,
X
,
x; then lim
n
→
∞
⁡
D
x
,
z
n
=
0 and so z
n
<
∞ except possibly for finite number of terms. Let m be the smallest natural number such that z
n
<
∞ for all n
≥
m. Now, let y
∈
X; if y
=
∞, then(8)
D
x
,
y
=
∞
=
limsup
n
→
∞
⁡
D
z
n
,
y
.On the other hand, if y
<
∞, then we get (9)
D
x
,
y
=
x
-
y
≤
x
-
z
n
+
z
n
-
y
∀
n
≥
m
.Thus, we find that (10)
D
x
,
y
=
x
-
y
≤
limsup
n
→
∞
⁡
D
z
n
,
y
.Similarly, if x
n
∈
C
R
D
,
X
,
x, then for any y
∈
X we have (11)
D
y
,
x
≤
limsup
n
→
∞
⁡
D
y
,
x
n
.Consequently, D satisfies condition (D2) with C
=
1. Therefore, X
,
D is a quasi-J
S-metric space. Now, let x
n
⊆
X be D-Cauchy sequence. Then there exists a smallest m
∈
N such that x
n
<
∞ for all n
≥
m. As the restriction of D to R is just the usual metric on R, x
n is D-convergent to some x
∈
X
. Thus, X
,
D is a complete quasi-J
S-metric space.Definition 8.
LetX be a nonempty set. A mapping q
:
X
×
X
→
[
0
,
∞
) is called quasi-metric on X, if the following conditions are fulfilled:(Q1)
for everyx
,
y
∈
X, we have q
x
,
y
=
0
=
q
y
,
x
⇔
x
=
y;
(Q2)
for everyx
,
y
,
z
∈
X, we have q
x
,
y
≤
q
x
,
z
+
q
z
,
y.
Here, the pair(
X
,
q
) is called quasi-metric space.Proposition 9.
Any quasi-metric space is a quasi-J
S-metric space with C
=
1.Proof is straightforward, so we omit it.Example 10.
LetX be a set and let f
:
X
→
[
0
,
∞
) be an arbitrary one to one function. Set(12)
q
x
,
y
=
max
⁡
f
y
-
f
x
,
0
∀
x
,
y
∈
X
.Then q is a quasi-metric space [2].In 2012, Shah and Hussain [3] introduced the concept of quasi-b-metric spaces and verified some fixed point theorems in quasi-b-metric spaces.Definition 11.
LetX be a nonempty set, let s
≥
1 be a given real number, and let d
:
X
×
X
→
[
0
,
∞
) be a mapping satisfying the following conditions:(QBM1) for every x
,
y
∈
X, we have d
x
,
y
=
0
⇔
x
=
y;
(QBM2) for every x
,
y
,
z
∈
X, we have d
x
,
y
≤
s
d
x
,
z
+
d
z
,
y.
Then,d is said to be a quasi-b-metric space, and X
,
d is called quasi-b-metric space.The following proposition followed immediately from the previous definition.Proposition 12.
Any quasi-b-metric space is a quasi-J
S-metric space with C
=
s.Proof.
Let(
X
,
d
) be a quasi-b-metric space. Since the first condition is straightforward, it is sufficient to show that d fulfils the property (D2) of Definition 1. Let x
∈
X and {
x
n
}
∈
C
L
(
d
,
X
,
x
). For every y
∈
X, by the property (QBM2), we have (13)
d
x
,
y
≤
s
d
x
,
x
n
+
s
d
x
n
,
y
,for each n. Thus we have (14)
d
x
,
y
≤
s
limsup
n
→
∞
⁡
d
x
n
,
y
.Analogously, for the case {
x
n
}
∈
C
R
(
d
,
X
,
x
), we derive that (15)
d
y
,
x
≤
s
limsup
n
→
∞
⁡
d
y
,
x
n
.In 2005, Zeyada et al. [4] introduced the concept of complete dislocated quasi-metric space and obtain some fixed point results on it.Definition 13.
