FG- COUPLED FIXED POINT THEOREMS IN PARTI-
ALLY ORDERED S∗METRIC SPACES
Prajisha E.
Assistant Professor, Department of Mathematics, Amrita Vishwa Vidyapeetham, Amritapuri (India).
E-mail:prajisha1991@gmail.com, prajishae@am.amrita.edu
ORCID:0000-0001-6677-3135
Shaini P.
Professor, Department of Mathematics,Central University of Kerala (India).
E-mail:shainipv@gmail.com
ORCID:0000-0001-9958-9211
Reception: 12/09/2022 Acceptance: 27/09/2022 Publication: 29/12/2022
Suggested citation:
Prajisha E. and Shaini P. (2022). FG- coupled fixed point theorems in partially ordered S* metric spaces.3C TIC.
Cuadernos de desarrollo aplicados a las TIC,11 (2), 81-97. https://doi.org/10.17993/3ctic.2022.112.81-97
https://doi.org/10.17993/3ctic.2022.112.81-97
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
81
ABSTRACT
This is a review paper based on a recent article on FG- coupled fixed points [17], in which the authors
established FG- coupled fixed point theorems in partially ordered complete
S∗
metric space. The results
were illustrated by suitable examples, too. An
S∗
metric is an n-tuple metric from n-product of a set
to the non negative reals. The theorems in [17] generalizes the main results of Gnana Bhaskar and
Lakshmikantham [5].
KEYWORDS
FG- Coupled Fixed Point, Mixed Monotone Property, Partially Ordered Set, S∗Metric
https://doi.org/10.17993/3ctic.2022.112.81-97
1 INTRODUCTION
In 1906, Maurice Frechet introduced the concept of metric as a generalization of distance. He defined
a metric on a set as a function from the bi-product of the set to the non-negative reals that satisfy
certain axioms. Later, several authors generalized the concept of metrics by either changing the domain
or co-domain of the metric function or by varying the properties of the metric function [3,4,8,11,15,16].
An n-tuple metric called
S∗
metric is the latest development in this direction. Since the existence
of fixed points is depending on the function as well as on its domain, studies started on fixed point
theory by considering those generalized metric spaces. Now a lot of fixed point and coupled fixed point
results are available under different types of metric spaces [2,6,7, 9, 13,14]. In [1] Abdellaoui, M.A. and
Dahmani, Z. introduced
S∗
metric and they have proved fixed point results in
S∗
metric spaces. But
the same concept can be seen in [10], under a different name. In [10] Mujahid Abbas, Bashir Ali, and
Yusuf I Suleiman coined the name A- metric for this concept, and they have proved common coupled
fixed point theorems with an illustrative example.
Recently, the concept of FG- coupled fixed points was introduced as a generalization of the concept of
coupled fixed points in [12]. Some of the famous coupled fixed point theorems are generalized to FG-
coupled fixed point theorems in [12,18, 19].
In [17], the authors established FG- coupled fixed point theorems in the setting of partially ordered
complete S∗metric spaces. This is a review paper of [17].
Some useful definitions and results are as follows:
Definition 1. [1,10] An S∗metric on a nonempty set X is a function
S∗:Xn→[0,∞)satisfying:
(i) S∗(x1,x
2,···,x
n)≥0,
(ii) S∗(x1,x
2,···,x
n)=0if and only if x1=x2=···=xn,
(iii) S∗(x1,x
2,···,x
n)≤S∗(x1,···,x
1,a)+S∗(x2,···,x
2,a)+···+S∗(xn,···,x
n,a)
for any x1,x
2,···,x
n,a∈X. The pair (X, S∗)is called S∗metric space.
Lemma 1. [1,10] Suppose that (X, S∗)is an S∗metric space. Then for all
x1,x
2∈X, we have S∗(x1,x
1,···,x
1,x
2)=S∗(x2,x
2,···,x
2,x
1)
Definition 2. [1, 10] We say that the sequence
{xp}p∈N
of the space X is convergent to
x
if
S∗(xp,x
p,···,x
p,x)→0as p→∞. We write limp→∞ xp=x
Definition 3. [1,10] We say that the sequence
{xp}p∈N
of the space X is of Cauchy if for each
ϵ>
0,
there exist p0∈Nsuch that for any p, q ≥p0,
S∗(xp,···,x
p,x
q)<ϵ
The space (X, S∗)is complete if all its Cauchy sequences are convergent.
Lemma 2. [1,10] Let (
X, S∗
)be an
S∗
metric space. If
{xp}p∈N
in X converges to
x
, then
x
is unique.
Definition 4. [12] Let
X
and
Y
be any two non-empty sets and
F
:
X×Y→X
and
G
:
Y×X→Y
be two mappings. An element (
x, y
)
∈X×Y
is said to be an
FG
- coupled fixed point if
F
(
x, y
)=
x
and G(y, x)=y.
Definition 5. [12] Let (
X, ⪯P1
)and (
Y,⪯P2
)be two partially ordered sets and
F
:
X×Y→X
and
G
:
Y×X→Y
be two mappings. We say that
F
and
G
have mixed monotone property if
F
and
G
are
increasing in first variable and monotone decreasing second variable, i.e., if for all (x, y)∈X×Y,
x1,x
2∈X, x1⪯P1x2implies F(x1,y)⪯P1F(x2,y)and G(y, x2)⪯P2G(y, x1)and
y1,y
2∈Y,y1⪯P2y2implies F(x, y2)⪯P1F(x, y1)and G(y1,x)⪯P2G(y2,x).
https://doi.org/10.17993/3ctic.2022.112.81-97
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
82
ABSTRACT
This is a review paper based on a recent article on FG- coupled fixed points [17], in which the authors
established FG- coupled fixed point theorems in partially ordered complete
S∗
metric space. The results
were illustrated by suitable examples, too. An
S∗
metric is an n-tuple metric from n-product of a set
to the non negative reals. The theorems in [17] generalizes the main results of Gnana Bhaskar and
Lakshmikantham [5].
KEYWORDS
FG- Coupled Fixed Point, Mixed Monotone Property, Partially Ordered Set, S∗Metric
https://doi.org/10.17993/3ctic.2022.112.81-97
1 INTRODUCTION
In 1906, Maurice Frechet introduced the concept of metric as a generalization of distance. He defined
a metric on a set as a function from the bi-product of the set to the non-negative reals that satisfy
certain axioms. Later, several authors generalized the concept of metrics by either changing the domain
or co-domain of the metric function or by varying the properties of the metric function [3,4,8,11,15,16].
An n-tuple metric called
S∗
metric is the latest development in this direction. Since the existence
of fixed points is depending on the function as well as on its domain, studies started on fixed point
theory by considering those generalized metric spaces. Now a lot of fixed point and coupled fixed point
results are available under different types of metric spaces [2,6,7, 9, 13,14]. In [1] Abdellaoui, M.A. and
Dahmani, Z. introduced
S∗
metric and they have proved fixed point results in
S∗
metric spaces. But
the same concept can be seen in [10], under a different name. In [10] Mujahid Abbas, Bashir Ali, and
Yusuf I Suleiman coined the name A- metric for this concept, and they have proved common coupled
fixed point theorems with an illustrative example.
Recently, the concept of FG- coupled fixed points was introduced as a generalization of the concept of
coupled fixed points in [12]. Some of the famous coupled fixed point theorems are generalized to FG-
coupled fixed point theorems in [12,18, 19].
In [17], the authors established FG- coupled fixed point theorems in the setting of partially ordered
complete S∗metric spaces. This is a review paper of [17].
Some useful definitions and results are as follows:
Definition 1. [1,10] An S∗metric on a nonempty set X is a function
S∗:Xn→[0,∞)satisfying:
(i) S∗(x1,x
2,···,x
n)≥0,
(ii) S∗(x1,x
2,···,x
n)=0if and only if x1=x2=···=xn,
(iii) S∗(x1,x
2,···,x
n)≤S∗(x1,···,x
1,a)+S∗(x2,···,x
2,a)+···+S∗(xn,···,x
n,a)
for any x1,x
2,···,x
n,a∈X. The pair (X, S∗)is called S∗metric space.
Lemma 1. [1,10] Suppose that (X, S∗)is an S∗metric space. Then for all
x1,x
2∈X, we have S∗(x1,x
1,···,x
1,x
2)=S∗(x2,x
2,···,x
2,x
1)
Definition 2. [1, 10] We say that the sequence
{xp}p∈N
of the space X is convergent to
x
if
S∗(xp,x
p,···,x
p,x)→0as p→∞. We write limp→∞ xp=x
Definition 3. [1,10] We say that the sequence
{xp}p∈N
of the space X is of Cauchy if for each
ϵ>
0,
there exist p0∈Nsuch that for any p, q ≥p0,
S∗(xp,···,x
p,x
q)<ϵ
The space (X, S∗)is complete if all its Cauchy sequences are convergent.
Lemma 2. [1,10] Let (
X, S∗
)be an
S∗
metric space. If
{xp}p∈N
in X converges to
x
, then
x
is unique.
Definition 4. [12] Let
X
and
Y
be any two non-empty sets and
F
:
X×Y→X
and
G
:
Y×X→Y
be two mappings. An element (
x, y
)
∈X×Y
is said to be an
FG
- coupled fixed point if
F
(
x, y
)=
x
and G(y, x)=y.
Definition 5. [12] Let (
X, ⪯P1
)and (
Y,⪯P2
)be two partially ordered sets and
F
:
X×Y→X
and
G
:
Y×X→Y
be two mappings. We say that
F
and
G
have mixed monotone property if
F
and
G
are
increasing in first variable and monotone decreasing second variable, i.e., if for all (x, y)∈X×Y,
x1,x
2∈X, x1⪯P1x2implies F(x1,y)⪯P1F(x2,y)and G(y, x2)⪯P2G(y, x1)and
y1,y
2∈Y,y1⪯P2y2implies F(x, y2)⪯P1F(x, y1)and G(y1,x)⪯P2G(y2,x).
https://doi.org/10.17993/3ctic.2022.112.81-97
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
83
Note 1. [12] Let
F
:
X×Y→X
and
G
:
Y×X→Y
be two mappings, then for
n≥
1
,F
n
(
x, y
)=
F
(
Fn−1
(
x, y
)
,G
n−1
(
y, x
)) and
Gn
(
y, x
)=
G
(
Gn−1
(
y, x
)
,Fn−1
(
x, y
)), and
F0
(
x, y
)=
x
and
G0
(
y, x
)=
yfor all x∈Xand y∈Y.
Note 2. Let (
X, ⪯P1
)and (
Y,⪯P2
)be two partially ordered sets, then we define the partial order
≤12
on
X×Yand the partial order ≤21 on Y×Xas follows:
For all x, u ∈Xand y, v ∈Y
(x, y)≤12 (u, v)⇔x⪯P1uand v⪯P2y
(y, x)≤21 (v, u)⇔y⪯P2vand u⪯P1x
2 Main Results in [17]
Mainly two
FG
-coupled fixed point theorems are discussed in [17], first one deals with the existence of
FG
-coupled fixed point and the second deals with both the existence and uniqueness of
FG
-coupled
fixed point. They are as follow:
Theorem 1. [17] Let (
X, S∗
x,⪯P1
)and (
Y,S∗
y,⪯P2
)be two partially ordered complete
S∗
metric spaces
and
F
:
X×Y→X
and
G
:
Y×X→Y
be two mappings with mixed monotone property and satisfy
the following:
S∗
xF(x, y),···,F(x, y),F(u, v)
≤a1S∗
x(x, ···, x, u)+a2S∗
xx, ···, x, F (x, y)+a3S∗
xx, ···, x, F (u, v)
+a4S∗
xu, ···, u, F (x, y)+a5S∗
xu, ···, u, F (u, v),∀(x, y)≤12 (u, v)(1)
and
S∗
yG(y, x),···,G(y, x),G(v, u)
≤b1S∗
y(y, ···,y,v)+b2S∗
yy, ···, y, G(y, x)+b3S∗
yy, ···, y, G(v, u)
+b4S∗
yv, ···, v, G(y, x)+b5S∗
yv, ···, v, G(v, u),∀(y, x)≤21 (v, u)(2)
for the non negative ai,b
i,i=1,2,3,4,5with
a1+a2+na3+a5<1,b
1+b2+nb4+b5<1,a
3+a5<1,b
2+b4<1.
