Some Fixed-Point Results on
-Cone Metric Spaces
Jerolina Fernandez
Department of Mathematics and Statistics, The Bhopal School of Social Sciences, Bhopal, India
Keywords: Fixed Point,
-Cone Metric Space, Expansive Mapping.
Abstract: In mathematical analysis, diverse generalizations of metric spaces like 2-Metric, D-metric, G-metric, S-
metric, b-metric, Cone metric, and N-cone metric spaces have been studied. Malviya et al. (2012) introduced
N-cone metric spaces, a generalization of cone and S-metric spaces, exploring their properties in fixed-point
theory. This paper extends and revises results from Wang et al. (1984) in this novel context. Theorems and
corollaries demonstrate the uniqueness and existence of fixed points under specified conditions. These
findings enrich the understanding of generalized metric spaces and their applications in mathematical analysis.
1 INTRODUCTION
In the literature of Mathematical Analysis there are
various generalization of metric spaces like as 2-
Metric space (‘Gahler 1963, 1966’), D-metric space
(‘Mustafa Z. and Sims B. [2003, 2006]’), G-metric
Space (‘Dhage B.C., 1992’), S-metric Space (‘Shaban
S. et al., 2012’), b-metric Space (‘Bakhtin, I.A.,
1989’), Cone metric space (‘Huang et al., 2007’) etc.
In 2012, (‘Malviya et al. [accepted in FILOMAT]’)
defined a new structure namely N-cone metric space,
which was the generalization of cone metric space
and S-metric space, and studied various properties
and their applications in fixed point theory. In this
paper we extend and modify the results of (‘Wang et
al., 1984’) in this new setting.
Definition 1.1 (Gahlers 1963, 1966). Let be a
nonempty set. A generalized metric (or 2-metric) on
is a function d:
that satisfies the following
conditions for all .
d
d
if and only if
d(x, y, z) = d(p{x, y, z}), (symmetry),where p is
permutation function,
d
d(x, y, a) + d(x, a, z) + d(a, y, z) for all
.
Then the function d is called a 2- metric and the pair
(X, d) is called a 2-metric space.
Definition 1.2 (Mustafa 2003, 2006). Let be a
nonempty set. A - metric on is a function
that satisfies the following conditions for all
.
for all
.
Then the function is called an G- metric and the pair
is called a -metric space.
Definition 1.3 (Sedghi, 2012). Let be a nonempty
set. An -metric on is a function
that satisfies the following conditions for all
.
if and only if
Then the function is called an - metric and the pair
is called an -metric space
Definition 1.4 (Bakhtin, 1989). Let X be a nonempty
set and
≥ 1 a given real number. A
function d : X × X→ R
+
is a -metric on X if, for
all x, y, z X, the following conditions hold:
(1) d(
,
) = 0 if and only if
=
,
(2) d(
,
) = d(
,
),
(3) d(, ) ≤ s [d(, ) +d (, )].
In this case, the pair (X, ) is called a -metric space
178
Fernandez, J.
Some Fixed-Point Results on N_b-Cone Metric Spaces.
DOI: 10.5220/0012609000003739
Paper published under CC license (CC BY-NC-ND 4.0)
In Proceedings of the 1st International Conference on Artificial Intelligence for Internet of Things: Accelerating Innovation in Industry and Consumer Electronics (AI4IoT 2023), pages 178-181
ISBN: 978-989-758-661-3
Proceedings Copyright © 2024 by SCITEPRESS – Science and Technology Publications, Lda.
Definition 1.5 (Huang and Zhang, 2007). Let X be a
nonempty set and E be the real Banach space.
Suppose the mapping d:X×X→ E satisfies
1.0 d(x, y) for all x, y X and d(x, y) = 0 if and only
if x = y;
2. d(x, y) = d(y, x) for all x, y X;
3. d(x, y) d(x, z) + d(z, y) for all x, y, z X.
Then d is called a cone metric on X and (X, d) is called
a cone metric space.
Definition 1.6 (Malviya N., Fisher, 2003). Let be a
nonempty set. An -cone metric on is a
function
, that satisfies the following
conditions for all .
if and only if
Then the function is called an -cone metric and
the pair is called an -cone metric space.
2 MAIN RESULTS
Definition 2.1. Let be a nonempty set, E is the real
Banach space and s ≥ 1 be a given real
number. An
-cone metric on is a
function
, that satisfies the following
conditions for all .
if and only if
Then the function
is called an
-cone metric and
the pair
is called an
-cone metric space.
Definition 2.2. If
is an
-cone metric space,
then it is called symmetric if for all we
have
.
Definition 2.3. Let
be an
-cone metric
space. Let
be a sequence in and . If for
every with there is such that for
all
, then
is said to be
convergent,
converges to and is the limit of
. We denote this by
as
.