LetX be a nonempty set and d
:
X
×
X
→
[
0
,
∞
). d is said to be dislocated quasi-metric (or quasi-metric-like) if it satisfies the following conditions for every x
,
y
,
z
∈
X:(QML1) d
x
,
y
=
0
=
d
y
,
x
⇒
x
=
y;
(QML2) d
x
,
y
≤
d
x
,
z
+
d
z
,
y.
In this caseX
,
d is called dislocated quasi-metric space (or quasi-metric-like space). If in addition d satisfies(QML3) d
x
,
y
=
d
y
,
x for every x
,
y
∈
X,then it is called dislocated metric.Proposition 14.
Any dislocated quasi-metric space is a quasi-J
S-metric space.Example 15.
LetX
=
[
0,1
] and define q
:
X
×
X
→
[
0
,
∞
) by(16)
q
x
,
y
=
x
-
y
+
x
for  each
x
,
y
∈
X
.Then X
,
q is a dislocated quasi-metric.In 1988, Kozlowski introduced the notion of modular spaces [5]; before we generalize this notion to the quasi form we need the following definitions.Definition 16.
LetX be a linear space over R. A function ρ
:
X
→
[
0
,
∞
] is said to be quasi-modular if the following conditions hold:(q
ρ
1)
ρ
(
x
)
=
0
⇔
x
=
0;
(q
ρ
2)
for everyx
,
y
∈
X, we have(17)
ρ
α
x
+
β
y
≤
ρ
x
+
ρ
y
,whenever α
,
β
≥
0 and α
+
β
=
1. If in addition ρ satisfies (18)
ρ
-
x
=
ρ
x
for  every
x
∈
X
,then ρ is called modular on X.Definition 17.
LetX be a linear space and let ρ
:
X
→
[
0
,
∞
] be a quasi-modular space on X. The set(19)
X
ρ
=
x
∈
X
:
lim
λ
→
0
⁡
ρ
λ
x
=
0is called a quasi-modular space.The convergence in quasi-modular spaces is defined as follows.Definition 18.
LetX
ρ be a quasi-modular space, let x
n be a sequence in X
ρ, and x
∈
X
ρ.(i)
x
n is said to be left ρ-convergent to x
∈
X if lim
n
→
∞
⁡
ρ
x
-
x
n
=
0.
(ii)
x
n is said to be right ρ-convergent to x
∈
X if lim
n
→
∞
⁡
ρ
x
n
-
x
=
0.
(iii)
x
n is said to be ρ-convergent to x
∈
X if it is both left and right ρ-convergent to x.Definition 19.
LetX
ρ be a quasi-modular space and let x
n be a sequence in X
ρ.(i)
x
n is said to be left ρ-Cauchy if lim
m
,
n
→
∞
⁡
ρ
x
m
+
n
-
x
n
=
0.
(ii)
x
n is said to be right ρ-Cauchy if lim
m
,
n
→
∞
⁡
ρ
x
n
-
x
m
+
n
=
0.
(iii)
x
n is said to be ρ-Cauchy if it is both left and right ρ-Cauchy.Definition 20.
(i) A quasi-modular spaceX
ρ is said to be left (right) ρ-complete if every left (right) ρ-Cauchy sequence converges to some x
∈
X
ρ.
(ii) A quasi-modular spaceX
ρ is said to be ρ-complete if and only if it is both left and right ρ-complete.Definition 21.
LetX
ρ be a quasi-modular space.(i)
ρ is said to have left Fatou property if for every y
∈
X
ρ
(20)
ρ
y
-
x
≤
liminf
n
→
∞
⁡
ρ
y
-
x
n
,
wheneverx
n
⊆
X
ρ is left ρ-convergent to x
∈
X
ρ.
(ii)
ρ is said to have right Fatou property if for every y
∈
X
ρ
(21)
ρ
x
-
y
≤
liminf
n
→
∞
⁡
ρ
x
n
-
y
,
wheneverx
n
⊆
X
ρ is right ρ-convergent to x
∈
X
ρ.
(iii)
ρ is said to have Fatou property if it has left and right Fatou property.LetX
ρ be a quasi-modular space. Define the mapping D
ρ
:
X
ρ
×
X
ρ
→
[
0
,
∞
] by(22)
D
ρ
x
,
y
=
ρ
x
-
y
∀
x
,
y
∈
X
ρ
.We have the following example of a quasi-J
S-space.Proposition 22.