Also suppose that either
(I) F and G are continuous or
(II) Xand Yhave the following properties:
(i) if {zk}is an increasing sequence in Xwith zk→z, then zk⪯P1zfor all k∈N
(ii) if {wk}is a decreasing sequence in Ywith wk→w, then w⪯P2wkfor all k∈N.
If there exist
x0∈X
and
y0∈Y
with (
x0,y
0
)
≤12 F
(
x0,y
0
)
,G
(
y0,x
0
)
, then there exist an
FG
-
coupled fixed point.
Proof. Given
x0∈X
and
y0∈Y
such that (
x0,y
0
)
≤12 F
(
x0,y
0
)
,G
(
y0,x
0
)
. If (
x0,y
0
)=
F(x0,y
0),G(y0,x
0)then (x0,y
0)is an FG- coupled fixed point.
Otherwise we have
(x0,y
0)<12 F(x0,y
0),G(y0,x
0)
Then by the definition of the partial order on X×Ywe have either
x0⪯P1F(x0,y
0)and G(y0,x
0)≺P2y0or x0≺P1F(x0,y
0)and G(y0,x
0)⪯P2y0.
Without loss of generality we assume that
x0⪯P1F(x0,y
0)and G(y0,x
0)≺P2y0(3)
https://doi.org/10.17993/3ctic.2022.112.81-97
Let x1=F(x0,y
0)and y1=G(y0,x
0).
By (3) we have,
x0⪯P1x1and y1≺P2y0
By the mixed monotone property of Fand Gwe have
F(x0,y
0)⪯P1F(x1,y
0)
⪯P1F(x1,y
1)(4)
and
G(y1,x
1)⪯P2G(y0,x
1)
⪯P2G(y0,x
0)(5)
Let x2=F(x1,y
1)and y2=G(y1,x
1).
By (4) and (5) we have,
x1⪯P1x2and y2⪯P2y1
Continuing this process by using the mixed monotone property of
F
and
G
and by using the definition of
partial order on
X×Y
we get sequences
{xm}
and
{ym}
in
X
and
Y
respectively as: for all
m∈N∪{
0
}
xm+1 =F(xm,y
m)and ym+1 =G(ym,x
m)(6)
with the property that for all m∈N∪{0}
xm⪯P1xm+1 and ym+1 ⪯P2ym(7)
That is by the definition of partial order on X×Yand Y×Xwe have,
(xm,y
m)≤12 (xm+1,y
m+1)
and
(ym,x
m)≥21 (ym+1,x
m+1)
Claim: For all k∈N
S∗
x(xk,···,x
k,x
k+1)≤αkS∗
x(x0,···,x
0,x
1)(8)
and
S∗
y(yk,···,y
k,y
k+1)≤βkS∗
y(y0,···,y
0,y
1)(9)
where
α=a1+a2+(n−1)a3
1−a3−a5
<1and β=b1+b5+(n−1)b4
1−b2−b4
<1(10)
Now, we prove the claim by the method of mathematical induction.
When k=1we have,
S∗
x(x1,···,x
1,x
2)
=S∗
xF(x0,y
0),···,F(x0,y
0),F(x1,y
1)
≤a1S∗
x(x0,···,x
0,x
1)+a2S∗
xx0,···,x
0,F(x0,y
0)+a3S∗
xx0,···,x
0,F(x1,y
1)
+a4S∗
xx1,···,x
1,F(x0,y
0)+a5S∗
xx1,···,x
1,F(x1,y
1)
=a1S∗
x(x0,···,x
0,x
1)+a2S∗
x(x0,···,x
0,x
1)+a3S∗
x(x0,···,x
0,x
2)
+a5S∗
x(x1,···,x
1,x
2)
=(a1+a2)S∗
x(x0,···,x
0,x
1)+a3S∗
x(x0,···,x
0,x
2)+a5S∗
x(x1,···,x
1,x
2)
≤(a1+a2)S∗
x(x0,···,x
0,x
1)+a3(n−1)S∗
x(x0,···,x
0,x
1)
+S∗
x(x2,···,x
2,x
1)+a5S∗
x(x1,···,x
1,x
2)
=(a1+a2)S∗
x(x0,···,x
0,x
1)+a3(n−1)S∗
x(x0,···,x
0,x
1)
+S∗
x(x1,···,x
1,x
2)+a5S∗
x(x1,···,x
1,x
2)
=(a1+a2+(n−1)a3)S∗
x(x0,···,x
0,x
1)+(a3+a5)S∗
x(x1,···,x
1,x
2)
https://doi.org/10.17993/3ctic.2022.112.81-97
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
84
Note 1. [12] Let
F
:
X×Y→X
and
G
:
Y×X→Y
be two mappings, then for
n≥
1
,F
n
(
x, y
)=
F
(
Fn−1
(
x, y
)
,G
n−1
(
y, x
)) and
Gn
(
y, x
)=
G
(
Gn−1
(
y, x
)
,Fn−1
(
x, y
)), and
F0
(
x, y
)=
x
and
G0
(
y, x
)=
yfor all x∈Xand y∈Y.
Note 2. Let (
X, ⪯P1
)and (
Y,⪯P2
)be two partially ordered sets, then we define the partial order
≤12
on
X×Yand the partial order ≤21 on Y×Xas follows:
For all x, u ∈Xand y, v ∈Y
(x, y)≤12 (u, v)⇔x⪯P1uand v⪯P2y
(y, x)≤21 (v, u)⇔y⪯P2vand u⪯P1x
2 Main Results in [17]
Mainly two
FG
-coupled fixed point theorems are discussed in [17], first one deals with the existence of
FG
-coupled fixed point and the second deals with both the existence and uniqueness of
FG
-coupled
fixed point. They are as follow:
Theorem 1. [17] Let (
X, S∗
x,⪯P1
)and (
Y,S∗
y,⪯P2
)be two partially ordered complete
S∗
metric spaces
and
F
:
X×Y→X
and
G
:
Y×X→Y
be two mappings with mixed monotone property and satisfy
the following:
S∗
xF(x, y),···,F(x, y),F(u, v)
≤a1S∗
x(x, ···, x, u)+a2S∗
xx, ···, x, F (x, y)+a3S∗
xx, ···, x, F (u, v)
+a4S∗
xu, ···, u, F (x, y)+a5S∗
xu, ···, u, F (u, v),∀(x, y)≤12 (u, v)(1)
and
S∗
yG(y, x),···,G(y, x),G(v, u)
≤b1S∗
y(y, ···,y,v)+b2S∗
yy, ···, y, G(y, x)+b3S∗
yy, ···, y, G(v, u)
+b4S∗
yv, ···, v, G(y, x)+b5S∗
yv, ···, v, G(v, u),∀(y, x)≤21 (v, u)(2)
for the non negative ai,b
i,i=1,2,3,4,5with
a1+a2+na3+a5<1,b
1+b2+nb4+b5<1,a
3+a5<1,b
2+b4<1.
Also suppose that either
(I) F and G are continuous or
(II) Xand Yhave the following properties:
(i) if {zk}is an increasing sequence in Xwith zk→z, then zk⪯P1zfor all k∈N
(ii) if {wk}is a decreasing sequence in Ywith wk→w, then w⪯P2wkfor all k∈N.
If there exist
x0∈X
and
y0∈Y
with (
x0,y
0
)
≤12 F
(
x0,y
0
)
,G
(
y0,x
0
)
, then there exist an
FG
-
coupled fixed point.
Proof. Given
x0∈X
and
y0∈Y
such that (
x0,y
0
)
≤12 F
(
x0,y
0
)
,G
(
y0,x
0
)
. If (
x0,y
0
)=
F(x0,y
0),G(y0,x
0)then (x0,y
0)is an FG- coupled fixed point.
Otherwise we have
(x0,y
0)<12 F(x0,y
0),G(y0,x
0)
Then by the definition of the partial order on X×Ywe have either
x0⪯P1F(x0,y
0)and G(y0,x
0)≺P2y0or x0≺P1F(x0,y
0)and G(y0,x
0)⪯P2y0.
Without loss of generality we assume that
x0⪯P1F(x0,y
0)and G(y0,x
0)≺P2y0(3)
https://doi.org/10.17993/3ctic.2022.112.81-97
Let x1=F(x0,y
0)and y1=G(y0,x
0).
By (3) we have,
x0⪯P1x1and y1≺P2y0
By the mixed monotone property of Fand Gwe have
F(x0,y
0)⪯P1F(x1,y
0)
⪯P1F(x1,y
1)(4)
and
G(y1,x
1)⪯P2G(y0,x
1)
⪯P2G(y0,x
0)(5)
Let x2=F(x1,y
1)and y2=G(y1,x
1).
By (4) and (5) we have,
x1⪯P1x2and y2⪯P2y1
Continuing this process by using the mixed monotone property of
F
and
G
and by using the definition of
partial order on
X×Y
we get sequences
{xm}
and
{ym}
in
X
and
Y
respectively as: for all
m∈N∪{
0
}
xm+1 =F(xm,y
m)and ym+1 =G(ym,x
m)(6)
with the property that for all m∈N∪{0}
xm⪯P1xm+1 and ym+1 ⪯P2ym(7)
That is by the definition of partial order on X×Yand Y×Xwe have,
(xm,y
m)≤12 (xm+1,y
m+1)
and
(ym,x
m)≥21 (ym+1,x
m+1)
Claim: For all k∈N
S∗
x(xk,···,x
k,x
k+1)≤αkS∗
x(x0,···,x
0,x
1)(8)
and
S∗
y(yk,···,y
k,y
k+1)≤βkS∗
y(y0,···,y
0,y
1)(9)
where
α=a1+a2+(n−1)a3
1−a3−a5
<1and β=b1+b5+(n−1)b4
1−b2−b4
<1(10)
Now, we prove the claim by the method of mathematical induction.
When k=1we have,
S∗
x(x1,···,x
1,x
2)
=S∗
xF(x0,y
0),···,F(x0,y
0),F(x1,y
1)
≤a1S∗
x(x0,···,x
0,x
1)+a2S∗
xx0,···,x
0,F(x0,y
0)+a3S∗
xx0,···,x
0,F(x1,y
1)
+a4S∗
xx1,···,x
1,F(x0,y
0)+a5S∗
xx1,···,x
1,F(x1,y
1)
=a1S∗
x(x0,···,x
0,x
1)+a2S∗
x(x0,···,x
0,x
1)+a3S∗
x(x0,···,x
0,x
2)
+a5S∗
x(x1,···,x
1,x
2)
=(a1+a2)S∗
x(x0,···,x
0,x
1)+a3S∗
x(x0,···,x
0,x
2)+a5S∗
x(x1,···,x
1,x
2)
≤(a1+a2)S∗
x(x0,···,x
0,x
1)+a3(n−1)S∗
x(x0,···,x
0,x
1)
+S∗
x(x2,···,x
2,x
1)+a5S∗
x(x1,···,x
1,x
2)
=(a1+a2)S∗
x(x0,···,x
0,x
1)+a3(n−1)S∗
x(x0,···,x
0,x
1)
+S∗
x(x1,···,x
1,x
2)+a5S∗
x(x1,···,x
1,x
2)
=(a1+a2+(n−1)a3)S∗
x(x0,···,x
0,x
1)+(a3+a5)S∗
x(x1,···,x
1,x
2)
https://doi.org/10.17993/3ctic.2022.112.81-97
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
85
which implies that
(1 −a3−a5)S∗
x(x1,···,x
1,x
2)≤(a1+a2+(n−1)a3)S∗
x(x0,···,x
0,x
1)
That is,
S∗
x(x1,···,x
1,x
2)≤a1+a2+(n−1)a3
1−a3−a5
S∗
x(x0,···,x
0,x
1)
Similarly we have,
S∗
y(y1,···,y
1,y
2)≤(b1+b5+(n−1)b4)S∗
y(y0,···,y
0,y
1)+(b2+b4)S∗
y(y1,···,y
1,y
2)
which implies that
(1 −b2−b4)S∗
y(y1,···,y
1,y
2)≤(b1+b5+(n−1)b4)S∗
y(y0,···,y
0,y
1)
That is,
S∗
y(y1,···,y
1,y
2)≤b1+b5+(n−1)b4
1−b2−b4
S∗
y(y0,···,y
0,y
1)
Thus the claim is true for k=1.
