Generalized almost Contractions on Extended Quasi-Cone B-Metric
Spaces
SILVANA LIFTAJ1, ERIOLA SILA2, ZAMIR SELKO3
1Department of Mathematics,
“Aleksander Moisiu” University,
Lagjja 1, Rruga e Currilave, Durrës, 2001, Shqipëri,
Kampusi i Ri Universitar, Rr. Miqësia, Spitallë, Durrës, 2009, Shqipëri,
ALBANIA
2Department of Mathematics,
University of Tirana,
Bulevardi “Zogi i pare”, Tirana, 1057,
ALBANIA
3Department of Mathematics,
“Aleksander Xhuvani” University,
Rruga”Ismail Zyma”, Elbasan,
ALBANIA
Abstract: - Fixed Point Theory is among the most valued research topics nowadays. Over the years, it has been
developed in three directions: by generalizing the metric space, by establishing new contractive conditions, and
by applying its results to various fields such as Differential Equations, Integral Equations, Economics, etc. In
this paper, we define a new class of cone metric spaces called the class of extended quasi-cone b-metric spaces.
Extended quasi-cone b-metric spaces generalize cone metric spaces and quasi-cone b-metric spaces. We have
studied topological issues, such as the right and left topologies, right (left) Cauchy, and convergent sequences.
Furthermore, there are determined generalized τ-almost contractions, which extend the almost contractions. The
highlight of this study is the investigation of the existence and uniqueness of a fixed point for some types of
generalized τ-almost contractions in extended quasi-cone b-metric space. We prove some corollaries and
theorems for known contractions in extended quasi-cone b-metric spaces. Our results generalize some known
theorems given in literature due to the new cone metric spaces and contractions. Concrete examples illustrate
theoretical outcomes. In addition, we show an application of the main results to Integral Equations, which
provides the applicative side of them.
Key-Words: - Extended quasi cone b-metric space, fixed point, Cauchy sequence, generalized almost
contraction, Convergent sequence, Right (left) open balls, Right (left) topology.
Received: July 23, 2023. Revised: October 25, 2023. Accepted: November 4, 2023. Published: November 29, 2023.
1 Introduction
The mathematician in, [1], defined the cone metric
by generalizing the metric by replacing the set of
nonnegative real numbers with a cone. Later, in, [2],
authors extended this concept to quasi-cone metric
and proved some great fixed-point results that
generalized Banach's contraction in quasi-cone
metric space. Initiating by, [3], in which were
studied the concept of b-metric space by modifying
the triangle inequality of a metric, the authors in,
[4], established cone b-metric space, investigated its
topological properties, and obtained some
approximating fixed-point results in these spaces.
Recently, in, [5], there is determined a new space
called extended b-metric space. Many authors have
surveyed fixed points on these spaces, [6], [7], [8],
[9], [10], [11].
Encouraged by their results, in this paper, we
study the extended quasi-cone b-metric space. We
have set out left and right Cauchy sequences,
convergent sequences, and some topological points
in this space. Also, we prove several fixed points
results for generalized almost contractions. In
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addition, we give an application in Integral
Equations of our main theorem.
2 Preliminaries
Definition 2.1. [1], Let be a subset of , where
is an ordered Banach space. The set is said to be
a cone if it satisfies the following conditions:
1. ,
2. for each and
,
3. if then .
The cone is called normal if for every,
then , where . is called
the normality constant of .
The authors in, [1], have defined a partial ordering
relation in cone as follows.
For each if and
if and ; for every only
if
Definition 2.2. [1], Let be a cone and a non–
empty set. The map is called a cone
metric if it satisfies the following conditions:
1. if and only if , for every
,
2. for every ,
3. for each
The ordered couple is called a cone metric
space.
The authors in, [2], generalized the cone metric
space to quasi-cone metric space as follows:
Definition 2.3. [2], Let be a cone and a
nonempty set. The map is called a
quasi-cone metric if it satisfies the following
conditions:
1. if and only if , for every
,
2. for each
.
The ordered couple is called a quasi-cone
metric space.
In, [5], there were extended b-metric spaces to
extended b-metric spaces by replacing the third
condition of the metric.
Definition 2.4. [5], Let be a nonempty set and
be a function. The mapping
is called extended b-metric
space if it satisfies the following conditions:
1. if and only if , for every
;
2. for every ,
3. for
each .
The pair is called an extended b-metric
space.
Inspired by authors in, [2], and, [5], we define a new
generalization of quasi-cone space as below:
Definition 2.5. Let be a cone,
be a function, and a nonempty set. The mapping
is called extended quasi-cone b-
metric if it satisfies the following conditions:
1. if and only if , for every
,
2. for
each
The pair is called an extended cone b-metric
space.
Example 2.6. Let be an ordered
Banach space where is the Euclidian norm and
is a cone, . Define
, and
,
,
and
.
We see that the function satisfies both condition
1 and 2 of extended quasi-cone b-metric space. As a
result, is an extended quasi-cone b-metric, and
the couple is an extended quasi-cone b-
metric space, but it is not a quasi-cone metric space.