LetX
ρ be a quasi-modular space such that ρ has Fatou property. Then D
ρ is quasi-J
S-metric on X
ρ with C
=
1.Proof.
Clearly,D
ρ satisfies (D1). Let us prove that D
ρ satisfies (D2). Let x
∈
X
ρ and let x
n
∈
C
L
D
ρ
,
X
ρ
,
x
. As ρ has Fatou property for any y
∈
X
ρ we have(23)
D
ρ
x
,
y
=
ρ
x
-
y
≤
liminf
n
→
∞
⁡
ρ
x
n
-
y
≤
limsup
n
→
∞
⁡
ρ
x
n
-
y
=
limsup
n
→
∞
⁡
D
ρ
x
n
,
y
.Similarly, if x
n
∈
C
R
D
ρ
,
X
ρ
,
x, then for every y
∈
X
ρ we have(24)
D
ρ
y
,
x
=
ρ
y
-
x
≤
liminf
n
→
∞
⁡
ρ
y
-
x
n
≤
limsup
n
→
∞
⁡
ρ
y
-
x
n
=
limsup
n
→
∞
⁡
D
ρ
y
,
x
n
.Thus, D
ρ is a quasi-J
S-metric space on X
ρ.Consequently, we have the following result.Proposition 23.
LetX
ρ be a quasi-modular space where ρ has Fatou property. Then(i)
a sequencex
n
⊆
X
ρ is left ρ-convergent (right ρ-convergent or ρ-convergent) to some x
∈
X
ρ if and only if it is left D
ρ-convergent (right D
ρ-convergent or D
ρ-convergent) to x;
(ii)
a sequencex
n
⊆
X
ρ is left ρ-Cauchy (right ρ-Cauchy or ρ-Cauchy) if and only if it is left D
ρ-Cauchy (right D
ρ-Cauchy or D
ρ-Cauchy);
(iii)
X
ρ is left ρ-complete (right ρ-complete or ρ-complete) if and only if it is left D
ρ-complete (right D
ρ-complete or D
ρ-complete).


## 2. The Banach Contraction Principle in a Quasi-J
S-Metric Space

The Banach contraction principle was extended to aJ
S-metric space by Jleli and Samet in [1]. We shall extend this principle to a quasi-J
S-metric space.Definition 24.
LetX
,
D be a quasi-J
S-metric space and let f
:
X
→
X be a function. We say that f is k-contraction if (25)
D
f
x
,
f
y
≤
k
D
x
,
y
,for every x
,
y
∈
X, where k
∈
0,1
.Proposition 25.
LetX
,
D be a quasi-J
S-metric space. Suppose that the function f
:
X
→
X is k-contraction for some k
∈
0,1
. Then any fixed point ω
∈
X of f with D
ω
,
ω
<
∞ satisfies D
ω
,
ω
=
0
.Proof.
Letω
∈
X be a fixed point of f with D
ω
,
ω
<
∞. Then, as f is k-contraction, we have(26)
D
ω
,
ω
=
D
f
ω
,
f
ω
≤
k
D
ω
,
ω
,which is possible only if D
ω
,
ω
=
0
.For eachx
∈
X let us define(27)
δ
D
,
f
,
x
=
sup
⁡
D
f
i
x
,
f
j
x
:
i
,
j
∈
N
0
,where f
0
x
=
x
.The following theorem is an extension of Banach contraction principle in the context of a quasi-J
S-metric space.Theorem 26.
Let(
X
,
D
) be a D-complete quasi-J
S-metric space and let f
:
X
→
X be a k-contraction mapping for some k
∈
0,1
. Suppose that there exists x
0
∈
X such that δ
D
,
f
,
x
0
<
∞
. Then, f has a fixed point ω
∈
X and f
n
x
0 is D-convergent to ω. Moreover, if ω
′ is another fixed point of f such that D
ω
,
ω
′
<
∞ and D
ω
′
,
ω
<
∞, then ω
=
ω
′.Proof.