Now assume the claim for k≤mand check for k=m+1.
Consider,
S∗
x(xm+1,···,x
m+1,x
m+2)
=S∗
xF(xm,y
m),···,F(xm,y
m),F(xm+1,y
m+1)
≤a1S∗
x(xm,···,x
m,x
m+1)+a2S∗
xxm,···,x
m,F(xm,y
m)
+a3S∗
xxm,···,x
m,F(xm+1,y
m+1)+a4S∗
xxm+1,···,x
m+1,F(xm,y
m)
+a5S∗
xxm+1,···,x
m+1,F(xm+1,y
m+1)
=a1S∗
x(xm,···,x
m,x
m+1)+a2S∗
x(xm,···,x
m,x
m+1)
+a3S∗
x(xm,···,x
m,x
m+2)+a5S∗
x(xm+1,···,x
m+1,x
m+2)
=(a1+a2)S∗
x(xm,···,x
m,x
m+1)+a3S∗
x(xm,···,x
m,x
m+2)+a5S∗
x(xm+1,···,x
m+1,x
m+2)
≤(a1+a2)S∗
x(xm,···,x
m,x
m+1)+a3(n−1)S∗
x(xm,···,x
m,x
m+1)
+S∗
x(xm+2,···,x
m+2,x
m+1)+a5S∗
x(xm+1,···,x
m+1,x
m+2)
=(a1+a2)S∗
x(xm,···,x
m,x
m+1)+a3(n−1)S∗
x(xm,···,x
m,x
m+1)
+S∗
x(xm+1,···,x
m+1,x
m+2)+a5S∗
x(xm+1,···,x
m+1,x
m+2)
=(a1+a2+(n−1)a3)S∗
x(xm,···,x
m,x
m+1)+(a3+a5)S∗
x(xm+1,···,x
m+1,x
m+2)
which implies that
(1 −a3−a5)S∗
x(xm+1,···,x
m+1,x
m+2)
≤(a1+a2+(n−1)a3)S∗
x(xm,···,x
m,x
m+1)
≤(a1+a2+(n−1)a3)a1+a2+(n−1)a3
1−a3−a5mS∗
x(x0,···,x
0,x
1)
Therefore,
S∗
x(xm+1,···,x
m+1,x
m+2)≤a1+a2+(n−1)a3
1−a3−a5m+1 S∗
x(x0,···,x
0,x
1)
Similarly we have,
S∗
y(ym+1,···,y
m+1,y
m+2)≤(b1+b5+b4(n−1))S∗
y(ym,···,y
m,y
m+1)
+(b2+b4)S∗
y(ym+1,···,y
m+1,y
m+2)
https://doi.org/10.17993/3ctic.2022.112.81-97
which implies that
(1 −b2−b4)S∗
y(ym+1,···,y
m+1,y
m+2)
≤(b1+b5+b4(n−1))S∗
y(ym,···,y
m,y
m+1)
≤(b1+b5+b4(n−1))b1+b5+(n−1)b4
1−b2−b4mS∗
y(y0,···,y
0,y
1)
Therefore,
S∗
y(ym+1,···,y
m+1,y
m+2)≤b1+b5+(n−1)b4
1−b2−b4m+1S∗
y(y0,···,y
0,y
1)
Thus the claim is true for all k∈N.
Next we prove that {xm}is a Cauchy sequence in Xand {ym}is a Cauchy sequence in Y.
Let p, q ∈Nwith p<q.
Consider,
S∗
x(xp,···,x
p,x
q)
≤(n−1)S∗
x(xp,···,x
p,x
p+1)+S∗
x(xq,···,x
q,x
p+1)
=(n−1)S∗
x(xp,···,x
p,x
p+1)+S∗
x(xp+1,···,x
p+1,x
q)
≤(n−1)S∗
x(xp,···,x
p,x
p+1)+(n−1)S∗
x(xp+1,···,x
p+1,x
p+2)
+S∗
x(xq,···,x
q,x
p+2)
=(n−1)S∗
x(xp,···,x
p,x
p+1)+S∗
x(xp+1,···,x
p+1,x
p+2)
+S∗
x(xp+2,···,x
p+2,x
q)
...
...
...
≤(n−1)S∗
x(xp,···,x
p,x
p+1)+S∗
x(xp+1,···,x
p+1,x
p+2)
+···+S∗
x(xq−2,···,x
q−2,x
q−1)+S∗
x(xq,···,x
q,x
q−1)
=(n−1)
q−2
i=p
S∗
x(xi,···,x
i,x
i+1)+S∗
x(xq−1,···,x
q−1,x
q)
≤(n−1)
q−2
i=p
αiS∗
x(x0,···,x
0,x
1)+αq−1S∗
x(x0,···,x
0,x
1)
=(n−1)S∗
x(x0,···,x
0,x
1)
q−2
i=p
αi+αq−1S∗
x(x0,···,x
0,x
1)
≤(n−1) αp
1−αS∗
x(x0,···,x
0,x
1)+αq−1S∗
x(x0,···,x
0,x
1)
→0as p, q →∞since α<1
Thus, {xm}is a Cauchy sequence in X.
https://doi.org/10.17993/3ctic.2022.112.81-97
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
86
which implies that
(1 −a3−a5)S∗
x(x1,···,x
1,x
2)≤(a1+a2+(n−1)a3)S∗
x(x0,···,x
0,x
1)
That is,
S∗
x(x1,···,x
1,x
2)≤a1+a2+(n−1)a3
1−a3−a5
S∗
x(x0,···,x
0,x
1)
Similarly we have,
S∗
y(y1,···,y
1,y
2)≤(b1+b5+(n−1)b4)S∗
y(y0,···,y
0,y
1)+(b2+b4)S∗
y(y1,···,y
1,y
2)
which implies that
(1 −b2−b4)S∗
y(y1,···,y
1,y
2)≤(b1+b5+(n−1)b4)S∗
y(y0,···,y
0,y
1)
That is,
S∗
y(y1,···,y
1,y
2)≤b1+b5+(n−1)b4
1−b2−b4
S∗
y(y0,···,y
0,y
1)
Thus the claim is true for k=1.
Now assume the claim for k≤mand check for k=m+1.
Consider,
S∗
x(xm+1,···,x
m+1,x
m+2)
=S∗
xF(xm,y
m),···,F(xm,y
m),F(xm+1,y
m+1)
≤a1S∗
x(xm,···,x
m,x
m+1)+a2S∗
xxm,···,x
m,F(xm,y
m)
+a3S∗
xxm,···,x
m,F(xm+1,y
m+1)+a4S∗
xxm+1,···,x
m+1,F(xm,y
m)
+a5S∗
xxm+1,···,x
m+1,F(xm+1,y
m+1)
=a1S∗
x(xm,···,x
m,x
m+1)+a2S∗
x(xm,···,x
m,x
m+1)
+a3S∗
x(xm,···,x
m,x
m+2)+a5S∗
x(xm+1,···,x
m+1,x
m+2)
=(a1+a2)S∗
x(xm,···,x
m,x
m+1)+a3S∗
x(xm,···,x
m,x
m+2)+a5S∗
x(xm+1,···,x
m+1,x
m+2)
≤(a1+a2)S∗
x(xm,···,x
m,x
m+1)+a3(n−1)S∗
x(xm,···,x
m,x
m+1)
+S∗
x(xm+2,···,x
m+2,x
m+1)+a5S∗
x(xm+1,···,x
m+1,x
m+2)
=(a1+a2)S∗
x(xm,···,x
m,x
m+1)+a3(n−1)S∗
x(xm,···,x
m,x
m+1)
+S∗
x(xm+1,···,x
m+1,x
m+2)+a5S∗
x(xm+1,···,x
m+1,x
m+2)
=(a1+a2+(n−1)a3)S∗
x(xm,···,x
m,x
m+1)+(a3+a5)S∗
x(xm+1,···,x
m+1,x
m+2)
which implies that
(1 −a3−a5)S∗
x(xm+1,···,x
m+1,x
m+2)
≤(a1+a2+(n−1)a3)S∗
x(xm,···,x
m,x
m+1)
≤(a1+a2+(n−1)a3)a1+a2+(n−1)a3
1−a3−a5mS∗
x(x0,···,x
0,x
1)
Therefore,
S∗
x(xm+1,···,x
m+1,x
m+2)≤a1+a2+(n−1)a3
1−a3−a5m+1 S∗
x(x0,···,x
0,x
1)
Similarly we have,
S∗
y(ym+1,···,y
m+1,y
m+2)≤(b1+b5+b4(n−1))S∗
y(ym,···,y
m,y
m+1)
+(b2+b4)S∗
y(ym+1,···,y
m+1,y
m+2)
https://doi.org/10.17993/3ctic.2022.112.81-97
which implies that
(1 −b2−b4)S∗
y(ym+1,···,y
m+1,y
m+2)
≤(b1+b5+b4(n−1))S∗
y(ym,···,y
m,y
m+1)
≤(b1+b5+b4(n−1))b1+b5+(n−1)b4
1−b2−b4mS∗
y(y0,···,y
0,y
1)
Therefore,
S∗
y(ym+1,···,y
m+1,y
m+2)≤b1+b5+(n−1)b4
1−b2−b4m+1S∗
y(y0,···,y
0,y
1)
Thus the claim is true for all k∈N.
Next we prove that {xm}is a Cauchy sequence in Xand {ym}is a Cauchy sequence in Y.
Let p, q ∈Nwith p<q.
Consider,
S∗
x(xp,···,x
p,x
q)
≤(n−1)S∗
x(xp,···,x
p,x
p+1)+S∗
x(xq,···,x
q,x
p+1)
=(n−1)S∗
x(xp,···,x
p,x
p+1)+S∗
x(xp+1,···,x
p+1,x
q)
≤(n−1)S∗
x(xp,···,x
p,x
p+1)+(n−1)S∗
x(xp+1,···,x
p+1,x
p+2)
+S∗
x(xq,···,x
q,x
p+2)
=(n−1)S∗
x(xp,···,x
p,x
p+1)+S∗
x(xp+1,···,x
p+1,x
p+2)
+S∗
x(xp+2,···,x
p+2,x
q)
...
...
...
≤(n−1)S∗
x(xp,···,x
p,x
p+1)+S∗
x(xp+1,···,x
p+1,x
p+2)
+···+S∗
x(xq−2,···,x
q−2,x
q−1)+S∗
x(xq,···,x
q,x
q−1)
=(n−1)
q−2
i=p
S∗
x(xi,···,x
i,x
i+1)+S∗
x(xq−1,···,x
q−1,x
q)
≤(n−1)
q−2
i=p
αiS∗
x(x0,···,x
0,x
1)+αq−1S∗
x(x0,···,x
0,x
1)
=(
n−1)S∗
x(x0,···,x
0,x
1)
q−2
i=p
αi+αq−1S∗
x(x0,···,x
0,x
1)
≤(n−1) αp
1−αS∗
x(x0,···,x
0,x
1)+αq−1S∗
x(x0,···,x
0,x
1)
→0as p, q →∞since α<1
Thus, {xm}is a Cauchy sequence in X.
https://doi.org/10.17993/3ctic.2022.112.81-97
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
87
Similarly we have,
S∗
y(yp,···,y
p,y
q)≤(n−1)
q−2
i=p
S∗
y(yi,···,y
i,y
i+1)+S∗
y(yq−1,···,y
q−1,y
q)
≤(n−1)
q−2
i=p
βiS∗
y(y0,···,y
0,y
1)+βq−1S∗
y(y0,···,y
0,y
1)
=(n−1) S∗
y(y0,···,y
0,y
1)
q−2
i=p
βi+βq−1S∗
y(y0,···,y
0,y
1)
≤(n−1) βp
1−βS∗
y(y0,···,y
0,y
1)+βq−1S∗
y(y0,···,y
0,y
1)
→0as p, q →∞since β<1
Thus, {ym}is a Cauchy sequence in Y.
