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# A new approach to the study of fixed point theory for simulation functions in G-Metric spaces

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## Abstract

In this paper first of all, we introduce the mapping : [0;1) X [0;1) R,called the simulation function and the notion of Z-contraction with respect to which generalize several known types of contractions. Secondly, we prove certain xed point theorems using simulation functions in G-Metric spaces. An example is also given to support our result.
Bol. Soc. Paran. Mat. (3s.)v. 37 2(2019): 113119.
c
SPM –ISSN-2175-1188 on line ISSN-00378712 in press
SPM: www.spm.uem.br/bspm doi:10.5269/bspm.v37i2.34690
A New Approach to the Study of Fixed Point Theorems for Simulation
Functions in G-Metric Spaces
Manoj Kumar and Rashmi Sharma
abstract: In this paper ﬁrst of all, we introduce the mapping ζ: [0,)×[0,)
R, called the simulation function and the notion of Z-contraction with respect to
ζwhich generalize several known types of contractions. Secondly, we prove certain
ﬁxed point theorems using simulation functions in G-Metric spaces. An example is
also given to support our results.
Key Words: Simulation function; Contraction mapping; Z-contraction; Fixed
point; G-Metric spaces.
Contents
1 Introduction 113
2 Main Results 114
1. Introduction
Let (X, d) be a metric space and T:XXbe a mapping, then Tis called a
contraction(Banach Contraction) on Xif
d(T x, T y)λd(x, y)
for all x, y X.
Where λis a real such that λ[0,1). A point xXis called a ﬁxed point of T
if T x =x. The well-known Banach Contraction Principle[1] ensures the existence
and uniqueness of a ﬁxed point of a contraction on a complete metric space. After
this principle, several authors generalized this principle by introducing the various
contractions on metric spaces[2, 3-9]. In this work, we introduce a mapping namely
simulation funtion and the notion of Z-contraction. Among all the generalized
metric spaces, the notion of G-Metric spaces was introduced by Mustafa and Sims
in[10], where in the authors discuss the topological properties of this space and
proved the analog of the Banach Contraction Principle in the context of G-Metric
spaces.
Deﬁnition 1.1. AG-Metric space (X, G) is said to be symmetric if G(x, y, y) =
G(y, x, x) for all x, y X.
Example 1.2. Let (X, d) be the usual metric space then the function G:
X×X×X[0,) deﬁned by G(x, y, z) = max{d(x, y), d(y, z), d(z , x)}for all
x, y, z Xis a G-Metric space.
2010 Mathematics Subject Classiﬁcation: 47H10, 54H25, 54C30.
Submitted December 30, 2016. Published March 06, 2017
113 Typeset by BSP
Mstyle.
c
Soc. Paran. de Mat.
114 M. Kumar and R. Sharma
Deﬁnition 1.3. Let Xbe a nonempty set and G:X×X×X[0,) be a
function satisfying the following properties:
(G1)G(x, y, z) = 0 if x=y=z,
(G2)0 < G(x, x, y) for all x, y Xwith x6=y,
(G3)G(x, x, y)G(x, y, z ) for all x, y, z X, with z6=y,
(G4)G(x, y, z) = G(x, z, y) = G(y, z , x) = ... (symmetry in all three variables),
(G5)G(x, y, z)G(x, a, a) + G(a, y, z), for all x, y, z , a X(rectangle inequlity).
Then, the function Gis called a generalized metric, or, more speciﬁcally,a G-metric
on X, and the pair (X, G) is called a G-metric space.
Deﬁnition 1.4. Let (X, G),(X, G) be G-Metric spaces, then a function
f:XXis G-continuous at a point xXif and only if it is G-sequentially con-
tinuous at x, that is, whenever {xn}is G-convergent to x,{f(xn)}is G-convergent
to f(x).
Recently, Khojasteh et al. [11] introduced a new class of mappings called sim-
ulation functions. Later Argoubi et al. [12] slightly modiﬁed the deﬁnition of
simulation functions in the deﬁnition of simulation functions by withdrawing a
condition.
Let Zbe the set of simulation functions in the sense of Argoubi et al.[12].
Deﬁnition 1.5. A simulation function is a mapping ζ: [0,)×[0,)R
satisfying the following conditions:
(ζ1)ζ(t, s)< s tfor all t, s > 0
(ζ2) if {tn}and {sn}are sequences in (0,) such that
limn→∞{tn}=limn→∞{sn}=l(0,),
then
limn→∞supζ(tn, sn)<0.
2. Main Results
In this section, we deﬁne the simulation function, give some examples and prove
a related ﬁxed point result.
Deﬁnition 2.1. Let (X, G) be a G-Metric space, f:XXa mapping and
ζZ. Then fis called a Z-contraction with respect to ζif the following condition
is satisﬁed
ζ(G(f x, fy, f z), G(x, y, z)) 0 for all x, y, z X. (2.1)
Lemma 2.2. Let (X, G) be a G-Metric space and f:XXbe a Z-
contraction with respect to ζZ. Then, fis asymptotically regular at every
xX.