We shall prove thatf
n
x
0 is a D-Cauchy sequence. Let n
∈
N, as f is k-contraction, for each i
,
j
∈
N we have(28)
D
f
n
+
i
x
0
,
f
n
+
j
x
0
≤
k
D
f
n
-
1
+
i
x
0
,
f
n
-
1
+
j
x
0
,which implies that(29)
δ
D
,
f
,
f
n
x
0
≤
k
δ
D
,
f
,
f
n
-
1
x
0
.So, we obtain that (30)
δ
D
,
f
,
f
n
x
0
≤
k
n
δ
D
,
f
,
x
0
.Taking (30) into account and regarding the definition of δ, for every m
,
n
∈
N
0, we have(31)
D
f
n
x
0
,
f
n
+
m
x
0
≤
δ
D
,
f
,
f
n
x
0
≤
k
n
δ
D
,
f
,
x
0
.Using the fact that δ
D
,
f
,
x
0
<
∞ and k
∈
0,1 we have(32)
lim
m
,
n
→
∞
⁡
D
f
n
x
0
,
f
n
+
m
x
0
=
0
,which implies that f
n
x
0 is right D-Cauchy.
Analogously, we have(33)
D
f
n
+
m
x
0
,
f
n
x
0
≤
δ
D
,
f
,
f
n
x
0
≤
k
n
δ
D
,
f
,
x
0
,which implies (34)
lim
m
,
n
→
∞
⁡
D
f
n
+
m
x
0
,
f
n
x
0
=
0
.Thus, f
n
x
0 is left D-Cauchy and, hence, it is D-Cauchy sequence in X
,
D
. By completeness of X
,
D there exists ω
∈
X such that lim
n
→
∞
⁡
f
n
x
0
=
ω. Since f is k-contraction, we have (35)
D
f
n
+
1
x
0
,
f
ω
≤
k
D
f
n
x
0
,
ω
,
D
f
ω
,
f
n
+
1
x
0
≤
k
D
ω
,
f
n
x
0
.Hence lim
n
→
∞
⁡
D
f
n
+
1
x
0
,
f
ω
=
0
=
lim
n
→
∞
⁡
D
f
ω
,
f
n
+
1
x
0
=
0
. So f
ω is another D-limit for the sequence f
n
x
0
. By the uniqueness of the limit in a quasi-J
S-metric space (Proposition 4) we have ω
=
f
ω. Now, if ω
′ is another fixed point of f with D
ω
,
ω
′
<
∞, then, as f is k-contraction, we have (36)
D
ω
,
ω
′
=
D
f
ω
,
f
ω
′
≤
k
D
ω
,
ω
′
,which implies that D
ω
,
ω
′
=
0. Similarly, using the fact that D
ω
′
,
ω
<
∞, we can prove that D
ω
′
,
ω
=
0
. Therefore, ω
=
ω
′.Since any quasi-metric space and any quasi-b-metric space is a quasi-J
S-metric space, we derive the following results.Corollary 27.
LetX
,
q be a complete quasi-metric space and let f
:
X
→
X be k-contraction mapping for some k
∈
0,1. Suppose that there exists x
0
∈
X such that(37)
sup
⁡
q
f
i
x
0
,
f
j
x
0
:
i
,
j
∈
N
0
<
∞
.Then f has a unique fixed point ω
∈
X. Moreover, the sequence f
n
x
0 converges to ω.Corollary 28.
LetX
,
d be a complete quasi-b-metric space and let f
:
X
→
X be k-contraction mapping for some k
∈
0,1. Suppose that there exists x
0
∈
X such that(38)
sup
⁡
d
f
i
x
0
,
f
j
x
0
:
i
,
j
∈
N
0
<
∞
.Then f has a unique fixed point ω
∈
X. Moreover, the sequence f
n
x
0 converges to ω.We can obtain a similar result in the context of complete dislocated quasi-metric spaces.


## 3. Ćirić Type Contraction in a Quasi-J
S-Metric Space

In this section, we consider the existence and uniqueness of fixed point for Ćirić type contraction in the setting of quasi-J
S-metric space.Definition 29.