Since Xand Yare complete S∗metric spaces, there exist x∈Xand y∈Ysuch that
lim
p→∞ xp=xand lim
p→∞ yp=y(11)
Case (I): First assume that Fand Gare continuous.
Therefore by (6) and (11) we have,
x= lim
p→∞ xp+1 = lim
p→∞ F(xp,y
p)=F(x, y)
and
y= lim
p→∞ yp+1 = lim
p→∞ G(yp,x
p)=G(y, x)
That is F(x, y)=xand G(y, x)=y.
Thus (x, y)is an FG- coupled fixed point.
Case (II): Suppose that Xand Yhave the properties (i)and (ii)respectively.
By (7) we have,
{xm}
is an increasing sequence in
X
and
{ym}
is a decreasing sequence in
Y
and by
(11) we have limm→∞ xm=xand limm→∞ ym=y
Therefore by the hypothesis we have for all m∈N
xm⪯P1xand y⪯P2ym
Therefore by the definition of partial order on X×Yand Y×Xwe have
(xm,y
m)≤12 (x, y)and (y, x)≤21 (ym,x
m)
Consider,
S∗
xx, ···, x, F (x, y)
≤(n−1)S∗
x(x, ···, x, xm+1)+S∗
x(F(x, y),···,F(x, y),x
m+1)
=(n−1)S∗
x(x, ···, x, xm+1)+S∗
xF(x, y),···,F(x, y),F(xm,y
m)
=(n−1)S∗
x(x, ···, x, xm+1)+S∗
xF(xm,y
m),···,F(xm,y
m),F(x, y)
≤(n−1)S∗
x(x, ···, x, xm+1)+a1S∗
x(xm,···,x
m,x)+a2S∗
xxm,···,x
m,F(xm,y
m)
+a3S∗
xxm,···,x
m,F(x, y)+a4S∗
xx, ···, x, F (xm,y
m)+a5S∗
xx, ···, x, F (x, y)
=(n−1)S∗
x(x, ···, x, xm+1)+a1S∗
x(xm,···,x
m,x)+a2S∗
x(xm,···,x
m,x
m+1)
+a3S∗
xxm,···,x
m,F(x, y)+a4S∗
x(x, ···, x, xm+1)+a5S∗
xx, ···, x, F (x, y)
≤(n−1)S∗
x(x, ···, x, xm+1)+a1S∗
x(xm,···,x
m,x)+a2S∗
x(xm,···,x
m,x
m+1)
+a3(n−1)S∗
x(xm,···,x
m,x)+S∗
x(F(x, y),···,F(x, y),x)
+a4S∗
x(x, ···, x, xm+1)+a5S∗
xx, ···, x, F (x, y)https://doi.org/10.17993/3ctic.2022.112.81-97
By taking the limit as m→∞on both sides, and by using (11) and Lemma 1 we get
S∗
xx, ···, x, F (x, y)≤(a3+a5)S∗
xx, ···, x, F (x, y)
since a3+a5<1we get S∗
xx, ···, x, F (x, y)=0
Thus we get
F(x, y)=x(12)
Similarly,
S∗
yy, ···, y, G(y, x)
≤(n−1)S∗
y(y, ···,y,y
m+1)+S∗
y(G(y, x),···,G(y, x),y
m+1)
≤(n−1)S∗
y(y, ···, y, ym+1)+S∗
yG(y, x),···,G(y, x),G(ym,x
m)
≤(n−1)S∗
y(y, ···, y, ym+1)+b1S∗
y(y, ···,y,y
m)+b2S∗
yy, ···,y,G(y, x)
+b3S∗
yy, ···,y,G(ym,x
m)+b4S∗
yym,···,y
m,G(y, x)
+b5S∗
yym,···,y
m,G(ym,x
m)
≤(n−1)S∗
y(y, ···, y, ym+1)+b1S∗
y(y, ···,y,y
m)+b2S∗
yy, ···,y,G(y, x)
+b3S∗
y(y, ···, y, ym+1)+b4S∗
yym,···,y
m,G(y, x)+b5S∗
y(ym,···,y
m,y
m+1)
≤(n−1)S∗
y(y, ···, y, ym+1)+b1S∗
y(y, ···,y,y
m)+b2S∗
yy, ···,y,G(y, x)
+b3S∗
y(y, ···, y, ym+1)+b4(n−1)S∗
y(ym,···,y
m,y)+S∗(G(y, x),···,G(y, x),y)
+b5S∗
y(ym,···,y
m,y
m+1)
By taking the limit as m→∞on both sides, using (11) and Lemma 1 we get
S∗
yy, ···, y, G(y, x)≤(b2+b4)S∗
yy, ···, y, G(y, x)
since b2+b4<1we get S∗
yy, ···, y, G(y, x)=0
Thus we get
G(y, x)=y(13)
Therefore by (12) and (13), (x, y)is an FG- coupled fixed point.
Hence the proof.
By taking
n
=2,
X
=
Y
,
F
=
G
,
a2
=
b2
=
k
,
a5
=
b5
=
l
and the remaining
ai,b
i
=0,wegeta
coupled fixed fixed point theorem for Kannan type mapping. We give the result as a corollary as follows:
Corollary 1. Let (
X, d, ⪯
)be a partially ordered complete metric space and
F
:
X×X→X
be a
mapping having the mixed monotone property on Xsatisfying:
dF(x, y),F(u, v)≤kd
x, F (x, y)+ld
u, F (u, v)∀(x, y)≥(u, v)
for non negative k, l with k+l<1
Suppose that either
(I) Fis continuous or
(II) Xsatisfy the following:
(i) if {xk}is an increasing sequence in Xwith xk→x, then xk⪯xfor all k∈N
(ii) if {yk}is a decreasing sequence in Xwith yk→y, then y⪯ykfor all k∈N.
If there exist x0,y
0∈Xsuch that (x0,y
0)≤F(x0,y
0),F(y0,x
0)then Fhas a coupled fixed point.
By taking
n
=2,
X
=
Y
,
F
=
G
,
a3
=
b3
=
k
,
a4
=
b4
=
l
and the remaining
ai,b
i
=0,wegeta
coupled fixed point theorem for Chatterjea type mapping. We give the result as a corollary as follows:
https://doi.org/10.17993/3ctic.2022.112.81-97
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
88
Similarly we have,
S∗
y(yp,···,y
p,y
q)≤(n−1)
q−2
i=p
S∗
y(yi,···,y
i,y
i+1)+S∗
y(yq−1,···,y
q−1,y
q)
≤(n−1)
q−2
i=p
βiS∗
y(y0,···,y
0,y
1)+βq−1S∗
y(y0,···,y
0,y
1)
=(n−1) S∗
y(y0,···,y
0,y
1)
q−2
i=p
βi+βq−1S∗
y(y0,···,y
0,y
1)
≤(n−1) βp
1−βS∗
y(y0,···,y
0,y
1)+βq−1S∗
y(y0,···,y
0,y
1)
→0as p, q →∞since β<1
Thus, {ym}is a Cauchy sequence in Y.
Since Xand Yare complete S∗metric spaces, there exist x∈Xand y∈Ysuch that
lim
p→∞ xp=xand lim
p→∞ yp=y(11)
Case (I): First assume that Fand Gare continuous.
Therefore by (6) and (11) we have,
x= lim
p→∞ xp+1 = lim
p→∞ F(xp,y
p)=F(x, y)
and
y= lim
p→∞ yp+1 = lim
p→∞ G(yp,x
p)=G(y, x)
That is F(x, y)=xand G(y, x)=y.
Thus (x, y)is an FG- coupled fixed point.
Case (II): Suppose that Xand Yhave the properties (i)and (ii)respectively.
By (7) we have,
{xm}
is an increasing sequence in
X
and
{ym}
is a decreasing sequence in
Y
and by
(11) we have limm→∞ xm=xand limm→∞ ym=y
Therefore by the hypothesis we have for all m∈N
xm⪯P1xand y⪯P2ym
Therefore by the definition of partial order on X×Yand Y×Xwe have
(xm,y
m)≤12 (x, y)and (y, x)≤21 (ym,x
m)
Consider,
S∗
xx, ···, x, F (x, y)
≤(n−1)S∗
x(x, ···, x, xm+1)+S∗
x(F(x, y),···,F(x, y),x
m+1)
=(n−1)S∗
x(x, ···, x, xm+1)+S∗
xF(x, y),···,F(x, y),F(xm,y
m)
=(n−1)S∗
x(x, ···, x, xm+1)+S∗
xF(xm,y
m),···,F(xm,y
m),F(x, y)
≤(n−1)S∗
x(x, ···, x, xm+1)+a1S∗
x(xm,···,x
m,x)+a2S∗
xxm,···,x
m,F(xm,y
m)
+a3S∗
xxm,···,x
m,F(x, y)+a4S∗
xx, ···, x, F (xm,y
m)+a5S∗
xx, ···, x, F (x, y)
=(n−1)S∗
x(x, ···, x, xm+1)+a1S∗
x(xm,···,x
m,x)+a2S∗
x(xm,···,x
m,x
m+1)
+a3S∗
xxm,···,x
m,F(x, y)+a4S∗
x(x, ···, x, xm+1)+a5S∗
xx, ···, x, F (x, y)
≤(n−1)S∗
x(x, ···, x, xm+1)+a1S∗
x(xm,···,x
m,x)+a2S∗
x(xm,···,x
m,x
m+1)
+a3(n−1)S∗
x(xm,···,x
m,x)+S∗
x(F(x, y),···,F(x, y),x)
+a4S∗
x(x, ···, x, xm+1)+a5S∗
xx, ···, x, F (x, y)https://doi.org/10.17993/3ctic.2022.112.81-97
By taking the limit as m→∞on both sides, and by using (11) and Lemma 1 we get
S∗
xx, ···, x, F (x, y)≤(a3+a5)S∗
xx, ···, x, F (x, y)
since a3+a5<1we get S∗
xx, ···, x, F (x, y)=0
Thus we get
F(x, y)=x(12)
Similarly,
S∗
yy, ···, y, G(y, x)
≤(n−1)S∗
y(y, ···,y,y
m+1)+S∗
y(G(y, x),···,G(y, x),y
m+1)
≤(n−1)S∗
y(y, ···, y, ym+1)+S∗
yG(y, x),···,G(y, x),G(ym,x
m)
≤(n−1)S∗
y(y, ···, y, ym+1)+b1S∗
y(y, ···,y,y
m)+b2S∗
yy, ···,y,G(y, x)
+b3S∗
yy, ···,y,G(ym,x
m)+b4S∗
yym,···,y
m,G(y, x)
+b5S∗
yym,···,y
m,G(ym,x
m)
≤(n−1)S∗
y(y, ···, y, ym+1)+b1S∗
y(y, ···,y,y
m)+b2S∗
yy, ···,y,G(y, x)
+b3S∗
y(y, ···, y, ym+1)+b4S∗
yym,···,y
m,G(y, x)+b5S∗
y(ym,···,y
m,y
m+1)
≤(n−1)S∗
y(y, ···, y, ym+1)+b1S∗
y(y, ···,y,y
m)+b2S∗
yy, ···,y,G(y, x)
+b3S∗
y(y, ···, y, ym+1)+b4(n−1)S∗
y(ym,···,y
m,y)+S∗(G(y, x),···,G(y, x),y)
+b5S∗
y(ym,···,y
m,y
m+1)
By taking the limit as m→∞on both sides, using (11) and Lemma 1 we get
S∗
yy, ···, y, G(y, x)≤(b2+b4)S∗
yy, ···, y, G(y, x)
since b2+b4<1we get S∗
yy, ···, y, G(y, x)=0
Thus we get
G(y, x)=y(13)
Therefore by (12) and (13), (x, y)is an FG- coupled fixed point.
