International Journal of Innovative Technology and Exploring Engineering (IJITEE)
ISSN: 2278-3075, Volume-9 Issue-3, January 2020
1363
Published By:
Blue Eyes Intelligence Engineering
& Sciences Publication
Retrieval Number: B7502129219/2020©BEIESP
DOI: 10.35940/ijitee.B7502.019320
Abstract. In this communication, we establish new fixed point
theorem in G
b
-metric spaces. Moreover we examine the results for
existence as well as uniqueness, which are related to the G-metric
space. Our results generalize distinguished results and the
mapping satisfying such contraction mention in the literature. In
this sense, our results provide extension as well as improvement in
the results of G
b
-metric space. Also, we give some examples which
verify our results.
Keywords: Set-valued contraction, b-metric spaces, fixed point
theory, generalized G
b
-metric spaces.
2010 MSC: 54H25, 47S40, 54A40.
I. INTRODUCTION
The fixed point theory is very significant and practical in
mathematics. Due to its important and simplicity, several
authors extended it many different results of fixed point
theory. Amini [2] generalized its results in quasi contraction
maps and also identifies various results of fixed point theory.
TV. An [3] expressed the results of stone type theorem on
b-metric space and allow some deep understanding of Fuzzy
b-metric, set valued quasi contractions and Suzuki-type fixed
point results, as we can see in [8,9]. Nashine and Kadelburg
[5] found some results over contraction mapping which are
loyal for all notions in metric spaces. Arshad [4] derived some
different results of metric spaces which plays very important
role in fixed point in b-metric spaces. Subsequently, many
authors extend and generalized these theorems in different
directions. In this article, we give a new generalize metric
spaces introduce by Hussain et al.[6,7] and that covers a huge
class of topological spaces including dislocated metric spaces
with Fatou's property, G
b
-spaces, b-metric spaces, generalize
metric spaces.
In this work, we establish some results of [1] for generalize
G
b
-metric spaces, of course three variable x, y, z will be
consider for solution. Also, the obtained results are supported
by an application and examples for the existence and
uniqueness solution for integral type problems.
II. PRELIMINARIES
Revised Manuscript Received on January 05, 2020
* Correspondence Author
R. K Saini, Departments of Mathematical Sciences & Computer
Applications, B. U., Jhansi, U.P. India
Mukesh Kushwaha, Departments of Mathematical Sciences & Computer
Applications, B. U., Jhansi, U.P. India
Adesh Kumar Tripathi*, Department of Mathematics, Maharishi
Markandeshwar, Deemed to be University, Mullana, Haryana, India.
Definition.1 let X be a non-empty set and 1 s be a given
real number. A function
*
: d E E
is called b-metric
[1, 2, 3] if it satisfies the following properties for each
,, uvw E
a.
*
(,) 0 ; d uv u v
b.
* *
(,) ( , ); d uv d vu
c.
* * *
(, ) (,) (, ) d uw sd uv d uw
The pair
*
( , ) Ed is called a b-metric space.
Example.1 Let ( )
p
E l with
1
0 p
s
where
1
( ) :
p
p n n
n
l u u
define
*
: d E E
as
1
*
1
,
p
p
n n
n
d uv u v
where ,
n n
u u v v then
*
d
is a b -metric space with coefficient
1
2
p
s .
2. Let
1
0,
p
E L
s
be the set of all real function
*
1
, 0, u t t
s
such that
1
*
0
p
u t
with
1
0 p
s
define
*
: d E E
as:
1
1
* * *
0
,
p
p
d uv u t v t
Then
*
d is b-metric with coefficient
1
2
p
s . If we take
1 s then the results similar goes to b -metric space with the
concept of metric space for more details see [8, 9, 10]
Definition.2 Let
*
, Ed be a b -metric space. A sequence
n
u in E is called
(i) Cauchy iff
*
, 0
l k
d uv as , lk ;
(ii)Convergent iff there exists u E such
that
*
, 0
n
d u u as n i.e. lim
n
n
u u
;
(iii)The b-metric space , Ed is complete [12] if every
Cauchy sequence is convergent.
III. MAIN RESULTS
In this section we collect the previous information and to
show the generalization of G
b
-metric spaces also extended
some results of fixed point results
Definition.3 Let ( , ) Ed be a G-metric space. A sequence
n
u in E is said to be
New Fixed Point Results In Generalized
G
b
-Metric Spaces
R. K Saini, Mukesh Kushwaha, Adesh Kumar Tripathi