Remark 2.7. If
for each , where is constant,
then the pair is a quasi-cone b-metric space.
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Below, we present some topological aspects of
extended quasi-cone b-metric spaces.
Let be an ordered Banach space, a regular
cone, and normal in with normality constant
and an extended quasi-cone b-metric
space.
We take , where .
Definition 2.8. The set
(
) is called the right (left) open ball with center and
radius .
Definition 2.9. The set
(
) is called the right (left) closed ball with center
and radius .
Theorem 2.10. Let be an extended quasi-
cone b-metric space. The family
G} is a
topology in .
The topology is is called right topology induced
by the extended quasi-cone b-metric .
Similarly, we can define the left topology induced
by the extended quasi-cone b-metric .
We establish the following results for the right
topology in .
Definition 2.11. The set is called open if
.
Definition 2.12. The set is called the right
neighborhood of if there exists an open ball
centered in such that (.
Remark 2.13. The topology in satisfies
the First Axiom of Countability.
Theorem 2.14. The set is right open if and
only if, for each point , there exists
such that .
Definition 2.15. The set is right closed if its
complement in is right to open.
Theorem 2.16. The space is T1.
Proof. We have to show that for each , the set
is right closed, or the set is right open.
Taking , then .
Denote and since there exists
such that
. We prove that
. Considering
, then
. As a result, ,
and. Consequently, is T1.
Definition 2.17. Let be an extended quasi-
cone b-metric space and a sequence in.
1. The sequence is called right (left)
convergent to if, for every ,
there exists , such that for each
, it implies (
) or
.
2. The sequence is called bi-convergent
to if it is right and left convergent to
.
3. The sequence is called right (left)
Cauchy in if, for every , there
exists , such that for each
it yields (
or
).
4. The sequence is called bi-Cauchy in
if it is right and left Cauchy in .
5. The extended quasi-cone b-metric space
is called complete if every bi-
Cauchy sequence in is bi-convergent.
Remark 2.18. There exist sequences that are left
Cauchy or right Cauchy but not bi-Cauchy.
Example 2.19. Taking
and considering
for and
, the couple is an extended quasi-
cone b-metric space.
Let
be a sequence in . For ,
we have
, which
shows that the sequence
is not right Cauchy.
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For , we have
which proves that the
sequence
is left Cauchy.
Authors in, [12], presented a new class of weak
contractions called almost contraction:
Let be a metric space and a
function. is called almost contraction or -
contraction if it satisfies the following inequality:
for each , and .
In, [13], extended this contraction to generalized
almost contraction.
Let be a metric space and a
function. is called generalized almost contraction
if it completes the following inequality:
for each , and .
He proved the existence and the uniqueness of
fixed points on the respective contractions in metric
space.
Definition 2.20. [14], The function is
called a comparison function if it satisfies the
following conditions:
1. for each ,
2.
for each .
Initiating from above, we determine a new class
of generalized almost contractions called τ-almost
contraction, as follows:
Denote
and
.
The function is called a generalized τ-almost
contraction if it satisfies the following inequality:
(1)
for every where is a comparison
function, and .
3 Main Results
3.1 Fixed Point Results
In this section, we show some fixed-point results for
generalized τ-almost contractions in extended quasi-
cone b-metric space.
Lemma 3.1.1. Let be a complete and
Hausdorff extended quasi-cone b-metric space, with
the normality constant of cone , and a
generalized τ-almost contraction. Let be a
point in such that is bounded. Then, for
each , the inequalities
(2) and (3) hold,
where .
Proof. Since the orbit is bounded, there exits
such that .
To prove that , for each
, we use the mathematical induction method.
For , we have
.
Since the orbit of a point is bounded,
, we have
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.
As a result, the inequality
holds.
Suppose that for
is true.
Let's prove the inequality for .
We see that the extended quasi-cone metric
satisfies the following inequality:
.
Considering the values of
, we
have the following cases:
If
, then we have:
, we get
If then
.
Consequently, for every the inequality
holds.
In addition, completes the following
inequality:
So, the inequality is
true for each .
Theorem 3.1.2. Let be a complete and
Hausdorff extended quasi cone b-metric space, with
the normality constant of cone , and is
generalized τ-almost contraction. Let be a
point in such that is bounded,
and
. Then, the
function has a unique fixed point in .
Proof. Defining the sequence , we
consider the following cases.
If there exists , , then the
function has a fixed point .
Suppose that , for each .
Using Lemma 3.1.1, we have that for all
.,
.
Firstly, we prove that the sequence is
bi-Cauchy (right and left Cauchy).
Taking , and using step-by-step the third
condition of extended quasi-cone metric, yields
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.
Taking the norm of both sides and using the
normality property of the cone, we obtain
As a result, we have shown that:
,
where is the sequence of partial sums of
series
.
This series is convergent since it satisfies the
D'Alembert Criterion.
