Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
10,
No.
4,
August
2020,
pp.
3839
3853
ISSN:
2088-8708,
DOI:
10.11591/ijece.v10i4.pp3839-3853
r
3839
'
-contraction
and
some
fixed
point
r
esults
via
modified
!
-distance
mappings
in
the
frame
of
complete
quasi
metric
spaces
and
applications
K.
Aboday
eh
1
,
T
.
Qawasmeh
2
,
W
.
Shatanawi
1,3
,
A.
T
allafha
4
1,3
Department
of
Mathematics
and
general
sciences,
Prince
Sultan
Uni
v
ersity
Riyadh,
Saudi
Arabia
2
Department
of
Mathematics,
F
aculty
of
Science
and
Information
T
echnology
,
Jadara
Uni
v
ersity
,
Irbid,
Jordan
3
Department
of
Medical
Research,
China
Medical
Uni
v
ersity
Hospital,
China
Medical
Uni
v
ersi
ty
,
T
aichung,
T
aiw
an
4
Department
of
mathematics,
School
of
Science,
Uni
v
ersity
of
Jordan,
Amman,
Jordan
Article
Inf
o
Article
history:
Recei
v
ed
Jun
14,
2019
Re
vised
Jan
12,
2020
Accepted
Feb
2,
2020
K
eyw
ords:
'
-contraction
Common
fix
ed
point
Modified
!
-distance
Quasi
metric
space
Ultra
distance
function
ABSTRA
CT
In
this
Article,
we
intr
oduce
the
notion
of
an
'
-contraction,
which
is
based
on
modified
!
-distance
mappings,
and
emplo
y
this
ne
w
definition
to
pro
v
e
some
fix
ed
point
result.
Moreo
v
er
,
to
highlight
the
significance
of
our
w
ork,
we
present
an
interesting
e
xample
along
with
an
application.
Copyright
c
2020
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Kamaleldin
Abodayeh,
Prince
Sultan
Uni
v
ersity
,
Riyadh
11586,
Saudi
Arabia.
Email:
kamal@psu.edu.sa
1.
INTR
ODUCTION
One
of
the
most
important
methods
in
mathematics
used
t
o
discuss
the
e
xistence
and
uniqueness
of
a
solution
of
such
equations
is
the
Banach
contraction
principle
[1].
It
is
consi
dered
as
a
v
aluable
tool
in
fix
ed
point
theory
.
Since
then,
man
y
mathematicians
in
v
estig
ated
the
Banach
contraction
princi
p
l
e
in
man
y
directions.
In
[2],
Abodayeh
et
al.
utili
zed
the
concept
of
distance
to
gi
v
e
some
ne
w
generalizations
of
Banach
contraction
principle.
Shatana
wi,
M.
Postolache
in
[3,
4]
studied
some
common
fix
ed
points
of
such
mappings.
F
or
more
generali
zations
of
Banach
fix
ed
point
theory
,
see
[5–18].
In
1931
W
ilson
[19]
introduced
the
notion
of
quasi
metric
space
as
belo
w:
Definition
1
[19]
W
e
call
the
function
q
:
E
E
!
[0
;
1
)
a
quasi
metric
if
it
satisfies:
(i)
q
(
e
1
;
e
2
)
=
0
,
e
1
=
e
2
;
(ii)
q
(
e
1
;
e
2
)
(
e
1
;
e
3
)
+
q
(
e
3
;
e
1
)
for
all
e
1
;
e
2
;
e
3
2
E
.
The
pair
(
E
;
q
)
is
called
a
quasi
metric
space.
F
or
some
w
ork
in
quasi
metric
spaces,
see
[20–23]
If
the
symmetry
condition
is
added
to
(
E
;
q
)
(i.e.
q
(
e
1
;
e
2
)
=
q
(
e
2
;
e
1
)
for
all
e
1
;
e
2
2
E
),
then
the
space
(
E
;
q
)
is
a
metric
space.
J
ournal
homepage:
http://ijece
.iaescor
e
.com/inde
x.php/IJECE
Evaluation Warning : The document was created with Spire.PDF for Python.
3840
r
ISSN:
2088-8708
Henceforth,
we
denote
by
(
E
;
q
)
a
quasi
metric
space.
T
o
generate
a
metric
d
on
E
.
Define
d
:
E
E
!
[0
;
1
)
by
d
=
max
f
q
(
e
1
;
e
2
)
;
q
(
e
2
;
e
1
)
g
:
The
concepts
of
completeness
and
con
v
er
gence
of
quasi
metric
spaces
are
gi
v
en
belo
w:
Definition
2
[24,
25]
A
sequence
(
e
s
)
con
ver
g
es
to
e
2
E
if
lim
s
!1
q
(
e
s
;
e
)
=
lim
s
!1
q
(
e
;
e
s
)
=
0
.
Definition
3
[25]
Let
(
e
s
)
be
a
sequence
in
E
.