Letf
:
X
→
X be a function and k
∈
0,1
. We say that f is a generalized k-quasi-contraction mapping if it satisfies the following condition:(39)
Q
:
D
f
x
,
f
y
≤
k
max
⁡
D
x
,
y
,
D
x
,
f
x
,
D
y
,
f
y
,
D
x
,
f
y
,
D
y
,
f
x
,for every x
,
y
∈
X.Proposition 30.
Suppose thatf is a generalized k-quasi-contraction mapping for some k
∈
0,1. If f has a fixed point ω
∈
X with D
ω
,
ω
<
∞, then D
ω
,
ω
=
0.Theorem 31.
LetX
,
D be a D-complete quasi-J
S-metric space with constant C and let f
:
X
→
X be a generalized k-quasi-contraction mapping for some k
∈
0,1. Suppose that there exists x
0
∈
X such that δ
D
,
f
,
x
0
<
∞
. Then f
n
x
0 converges to some ω
∈
X. If D
x
0
,
f
ω
<
∞, D
ω
,
f
ω
<
∞, and C
<
1, then ω is a fixed point of f. Moreover, if ω
′ is another fixed point of f with D
ω
,
ω
′
<
∞, D
ω
′
,
ω
<
∞, and D
ω
′
,
ω
′
<
∞, then ω
=
ω
′.Proof.
Letn
∈
N, since f is generalized k-quasi-contraction mapping, for all i
,
j
∈
N
0; we have(40)
D
f
n
+
i
x
0
,
f
,
n
+
j
x
0
≤
k
max
⁡
D
f
n
-
1
+
i
x
0
,
f
n
-
1
+
j
x
0
,
D
f
n
-
1
+
i
x
0
,
f
n
+
i
x
0
,
D
f
n
-
1
+
i
x
0
,
f
n
+
j
x
0
,
D
f
n
-
1
+
j
x
0
,
f
n
+
j
x
0
,
D
f
n
-
1
+
j
x
0
,
f
n
+
i
x
0
.As(41)
δ
D
,
f
,
f
n
-
1
x
0
=
sup
⁡
D
f
n
-
1
+
i
x
,
f
n
-
1
+
j
x
:
i
,
j
∈
N
0
,we have(42)
max
⁡
D
f
n
-
1
+
i
x
0
,
f
n
-
1
+
j
x
0
,
D
f
n
-
1
+
i
x
0
,
f
n
+
i
x
0
,
D
f
n
-
1
+
i
x
0
,
f
n
+
j
x
0
,
D
f
n
-
1
+
j
x
0
,
f
n
+
j
x
0
,
D
f
n
-
1
+
j
x
0
,
f
n
+
i
x
0
≤
δ
D
,
f
,
f
n
-
1
x
0
.Therefore, (40) yields δ
D
,
f
,
f
n
x
0
≤
k
δ
D
,
f
,
f
n
-
1
x
0
. Hence for any n
∈
N
0 we have (43)
δ
D
,
f
,
f
n
x
0
≤
k
n
δ
D
,
f
,
x
0
.Then for every m
,
n
∈
N
0 we find (44)
D
f
n
x
0
,
f
n
+
m
x
0
≤
δ
D
,
f
,
f
n
x
0
≤
k
n
δ
D
,
f
,
x
0
,
D
f
n
+
m
x
0
,
f
n
x
0
≤
δ
D
,
f
,
f
n
x
0
≤
k
n
δ
D
,
f
,
x
0
.Since δ
D
,
f
,
x
0
<
∞ and k
<
1, we derive (45)
lim
m
,
n
→
∞
⁡
D
f
n
x
0
,
f
n
+
m
x
0
=
lim
m
,
n
→
∞
⁡
D
f
n
+
m
x
0
,
f
n
x
0
=
0
.This implies that f
n
x
0 is both left and right D-Cauchy sequence and hence D-Cauchy sequence. By completeness of X there exists some ω
∈
X such that f
n
x
0 converges to ω; that is, lim
n
→
∞
⁡
D
f
n
x
0
,
ω
=
lim
n
→
∞
⁡
D
ω
,
f
n
x
0
=
0.
