Collective Sensing: a Fixed-Point Approach in the Metric Space 1
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Abstract—In this paper we established some fixed point
theorems in the frame of a ( )vb s metric space which is
generalize of b-metric space and v-generalized metric space.
Our results extended and improved the results of the classical
results by Branchini and Sehgal.
Index Terms— Fixed point theory, metric space, b-metric
space, wt-distance
I. INTRODUCTION
HE origin of fixed point theory known as the Banach
contraction principle, which is the starting point of
metric fixed point theory. The study of metric fixed point
theory has been on the past two decades. Many fixed point
theorems were obtained in the notion of metric space (or the
standard metric space) and its generalization (see in [1], [3],
[6], [8], [11] and [12]). One generalization of the metric
space is b-metric space which was introduced by Bakhtin [2]
and Czerwik [5]. Branciari [3] introduced the concept of
v generalized metric space. Furthermore, Mitrovic and
Radenovic [10] introduced ( )vb s -metric spaces which
generalizes the concept of b-metric spaces and v-generalize
metric space. Applications of metric fixed point theory were
given in different areas like variational and linear are
inequalities and approximation theory. A number of
researchers had defined contractive type mapping on a
metric space. The classical by Banach, Kannan, Reich,
Chetterea, and Sehgal gave a substantial new contractive
mapping to prove the fixed point theorem. Hussan et. al. [7]
introduced a new concept of wt-distance on b-metric space
and proved some fixed point theorem by using wt-distance
in a partially order b-metric space. Moreover, Some fixed
point theorems on the setting of Banach and Reich
contractions in vb (s) -metric spaces were given by Mitrovic
and Radenovic [10]. In this paper, we present some fixed
point theorems in the setting of Sehgal contraction in vb (s)
-metric spaces.
II. PRELIMINARIES
Definition 1 ([2]). Let X be a nonempty set and 1s a
Manuscript received March 21, 2018; revised March 31, 2018. This
work was financial supported in part by the Faculty of Science, Maejo
University, Chiang Mai, Thailand, 50290, under Grant MJU2-60-069
Y. Khongtham is with the Faculty of Science, Maejo University, Chiang
Mai, Thailand, (Phone: +66-53873881; fax: +66-53878225; e-mail:
yaowa.k@ mju.ac.th).
given reed number. A function : [0, )p X X R is
called b metric if for all , ,x y z X the following condition
are satisfied:
1) ( , ) 0p x y if and only if ;x y
2) ( , ) ( , );p x y p y x
3) ( , ) [ ( , ) ( , )]p x z s p x y p y z
A triplet ( , , )X p s is called a b metric space with
coefficient s. (in short bMS)
Definition 2 ([5]). Let X be a nonempty set and 1s a
given real number. A mapping : [0, ]p X X is called
a rectangular metric space is for all ,x y X satisfy.
1) ( , ) 0p x y if and any if ;x y
2) ( , ) ( , )p x y p y x
3) ( , ) [ ( , ) ( , ) ( , )]p x y s p x u p u v p v y for all distinct
points , , .u v X x y
The pair ( , )X p is called a rectangular b-metric space (in
short RbMS)
Definition 3 ([3]). Let X be a nonempty set and 1s a
given read number, and .v N A function from
[0, )X X is called v generalized metric if for all
,x y X and for all distinct founts 1 2, ,..., vt t t X , each of
them different from x and ,y the following conditions are
satisfied:
1) ( , ) 0vp x y if and only if ;x y
2) ( , ) ( , );v vp x y p y x
3) there exists a real number 1s such that
1 1 2( , ) [ ( , ) ( , ) ... ( , ).v v v v vp x y s p x t p t t p t y
Then ( , )vX p is called a v generalized metric space.
Definition 4 ([11]). Let X be a nonempty set and 1s a
given read number, and .v N A function vp from
[0, )X X is called ( )vb s -metric space if for all
,x y X and for all distinct founts 1 2, ,..., vt t t X , each of
them different from x and ,y the following conditions are
satisfied:
1) ( , ) 0vp x y if and only if ;x y
2) ( , ) ( , );v vp x y p y x
3) there exists a real number 1s such that
Contraction on Some Fixed Point Theorem
in ( )vb s Metric Spaces
Yaowaluck Khongtham
T
Proceedings of the World Congress on Engineering 2018 Vol I WCE 2018, July 4-6, 2018, London, U.K.
