Common Fixed Point Results Related to Generalized F-Contractions in
Extended Cone b-Metric Spaces
ZAMIR SELKO1, ERIOLA SILA2
1Department of Mathematics,
“Aleksandër Xhuvani” University,
Elbasan,
ALBANIA
2Department of Mathematics,
University of Tirana,
Tirane,
ALBANIA
Abstract: - This paper presents a new class of F-functions defined on a cone and proves some theorems
showing the uniqueness and existence of common fixed points for two functions satisfying a generalized F
nonlinear contractions condition in extended cone b-metric spaces. Several examples illustrate the main
theorems and demonstrate the applicable side of theoretical results.
Key-Words: - Generalized F-contraction, Fixed point, Cauchy sequences, Extended cone metric space,
Convergent sequences.
Received: November 25, 2023. Revised: March 17, 2024. Accepted: April 11, 2024. Published: May 10, 2024.
1 Introduction
In 2007, [1], restructured the concept of metric
spaces by introducing cone metric spaces, wherein
the traditional real numbers were replaced with an
ordering Banach space. Through their pioneering
work, they established several fixed-point theorems
for contractive mappings within these spaces,
effectively extending analogous results previously
established in conventional metric spaces. This
innovative approach not only broadened the scope
of metric space theory but also provided a fresh
perspective on the convergence properties of
mappings in the realm of cone metric spaces.
Based on the metric cone spaces, many authors have
generalized them and studied the results of fixed
points in them, as in [2], [3], [4].
Among the generalized cone metric spaces are
the extended b-metric cone spaces conceived by [5],
and [6]. The authors relied on the concept of
extended b-metric spaces given by [7] and studied in
them the existence and uniqueness of fixed points
for Kannan contractions, [8]. Recently, these spaces
have been placed as the focus of study for some
mathematicians, such as in [9] and [10].
In 2012, [11], introduced a new contraction to
fixed point theory by introducing the concept of F-
contraction. This new contraction was studied by
many other authors in different metric spaces such
as in [12], [13], [14], [15], [16], [17], [18], [19],
[20].
In this paper, a generalization of F-contractions
in extended cone metric spaces is given and the
existence and uniqueness of common fixed points
for two functions that complete this generalized F-
contraction are studied. Also, a result on the
existence and uniqueness of a fixed point for a
contraction where an ultra-altering function is used
is verified. The methodology used throughout this
paper is proof. To prove the main results, we use
Cauchy and convergent sequences, respectively.
Concrete examples accompany the main results of
the paper. In addition, our results generalize some
theorems of given references.
2 Preliminaries
Definition 2.1. [1] Let be a non-empty subset of
where is an ordered Banach space. The set is
called cone if and only if:
(i) is closed, nonempty, and
(ii)
(iii) implies
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When a cone is given, a partial ordering
concerning P is defined by the relation if
and only if To indicate but
we denote while will stand for
where int P denotes the interior of P.
The cone is called normal if, there is a positive
real number such that, for all we have:
The last positive number satisfying the above is
called the normal constant of
The cone P is called regular if every increasing
sequence that is bounded from above is convergent.
That is, if is a sequence such that
for some such that
∞ Equivalently, the cone P is
regular if and only if any decreased sequence that is
bounded from below is convergent. A regular cone
is a normal cone.
In the following, we always suppose E is a
Banach space, P is a cone in E with and
is partial ordering concerning P.
Giving generalizations of metric spaces has
been an open challenge for mathematicians. One of
the most interesting generalizations of metric spaces
was introduced in [1].
Definition 2.2. [2], Let be a cone and X a non-
empty set. The function is called a
cone metric if it satisfies the following conditions:
(c1) for and
iff
(c2) for all
(c3) for all
The pair is called a cone metric space.
The concept of cone metric space and fixed
point theory on these spaces has been developed by
many authors in their works.
In 1998, [21], introduced the following
interesting concept.