Note that as in (44) for any m
,
n
∈
N
0 we have (46)
D
f
n
x
0
,
f
m
x
0
≤
δ
D
,
f
,
f
n
x
0
≤
k
n
δ
D
,
f
,
x
0
,
D
f
m
x
0
,
f
n
x
0
≤
δ
D
,
f
,
f
n
x
0
≤
k
n
δ
D
,
f
,
x
0
.
Now, assume thatD
ω
,
f
ω
<
∞ and D
x
0
,
f
ω
<
∞. Then, by using (46) and condition (D2) there exists C
>
0 such that (47)
D
ω
,
f
n
x
0
≤
C
limsup
m
→
∞
⁡
D
f
m
x
0
,
f
n
x
0
≤
C
k
n
δ
D
,
f
,
x
0
,
D
f
n
x
0
,
ω
≤
C
limsup
m
→
∞
⁡
D
f
n
x
0
,
f
m
x
0
≤
C
k
n
δ
D
,
f
,
x
0for every n
∈
N
0.
On the other hand, asf is generalized k-quasi-contraction mapping we have(48)
D
f
x
0
,
f
ω
≤
k
max
⁡
D
x
0
,
ω
,
D
x
0
,
f
x
0
,
D
ω
,
f
ω
,
D
f
x
0
,
ω
,
D
x
0
,
f
ω
.By using (46) and (47) we have(49)
D
x
0
,
ω
≤
C
k
0
δ
D
,
f
,
x
0
=
C
δ
D
,
f
,
x
0
,
D
x
0
,
f
x
0
≤
k
0
δ
D
,
f
,
x
0
=
δ
D
,
f
,
x
0
,
D
f
x
0
,
ω
≤
C
k
δ
D
,
f
,
x
0
<
C
δ
D
,
f
,
x
0
. Hence, we derive (50)
D
f
x
0
,
f
ω
≤
max
⁡
k
C
δ
D
,
f
,
x
0
,
k
δ
D
,
f
,
x
0
,
k
D
ω
,
f
ω
,
k
D
x
0
,
f
ω
.Again by using the fact that f is generalized k-quasi-contraction mapping and the technique used above, we observe that (51)
D
f
2
x
0
,
f
ω
≤
max
⁡
k
2
C
δ
D
,
f
,
x
0
,
k
2
δ
D
,
f
,
x
0
,
k
D
ω
,
f
ω
,
k
2
D
x
0
,
f
ω
.
By continuing in the same manner, we deduce(52)
D
f
n
x
0
,
f
ω
≤
max
⁡
k
n
C
δ
D
,
f
,
x
0
,
k
n
δ
D
,
f
,
x
0
,
k
D
ω
,
f
ω
,
k
n
D
x
0
,
f
ω
,for every n
∈
N. Now, as D
x
0
,
f
ω
<
∞ and δ
D
,
f
,
x
0
<
∞, we have(53)
limsup
n
→
∞
⁡
D
f
n
x
0
,
f
ω
≤
k
D
ω
,
f
ω
.Regarding the condition (D2) and the fact that C
<
1 and k
<
1, we get(54)
D
ω
,
f
ω
≤
C
limsup
n
→
∞
⁡
D
f
n
x
0
,
f
ω
≤
C
k
D
ω
,
f
ω
<
D
ω
,
f
ω
.Thus, D
ω
,
f
ω
=
0. By analogy, as in the above, we can conclude that(55)
D
f
ω
,
ω
<
D
ω
,
f
ω
=
0
.Hence, we have D
ω
,
f
ω
=
D
f
ω
,
ω
=
0. Therefore, we get ω
=
f
ω; that is, ω is a fixed point of f.