Then
we
call
(i)
(
e
s
)
left-Cauc
hy
if
for
any
>
0
,
ther
e
e
xists
N
0
2
N
suc
h
that
q
(
e
s
;
e
t
)
<
for
all
s
t
>
N
0
.
(ii)
(
e
s
)
right-Cauc
hy
if
for
any
>
0
,
ther
e
e
xists
N
0
2
N
suc
h
that
q
(
e
s
;
e
t
)
<
for
all
t
s
>
N
0
.
Definition
4
[24,
25]
A
sequence
(
e
s
)
in
E
is
called
a
Cauc
hy
sequence
if
(i)
If
for
any
>
0
;
ther
e
e
xists
N
0
2
N
suc
h
that
q
(
e
s
;
e
t
)
for
all
s;
t
>
N
0
;
or
(ii)
(
e
s
)
is
right
and
left
Cauc
hy
.
Definition
5
[24,
25]
W
e
say
(
E
;
q
)
is
complete
if
e
very
Cauc
hy
sequence
(
e
s
)
in
E
is
con
ver
g
ent.
In
2016,
Ale
gre
and
Marin
[26]
introduced
the
notion
of
modified
!
-distance
mappings
on
(
E
;
q
)
.
Definition
6
[26]
A
modified
!
-distance
(shortly
m
!
-distance
)
on
(
E
;
q
)
is
a
function
:
E
E
!
[0
;
1
)
,
whic
h
satisfies
the
following:
(W1)
(
e
1
;
e
2
)
(
e
1
;
e
3
)
+
(
e
3
;
e
2
)
for
all
e
1
;
e
2
;
e
3
2
E
;
(W2)
(
e;
:
)
:
E
!
[0
;
1
)
is
lower
semi-continuous
for
all
e
2
E
;
and
(mW3)
for
eac
h
%
>
0
ther
e
e
xists
>
0
suc
h
that
if
(
e
1
;
e
2
)
and
(
e
2
;
e
3
)
,
then
q
(
e
1
;
e
3
)
%
for
all
e
1
;
e
2
;
e
3
2
E
.
Henceforth,
we
denote
by
an
m
!
-distance
mapping.
Definition
7
[26]
if
is
lower
semi-continuous
on
the
fir
st
and
second
coor
dinates,
then
is
called
a
s
tr
ong
m
!
-distance
.
Remark
1
[26]
Every
quasi
metric
q
on
E
is
m
!
-distance
.
Lemma
1
[33]
Let
(
%
s
)
,
(
s
)
be
two
sequences
of
nonne
gative
r
eal
number
s
that
con
ver
g
e
to
zer
o.
Then
we
have
the
following:
(i)
If
(
e
s
;
e
t
)
%
s
for
all
s;
t
2
N
with
t
s
,
then
(
e
s
)
is
right
Cauc
hy
in
(
E
;
q
)
.
(ii)
If
(
e
s
;
e
t
)
t
for
all
s;
t
2
N
with
t
s
,
then
(
e
s
)
is
left
Cauc
hy
in
(
E
;
q
)
.
Remark
2
[33]
The
abo
ve
lemma
show
that
if
lim
s;t
!1
p
(
e
s
;
e
t
)
=
0
,
then
(
e
s
)
is
Cauc
hy
in
(
E
;
q
)
.
F
or
more
results
in
fix
ed
point
theory
in
!
and
modified
!
distances,
we
ask
the
readers
to
consider
[20,
27–31,
33,
34].
Definition
8
[35]
A
self
function
'
on
[0
;
1
)
is
said
to
be
an
ultr
a
distance
function
if
'
satisfies
'
(
)
=
0
,
=
0
and
if
(
s
)
is
a
sequence
in
[0
;
1
)
suc
h
that
lim
s
!
+
1
'
(
s
)
=
0
,
then
lim
s
!
+
1
s
=
0
.
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
4,
August
2020
:
3839
–
3853
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
3841
2.
MAIN
RESUL
TS
The
definition
of
'
-contraction
on
a
pair
of
self
mappings
is
defined
as
follo
ws:
Definition
9
Equipped
(
E
;
q
)
with
and
let
F
;
T
be
two
self
mappings
on
E
.
Then
the
pair
(
F
;
T
)
is
called
'
-contr
action
if
ther
e
e
xists
an
ultr
a
distance
function
'
and
a
given
>
0
suc
h
that
for
all
e
1
;
e
2
2
E
we
have:
'
(
F
e
1
;
T
e
2
)
(
e
1
;
e
2
)
+
(
e
1
;
F
e
1
)
max
'
(
e
1
;
F
e
1
)
;
'
(
e
2
;
T
e
2
)
:
And
'
(
T
e
1
;
F
e
2
)
(
e
1
;
e
2
)
+
(
e
1
;
T
e
1
)
max
'
(
e
1
;
T
e
1
)
;
'
(
e
2
;
F
e
2
)
:
Ne
xt,
we
introduce
our
first
result:
Theor
em
2
Equipped
(
E
;
q
)
with
and
let
F
;
T
be
two
self
mappings
on
E
suc
h
that
the
pair
(
F
;
T
)
is
an
'
-contr
action.