Lemma 1. Let
be an
-cone metric space
and be a normal cone with normal constant . Let
be a sequence in . If
converges to and
also convergesto then . That is the limit
of
, if exists is unique.
Definition 2.4. Let
be an
-cone metric
space and
be a sequence in . If for any
with there is such that for all
then
is called a Cauchy
sequence in .
Definition 2.5. Let
be an
-cone metric
space. If every Cauchy sequence in is convergent in
, then is called a complete
-cone metric space.
Lemma 2. Let
be an
-cone metric space
and
be a sequence in . If
converges to ,
then
is a Cauchy sequence.
Definition 2.6. Let
and
be
-cone
metric spaces. Then a function
is said to be
continuous at a point if and only if it is
sequentially continuous at , that is whenever
is
convergent to we have
is convergent to.
Lemma 3. Let
be an
-cone metric space
and be a normal cone with normal constant . Let
and
be two sequences in and suppose that
as . Then
as
Remark 1. If
in an
-cone metric space in
then every subsequence of
converges to in
.
Proposition 1. Let
be an
-cone metric
space and be a cone in a real Banach space . If
then .
Lemma 4. Let
be an
-cone metric space,
be an
-cone in a real Banach space and
. If
and
in and
then
Expansive Map: We define expansive map in
-
cone metric space as follows
Definition 2.7. Let
be an
-cone metric
space. A map is said to be an expansive
mapping if there exists a constant such that
for all
Example 1. Let
be an
-cone metric space.
Define a self map by where
, for all . Clearly is an expansive map in .
Some Fixed-Point Results on N_b-Cone Metric Spaces
179
Theorem 1. Let
be a complete symmetric
-
cone metric space with respect to a cone contained
in a real Banach space . Let and be two surjective
continuous self map of satisfying.
for every where ,
. Then and have a unique common fixed
point in .
Proof: We define a sequence
as follows for
If
for some then we see that
is a fixed point of and . Therefore, we suppose
that no two consecutive terms of sequence
are
equal.
Now we put
and
in (1.1.1) we
get
where
Similarly, we can calculate
Where
and so on.
for
where
then .
Now we shall prove that
is a Cauchy sequence.
For this for every positive integer , we have
which implies that
as
.
Sinc as .
Therefore
is a Cauchy sequence in , which is
complete space, so
.
Existence of Fixed Point: Since mappings are
continuous therefore existence of fixed point follows
very easily. As shown below
Similarly
which shows that is a common fixed point of and
.
Uniqueness: Let be another common fixed point of
and , that is
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180
This completes the proof of the Theorem 1.
Corollary 2. Let
be a complete symmetric
-cone metric space with respect to a cone
contained in a real Banach space . Let and be
two surjective continuous self maps of satisfying.
where . Then and have a unique common
fixed point in .
Proof: If we put in Theorem 1, then we
get above Corollary 2.
Corollary 3. Let
be a complete symmetric
-cone metric space with respect to a cone
contained in a real Banach space . Let be a
continuous surjective self map of satisfying.
where . Then has a unique fixed point in .
Proof: If we put in Corollary 2 then we get
above Corollary 3 which is an extension of Theorem
1 of Wang et al. (Wang et al., 1984) in
-cone metric
space.
Corollary 4. Let
be a complete symmetric
-cone metric space and be a continuous
surjection. Suppose that there exists a positive integer
and a real number such
that
for all
. Then has a unique fixed point in .
Proof: From Corollary 3,
has a unique fixed point
. But
, so is also a fixed
point of
. Hence , is a fixed point of .
Since the fixed point of is also fixed point of
, the
fixed point of is unique.
Corollary 5. Let
be a complete symmetric
-cone metric space with respect to a cone
contained in a real Banach space . Let and be
two continuous surjective self maps of satisfying.
for every where and
. Then and have a unique common fixed point
in .
Proof: The proof is similar to proof of the Theorem 1.
Corollary 6. Let
be a complete symmetric
-cone metric space with respect to a cone
contained in a real Banach space . Let be
surjective continuous self map of satisfying.
for every where and
. Then has a unique fixed point in .
Proof: If we put in Corollary 5 then we get
above Corollary 6 which is an extension of Theorem
2 of Wang et al. [10] in
-cone metric space.
The following example demonstrates Corollary 3.
Example 2. Let
and and
is defined by
where are positive constants. Then
is
a symmetric
-cone metric space. Define a self map
on as follows for all . Clearly is
an expansive mapping. If we take then
condition (1.1.5) holds trivially good and is the
unique fixed point of the map .
Remark 2. In Corollary 6, we proved the fixed point
is unique by using only and there is no need of
and , so it extend and unify the Theorem
2 of Wang et al. [10] in
-cone metric space.
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