Now, ifω
′ is another fixed point of f with D
ω
′
,
ω
<
∞, D
ω
,
ω
′
<
∞, and D
ω
′
,
ω
′
<
∞, then as f is generalized k-quasi contraction mapping we have(56)
D
ω
′
,
ω
=
D
f
ω
′
,
f
ω
≤
k
max
⁡
D
ω
′
,
ω
,
D
ω
′
,
ω
′
,
D
ω
,
ω
,
D
ω
,
ω
′as D
(
ω
,
f
(
ω
)
)
=
D
(
ω
,
ω
)
<
∞ and D
(
ω
′
,
ω
′
)
<
∞. So, by Proposition 30, we have (57)
D
ω
,
ω
=
D
ω
′
,
ω
′
=
0
,which implies that D
ω
′
,
ω
≤
k
max
⁡
{
D
ω
′
,
ω
,
D
(
ω
,
ω
′
)
}. In a similar way, we derive that (58)
D
ω
,
ω
′
≤
k
max
⁡
D
ω
′
,
ω
,
D
ω
,
ω
′
,so that (59)
max
⁡
D
ω
′
,
ω
,
D
ω
,
ω
′
≤
k
max
⁡
D
ω
′
,
ω
,
D
ω
,
ω
′
.Since D
ω
,
ω
′
<
∞, D
ω
′
,
ω
<
∞, and k
∈
0,1, we deduce that D
ω
′
,
ω
=
D
ω
,
ω
′
=
0 which yields ω
=
ω
′.


## 4. Fixed Point Theorems in Quasi-J
S-Metric Space with Partial Order

Definition 32.
LetX
,
D be a quasi-J
S-metric space with partial order ⪯ and let f
:
X
→
X be a mapping. We say that f is weakly continuous if the following condition holds: if x
n
⊆
X is D-convergent to x
∈
X, then there exists a subsequence x
n
k of x
n such that f
x
n
k is D-convergent to f
x as k
→
∞.Definition 33.
LetX be a nonempty set with partial order ⪯. A mapping f
:
X
→
X is said to be nondecreasing if (60)
x
⪯
y
⟹
f
x
⪯
f
y
where
x
,
y
∈
X
.Definition 34.
The pairX
,
⪯ is said to be D-left regular (resp., D-right regular) if the following condition holds: for every sequence x
n
⊆
X satisfies x
n
+
1
⪯
x
n (x
n
⪯
x
n
+
1) for each n
∈
N, with x
n being D-convergent to x
∈
X; then there exists a subsequence x
n
k of x
n such that x
⪯
x
n
k (x
n
k
⪯
x) for every k
∈
N.
The pairX
,
⪯ is said to be D-regular if and only if it is left and right D-regular.Definition 35.
A functionf
:
X
→
X is said to be weakly k-contraction if x
⪯
y or y
⪯
x implies (61)
D
f
x
,
f
y
⪯
k
D
x
,
y
.That is, whenever x
,
y
∈
X are comparable condition (61) is satisfied.Theorem 36.
LetX
,
D be a quasi-J
S-metric space with partial order ⪯ and let f
:
X
→
X be a function. Assume that the following conditions hold:(i)
X
,
D is D-complete;
(ii)
f is weakly continuous;
(iii)
f is weakly k-contraction for some k
∈
0,1;
(iv)
f is nondecreasing;
(v)
there existsx
0
∈
X such that δ
D
,
f
,
x
0
<
∞ and x
0
⪯
f
x
0.
Thenf has a fixed point ω
∈
X and f
n
x
0 is D-convergent to ω. Moreover, if D
ω
,
ω
<
∞, then D
ω
,
ω
=
0.Proof.
Sincef is nondecreasing and x
0
⪯
f
x
0, then for every n
∈
N
0 we have(62)
f
n
x
0
⪯
f
n
+
1
x
0and by the transitivity of ⪯, for every p
,
q
∈
N
0, we have(63)
p
≤
q
⟹
f
p
x
0
⪯
f
q
x
0
.Therefore, for each n
∈
N,  i
,
j
∈
N
0, f
n
+
i
x
0 and f
n
+
j
x
0 are always comparable. As f is weak k-contraction for each n
∈
N,  i
,
j
∈
N
0, we have(64)
D
f
n
+
i
x
0
,
f
n
+
j
x
0
⪯
k
D
f
n
-
1
+
i
x
0
,
f
n
-
1
+
j
x
0
.Hence(65)
δ
D
,
f
,
f
n
x
0
≤
k
δ
D
,
f
,
f
n
-
1
x
0
.