Hence the proof.
By taking
n
=2,
X
=
Y
,
F
=
G
,
a2
=
b2
=
k
,
a5
=
b5
=
l
and the remaining
ai,b
i
=0,wegeta
coupled fixed fixed point theorem for Kannan type mapping. We give the result as a corollary as follows:
Corollary 1. Let (
X, d, ⪯
)be a partially ordered complete metric space and
F
:
X×X→X
be a
mapping having the mixed monotone property on Xsatisfying:
dF(x, y),F(u, v)≤kd
x, F (x, y)+ld
u, F (u, v)∀(x, y)≥(u, v)
for non negative k, l with k+l<1
Suppose that either
(I) Fis continuous or
(II) Xsatisfy the following:
(i) if {xk}is an increasing sequence in Xwith xk→x, then xk⪯xfor all k∈N
(ii) if {yk}is a decreasing sequence in Xwith yk→y, then y⪯ykfor all k∈N.
If there exist x0,y
0∈Xsuch that (x0,y
0)≤F(x0,y
0),F(y0,x
0)then Fhas a coupled fixed point.
By taking
n
=2,
X
=
Y
,
F
=
G
,
a3
=
b3
=
k
,
a4
=
b4
=
l
and the remaining
ai,b
i
=0,wegeta
coupled fixed point theorem for Chatterjea type mapping. We give the result as a corollary as follows:
https://doi.org/10.17993/3ctic.2022.112.81-97
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
89
Corollary 2. Let (
X, d, ⪯
)be a partially ordered complete metric space and
F
:
X×X→X
be a
mapping having the mixed monotone property on Xsatisfying:
dF(x, y),F(u, v)≤kd
x, F (u, v)+ld
u, F (x, y)∀(x, y)≥(u, v)
for k, l ∈[0,1
2)
Suppose that either
(I) Fis continuous or
(II) Xsatisfy the following:
(i) if {xk}is an increasing sequence in Xwith xk→x, then xk⪯xfor all k∈N
(ii) if {yk}is a decreasing sequence in Xwith yk→y, then y⪯ykfor all k∈N.
If there exist x0,y
0∈Xsuch that (x0,y
0)≤F(x0,y
0),F(y0,x
0)then Fhas a coupled fixed point.
Remark 1. By putting different values to the constants
ai,b
i
;
i
=1
,
2
,
3
,
4
,
5which satisfy the conditions
mentioned in Theorem 1 we get various FG- coupled fixed point theorems.
Remark 2. By varying the constants
ai,b
i
;
i
=1
,
2
,
3
,
4
,
5which satisfy the conditions mentioned in
Theorem 1 and by taking X=Yand F=Gwe get different coupled fixed point theorems.
Example 1. Let X= [0,1] and Y=[−1,0]
Consider the metric S∗defined on both Xand Yas
S∗(a1,···,a
n)=
n
i=1
i<j |ai−aj|
For x, u ∈X, consider the partial order ≤as x≤u⇔x=u
and for y, v ∈Y, define partial order ≤as y≤v⇔y=v.
Let F:X×Y→Xand G:Y×X→Ybe defined as
F(x, y)=x−y
2and G(y, x)=2y−x
3
As per the partial order defined on
X
and
Y
it can be easily verified that
F
and
G
are mixed monotone
mappings and satisfy the conditions (1) and (2).
Here {(x, −x):x∈[0,1]}is the set of all FG- coupled fixed points.
Theorem 2. [17] Let (
X, S∗
x,⪯P1
)and (
Y,S∗
y,⪯P2
)be two partially ordered complete
S∗
metric spaces
and F:X×Y→Xand G:Y×X→Ybe two mappings with mixed monotone property satisfying:
S∗
xF(x, y),···,F(x, y),F(u, v)≤aS∗
x(x, ···, x, u)+bS∗
y(y, ···, y, v),∀(x, y)≤12 (u, v)(14)
and
S∗
yG(y, x),···,G(y, x),G(v, u)≤aS∗
y(y, ···, y, v)+bS∗
x(x, ···, x, u),∀(y, x)≤21 (v, u)(15)
for non negative a, b with a+b<1. Also suppose that either
(I) F and G are continuous or
(II) Xand Yhave the following properties:
(i) if {zk}is an increasing sequence in Xwith zk→z, then zk⪯P1zfor all k∈N
(ii) if {wk}is a decreasing sequence in Ywith wk→w, then w⪯P2wkfor all k∈N.
https://doi.org/10.17993/3ctic.2022.112.81-97
If there exist
x0∈X
and
y0∈Y
with (
x0,y
0
)
≤12 F
(
x0,y
0
)
,G
(
y0,x
0
)
, then there exist an
FG
-
coupled fixed point in X×Y.
Moreover unique FG- coupled fixed point exists if
(III)
for every (
x, y
)
,
(
x1,y
1
)
∈X×Y
there exist a (
u, v
)
∈X×Y
that is comparable to both
(x, y)and (x1,y
1).
Proof. Following as in the prof of Theorem 1 we can construct an increasing sequence
{xm}m∈N
in
Xand a decreasing sequence {ym}m∈Nin Ydefined as:
xm+1 =F(xm,y
m)and ym+1 =G(ym,x
m)(16)
with the property that
(xm,y
m)≤12 (xm+1,y
m+1)
and
(ym,x
m)≤21 (ym+1,x
m+1)
By (16) we have
xm+1 =F(xm,y
m)=Fm+1(x0,y
0)and ym+1 =G(ym,x
m)=Gm(y0,x
0)(17)
Claim: For p∈N,
S∗
x(xp,···,x
p,x
p+1)≤(a+b)pS∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)(18)
and
S∗
y(yp,···,y
p,y
p+1)≤(a+b)pS∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)(19)
Now we prove the claim by the method of mathematical induction.
When p=1we have,
S∗
x(x1,···,x
1,x
2)=S∗
xF(x0,y
0),···,F(x0,y
0),F(x1,y
1)
≤aS
∗
x(x0,···,x
0,x
1)+bS
∗
y(y0,···,y
0,y
1)
≤(a+b)[S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
and
S∗
y(y1,···,y
1,y
2)=S∗
y(y2,···,y
2,y
1)
=S∗
yG(y1,x
1),···,G(y1,x
1),G(y0,x
0)
≤aS
∗
y(y1,···,y
1,y
0)+bS
∗
x(x1,···,x
1,x
0)
=aS
∗
y(y0,···,y
0,y
1)+bS
∗
x(x0,···,x
0,x
1)
≤(a+b)S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)
Therefore the claim is true for p=1.
Now assume the claim for p≤mand check for p=m+1.
Consider,
S∗
x(xm+1,···,x
m+1,x
m+2)= S∗
xF(xm,y
m),···,F(xm,y
m),F(xm+1,y
m+1)
≤aS
∗
x(xm,···,x
m,x
m+1)+bS
∗
y(ym,···,y
m,y
m+1)
≤a(a+b)m[S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
+b(a+b)m[S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
=(a+b)m+1 [S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
Similarly,
S∗
y(ym+1,···,y
m+1,y
m+2)≤(a+b)m+1 [S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
https://doi.org/10.17993/3ctic.2022.112.81-97
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
90
Corollary 2. Let (
X, d, ⪯
)be a partially ordered complete metric space and
F
:
X×X→X
be a
mapping having the mixed monotone property on Xsatisfying:
dF(x, y),F(u, v)≤kd
x, F (u, v)+ld
u, F (x, y)∀(x, y)≥(u, v)
for k, l ∈[0,1
2)
Suppose that either
(I) Fis continuous or
(II) Xsatisfy the following:
(i) if {xk}is an increasing sequence in Xwith xk→x, then xk⪯xfor all k∈N
(ii) if {yk}is a decreasing sequence in Xwith yk→y, then y⪯ykfor all k∈N.
If there exist x0,y
0∈Xsuch that (x0,y
0)≤F(x0,y
0),F(y0,x
0)then Fhas a coupled fixed point.
Remark 1. By putting different values to the constants
ai,b
i
;
i
=1
,
2
,
3
,
4
,
5which satisfy the conditions
mentioned in Theorem 1 we get various FG- coupled fixed point theorems.
Remark 2. By varying the constants
ai,b
i
;
i
=1
,
2
,
3
,
4
,
5which satisfy the conditions mentioned in
Theorem 1 and by taking X=Yand F=Gwe get different coupled fixed point theorems.
Example 1. Let X= [0,1] and Y=[−1,0]
Consider the metric S∗defined on both Xand Yas
S∗(a1,···,a
n)=
n
i=1
i<j |ai−aj|
For x, u ∈X, consider the partial order ≤as x≤u⇔x=u
and for y, v ∈Y, define partial order ≤as y≤v⇔y=v.
Let F:X×Y→Xand G:Y×X→Ybe defined as
F(x, y)=x−y
2and G(y, x)=2y−x
3
As per the partial order defined on
X
and
Y
it can be easily verified that
F
and
G
are mixed monotone
mappings and satisfy the conditions (1) and (2).
Here {(x, −x):x∈[0,1]}is the set of all FG- coupled fixed points.
Theorem 2. [17] Let (
X, S∗
x,⪯P1
)and (
Y,S∗
y,⪯P2
)be two partially ordered complete
S∗
metric spaces
and F:X×Y→Xand G:Y×X→Ybe two mappings with mixed monotone property satisfying:
S∗
xF(x, y),···,F(x, y),F(u, v)≤aS∗
x(x, ···, x, u)+bS∗
y(y, ···, y, v),∀(x, y)≤12 (u, v)(14)
and
S∗
yG(y, x),···,G(y, x),G(v, u)≤aS∗
y(y, ···, y, v)+bS∗
x(x, ···, x, u),∀(y, x)≤21 (v, u)(15)
for non negative a, b with a+b<1. Also suppose that either
(I) F and G are continuous or
(II) Xand Yhave the following properties:
(i) if {zk}is an increasing sequence in Xwith zk→z, then zk⪯P1zfor all k∈N
(ii) if {wk}is a decreasing sequence in Ywith wk→w, then w⪯P2wkfor all k∈N.
https://doi.org/10.17993/3ctic.2022.112.81-97
If there exist
x0∈X
and
y0∈Y
with (
x0,y
0
)
≤12 F
(
x0,y
0
)
,G
(
y0,x
0
)
, then there exist an
FG
-
coupled fixed point in X×Y.
Moreover unique FG- coupled fixed point exists if
(III)
for every (
x, y
)
,
(
x1,y
1
)
∈X×Y
there exist a (
u, v
)
∈X×Y
that is comparable to both
(x, y)and (x1,y
1).
Proof. Following as in the prof of Theorem 1 we can construct an increasing sequence
{xm}m∈N
in
Xand a decreasing sequence {ym}m∈Nin Ydefined as:
xm+1 =F(xm,y
m)and ym+1 =G(ym,x
m)(16)
with the property that
(xm,y
m)≤12 (xm+1,y
m+1)
and
(ym,x
m)≤21 (ym+1,x
m+1)
By (16) we have
xm+1 =F(xm,y
m)=Fm+1(x0,y
0)and ym+1 =G(ym,x
m)=Gm(y0,x
0)(17)
Claim: For p∈N,
S∗
x(xp,···,x
p,x
p+1)≤(a+b)pS∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)(18)
and
S∗
y(yp,···,y
p,y
p+1)≤(a+b)pS∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)(19)
Now we prove the claim by the method of mathematical induction.