Also,
assume
(
e
j
+1
;
e
j
)
=
0
or
(
e
j
;
e
j
+1
)
=
0
,
for
some
j
2
N
[
f
0
g
.
Then
e
j
is
a
unique
common
fixed
point
of
F
and
T
in
E
.
Pr
oof
.
Let
e
0
2
E
.
W
e
create
a
sequence
(
e
j
)
in
E
inducti
v
ely
by
taking
F
e
2
j
=
e
2
j
+1
and
T
e
2
j
+1
=
e
2
j
+2
for
all
j
2
N
[
f
0
g
.
T
o
pro
v
e
the
result,
we
ha
v
e
to
consider
the
follo
wing
cases:
Case(1):
(
e
j
;
e
j
+1
)
=
0
.
If
j
is
e
v
en,
then
j
=
2
k
for
some
k
2
N
[
f
0
g
,
so
we
ha
v
e
(
e
2
k
;
e
2
k
+1
)
=
0
and
so
'
(
e
2
k
;
e
2
k
+1
)
=
0
.
No
w
,
since
the
pair
(
F
;
T
)
is
an
'
-contraction,
we
get:
'
(
e
2
k
+1
;
e
2
k
+2
)
=
'
(
F
e
2
k
;
T
e
2
k
+1
)
(
e
2
k
;e
2
k
+1
)
+
(
e
2
k
;F
e
2
k
)
max
'
(
e
2
k
;
F
e
2
k
)
;
'
(
e
2
k
+1
;
T
e
2
k
+1
)
=
(
e
2
k
;e
2
k
+1
)
+
(
e
2
k
;e
2
k
+1
)
max
'
(
e
2
k
;
e
2
k
+1
)
;
'
(
e
2
k
+1
;
e
2
k
+2
)
=
0
:
By
the
definition
of
'
,
we
ha
v
e
(
e
2
k
+1
;
e
2
k
+2
)
=
0
:
(1)
From
the
assumption
we
ha
v
e
(
e
2
k
;
e
2
k
+1
)
=
0
and
by
(1)
we
get
that
(
e
2
k
;
e
2
k
+2
)
=
0
:
(2)
Also,
by
using
mW3
of
the
definition
of
,
we
get
that
q
(
e
2
k
;
e
2
k
+2
)
=
0
:
(3)
'
(
e
2
k
+2
;
e
2
k
+1
)
=
'
(
T
e
2
k
+1
;
F
e
2
k
)
(
e
2
k
+1
;e
2
k
)
+
(
e
2
k
+1
;T
e
2
k
+1
)
max
'
(
e
2
k
+1
;
e
2
k
+2
)
;
'
(
e
2
k
;
e
2
k
+1
)
=
(
e
2
k
+1
;e
2
k
)
+
(
e
2
k
+1
;e
2
k
+2
)
max
'
(
e
2
k
+1
;
e
2
k
+2
)
;
'
(
e
2
k
;
e
2
k
+1
)
=
0
:
Therefore,
(
e
2
k
+2
;
e
2
k
+1
)
=
0
:
(4)
'
-contr
action
and
some
fixed
point
r
esults
...
(K.
Abodayeh)
Evaluation Warning : The document was created with Spire.PDF for Python.
3842
r
ISSN:
2088-8708
Also,
using
the
Equations
(2),
(4)
and
mW3
of
the
definition
of
,
we
get
that
q
(
e
2
k
+1
;
e
2
k
)
=
0
:
(5)
Hence,
e
2
k
=
e
2
k
+1
=
e
2
k
+2
and
so
e
j
is
a
common
fix
ed
point
of
F
and
T
in
E
.
If
j
is
odd,
then
j
=
2
k
+
1
,
for
some
k
2
N
[
f
0
g
.
Then
we
ha
v
e
(
e
2
k
+1
;
e
2
k
+2
)
=
0
and
hence
'
(
e
2
k
+1
;
e
2
k
+2
)
=
0
.
'
(
e
2
k
+2
;
e
2
k
+3
)
=
'
(
T
e
2
k
+1
;
F
e
2
k
+2
)
(
e
2
k
+1
;e
2
k
+2
)
+
(
e
2
k
+1
;T
e
2
k
+1
)
max
'
(
e
2
k
+1
;
e
2
k
+2
)
;
'
(
e
2
k
+2
;
e
2
k
+3
)
=
(
e
2
k
+1
;e
2
k
+2
)
+
(
e
2
k
+1
;e
2
k
+2
)
max
'
(
e
2
k
+1
;
e
2
k
+2
)
;
'
(
e
2
k
+2
;
e
2
k
+3
)
=
(
e
2
k
+1
;e
2
k
+2
)
+
(
e
2
k
+1
;e
2
k
+2
)
'
(
e
2
k
+2
;
e
2
k
+3
)
.
Let
L
=
(
e
2
k
+1
;e
2
k
+2
)
+
(
e
2
k
+1
;e
2
k
+2
)
.