Thus, for everyn
∈
N
0 we have(66)
δ
D
,
f
,
f
n
x
0
≤
k
n
δ
D
,
f
,
x
0
.
Using the above inequality we have for everym
,
n
∈
N
0
(67)
D
f
n
x
0
,
f
n
+
m
x
0
≤
δ
D
,
f
,
f
n
x
0
≤
k
n
δ
D
,
f
,
x
0
,
D
f
n
+
m
x
0
,
f
n
x
0
≤
δ
D
,
f
,
f
n
x
0
≤
k
n
δ
D
,
f
,
x
0
.Since δ
D
,
f
,
x
0
<
∞ and k
∈
0,1 we have(68)
lim
m
,
n
→
∞
⁡
D
f
n
x
0
,
f
n
+
m
x
0
=
lim
m
,
n
→
∞
⁡
D
f
n
+
m
x
0
,
f
n
x
0
=
0
,which means that f
n
x
0 is right and left D-Cauchy and hence D-Cauchy sequence. By the completeness of X
,
D there exists ω
∈
X such that f
n
x
0 is D-convergent to ω. Since f is weakly continuous, there exists a subsequence f
n
k
x
0 of f
n
x
0 such that f
n
k
+
1
x
0 is D-convergent to f
ω as k
→
∞. By the uniqueness of the limit in a quasi-J
S-metric space we have ω
=
f
ω and ω is a fixed point of f.
Now, ifD
ω
,
ω
<
∞ then as ω
⪯
ω and f is weak k-contraction we have(69)
D
ω
,
ω
=
D
f
ω
,
f
ω
≤
k
D
ω
,
ω
,which is possible only if D
ω
,
ω
=
0.The weak continuity condition off in the previous theorem can be replaced by the regularity of the pair X
,
⪯ as in the following result.Theorem 37.
LetX
,
D be a quasi-J
S-metric space with partial order ⪯ and let f
:
X
→
X be a function. Assume that the following conditions hold:(i)
X
,
D is D-complete;
(ii)
X
,
⪯ is regular;
(iii)
f is weakly k-contraction for some k
∈
0,1;
(iv)
f is nondecreasing;
(v)
there existsx
0
∈
X such that δ
D
,
f
,
x
0
<
∞ and x
0
⪯
f
x
0.
Thenf has a fixed point ω
∈
X and f
n
x
0 is D-convergent to ω. Moreover, if D
ω
,
ω
<
∞, then D
ω
,
ω
=
0.Proof.
Following the steps of the previous proof we can prove thatf
n
x
0 is D-convergent to ω
∈
X
. Moreover, we have(70)
f
n
x
0
⪯
f
n
+
1
x
0for every n
∈
N
0. Since X
,
⪯ is regular it is right regular and so there exists a subsequence f
n
k
x
0 of f
n
x
0 such that f
n
k
x
0
⪯
ω for each k
∈
N. As f is weakly k-contraction we have(71)
D
f
n
k
+
1
x
0
,
f
ω
≤
D
f
n
k
x
0
,
ω
.Using the inequality above, we get(72)
lim
k
→
∞
⁡
D
f
n
k
+
1
x
0
,
f
ω
=
0
.Similarly, we can prove that (73)
lim
k
→
∞
⁡
D
f
ω
,
f
n
k
+
1
x
0
=
0
,which implies that f
ω is a D-limit for the sequence f
n
k
+
1
x
0. By Proposition 4,  f
ω
=
ω; and ω is a fixed point of f. As in the previous proof(74)
D
ω
,
ω
<
∞
⟹
D
ω
,
ω
=
0
.Example 38.
LetX
,
D be the complete quasi-J
S-metric space introduced in Example 7. Define the function f
:
X
→
X by (75)
f
x
=
∞
if
x
=
∞
x
2
o
t
h
e
r
w
i
s
e
.Then clearly f is a k-contraction mapping with k
=
1
/
2. Let x
0
=
0 then δ
D
,
f
,
x
0
<
∞. So by Theorem 26  f has a fixed point.



---

*Source: 6963041-2016-04-28.xml*