ISBN: 978-988-14047-9-4 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
WCE 2018
1 1 2( , ) [ ( , ) ( , ) ... ( , ).v v v v vp x y s p x t p t t p t y
Definition 5 ([9]). Let ( , )vX p be a ( )vb s metric space,
nz be a sequence in X and .z X Then
1) The sequence nz is said to be convergent in
( , )vX p and converges to ,z if for every 0 there exists
0k such that ( , )np z z for all 0n k and denote by
lim .nn
z z
2) The sequence nz is said to Cauchy sequence in
( , )vX p if for every 0 there exists 0k such that
( , )m np z z for all 0, .m n k
3) ( , )vX p is said to be a complete ( )vb s metric space if
every Cauchy sequence in X converges to some .z X
Definition 6 ([9]). Let vp a real value function on a
( )vb s metric is said to be s lower semi-continuous at a
point z in X if liminf ( )nz zn
p z
or
( ) liminf ( ),nz zn
p z sp z
when ever nz X for each n N and
lim nn
z z
.
Definition 7 ([11]). Let ( , )vX p be a ( )vb s metric space
with constant 1.s Then a function : [0, )vp XxX is
called a wt distance on X if the following conditions are
satisfied:
1) 1 1 2 1, ( , ) ( , ) ... ( , ) ( ,v v vp x y s p x t p t t p t t p t y
for every v distinct 1 2 3, , ,..., , ;\vt t t t X x y
2) for any , ( , ) : [0, )vx X p x X is s lower semi-
continuous;
3) for any 0 there exists 0 such that ( , )p z x
and ( , )vp z y imply ( , ) .p x y
Lemma 1 ([11]). Let ( , )vX p be a ( )vb s metric space with
constant 1s and let q be a wt distance on .X let
nz and nu be sequences in .X Let n and n be
sequences in [0, ) converging to 0, and let
, , .k u t X Then the following hold:
1) If ( , )n nq z u and ( , )n nq z t for any ,n then
;u t
2) If ( , )n n nq z u and ( , )n nq z t for any ,n then
nu converges to ;t
3) If ( , )n m nq z z for any ,n m with ,m n then is
a Cauchy sequence;
4) If ( , )n nq u z for any ,n then nz is a Cauchy
sequence.
III. MAIN RESULTS
Theorem 1. Let q be a wt distance on a ( )vb s metric
space (X, p) with constant s 1. Let 1T and 2T be mappings
from X into itself. Suppose that there exists 1
[0, )rs
such
that
1) 1 2 1 2 1 2max ( ), ( ) , ( ), ( )p T x T T x p T x TT x
1 2min p x,T (x) ,p x,T (x)
for every x X and
2) 1 2inf , min , , , : 0p x v p x T x p x T x x X
for every v X with v is not a common fixed point in .X
Moreover, if 1 2 ,zz T z T then ( , ) 0.p z z
Proof. Let 0z X be arbitrary and defined a sequence
nz by
1 1
2 1
,
, .
n
n
n
T z if n is oddz
T z if n is even
If n is odd, then by using 1), we have
1( , )n np z z
1 1 2,n np T z T z
1 1 2 1 1,n np T z T T z
1 1 2 1 1 2 1 1 2 1max , , ,n n n np T z T T z p T z TT z
1 1 1 1 2 1min , , ,n n n np z T z p z T z
1 1 1,n np z T z
1, .n np z z
If n is even, then by 1), we have
1( , )n np z z
2 1 1,n np T z T z
2 1 1 2 1,n np T z TT z
1 1 2 1 1 2 1 1 2 1max , , ,n n n np T z T T z p T z TT z
1 1 1 1 2 1min , , ,n n n np z T z p z T z
1 2 1,n np z T z
1, .n np z z
Thus for any positive integer ,n we obtain
1, 1 , .n n np z zn p z z (3)
1 0 1, , .n
n np z z p z z (4)
For ,m n with ,m n we have by repeated of (4) that
1 1 2 3 2, , , ... ,n m n n n n n v n vp z z s p z z p z z p z z
2 0 0 0 0, , , .n v n n n n m n m n mp z z p z z p z z
Then
Proceedings of the World Congress on Engineering 2018 Vol I WCE 2018, July 4-6, 2018, London, U.K.
ISBN: 978-988-14047-9-4 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
WCE 2018
1 3
0 1, ... ,n n n v
n mp z z s p z z
2 0 0 0 0, , ,n v n n n n m n m n msp z z p z z p z z
1 3
0 1... ,n n n vs p z z
02 0, ,
nn
v n n ms p z z s p z z
000
( , ) 1 , .nm
n n ms p z z s p z z
1 300 11 , ... ,
n n n n v
n ms p z z s p z z
2 00 0, , .n m
v n ns p z z s p z z
We have that nz is Cauchy sequence in .X Since X is
complete, nz converge to some point .z X Let n be
fixed. Then since lim nn
z z
and ,np z is s lower semi
continuous, we have
, liminf ,n n mn
p z z sp z z
2
2 3
0 101 ... ,
1
nv
n
sp z z
s
2 2
2 00 00 0, , .