Definition 2.3. [21], Let X be a non-empty set and
be a given real number. A function
is called a b-metric if, for all it
satisfies the conditions:
(b1)
(b2)
(b3)
The pair is called b-metric space with
parameter s.
Authors in [6], introduced a new type of
generalized metric space by taking a two-variables
function instead of the parameter s.
Definition 2.4. [6], Let X be a nonempty set and
∞ A function
is an extended b-metric, if for all
it satisfies:
( 1)d
iff
( 2)d
for all
( 3)d
for
all
The pair is called an extended b-metric
space.
Authors in [5], in their generalization, extended
the domain of the function from to
thus giving this definition as follows.
Definition 2.5. [5], Let X be a non-empty set, and
∞ Let ℝ be a
function that satisfies the following conditions:
( 1)d
for all x, y and iff
( 2)d
for all x, y in X,
( 3)d
for
all x, y, z in X.
The function is called extended cone metric
on X and the pair is called extended cone
metric space.
Convergence, Cauchy sequences, continuity,
and completeness on extended cone metric spaces
are defined as follows:
Definition 2.6. [5], Consider a sequence in an
extended cone metric space (X, d) and let P be a
normal cone in E with normal constant K.
Then
(i) converges to x if for every with c > 0,
there exists such that for all
< c. Denoted by
∞ ∞
(ii) is said to be Cauchy in X if for every
with c > 0, there exists a positive integer such that
for all < c.
(iii) The mapping is said to be continuous
at a point if for every sequence {}
converging to x it follows that
∞
∞
(iv) (X, d) is said to be a complete cone metric space
if every Cauchy sequence is convergent in X.
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3 Main Results
In this section, we present a new class of functions
denoted where every element satisfies
the following conditions:
(1) is strictly increasing
(2) For every sequence the following
equivalence holds:
∞∞
∞
(3) For every sequence where
∞
there exists a number such that
∞∞
(4) If is a sequence that
then
Theorem 3.1. Let be a complete extended
cone metric space and two functions
that satisfy the following implication:
If then
(1)
where
and is a convergent function used in extended b-
metric for each .
Then S and T have a unique common fixed point.
Proof. Let be an arbitrary point in X. Define the
sequence by taking
21
,x Sx
and so on, or, more generally,
and
Beginning with the inequality expressed in
inequality (1), we proceed:
(2)
We observe that and
are respectively:
(3)
meanwhile,
Applying equalities (3) and (4) over (1) we derive
the following inequality
(6)
for all (6)
We distinguish the following cases:
Case 1.
If
inequality (5) takes the form:
Thus, for our last inequality is derived:
For the following assessments hold:
Continuing iteratively, it is observed that the
function F satisfies the following condition:
(7)
Taking the limit on both sides as tends toward
infinity, we ascertain that:
∞
∞
\Exploiting the condition (2) from the determination
of the function F, it follows that:
∞
∞
(8)
The fulfillment of this equation along with the
condition (3) of the function F implies that:
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∞
Now, we multiply both sides of the above inequality
(6) by
thus obtaining:
Taking limits on both sides of this inequality
and using the equalities (7) and (8) it follows that:
∞
By employing the definition of the convergent
sequence, we note that for, there exists
such that for all we have:
Therefore, we obtain this inequality:
for each
Below, we demonstrate that is a Cauchy
sequence on the extended cone metric space
To do so, for we can derive that:
.
Taking the norm of both sides, we obtain
∞
.
As a consequence:
∞
∞
Then, is a Cauchy sequence on Given
the completeness of , there exists a point
such that ∞
Subsequently, we must demonstrate that is a
common fixed point of the functions S and T. Since
F is strictly increasing, we can write that
From here it
follows that:
where
and
Taking the limits on both sides of (10) we derive
that:
∞
∞
or,
Consequently, Hence,
or
Similarly, if we see the inequality
we derive that
or As a consequence,
thus, is a common fixed point of
S and T. Finally, we must show that is the unique
common fixed point for S and T. Suppose that there
exists another point such
that We start from the inequation
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where,
and
After replacing
and in the last inequality, we
get:
which implies:
Thus This
inequality can be true only if or
Then the fixed point is unique.