Then
L
<
1
and
so
'
(
e
2
k
+2
;
e
2
k
+3
)
<
'
(
e
2
k
+2
;
e
2
k
+3
)
:
Thus,
'
(
e
2
k
+2
;
e
2
k
+3
)
=
0
.
By
the
definition
'
,
we
get
that
(
e
2
k
+2
;
e
2
k
+3
)
=
0
:
(6)
From
the
assumption,
we
ha
v
e
(
e
2
k
+1
;
e
2
k
+2
)
=
0
and
by
(6),
we
get
(
e
2
k
+1
;
e
2
k
+3
)
=
0
:
(7)
Also,
Condition
mW3
of
the
definition
of
implies
that
q
(
e
2
k
+1
;
e
2
k
+3
)
=
0
:
(8)
'
(
e
2
k
+3
;
e
2
k
+2
)
=
'
(
F
e
2
k
+2
;
T
e
2
k
+1
)
(
e
2
k
+2
;e
2
k
+1
)
+
(
e
2
k
+2
;F
e
2
k
+2
)
max
'
(
e
2
k
+2
;
F
e
2
k
+2
)
;
'
(
e
2
k
+1
;
T
e
2
k
+1
)
=
(
e
2
k
+2
;e
2
k
+1
)
+
(
e
2
k
+2
;e
2
k
+3
)
max
'
(
e
2
k
+2
;
e
2
k
+3
)
;
'
(
e
2
k
+1
;
e
2
k
+2
)
=
0
:
In
a
similar
manner
,
we
can
pro
v
e
that
if
(
e
j
+1
;
e
j
)
=
0
,
then
e
j
is
a
common
fix
ed
point
of
F
and
T
in
E
.
Ne
xt,
we
introduce
our
main
result:
Theor
em
3
Equipped
(
E
;
q
)
with
and
let
F
;
T
be
two
self
mappings
on
E
.
Assume
the
following
conditions
hold:
(
i
)
(
E
;
q
)
is
complete;
(
ii
)
The
pair
(
F
;
T
)
is
an
'
-contr
action
;
(
iii
)
F
and
T
ar
e
continuous;
(
iv
)
F
or
all
e
1
;
e
2
2
E
and
some
inte
g
er
L
we
have
(
e
1
;
e
2
)
L
.
Then
F
and
T
have
a
unique
common
fixed
point
in
E
.
Pr
oof
.
Let
e
0
2
E
.
Construct
a
sequence
(
e
n
)
in
E
inducti
v
ely
by
taking
F
e
2
n
=
e
2
n
+1
and
T
e
2
n
+1
=
e
2
n
+2
for
all
n
2
N
[
f
0
g
.
If
for
some
i
2
N
we
ha
v
e
(
e
i
;
e
i
+1
)
=
0
or
(
e
i
+1
;
e
i
)
=
0
,
then
by
Theorem
2,
e
i
is
a
unique
common
fix
ed
point
of
F
and
T
in
E
.
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
4,
August
2020
:
3839
–
3853
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
3843
No
w
,
assume
that
(
e
n
;
e
n
+1
)
6
=
0
and
(
e
n
+1
;
e
n
)
6
=
0
,
for
all
n
2
N
[
f
0
g
.
Since
the
pair
(
F
;
T
)
is
an
'
-contraction,
then
we
ha
v
e
'
(
e
2
n
+1
;
e
2
n
+2
)
=
'
(
F
e
2
n
;
T
e
2
n
+1
)
(
e
2
n
;e
2
n
+1
)
+
(
e
2
n
;F
e
2
n
)
max
'
(
e
2
n
;
F
e
2
n
)
;
'
(
e
2
n
+1
;
T
e
2
n
+1
)
=
(
e
2
n
;e
2
n
+1
)
+
(
e
2
n
;e
2
n
+1
)
max
'
(
e
2
n
;
e
2
n
+1
)
;
'
(
e
2
n
+1
;
e
2
n
+2
)
.
If
L
=
(
e
2
n
;e
2
n
+1
)
+
(
e
2
n
;e
2
n
+1
)
,
then
L
<
1
.
Also,
if
max
'
(
e
2
n
;
e
2
n
+1
)
;
'
(
e
2
n
+1
;
e
2
n
+2
)
=
'
(
e
2
n
+1
;
e
2
n
+2
)
,
we
get
that
'
(
e
2
n
+1
;
e
2
n
+2
)
L
max
'
(
e
2
n
;
e
2
n
+1
)
;
'
(
e
2
n
+1
;
e
2
n
+2
)
=
L'
(
e
2
n
+1
;
e
2
n
+2
)
<
'
(
e
2
n
+1
;
e
2
n
+2
)
.
(9)
Thus,
'
(
e
2
n
+1
;
e
2
n
+2
)
=
0
and
so
(
e
2
n
+1
;
e
2
n
+2
)
=
0
a
contradiction.