1 1
n m
v n nn n
s sp z z p z z
s s
Assume that v is not a common fixed point of 1T and
2 .T Then hypothesis 2), we have
1 20 inf , ) min , , , :p z v p z T z p z T z z X
1 2inf , min , , , :n n n n np z v p z T z p z T z n
2
3
0 10inf , 1 ...
1
nv
n
sp z z
s
2
2 00,
1
n
v nn
sp z z
s
2
0 100, , :
1
nmn n nn
sp z z p z z n
s
= 0,
which is a contradiction. Therefore, 1 2 .v T v T v If
1 2T v v T v for some ,v X then
( , )p v v max 1 2 1 2 1 2, , ,p T v T T v p T v TT v
1 2min , , ,p v T v p v T v
= min , , ,p v v p v v
= ,p v v
which give that , 0.p v v
Corollary 2. Let ( , )X p be complete ( )vb s metric space
with constant 1s and : .T X X Let q be a wt distance
on .X Suppose there exists 1
[0, )s
such that
2, ,q Tx T x q x Tx
for every x X and that
inf ( , ) ( , ) : 0p x y p x Tx x X
for every y X with .y Ty Then T has a fixed point in
.X Moreover, if ,z Tz then , 0.p z z
Proof. We have the desired result by taking 1 2T T T in
Theorem 1.
Theorem 3. Let ( , )X p be complete ( )vb s metric space
with constant 1s and : .T X X Suppose there exists
1[0, )
s such that
, max ( , ), ( , ), ( , ), ( , )p Tx Ty p x y p x Tx p y Ty p y Tx (5)
for every , .x y X Then T has a unique fixed point in .X
Proof. We treat the ( )vb s metric p as wt distance on .X
From (5), we have
2( , ) ( , ( ))p Tx T x p Tx T Tx
2max ( , ), ( , ), ( , ), ( , )p x Tx d x Tx d Tx T x p Tx Tx
2max ( , ), ( , ) .p x Tx p Tx T x (6)
Without loss of generality, we assume that 2 .Tx T x For,
otherwise, T has a fixed point. Since 1
,s
we obtain from
(6) that
2( , ) ( , ).p Tx T x p x Tx (7)
For every .x X Assume that there exists v X with
v Tv and inf ( , ) ( , ) : 0.p x v p x Tx x X Then there
exists a sequence nz in X such that
lim ( , ) ( , 0.n n nn
p z v p z Tz
So, we get lim , 0nn
p z v
and lim ( , ) 0.nn
p z Tz
So we
obtain from 1) in Lemma 1 that lim .nn
Tz v
We also have
1 1 2( , ) ( , ) ( , ) ( , ) ( , )n n n n n n np v Tv s p v Tz d Tz z d z z p z z
3 2 2... ( , ) ( , ) .n v n v n vp z z p z Tv
Then we have ( , ) ( , ).p v Tv sp v Tv Therefore
( , ) 0.p v Tv Then .v Tv This is a contradiction. Hence, if
Proceedings of the World Congress on Engineering 2018 Vol I WCE 2018, July 4-6, 2018, London, U.K.
ISBN: 978-988-14047-9-4 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
WCE 2018
,v Tv then inf ( , ) ( , ) : 0.p x y p x Tx x X By applying
Corollary 2, we obtain a fixed point of T in .X For
uniqueness, let v and *v are fixed points of ,T that is
( )T v v and * *( ) .T v v It follows from (5) that
* * * * *, max ( , ), ( , ), ( , ), ( , )p v v p v v p v Tv p v Tv p v Tv
* * * *max ( , ), ( , ), ( , ), ( , )p v v p v v p v v p v v
*( , ).p v v
Therefore, we have *( , ) 0,p v v i.e., *.v v
Remark 1. If we put 1s and ( , ) 0p x y in Theorem 1. we
have the fixed point result given by Branchini [4].
Remark 2. If we put 1s and 1v in Theorem 2 we have
the fixed point result given by Sehgal [13].
IV. CONCLUSION
Our theorem 2 extended many theorems in the previous
results given by many authors in the setting of metric space
(standard metric pace), b-metric space, bMS, RbMS, and
v generalized metric space as in remark 1. and remark 2.
Moreover, we shall extend this idea by using others
contractions.
ACKNOWLEDGMENT
I would like to thank the anonymous referee for valuable
comments.
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Proceedings of the World Congress on Engineering 2018 Vol I WCE 2018, July 4-6, 2018, London, U.K.
ISBN: 978-988-14047-9-4 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
WCE 2018