Example 3.2. Given the sets
and Define
by
where ,
and such that
, Also, let T and S be
respectively given by
and
Firstly, we derive that
Evaluating
the distances between respective points, we have
On the other hand, we notice that
,
,
,
and
,
Theorem 3.3. Let be a complete extended
cone metric space and a function that
satisfies the following implication
If then
(12)
where
and is a convergent function used in extended b-
metric for each .
Then T has a unique fixed point.
Proof. Taking the function in inequality (1)
we obtain the condition (12). As a result, the
function has a unique fixed point in .
Example 3.4 Let us take the sets
and Define by
,
and such that , .
Let T be given by
.
Initially we see that for a
Calculating the distances between respective points
we have ,
,
,
, and
,
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Then, make the following comparison between
and
.
Hence, the inequality (1) holds, then the functions
have a common fixed point .
Then, making the following comparison, we obtain
Since the inequality (12) holds, then the function
has a unique fixed point
.
Theorem 3.5 Let be a complete extended
cone metric space and two functions
that satisfy the following inequality
If then
(13)
for every , where is a function
which satisfies ,
and is a
convergent function used in extended b-metric for
each .
Then S and T have a unique common fixed point.
Proof. We use and the fact
that the function is strictly increasing in inequality
(13). As a consequence, we derive
for every . Using the same scheme of proof
as Theorem 3.1, the result is clear.
Theorem 3.6 Let be an extended cone b-
metric space and two functions that satisfy the
following inequality
(14)
for all and is a
sublinear altering, with
Then have a common fixed point.
Proof. Let's construct the sequence as in the
proofing procedure of Theorem 3.1 by choosing
an arbitrary point in X. Define the sequence by
taking
21
,x Sx
and
so on, or, more generally, and
We can see that for each the following
inequality holds
where
for all
Thus, we obtain this inequality
Replacing in the final inequality, we derive
which implies
For the following assessments, hold
To summarize, we have:
By using this procedure iteratively, we obtain:
Taking the limit when in inequality (17)
(18)
we have:
(19)
Now, by leveraging the third property of function ,
there exists such that:
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Multiplying (19) in both sides with:
we get
Since the map does not take negative values, as a
result, we conclude that
from which we can say
that for there exists such that, for all
with we get
or,
Next, let’s show that the sequence is a Cauchy
sequence. Supposing we have
Applying the property of we obtain:
which implies:
From the above result, it follows that
which means that is
Cauchy. By using the completeness of
there exists such that
Now, let us show that is a common fixed point of
S and T.
Applying the inequality from the condition of the
theorem for the triple we have
which implies:
Taking the limits on both sides in the inequality (23)
for we obtain:
Since
is increasing, it follows that
or
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Next, using the fact that we obtain
, and
Similarly, applying the inequality for the triple
we have
Following the same procedure as in the case of
the function S, we obtain showing
that is a common fixed point of S and T.
To complete the proof, we must show the
uniqueness of
If is another common fixed point of S
and T, then Using the condition of
theorem we have
which implies:
and
which follows
Since , the last
inequality holds only when which
implies , and the proof is done.
4 Conclusions
In this paper, we present some fixed point results for
functions that satisfy several inequalities related to
generalized F-contraction in extended cone b-metric
space are studied. Theorem 3.1 and Theorem 3.6 are
the highlights of this study. Theorem 3.1 is a
generalization of Theorem 2.4 in [16], since we
have proved the existence and uniqueness of a
common fixed point for two functions in extended
cone b-metric space, which is more general than
metric space. Theorem 3.3 generalizes Theorem 2.8
in [22], because of the use of maximum and
minimum respective distances in the function F in
extended cone b-metric space. Theorem 3.6 is an
extension of Theorem 2.1 in [23], using the
generalized function. In future work, authors
recommend the applications of their results to
Integral Equations to prove the existence and
uniqueness of various equations.
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Conflict of Interest
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