Therefore,
'
(
e
2
n
+1
;
e
2
n
+2
)
(
(
e
2
n
;
e
2
n
+1
)
+
(
e
2
n
;
e
2
n
+1
)
)
'
(
e
2
n
;
e
2
n
+1
)
:
(10)
'
(
e
2
n
+2
;
e
2
n
+1
)
=
'
(
T
e
2
n
+1
;
F
e
2
n
)
(
e
2
n
+1
;e
2
n
)
+
(
e
2
n
+1
;T
e
2
n
+1
)
max
'
(
e
2
n
+1
;
e
2
n
+2
)
;
'
(
e
2
n
;
e
2
n
+1
)
=
(
e
2
n
+1
;e
2
n
)
+
(
e
2
n
+1
;e
2
n
+2
)
max
'
(
e
2
n
+1
;
e
2
n
+2
)
;
'
(
e
2
n
;
e
2
n
+1
)
=
(
e
2
n
+1
;e
2
n
)
+
(
e
2
n
+1
;e
2
n
+2
)
'
(
e
2
n
;
e
2
n
+1
)
:
Also,
we
can
sho
w
that:
'
(
e
n
;
e
n
+1
)
(
(
e
n
1
;
e
n
)
+
(
e
n
1
;
e
n
)
)
'
(
e
n
1
;
e
n
)
:
(11)
And
'
(
e
n
+1
;
e
n
)
(
e
n
;
e
n
1
)
+
(
e
n
;
e
n
+1
)
'
(
e
n
1
;
e
n
)
:
(12)
No
w
,
'
(
e
n
;
e
n
+1
)
(
e
n
1
;e
n
)
+
(
e
n
1
;e
n
)
'
(
e
n
1
;
e
n
)
(
e
n
1
;e
n
)
+
(
e
n
1
;e
n
)
(
e
n
2
;e
n
1
)
+
(
e
n
2
;e
n
1
)
'
(
e
n
2
;
e
n
1
)
Q
n
i
=1
(
e
i
1
;e
i
)
+
(
e
i
1
;e
i
)
'
(
e
0
;
e
1
)
.
let
L
i
=
(
(
e
i
1
;e
i
)
+
(
e
i
1
;e
i
)
)
.
Then
L
i
<
1
for
all
i
2
f
1
;
2
;
;
n
g
,
so
we
ha
v
e
'
(
e
n
;
e
n
+1
)
n
1
Y
i
=1
L
i
(
'
(
e
n
;
e
n
+1
))
:
(13)
'
-contr
action
and
some
fixed
point
r
esults
...
(K.
Abodayeh)
Evaluation Warning : The document was created with Spire.PDF for Python.
3844
r
ISSN:
2088-8708
Letting
n
!
1
,
we
get
lim
n
!1
'
(
e
n
;
e
n
+1
)
=
0
:
(14)
Since
'
is
ultra
distance
function,
we
ha
v
e
lim
n
!1
(
e
n
;
e
n
+1
)
=
0
:
(15)
'
(
e
n
+1
;
e
n
)
(
e
n
;e
n
1
)
+
(
e
n
;e
n
+1
)
'
(
e
n
1
;
e
n
)
(
e
n
;e
n
1
)
+
(
e
n
;e
n
+1
)
(
e
n
2
;e
n
1
)
+
(
e
n
2
;e
n
1
)
'
(
e
n
2
;
e
n
1
)
(
e
n
;e
n
1
)
+
(
e
n
;e
n
+1
)
Q
n
1
i
=1
(
e
i
1
;e
i
)
+
(
e
i
;e
i
1
)
'
(
e
0
;
e
1
)
.
Let
L
i
=
(
(
e
i
1
;e
i
)
+
(
e
i
1
;e
i
)
)
.
Then
L
i
<
1
for
all
i
2
f
1
;
2
;
;
n
1
g
and
since
(
e
1
;
e
2
)
L
for
all
e
1
;
e
1
2
E
and
some
inte
ger
L
,
we
get
that
'
(
e
n
+1
;
e
n
)
L
n
1
Y
i
=1
L
i
(
'
(
e
0
;
e
1
))
:
(16)
Letting
n
!
1
,
we
get
that:
lim
n
!1
'
(
e
n
+1
;
e
n
)
=
0
:
(17)
The
definition
of
'
informs
us
lim
n
!1
(
e
n
+1
;
e
n
)
=
0
:
(18)
No
w
,
we
need
to
sho
w
that
(
e
s
)
is
a
Cauch
y
sequence
in
E
.
In
order
to
do
that,
we
first
pro
v
e
that
(
e
s
)
is
a
right
Cauch
y
sequence
i
n
(
E
;
q
)
.
F
or
each
s;
t
2
N
with
s
<
t
,
we
ha
v
e
the
follo
wing
cases:
Case
(1):
If
s
odd
and
t
e
v
en,
then
we
ha
v
e:
'
(
e
s
;
e
t
)
=
'
(
F
e
s
1
;
T
e
t
1
)
(
e
s
1
;e
t
1
)
+
(
e
s
1
;F
e
s
1
)
max
'
(
e
s
1
;
F
e
s
1
)
;
'
(
e
t
1
;
T
e
t
1
)
=
(
e
s
1
;e
t
1
)
+
(
e
s
1
;e
s
)
max
'
(
e
s
1
;
e
s
)
;
'
(
e
t
1
;
e
t
)
.