When p=1we have,
S∗
x(x1,···,x
1,x
2)=S∗
xF(x0,y
0),···,F(x0,y
0),F(x1,y
1)
≤aS
∗
x(x0,···,x
0,x
1)+bS
∗
y(y0,···,y
0,y
1)
≤(a+b)[S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
and
S∗
y(y1,···,y
1,y
2)=S∗
y(y2,···,y
2,y
1)
=S∗
yG(y1,x
1),···,G(y1,x
1),G(y0,x
0)
≤aS
∗
y(y1,···,y
1,y
0)+bS
∗
x(x1,···,x
1,x
0)
=aS
∗
y(y0,···,y
0,y
1)+bS
∗
x(x0,···,x
0,x
1)
≤(a+b)S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)
Therefore the claim is true for p=1.
Now assume the claim for p≤mand check for p=m+1.
Consider,
S∗
x(xm+1,···,x
m+1,x
m+2)= S∗
xF(xm,y
m),···,F(xm,y
m),F(xm+1,y
m+1)
≤aS
∗
x(xm,···,x
m,x
m+1)+bS
∗
y(ym,···,y
m,y
m+1)
≤a(a+b)m[S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
+b(a+b)m[S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
=(a+b)m+1 [S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
Similarly,
S∗
y(ym+1,···,y
m+1,y
m+2)≤(a+b)m+1 [S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
https://doi.org/10.17993/3ctic.2022.112.81-97
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
91
Thus the claim is true for all p∈N.
Next we prove that {xp}p∈Nand {yp}p∈Nare Cauchy sequences in Xand Yrespectively.
Consider for p, q ∈Nwith p<q,
S∗
x(xp,···,x
p,x
q)
≤(n−1)S∗
x(xp,···,x
p,x
p+1)+S∗
x(xq,···,x
q,x
p+1)
=(n−1)S∗
x(xp,···,x
p,x
p+1)+S∗
x(xp+1,···,x
p+1,x
q)
≤(n−1)S∗
x(xp,···,x
p,x
p+1)+(n−1)S∗
x(xp+1,···,x
p+1,x
p+2)
+S∗
x(xq,···,x
q,x
p+2)
=(n−1)S∗
x(xp,···,x
p,x
p+1)+S∗
x(xp+1,···,x
p+1,x
p+2)
+S∗
x(xp+2,···,x
p+2,x
q)
...
...
...
≤(n−1)S∗
x(xp,···,x
p,x
p+1)+S∗
x(xp+1,···,x
p+1,x
p+2)
+···+S∗
x(xq−2,···,x
q−2,x
q−1)+S∗
x(xq,···,x
q,x
q−1)
=(n−1)
q−2
i=p
S∗
x(xi,···,x
i,x
i+1)+S∗
x(xq−1,···,x
q−1,x
q)
≤(n−1)
q−2
i=p
(a+b)i[S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
+(a+b)q−1[S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
=(n−1) [S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
q−2
i=p
(a+b)i
+(a+b)q−1[S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
≤(n−1) (a+b)p
1−a−b[S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
+(a+b)q−1[S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
→0as p, q →∞,since a+b<1.
Thus, {xm}m∈Nis a Cauchy sequence in X.
Similarly we have,
S∗
y(yp,···,y
p,y
q)≤(n−1) (a+b)p
1−a−b[S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
+(a+b)q−1[S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
→0as p, q →∞,since a+b<1.
Thus, {ym}m∈Nis a Cauchy sequence in Y.
Since Xand Yare complete S∗metric spaces, there exist x∈Xand y∈Ysuch that
lim
p→∞ xp=xand lim
p→∞ yp=y(20)
As in the Theorem 1, by assuming the continuity of
F
and
G
we can prove that (
x, y
)
∈X×Y
is an
FG- coupled fixed point.
Now, suppose that Xand Yhave the properties (i)and (ii)respectively.
Since
{xm}
is increasing in
X
and
{ym}
is decreasing in
Y
and by using (20) we have
∀m∈Nxm⪯P1x
https://doi.org/10.17993/3ctic.2022.112.81-97
and ym⪰P2y
That is by the definition of partial order on X×Yand Y×Xwe have
(xm,y
m)≤12 (x, y)and (ym,x
m)≥21 (y, x)
Now consider,
S∗
xx, ···, x, F (x, y)
≤(n−1) S∗
xx, ···, x, F (xp,y
p)+S∗
xF(x, y),···,F(x, y),F(xp,y
p)
=(n−1) S∗
xx, ···, x, F (xp,y
p)+S∗
xF(xp,y
p),···,F(xp,y
p),F(x, y)
≤(n−1) S∗
x(x, ···, x, xp+1)+aS
∗
x(xp,···,x
p,x)+bS
∗
y(yp,···,y
p,y)
→0as p →∞
Thus, F(x, y)=x.
Similarly,
S∗
yy, ···, y, G(y, x)
≤(n−1) S∗
yy, ···, y, G(yp,x
p)+S∗
yG(y, x),···,G(y, x),G(yp,x
p)
=(n−1) S∗
y(y, ···,y,y
p+1)+S∗
yG(y, x),···,G(y, x),G(yp,x
p)
≤(n−1) S∗
y(y, ···,y,y
p+1)+aS
∗
y(y, ···, y, yp)+bS
∗
x(x, ···, x, xp)
→0as p →∞
Thus, G(y, x)=y.
That is, F(x, y)=xand G(y, x)=y.
Hence (x, y)is an FG- coupled fixed point.
Next we prove the uniqueness of FG- coupled fixed point.
Claim: for any two points (x1,y
1),(x2,y
2)∈X×Ywhich are comparable and for all k∈N
S∗
xFk(x1,y
1),···,Fk(x1,y
1),Fk(x2,y
2)≤(a+b)kS∗
x(x1,···,x
1,x
2)+S∗
y(y1,···,y
1,y
2)(21)
and
S∗
yGk(y1,x
1),···,G
k(y1,x
1),G
k(y2,x
2)≤(a+b)kS∗
x(x1,···,x
1,x
2)+S∗
y(y1,···,y
1,y
2)(22)
Without loss of generality assume that (x1,y
1)≤12 (x2,y
2).
That is by the definition of partial order we have x1⪯P1x2and y2⪯P2y1
By the mixed monotone property of Fand Gwe have
F(x1,y
1)⪯P1F(x2,y
1)
⪯P1F(x2,y
2)
and
G(y1,x
1)⪰p2G(y2,x
1)
⪰p2G(y2,x
2)
Again by the mixed monotone property of Fand Gwe have
F2(x1,y
1)=FF(x1,y
1),G(y1,x
1)
⪯P1FF(x2,y
2),G(y1,x
1)
⪯P1FF(x2,y
2),G(y2,x
2)
=F2(x2,y
2)
https://doi.org/10.17993/3ctic.2022.112.81-97
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
92
Thus the claim is true for all p∈N.
Next we prove that {xp}p∈Nand {yp}p∈Nare Cauchy sequences in Xand Yrespectively.
Consider for p, q ∈Nwith p<q,
S∗
x(xp,···,x
p,x
q)
≤(n−1)S∗
x(xp,···,x
p,x
p+1)+S∗
x(xq,···,x
q,x
p+1)
=(n−1)S∗
x(xp,···,x
p,x
p+1)+S∗
x(xp+1,···,x
p+1,x
q)
≤(n−1)S∗
x(xp,···,x
p,x
p+1)+(n−1)S∗
x(xp+1,···,x
p+1,x
p+2)
+S∗
x(xq,···,x
q,x
p+2)
=(n−1)S∗
x(xp,···,x
p,x
p+1)+S∗
x(xp+1,···,x
p+1,x
p+2)
+S∗
x(xp+2,···,x
p+2,x
q)
...
...
...
≤(n−1)S∗
x(xp,···,x
p,x
p+1)+S∗
x(xp+1,···,x
p+1,x
p+2)
+···+S∗
x(xq−2,···,x
q−2,x
q−1)+S∗
x(xq,···,x
q,x
q−1)
=(n−1)
q−2
i=p
S∗
x(xi,···,x
i,x
i+1)+S∗
x(xq−1,···,x
q−1,x
q)
≤(n−1)
q−2
i=p
(a+b)i[S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
+(a+b)q−1[S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
=(n−1) [S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
q−2
i=p
(a+b)i
+(a+b)q−1[S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
≤(n−1) (a+b)p
1−a−b[S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
+(a+b)q−1[S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
→0as p, q →∞,since a+b<1.
Thus, {xm}m∈Nis a Cauchy sequence in X.
Similarly we have,
S∗
y(yp,···,y
p,y
q)≤(n−1) (a+b)p
1−a−b[S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
+(a+b)q−1[S∗
x(x0,···,x
0,x
1)+S∗
y(y0,···,y
0,y
1)]
→0as p, q →∞,since a+b<1.
Thus, {ym}m∈Nis a Cauchy sequence in Y.
Since Xand Yare complete S∗metric spaces, there exist x∈Xand y∈Ysuch that
lim
p→∞ xp=xand lim
p→∞ yp=y(20)
As in the Theorem 1, by assuming the continuity of
F
and
G
we can prove that (
x, y
)
∈X×Y
is an
FG- coupled fixed point.
Now, suppose that Xand Yhave the properties (i)and (ii)respectively.
Since
{xm}
is increasing in
X
and
{ym}
is decreasing in
Y
and by using (20) we have
∀m∈Nxm⪯P1x
https://doi.org/10.17993/3ctic.2022.112.81-97
and ym⪰P2y
That is by the definition of partial order on X×Yand Y×Xwe have
(xm,y
m)≤12 (x, y)and (ym,x
m)≥21 (y, x)
Now consider,
S∗
xx, ···, x, F (x, y)
≤(n−1) S∗
xx, ···, x, F (xp,y
p)+S∗
xF(x, y),···,F(x, y),F(xp,y
p)
=(n−1) S∗
xx, ···, x, F (xp,y
p)+S∗
xF(xp,y
p),···,F(xp,y
p),F(x, y)
≤(n−1) S∗
x(x, ···, x, xp+1)+aS
∗
x(xp,···,x
p,x)+bS
∗
y(yp,···,y
p,y)
→0as p →∞
Thus, F(x, y)=x.
Similarly,
S∗
yy, ···, y, G(y, x)
≤(n−1) S∗
yy, ···, y, G(yp,x
p)+S∗
yG(y, x),···,G(y, x),G(yp,x
p)
=(n−1) S∗
y(y, ···,y,y
p+1)+S∗
yG(y, x),···,G(y, x),G(yp,x
p)
≤(n−1) S∗
y(y, ···,y,y
p+1)+aS
∗
y(y, ···, y, yp)+bS
∗
x(x, ···, x, xp)
→0as p →∞
Thus, G(y, x)=y.
That is, F(x, y)=xand G(y, x)=y.
Hence (x, y)is an FG- coupled fixed point.
Next we prove the uniqueness of FG- coupled fixed point.
Claim: for any two points (x1,y
1),(x2,y
2)∈X×Ywhich are comparable and for all k∈N
S∗
xFk(x1,y
1),···,Fk(x1,y
1),Fk(x2,y
2)≤(a+b)kS∗
x(x1,···,x
1,x
2)+S∗
y(y1,···,y
1,y
2)(21)
and
S∗
yGk(y1,x
1),···,G
k(y1,x
1),G
k(y2,x
2)≤(a+b)kS∗
x(x1,···,x
1,x
2)+S∗
y(y1,···,y
1,y
2)(22)
Without loss of generality assume that (x1,y
1)≤12 (x2,y
2).