.
In addition
and
.
So, we proved that the sequence is left
Cauchy.
Using the same method, we demonstrate that the
sequence is right Cauchy and it is Cauchy.
Since the space is complete, the sequence
converges to a point , and
.
We must show that is a fixed point of mapping .
Taking the limits of both sides, we derive to
.
Similarly, we can show that
.
Consequently, the sequence
converges to the point . Since the
space is Hausdorff, it yields that .
In the end, we prove that is the unique fixed
point of .
Suppose that there exists another fixed point
, of , .
We see that
.
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This inequality shows that , and
.
The following example illustrates Theorem
3.1.2.
Example 3.1.3 Considering , the
cone with normality
constant , ,
, and
,
the couple is an extended quasi cone b metric
space.
Define
and
,
which is a comparison
function.
Considering
, we obtain
.
Furthermore, we achieve
.
In addition, taking into consideration
, the following equalities hold:
,
,
,
and
.
The next step is to demonstrate that function
satisfies the inequality (1) of Lemma 3.1.1 by taking
into account the following cases:
If
, we obtain
For
, we get
.
Considering
, it yields
.
Taking
, we have
.
For
, we get the following
inequality
.
If
, it comes
.
As a result, we have that for every
, the inequality
holds.
The function satisfies the conditions of
Theorem 3.1.2, and it has a unique fixed point
.
Corollary 3.1.4. Let be a complete and
Hausdorff extended quasi cone b-metric space, with
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the constant of normality of cone , and a
function that satisfies the generalized almost
contraction:
(4)
for every where , and .
Let be a point in such that is
bounded and
. Then, the function has a unique fixed point in
.
Proof. Taking in Theorem
3.1.2, we prove Corollary 3.1.4.
Remark 3.1.5. Corollary 3.1.4 extends the result of
[14], in extended quasi-cone metric space.
Theorem 3.1.6. Let be a complete and
Hausdorff extended quasi cone b-metric space, with
the constant of normality of cone , and X→X a
function that satisfies the nonlinear contraction:
(5)
for every where is a comparison
function,
Let be a point in such that is
bounded
and
. Then, the
function has a unique fixed point in .
Proof. For every , we have that
.
As a result, the function satisfies the
inequality (1), and it has a unique fixed point in .
Example 3.1.7. Let be the segment [0,1],
and } is a
cone. Determine ,
,
where , .
The pair is an extended quasi-cone
metric space. Let ,
be a function, and ,
is a comparison function.
The function satisfies the conditions of
Theorem 3.1.6. Consequently, it has a unique
fixed point in .
Corollary 3.1.8. Let be a complete and
Hausdorff extended quasi cone b-metric space, with
a constant of normality of cone , and a
function that satisfies the contraction:
(6)
for every where . Let be
a point in such that is bounded
and
. Then,
the function has a unique fixed point in .
Remark 3.1.9. Corollary 3.1.8 extends the result in,
[11], on extended quasi b-cone metric space.
3.2 An Application to Integral Equation
Fixed Point Theory has a huge application to
Integral equations, where it guarantees the existence
and uniqueness of the solution. These applications
are studied by many authors who have contributed
to Fixed Point Theory, [15], [16], [17].
Considering
and given by
where , the
couple is a completely extended quasi-cone
b-metric space.
Theorem 3.2.1. The integral equation
, where
.
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and is a continuous function,
and are
continuous functions which satisfy the following
conditions:
1.
for all
2. ;
has a unique solution in .
Proof. Define the mapping given by
.
Below, we show that the mapping satisfies the
conditions of Corollary 3.1.5.
Firstly, we see that:
Hence, we have
.
Consequently, the function completes the
conditions of Corollary 3.1.5, and it has a unique
fixed point. This result leads to proof of the
existence and uniqueness of the solution of integral
equation
.
4 Conclusion
In this paper, we have defined new extended quasi-
cone b-metric spaces that generalize quasi-cone b-
metric spaces and cone metric spaces. There are
proven several topological properties of these
spaces. The highlight of this paper is Theorem 3.1.2,
which guarantees- the existence and uniqueness of a
fixed point for a generalized almost contraction in
extended quasi-cone b-metric space. This crucial
result extends Theorem 2 in, [5], Theorem 1 in,
[12], Theorem 2.1 in, [14], on ordered metric
spaces, and Theorem 3 in, [18]. The main
theoretical result is associated with an example that
shows its applicable side. Theorem 3.2.1 shows an
application of Fixed-Point Theory to Integral
Equations. It proves that there exists a unique
solution for an integral equation. According to this
application, this study contributes to Integral
Equations
As a further study of this paper, the readers can
see with interest the application of the main results
section to Differential Equations.
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9.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Silvana Liftaj has given the idea and proved a
significant part of section 3.1.
- Eriola Sila has defined the extended quasi-cone b-
metric space, topology, and Example 3.1.3.
- Zamir Selko has found the application of fixed
point results to Integral Equations.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that they are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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