=
(
e
s
1
;e
t
1
)
+
(
e
s
1
;e
s
)
'
(
e
s
1
;
e
s
)
:
Let
L
i
=
(
(
e
i
1
;e
i
)
+
(
e
i
1
;e
i
)
)
.
Since
(
e
1
;
e
2
)
L
for
all
e
1
;
e
2
2
E
and
some
inte
ger
L
,
we
ha
v
e
'
(
e
s
;
e
t
)
L
s
1
Y
i
=1
L
i
(
'
(
e
0
;
e
1
))
:
(19)
Letting
s;
t
!
1
,
we
ha
v
e
lim
'
s;t
!1
(
(
e
s
;
e
t
))
=
0
.
Thus,
lim
'
s;t
!1
(
e
s
;
e
t
)
=
0
:
(20)
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
4,
August
2020
:
3839
–
3853
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
3845
Case
(2):
If
s
e
v
en
and
t
odd,
then
we
ha
v
e:
'
(
e
s
;
e
t
)
=
'
(
T
e
s
1
;
F
e
t
1
)
(
e
s
1
;e
t
1
)
+
(
e
s
1
;T
e
s
1
)
max
'
(
e
s
1
;
T
e
s
1
)
;
'
(
e
t
1
;
F
e
t
1
)
=
(
e
s
1
;e
t
1
)
+
(
e
s
1
;e
s
)
max
'
(
e
s
1
;
e
s
)
;
'
(
e
t
1
;
e
t
)
.
=
(
e
s
1
;e
t
1
)
+
(
e
s
1
;e
s
)
'
(
e
s
1
;
e
s
)
:
Let
L
i
=
(
(
e
i
1
;e
i
)
+
(
e
i
1
;e
i
)
)
.
Since
(
e
1
;
e
2
)
L
for
all
e
1
;
e
2
2
E
and
some
inte
ger
L
,
then
we
get
that
'
(
e
s
;
e
t
)
L
s
1
Y
i
=1
L
i
(
'
(
e
0
;
e
1
))
:
(21)
Letting
s;
t
!
1
,
we
ha
v
e
lim
'
s;t
!1
(
(
e
s
;
e
t
))
=
0
.
So,
lim
'
s;t
!1
(
e
s
;
e
t
)
=
0
:
(22)
Case
(3):
If
s
and
t
are
odd,
we
get
(
e
s
;
e
t
)
(
e
s
;
e
s
+1
)
+
(
e
s
+1
;
e
t
)
:
(23)
Hence,
lim
s;t
!1
(
e
s
;
e
t
)
=
0
:
(24)
Case
(4):
If
s
and
t
are
e
v
en,
we
get
(
e
s
;
e
t
)
(
e
s
;
e
t
1
)
+
(
e
t
1
;
e
t
)
:
(25)
Hence,
lim
s;t
!1
(
e
s
;
e
t
)
=
0
:
(26)
Using
Lemma
1,
we
get
that
(
e
s
)
is
a
right
Cauch
y
sequence
in
(
E
;
q
)
.
Similarly
,
we
can
pro
v
e
that
(
e
s
)
is
a
left
Cauch
y
sequence
in
E
.
Hence,
(
e
s
)
is
a
Cauch
y
sequence
in
E
.
The
completeness
of
(
E
;
q
)
implies
that
there
e
xists
an
element
e
2
E
such
that
(
e
s
)
!
e
.
If
F
is
a
continuous
function
then
e
s
+1
=
F
e
s
!
F
e
.
By
the
uniqueness
of
limit,
we
get
that
F
e
=
e
.
In
a
similar
manner
,
we
can
pro
v
e
that
T
e
=
e
when
T
is
a
continuous
function.
T
o
pro
v
e
the
uniqueness
of
e
.
First
we
sho
w
that
(
e
;
e
)
=
0
.
'
(
e
;
e
)
=
'
(
F
e
;
T
e
)
(
e
;e
)
+
(
e
;F
e
max
'
(
e
;
F
e
)
;
'
(
e
;
T
e
)
=
(
e
;e
)
+
(
e
;e
)
max
'
(
e
;
e
)
;
'
(
e
;
e
)
=
0
.
Therefore,
(
e
;
e
)
=
0
.
'
-contr
action
and
some
fixed
point
r
esults
...
(K.
Abodayeh)
Evaluation Warning : The document was created with Spire.PDF for Python.
3846
r
ISSN:
2088-8708
Assume
that
there
e
xists
2
E
such
that
F
=
T
=
.
Then
'
(
e
;
)
=
'
(
F
e
;
T
)
(
e
;
)
+
(
e
;F
e
max
'
(
e
;
F
e
)
;
'
(
;
T
)
=
(
e
;
)
+
(
e
;e
)
max
'
(
e
;
e
)
;
'
(
;
)
=
0
.