That is by the definition of partial order we have x1⪯P1x2and y2⪯P2y1
By the mixed monotone property of Fand Gwe have
F(x1,y
1)⪯P1F(x2,y
1)
⪯P1F(x2,y
2)
and
G(y1,x
1)⪰p2G(y2,x
1)
⪰p2G(y2,x
2)
Again by the mixed monotone property of Fand Gwe have
F2(x1,y
1)=FF(x1,y
1),G(y1,x
1)
⪯P1FF(x2,y
2),G(y1,x
1)
⪯P1FF(x2,y
2),G(y2,x
2)
=F2(x2,y
2)
https://doi.org/10.17993/3ctic.2022.112.81-97
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
93
and
G2(y1,x
1)=GG(y1,x
1),F(x1,y
1)
⪰P2GG(y2,x
2),F(x1,y
1)
⪰P2GG(y2,x
2),F(x2,y
2)
=G2(y2,x
2)
Continuing like this we get ∀m∈N∪{0},
Fm(x1,y
1)⪯P1Fm(x2,y
2)and Gm(y1,x
1)⪰P2Gm(y2,x
2)
That is by the definition of partial order on X×Yand Y×Xwe have ∀m∈N∪{0}
Fm(x1,y
1),G
m(y1,x
1)≤12 Fm(x2,y
2),G
m(y2,x
2)
and Gm(y1,x
1),Fm(x1,y
1)≥21 Gm(y2,x
2),Fm(x2,y
2)
Now, we prove the claim by the method of mathematical induction.
When k=1we have,
S∗
xF(x1,y
1),···,F(x1,y
1),F(x2,y
2)
≤aS
∗
x(x1,···,x
1,x
2)+bS
∗
y(y1,···,y
1,y
2)
≤(a+b)[S∗
x(x1,···,x
1,x
2)+S∗
y(y1,···,y
1,y
2)]
and
S∗
yG(y1,x
1),···,G(y1,x
1),G(y2,x
2)
=S∗
yG(y2,x
2),···,G(y2,x
2),G(y1,x
1)
≤aS
∗
y(y2,···,y
2,y
1)+bS
∗
x(x2,···,x
2,x
1)
≤aS
∗
y(y1,···,y
1,y
2)+bS
∗
x(x1,···,x
1,x
2)
≤(a+b)[S∗
x(x1,···,x
1,x
2)+S∗
y(y1,···,y
1,y
2)]
Therefore claim is true for k=1.
Now assume the claim for k≤mand check for k=m+1.
Consider,
S∗
xFm+1(x1,y
1),···,Fm+1(x1,y
1),Fm+1(x2,y
2)
=S∗
xFFm(x1,y
1),G
m(y1,x
1),···,FFm(x1,y
1),G
m(y1,x
1),FFm(x2,y
2),G
m(y2,x
2)
≤aS∗
xFm(x1,y
1),···,Fm(x1,y
1),Fm(x2,y
2)+bS∗
yGm(y1,x
1),···,G
m(y1,x
1),G
m(y2,x
2)
≤a(a+b)m[S∗
x(x1,···,x
1,x
2)+S∗
y(y1,···,y
1,y
2)]
+b(a+b)m[S∗
x(x1,···,x
1,x
2)+S∗
y(y1,···,y
1,y
2)]
=(a+b)m+1[S∗
x(x1,···,x
1,x
2)+S∗
y(y1,···,y
1,y
2)]
Similarly we get,
S∗
yGm+1(y1,x
1),···,G
m+1(y1,x
1),G
m+1(y2,x
2)≤(a+b)m+1[S∗
x(x1,···,x
1,x
2)
+S∗
y(y1,···,y
1,y
2)]
Thus the claim is true for all k∈N.
Suppose that (x, y),(x∗,y
∗)be any two FG- coupled fixed points.
That is
F(x, y)=xand G(y, x)=y(23)
https://doi.org/10.17993/3ctic.2022.112.81-97
and
F(x∗,y
∗)=x∗and G(y∗,x
∗)=y∗(24)
Case 1: If (x, y)and (x∗,y
∗)are comparable, then
S∗
x(x, ···, x, x∗)=S∗
xF(x, y),···,F(x, y),F(x∗,y
∗)
≤aS
∗
x(x, ···, x, x∗)+bS
∗
y(y, ···, y, y∗)(25)
and
S∗
y(y, ···, y, y∗)=S∗
yG(y, x),···,G(y, x),G(y∗,x
∗)
≤aS
∗
y(y, ···,y,y
∗)+bS
∗
x(x, ···, x, x∗)(26)
Adding (25) and (26) we get
S∗
x(x, ···, x, x∗)+S∗
y(y, ···, y, y∗)≤(a+b)[S∗
x(x, ···, x, x∗)+S∗
y(y, ···, y, y∗)]
which implies that S∗
x(x, ···, x, x∗)+S∗
y(y, ···,y,y
∗)=0since a+b<1.
Therefore we have, x=x∗and y=y∗.
Case 2: Suppose (x, y)and (x∗,y
∗)are not comparable.
Then by the hypothesis there exist (u, v)∈X×Ywhich is comparable to both (x, y)and (x∗,y
∗).
Consider,
S∗
x(x, ···, x, x∗)=S∗
xFk(x, y),···,Fk(x, y),Fk(x∗,y
∗)
≤(n−1) S∗
xFk(x, y),···,Fk(x, y),Fk(u, v)+S∗
xFk(x∗,y
∗),···,Fk(x∗,y
∗),Fk(u, v)
≤(n−1) (a+b)k[S∗
x(x, ···, x, u)+S∗
y(y, ···, y, v)]
+(a+b)k[S∗
x(x∗,···,x
∗,u)+S∗
y(y∗,···,y
∗,v)]
→0as k →∞since a+b<1
Thus we have x=x∗.
Consider,
S∗
y(y, ···,y,y
∗)
=S∗
yGk(y, x),···,G
k(y, x),G
k(y∗,x
∗)
≤(n−1) S∗
yGk(y, x),···,G
k(y, x),G
k(v, u)+S∗
yGk(y∗,x
∗),···,G
k(y∗,x
∗),G
k(v, u)
≤(n−1) (a+b)k[S∗
x(x, ···, x, u)+S∗
y(y, ..., y, v)]
+(a+b)k[S∗
x(x∗,···,x
∗,u)+S∗
y(y∗,···,y
∗,v)]
→0as k →∞since a+b<1
Thus by Definition 1 (ii) we have y=y∗.
Therefore, x=x∗and y=y∗
Hence the proof.
Remark 3. By taking
n
=2,
a
=
b
=
k
2
,
X
=
Y
and
F
=
G
and assuming condition (I) and (II) of
the above theorem we get the Theorems 2.1 and 2.2 of Bhaskar and Lakshmikantham [?] respectively as
corollaries to our results.
Remark 4. By taking
n
=2,
a
=
b
=
k
2
,
X
=
Y
and
F
=
G
and assuming condition (I) and (III) of
the above theorem we get the Theorems 2.4 of Bhaskar and Lakshmikantham [?] as a corollary to our
results.
We illustrate the above theorem with the following example.
https://doi.org/10.17993/3ctic.2022.112.81-97
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
94
and
G2(y1,x
1)=GG(y1,x
1),F(x1,y
1)
⪰P2GG(y2,x
2),F(x1,y
1)
⪰P2GG(y2,x
2),F(x2,y
2)
=G2(y2,x
2)
Continuing like this we get ∀m∈N∪{0},
Fm(x1,y
1)⪯P1Fm(x2,y
2)and Gm(y1,x
1)⪰P2Gm(y2,x
2)
That is by the definition of partial order on X×Yand Y×Xwe have ∀m∈N∪{0}
Fm(x1,y
1),G
m(y1,x
1)≤12 Fm(x2,y
2),G
m(y2,x
2)
and Gm(y1,x
1),Fm(x1,y
1)≥21 Gm(y2,x
2),Fm(x2,y
2)
Now, we prove the claim by the method of mathematical induction.
When k=1we have,
S∗
xF(x1,y
1),···,F(x1,y
1),F(x2,y
2)
≤aS
∗
x(x1,···,x
1,x
2)+bS
∗
y(y1,···,y
1,y
2)
≤(a+b)[S∗
x(x1,···,x
1,x
2)+S∗
y(y1,···,y
1,y
2)]
and
S∗
yG(y1,x
1),···,G(y1,x
1),G(y2,x
2)
=S∗
yG(y2,x
2),···,G(y2,x
2),G(y1,x
1)
≤aS
∗
y(y2,···,y
2,y
1)+bS
∗
x(x2,···,x
2,x
1)
≤aS
∗
y(y1,···,y
1,y
2)+bS
∗
x(x1,···,x
1,x
2)
≤(a+b)[S∗
x(x1,···,x
1,x
2)+S∗
y(y1,···,y
1,y
2)]
Therefore claim is true for k=1.
Now assume the claim for k≤mand check for k=m+1.
Consider,
S∗
xFm+1(x1,y
1),···,Fm+1(x1,y
1),Fm+1(x2,y
2)
=S∗
xFFm(x1,y
1),G
m(y1,x
1),···,FFm(x1,y
1),G
m(y1,x
1),FFm(x2,y
2),G
m(y2,x
2)
≤aS∗
xFm(x1,y
1),···,Fm(x1,y
1),Fm(x2,y
2)+bS∗
yGm(y1,x
1),···,G
m(y1,x
1),G
m(y2,x
2)
≤a(a+b)m[S∗
x(x1,···,x
1,x
2)+S∗
y(y1,···,y
1,y
2)]
+b(a+b)m[S∗
x(x1,···,x
1,x
2)+S∗
y(y1,···,y
1,y
2)]
=(a+b)m+1[S∗
x(x1,···,x
1,x
2)+S∗
y(y1,···,y
1,y
2)]
Similarly we get,
S∗
yGm+1(y1,x
1),···,G
m+1(y1,x
1),G
m+1(y2,x
2)≤(a+b)m+1[S∗
x(x1,···,x
1,x
2)
+S∗
y(y1,···,y
1,y
2)]
Thus the claim is true for all k∈N.
Suppose that (x, y),(x∗,y
∗)be any two FG- coupled fixed points.
That is
F(x, y)=xand G(y, x)=y(23)
https://doi.org/10.17993/3ctic.2022.112.81-97
and
F(x∗,y
∗)=x∗and G(y∗,x
∗)=y∗(24)
Case 1: If (x, y)and (x∗,y
∗)are comparable, then
S∗
x(x, ···, x, x∗)=S∗
xF(x, y),···,F(x, y),F(x∗,y
∗)
≤aS
∗
x(x, ···, x, x∗)+bS
∗
y(y, ···, y, y∗)(25)
and
S∗
y(y, ···, y, y∗)=S∗
yG(y, x),···,G(y, x),G(y∗,x
∗)
≤aS
∗
y(y, ···,y,y
∗)+bS
∗
x(x, ···, x, x∗)(26)
Adding (25) and (26) we get
S∗
x(x, ···, x, x∗)+S∗
y(y, ···, y, y∗)≤(a+b)[S∗
x(x, ···, x, x∗)+S∗
y(y, ···, y, y∗)]
which implies that S∗
x(x, ···, x, x∗)+S∗
y(y, ···,y,y
∗)=0since a+b<1.
Therefore we have, x=x∗and y=y∗.
Case 2: Suppose (x, y)and (x∗,y
∗)are not comparable.
Then by the hypothesis there exist (u, v)∈X×Ywhich is comparable to both (x, y)and (x∗,y
∗).
Consider,
S∗
x(x, ···, x, x∗)=S∗
xFk(x, y),···,Fk(x, y),Fk(x∗,y
∗)
≤(n−1) S∗
xFk(x, y),···,Fk(x, y),Fk(u, v)+S∗
xFk(x∗,y
∗),···,Fk(x∗,y
∗),Fk(u, v)
≤(n−1) (a+b)k[S∗
x(x, ···, x, u)+S∗
y(y, ···, y, v)]
+(a+b)k[S∗
x(x∗,···,x
∗,u)+S∗
y(y∗,···,y
∗,v)]
→0as k →∞since a+b<1
Thus we have x=x∗.
Consider,
S∗
y(y, ···,y,y
∗)
=S∗
yGk(y, x),···,G
k(y, x),G
k(y∗,x
∗)
≤(n−1) S∗
yGk(y, x),···,G
k(y, x),G
k(v, u)+S∗
yGk(y∗,x
∗),···,G
k(y∗,x
∗),G
k(v, u)
≤(n−1) (a+b)k[S∗
x(x, ···, x, u)+S∗
y(y, ..., y, v)]
+(a+b)k[S∗
x(x∗,···,x
∗,u)+S∗
y(y∗,···,y
∗,v)]
→0as k →∞since a+b<1
Thus by Definition 1 (ii) we have y=y∗.