Thus,
we
ha
v
e
(
e
;
)
=
0
since
(
e
;
e
)
=
0
we
get
that
q
(
e
;
)
=
0
and
so
e
=
.
Cor
ollary
4
A
complete
(
E
;
q
)
Equipped
with
and
let
F
;
T
be
two
self
continuous
mappings
on
E
.
Assume
the
following
conditions
hold:
(
i
)
F
or
all
e
1
;
e
2
2
E
and
a
given
>
0
and
an
ultr
a
distance
function
'
we
have:
'
(
F
e
1
;
T
e
2
)
(
e
1
;
e
2
)
2(
+
(
e
1
;
F
e
1
))
'
(
e
1
;
F
e
1
)
+
'
(
e
2
;
T
e
2
)
:
And
'
(
T
e
1
;
F
e
2
)
(
e
1
;
e
2
)
2(
+
(
e
1
;
T
e
1
))
'
(
e
1
;
T
e
1
)
+
'
(
e
2
;
F
e
2
)
:
(
ii
)
F
or
all
e
1
;
e
2
2
E
we
have
(
e
1
;
e
2
)
L
for
some
inte
g
er
L
.
Then
F
and
T
have
a
unique
common
fixed
point
in
E
.
Pr
oof
.
'
(
F
e
1
;
T
e
2
)
(
e
1
;e
2
)
2(
+
(
e
1
;F
e
1
))
'
(
e
1
;
F
e
1
)
+
'
(
e
2
;
T
e
2
)
(
e
1
;e
2
)
+
(
e
1
;F
e
1
)
max
'
(
e
1
;
F
e
1
)
;
'
(
e
2
;
T
e
2
)
:
Similarly
,
we
can
pro
v
e
that:
'
(
T
e
1
;
F
e
2
)
(
e
1
;
e
2
)
2(
+
(
e
1
;
T
e
1
)
'
(
e
1
;
T
e
1
)
+
'
(
e
2
;
F
e
2
)
:
Cor
ollary
5
A
complete
(
E
;
q
)
Equipped
with
and
let
F
;
T
be
two
self
continuous
mappings
on
E
.
Assume
the
following
conditions
hold:
(
i
)
F
or
all
e
1
;
e
2
2
E
and
for
a
given
>
0
and
an
ultr
a
distance
function
'
and
k
2
[0
;
1)
we
have:
'
(
F
e
1
;
T
e
2
)
k
'
(
e
1
;
e
2
)
:
And
'
(
T
e
1
;
F
e
2
)
k
'
(
e
1
;
e
2
)
:
(
ii
)
F
or
all
e
1
;
e
2
2
E
we
have
(
e
1
;
e
2
)
L
for
some
inte
g
er
L
.
Then
F
and
T
have
a
unique
common
fixed
point
in
E
.
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
4,
August
2020
:
3839
–
3853
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
3847
Pr
oof
.
Let
'
(
)
=
and
let
k
=
(
(
e
1
;F
e
1
)
+
(
e
1
;F
e
1
)
)
.
Then
k
2
[0
;
1)
.
No
w
,
'
(
F
e
1
;
T
e
2
)
=
(
F
e
1
;
T
e
2
)
(
e
1
;F
e
1
)
+
(
e
1
;F
e
1
)
(
e
1
;
e
2
)
=
(
e
1
;e
2
)
+
(
e
1
;F
e
1
)
(
e
1
;
F
e
1
)
=
(
e
1
;e
2
)
+
(
e
1
;F
e
1
)
'
(
e
1
;
F
e
1
)
(
e
1
;e
2
)
+
(
e
1
;F
e
1
)
max
'
(
e
1
;
F
e
1
)
;
'
(
e
2
;
T
e
2
)
:
Similarly
,
we
can
pro
v
e
that:
'
(
T
e
1
;
F
e
2
)
k
'
(
e
1
;
e
2
)
:
If
we
tak
e
F
=
T
in
Corollary
5,
we
get
the
follo
wing
result:
Cor
ollary
6
A
complete
(
E
;
q
)
Equipped
with
and
let
F
be
a
self
continuous
mapping
on
E
.
Assume
the
following
conditions
hold:
(
i
)
F
or
all
e
1
;
e
2
2
E
and
for
a
given
>
0
and
an
ultr
a
distance
function
'
and
k
2
[0
;
1)
we
have:
'
(
F
e
1
;
F
e
2
)
k
'
(
e
1
;
e
2
)
:
(
ii
)
F
or
all
e
1
;
e
2
2
E
we
have
(
e
1
;
e
2
)
L
for
some
inte
g
er
L
.
Then
F
has
a
unique
common
fixed
point
in
E
.
Example
1
Let
E
=
0
;
1
;
;
m
wher
e
m
2
N
.
Define
F
;
T
on
E
as
follows:
F
(
e
1
)
=
8
<
:
0
if
e
1
2
f
0
;
1
g
;
1
if
e
1
2
f
2
;
3
;
;
5
g
;
2
if
e
1
2
f
6
;
7
;
;
m
g
.