Therefore, x=x∗and y=y∗
Hence the proof.
Remark 3. By taking
n
=2,
a
=
b
=
k
2
,
X
=
Y
and
F
=
G
and assuming condition (I) and (II) of
the above theorem we get the Theorems 2.1 and 2.2 of Bhaskar and Lakshmikantham [?] respectively as
corollaries to our results.
Remark 4. By taking
n
=2,
a
=
b
=
k
2
,
X
=
Y
and
F
=
G
and assuming condition (I) and (III) of
the above theorem we get the Theorems 2.4 of Bhaskar and Lakshmikantham [?] as a corollary to our
results.
We illustrate the above theorem with the following example.
https://doi.org/10.17993/3ctic.2022.112.81-97
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
95
Example 2. Let X= [0,∞)and Y=(−∞,0] with the usual order in R
Consider the S∗metric on both Xand Yas
S∗(a1,···,a
n)=
n
i=1
i<j |ai−aj|
Define F:X×Y→Xand G:Y×X→Yby
F(x, y)=2x−3y
7nand G(y, x)=2y−3x
7n
For x, u ∈Xand y, v ∈Ywith x≤uand y≥vwe have
2x−3y
7n≤2u−3y
7n,2y−3x
7n≥2y−3u
7n
and 2x−3y
7n≤2x−3v
7n,2y−3x
7n≥2v−3x
7n
That is,
F(x, y)≤F(u, y),G(y, x)≥G(y, u)and F(x, y)≤F(x, v),G(y, x)≥G(v, x)
Therefore Fand Gare mixed monotone mappings.
Next we show that Fand Gsatisfy the contractive type conditions (14) and (15)
S∗(F(x, y),···,F(x, y),F(u, v))=(n−1) 2x−3y
7n−2u−3v
7n
≤2
7n(n−1)|x−u|+3
7n(n−1)|y−v|
=2
7nS∗(x, ···, x, u)+ 3
7nS∗(y, ···, y, v)
and
S∗(G(y, x),···,G(y, x),G(v, u))=(n−1) 2y−3x
7n−2v−3u
7n
≤2
7n(n−1)|y−v|+3
7n(n−1)|x−u|
=2
7nS∗(y, ···,y,v)+ 3
7nS∗(x, ···, x, u)
Therefore Fand Gsatisfy the contractive type conditions (14) and (15) for a=2
7nand b=3
7n.
Here (0,0) is the unique FG- coupled fixed point.
REFERENCES
[1]
Abdellaoui, M.A. and Dahmani, Z. (2016). New Results on Generalized Metric Spaces, Malay-
sian Journal of Mathematical Sciences 10(1): 69 - 81.
[2]
Ajay Singh and Nawneet Hooda (2014). Coupled Fixed Point Theorems in S-metric Spaces.
International Journal of Mathematics and Statistics Invention, 2 (4), 33 – 39.
[3]
Dhage B. C. (1992). Generalized metric spaces mappings with fixed point. Bull. Calcutta Math.
Soc. 84, 329 - 336.
[4] Gahler, S (1963). 2-metriche raume und ihre topologische strukture. Math. Nachr. 26,115 - 148.
[5]
Gnana Bhaskar T.,Lakshmikantham V. (2006). Fixed point theorems in partially ordered
metric spaces and applications. Nonlinear Analysis 65, 1379 - 1393.
https://doi.org/10.17993/3ctic.2022.112.81-97
[6]
Hans Raj and Nawneet Hooda (2014). Coupled fixed point theorems in S- metric spaces
with mixed g- monotone property. International Journal of Emerging Trends in Engineering and
Development, 4 (4), 68 – 81.
[7]
Hemant Kumar Nashine (2012). Coupled common fixed point results in ordered G-metric spaces.
J. Nonlinear Sci. Appl. 1, 1 – 13.
[8]
Huang Long- Guang,Zhang Xian (2007). Cone metric spaces and fixed point theorems of
contractive mappings. Journal of Mathematical Analysis and Applications 332, 1468 – 1476.
[9]
Erdal Karapnar,Poom Kumam and Inci M Erhan (2012). Coupled fixed point theorems on
partially ordered G-metric spaces. Fixed Point Theory and Applications, 2012:174.
[10]
Mujahid Abbas,Bashir Ali and Yusuf I Suleiman (2015). Generalized coupled common
fixed point results in partially ordered A-metric spaces. Fixed Point Theory and Applications,64 DOI
10.1186/s13663- 015-0309-2.
[11]
Mustafa, Z. and Sims B.(2006). A new approach to generalized metric spaces. J. Nonlinear
Convex Anal., 7(2), 289 - 297.
[12]
Prajisha E. and Shaini P. (2019). FG- coupled fixed point theorems for various contractions in
partially ordered metric spaces. Sarajevo Journal of Mathematics, vol.15 (28), No.2, 291 – 307
[13]
K. Prudhvi (2016). Some Fixed Point Results in S-Metric Spaces. Journal of Mathematical
Sciences and Applications, Vol. 4, No. 1, 1-3.
[14]
Sabetghadam F.,Masiha H.P. and Sanatpour A.H. (2009). Some coupled fixed point
theorems in cone metric spaces. Fixed Point Theory and Applications, 8 doi:10.1155/2009/125426.
Article ID 125426.
[15]
Sedghi S.,Shobe N. and Aliouche A.(2012). A generalization of fixed point theorem in
S-metric spaces. Mat. Vesnik, 64, 258 – 266.
[16]
Sedghi S.,Shobe N. and Zhou H. (20007). A common fixed point theorem in
D∗
metric space.
Fixed Point Theory Appl.,1 - 13.
[17]
Prajisha E. and Shaini P. (2017). FG- coupled fixed point theorems in generalized metric spaces.
Mathematical Sciences International Research Journal, Volume 6 (Spl Issue), 24-29.
[18]
Karichery Deepa, and Shaini Pulickakunnel (2018). FG-coupled fixed point theorems for
contractive type mappings in partially ordered metric spaces. Journal of Mathematics and Applications
41.
[19]
Prajisha E. and Shaini P. (2017). FG- coupled fixed point theorems in cone metric spaces.
Carpathian Math. Publ., 9 (2), 163–170.
https://doi.org/10.17993/3ctic.2022.112.81-97
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
96
Example 2. Let X= [0,∞)and Y=(−∞,0] with the usual order in R
Consider the S∗metric on both Xand Yas
S∗(a1,···,a
n)=
n
i=1
i<j |ai−aj|
Define F:X×Y→Xand G:Y×X→Yby
F(x, y)=2x−3y
7nand G(y, x)=2y−3x
7n
For x, u ∈Xand y, v ∈Ywith x≤uand y≥vwe have
2x−3y
7n≤2u−3y
7n,2y−3x
7n≥2y−3u
7n
and 2x−3y
7n≤2x−3v
7n,2y−3x
7n≥2v−3x
7n
That is,
F(x, y)≤F(u, y),G(y, x)≥G(y, u)and F(x, y)≤F(x, v),G(y, x)≥G(v, x)
Therefore Fand Gare mixed monotone mappings.
Next we show that Fand Gsatisfy the contractive type conditions (14) and (15)
S∗(F(x, y),···,F(x, y),F(u, v))=(n−1) 2x−3y
7n−2u−3v
7n
≤2
7n(n−1)|x−u|+3
7n(n−1)|y−v|
=2
7nS∗(x, ···, x, u)+ 3
7nS∗(y, ···, y, v)
and
S∗(G(y, x),···,G(y, x),G(v, u))=(n−1) 2y−3x
7n−2v−3u
7n
≤2
7n(n−1)|y−v|+3
7n(n−1)|x−u|
=2
7nS∗(y, ···,y,v)+ 3
7nS∗(x, ···, x, u)
Therefore Fand Gsatisfy the contractive type conditions (14) and (15) for a=2
7nand b=3
7n.
Here (0,0) is the unique FG- coupled fixed point.
REFERENCES
[1]
Abdellaoui, M.A. and Dahmani, Z. (2016). New Results on Generalized Metric Spaces, Malay-
sian Journal of Mathematical Sciences 10(1): 69 - 81.
[2]
Ajay Singh and Nawneet Hooda (2014). Coupled Fixed Point Theorems in S-metric Spaces.
International Journal of Mathematics and Statistics Invention, 2 (4), 33 – 39.
[3]
Dhage B. C. (1992). Generalized metric spaces mappings with fixed point. Bull. Calcutta Math.
Soc. 84, 329 - 336.
[4] Gahler, S (1963). 2-metriche raume und ihre topologische strukture. Math. Nachr. 26,115 - 148.
[5]
Gnana Bhaskar T.,Lakshmikantham V. (2006). Fixed point theorems in partially ordered
metric spaces and applications. Nonlinear Analysis 65, 1379 - 1393.
https://doi.org/10.17993/3ctic.2022.112.81-97
[6]
Hans Raj and Nawneet Hooda (2014). Coupled fixed point theorems in S- metric spaces
with mixed g- monotone property. International Journal of Emerging Trends in Engineering and
Development, 4 (4), 68 – 81.
[7]
Hemant Kumar Nashine (2012). Coupled common fixed point results in ordered G-metric spaces.
J. Nonlinear Sci. Appl. 1, 1 – 13.
[8]
Huang Long- Guang,Zhang Xian (2007). Cone metric spaces and fixed point theorems of
contractive mappings. Journal of Mathematical Analysis and Applications 332, 1468 – 1476.
[9]
Erdal Karapnar,Poom Kumam and Inci M Erhan (2012). Coupled fixed point theorems on
partially ordered G-metric spaces. Fixed Point Theory and Applications, 2012:174.
[10]
Mujahid Abbas,Bashir Ali and Yusuf I Suleiman (2015). Generalized coupled common
fixed point results in partially ordered A-metric spaces. Fixed Point Theory and Applications,64 DOI
10.1186/s13663- 015-0309-2.
[11]
Mustafa, Z. and Sims B.(2006). A new approach to generalized metric spaces. J. Nonlinear
Convex Anal., 7(2), 289 - 297.
[12]
Prajisha E. and Shaini P. (2019). FG- coupled fixed point theorems for various contractions in
partially ordered metric spaces. Sarajevo Journal of Mathematics, vol.15 (28), No.2, 291 – 307
[13]
K. Prudhvi (2016). Some Fixed Point Results in S-Metric Spaces. Journal of Mathematical
Sciences and Applications, Vol. 4, No. 1, 1-3.
[14]
Sabetghadam F.,Masiha H.P. and Sanatpour A.H. (2009). Some coupled fixed point
theorems in cone metric spaces. Fixed Point Theory and Applications, 8 doi:10.1155/2009/125426.
Article ID 125426.
[15]
Sedghi S.,Shobe N. and Aliouche A.(2012). A generalization of fixed point theorem in
S-metric spaces. Mat. Vesnik, 64, 258 – 266.
[16]
Sedghi S.,Shobe N. and Zhou H. (20007). A common fixed point theorem in
D∗
metric space.
Fixed Point Theory Appl.,1 - 13.
[17]
Prajisha E. and Shaini P. (2017). FG- coupled fixed point theorems in generalized metric spaces.
Mathematical Sciences International Research Journal, Volume 6 (Spl Issue), 24-29.
[18]
Karichery Deepa, and Shaini Pulickakunnel (2018). FG-coupled fixed point theorems for
contractive type mappings in partially ordered metric spaces. Journal of Mathematics and Applications
41.
[19]
Prajisha E. and Shaini P. (2017). FG- coupled fixed point theorems in cone metric spaces.
Carpathian Math. Publ., 9 (2), 163–170.
https://doi.org/10.17993/3ctic.2022.112.81-97
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
97