T
(
e
2
)
=
8
<
:
0
if
e
2
2
f
0
;
1
;
;
5
g
;
1
if
e
2
2
f
6
;
7
;
;
10
g
;
2
if
e
2
2
f
11
;
12
;
;
m
g
.
Then
F
and
T
have
a
unique
fixed
point
in
E
.
Pr
oof
.
T
o
show
that
F
and
T
have
a
unique
fixed
point
in
E
.
Define
;
q
:
E
E
!
[0
;
1
)
suc
h
that
q
(
e
1
;
e
2
)
=
2
3
e
1
+
1
3
e
2
:
(
e
1
;
e
2
)
=
2
e
1
+
e
2
:
Also
define
'
(
)
:
[0
;
1
)
!
[0
;
1
)
as
follows:
'
(
)
=
(1
=
4)
if
2
[0
;
m
]
;
(1
=
4)(
2
+
2)
if
>
m
.
Then
1.
F
and
T
ar
e
continuous
functions.
'
-contr
action
and
some
fixed
point
r
esults
...
(K.
Abodayeh)
Evaluation Warning : The document was created with Spire.PDF for Python.
3848
r
ISSN:
2088-8708
2.
'
is
an
ultr
a
distance
function.
3.
(
E
;
q
)
is
a
complete
quasi
metric
space
.
4.
is
an
m
!
-distance
mapping
.
5.
The
pair
(
F
;
T
)
is
'
-contr
action
with
(
=
1)
i.e
.,
8
e
1
;
e
2
2
E
we
have
'
(
F
e
1
;
T
e
2
)
(
(
e
1
;
e
2
)
1
+
(
e
1
;
F
e
1
)
)
max
'
(
e
1
;
F
e
1
)
;
'
(
e
2
;
T
e
2
)
:
And
'
(
T
e
1
;
F
e
2
)
(
(
e
1
;
e
2
)
1
+
(
e
1
;
T
e
1
)
)
max
'
(
e
1
;
T
e
1
)
;
'
(
e
2
;
F
e
2
)
:
Now
,
it
is
an
easy
matter
to
c
hec
k
out
that
F
and
T
ar
e
continuous
functions.
In
addi
tion,,
it
is
obviously
that
'
is
an
ultr
a
distance
function,
is
an
m
!
-distance
mapping
and
(
E
;
q
)
is
a
quasi
metric
space
.
T
o
show
that
q
is
complete
,
let
(
e
s
)
be
a
Cauc
hy
sequence
in
E
.
Then
for
eac
h
s;
t
2
N
we
have
lim
s;t
!1
q
(
e
s
;
e
t
)
=
0
we
conclude
that
e
s
=
e
t
for
all
s;
t
2
N
b
ut
not
for
finitely
many
.
Ther
efor
e
,
(
e
s
)
is
a
con
ver
g
ent
sequence
in
E
.
Consequently
,
(
E
;
q
)
is
a
complete
quasi
metric
space
.
T
o
pr
o
ve
that
the
pair
(
F
;
T
)
is
'
-contr
action
with
(
=
1
),
we
need
to
consider
the
following
cases:
Case
(1):
If
e
1
2
f
0
;
1
g
,
then
we
have
the
following
subcases:
Subcase
(1):
If
e
2
2
f
0
;
1
;
;
5
g
,
then
'
(
F
e
1
;
T
e
2
)
=
'
(0
;
0)
=
0
:
Subcase
(2):
If
e
2
2
f
6
;
7
;
;
10
g
,
then
'
(
F
e
1
;
T
e
2
)
=
'
(0
;
1)
=
'
(1)
=
1
4
:
(
e
1
;e
2
)
1+
(
e
1
;F
e
1
)
max
'
(
e
1
;
F
e
1
)
;
'
(
e
2
;
T
e
2
)
=
(
e
1
;e
2
)
1+
(
e
1
;
0)
1
4
(
e
2
;
1)
=
2
e
1
+
e
2
2
e
1
+1
1
4
(2
e
2
+
1)
13
4
2
e
1
+6
2
e
1
+1
(
8
3
)(
13
4
)
1
4
:
Subcase
(3):
If
e
2
2
f
11
;
12
;
;
m
g
,
then
we
g
et
that
'
(
F
e
1
;
T
e
2
)
=
'
(0
;
2)
=
'
(2)
=
2
4
:
(
e
1
;e
2
)
1+
(
e
1
;F
e
1
)
max
'
(
e
1
;
F
e
1
)
;
'
(
e
2
;
T
e
2
)
=
(
e
1
;e
2
)
1+
(
e
1
;
0)
1
4
(
e
2
;
2)
=
2
e
1
+
e
2
2
e
1
+1
1
4
(2
e
2
+
2)
6
2
e
1
+11
2
e
1
+1
26
2
4
:
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
4,
August
2020
:
3839
–
3853
Evaluation Warning : The document was created with Spire.PDF for Python.