Proof: Let xXbe arbitrary. If for some pN
we have fpx=fp+1x, that is f y =y, where y=fp1x, that is fz =z, where
z=fp1x
Study of Fixed Point Theorems for Simulation Functions in G-Metric Spaces115
then, fny=fn1fy =fn1y=... =f y =yfor all nN. Now for suﬃcient large
nN, we obtain
G(fnx, f n+1x, fn+1x) = G(fnp+1 fp1x, f np+2fp1x, f np+2fp1x)
=G(fnp+1y, f np+2y, f np+2y)
=G(y, y, y ) = 0
Therefore, limn→∞ G(fnx, f n+1x, f n+1x) = 0
Suppose, fnx6=fn1xfor all nN, then it follows from (1) that
0ζ(G(fn+1x, f nx, f nx), G(fnx, f n1x, fn1x))
=ζ(G(ffnx, f fn1x, f f n1x), G(fnx, f n1x, fn1x))
G(fnx, f n1x, fn1x)G(fn+1 x, f nx, fnx)
The above inequality show that {G(fnx, f n1x, fn1x)}is a monotonically de-
creasing sequence of non-negative reals and so it must be convergent.
Let limn→∞ G(fnx, f n+1x, fn+1x) = r0. If r > 0 then since fis Z-
contraction with respect to ζZtherefore, we have
0limn→∞supζ(G(fn+1 x, f nx, fnx), G(fnx, f n1x, fn1x)) <0.
This, contradiction shows that r= 0, that is, limn→∞ G(fnx, f n+1x, f n+1x) = 0.
Thus, fis an asymptotically regular mapping at x.
Lemma 2.3. Let (X, G) be a G-Metric space and f:XXbe a Z-
contraction with respect to ζ. Then the Picard sequence {xn}generated by f
with initial value x0Xis a bounded sequence, where xn=f xn1for all nN.
Proof: Let x0Xbe arbitrary and {xn}be the Picard sequence, that is,
xn=fxn1for all nN. On the contrary, assume that {xn}is not bounded.
Without loss of generality we can assume that xn+p6=xnfor all n, p N. Since
{xn}is not bounded, there exists a subsequence {xn}such that n1= 1 and each
kN,nk+1 is the minimum integer such that
G(xn(k)+1, xn(k), xn(k))>1
and
G(xm, xn(k), xn(k))1
for nkmn(k)+1 1. Therefore, by the triangular inequality, we have
1<G(xn(k)+1, xn(k), xn(k))
G(xn(k)+1, xn(k)+1 1, xn(k)+1 1) + G(xn(k)+1 1, xn(k), xn(k))
G(xn(k)+1, xn(k)+1 1, xn(k)+1 1) + 1.
Letting k→ ∞ and using Lemma 2.2 we get
limk→∞G(xn(k)+1, xn(k), xn(k)) = 1
116 M. Kumar and R. Sharma
By (1), we get G(xn(k)+1, xn(k), xn(k))G(xn(k)+1 1, xn(k)1, xn(k)1), therefore
using the triangular inequality we obtain
1< G(xn(k)+1, xn(k), xn(k))G(xn(k)+1 1, xn(k)1, xn(k)1)
G(xn(k)+1 1, xn(k), xn(k)) + G(xn(k), xn(k)1, xn(k)1)
1 + G(xn(k), xn(k)1, xn(k)1)
Letting k→ ∞ and using Lemma 2.2, we obtain
limk→∞ G(xn(k)+1 1, xn(k)1, xn(k)1) = 1
Now, since fis a Z-contraction with respect to ζZtherefore, we have
0limk→∞supζ(G(f xn(k)+1 1, f xn(k)1, fxn(k)1))
=limk→∞supζ(G(xn(k)+1 , xn(k), xn(k)), G(xn(k)+1 1, xn(k)1, xn(k)1)) <0
Theorem 2.4. Let (X, G) be a complete G-Metric space and f:XXbe a
Z-contraction with respect to ζ. Then, fhas a unique ﬁxed point uin Xand for
every x0Xthe Picard sequence {xn}where xn=f xn1for all nNconverges
to the ﬁxed point of f.
Proof: Let x0Xbe arbitrary and {xn}be the Picard sequence, that is,
xn=fxn1for all nN. We shall show that this sequence is a Cauchy sequence.
For this, let
Cn=sup{G(xi, xj, xj) : i, j n}
Note that the sequence {xn}is a monotonically decreasing sequence of positive
reals and by Lemma 2.3 the sequence {xn}is bounded, threrefore Cn<for all
nN. Thus, {Cn}is monotonic bounded sequence, therefore convergent, that is,
there exists C0 such that limn→∞Cn=C. We shall show that C= 0. If C > 0
then by the deﬁnition Cn, for every kNthere exists mk> nkkand
Ck1
k< G(xm(k), xn(k), xn(k))Ck
Hence,
limk→∞G(xm(k), xn(k), xn(k))Ck(2.2)
Using (1) and the triangular inequality, we obtain
G(xm(k), xn(k), xn(k))G(xm(k)1, xn(k)1, xn(k)1)
G(xm(k)1, xm(k), xm(k)) + G(xm(k), xn(k), xn(k))
+G(xn(k), xn(k)1, xn(k)1)
Study of Fixed Point Theorems for Simulation Functions in G-Metric Spaces117
Using Lemma 2.2, (2) and letting k→ ∞ in the above inequality we get
limk→∞ G(xm(k)1, xn(k)1, xn(k)1) = C. (2.3)
Since Tis a Z-contraction with respect to ζZtherefore using (1), (2), (3) and
(ζ2), we get
0limk→∞supζ(G(xm(k)1, xn(k)1, xn(k)1), G(xm(k), xn(k), xn(k))) <0
This contradiction proves that C= 0 and so {xn}is a Cauchy sequence. Since Xis
a complete G-Metric space, there exists uXsuch that limn→∞xn=u. We shall
show that the point uis a ﬁxed point of f. Suppose f u 6=uthen G(u, f u, fu)>0.
Again, using (1), ζ1,ζ2, we have
0limn→∞supζ(G(f xn, f u, f u), G(xn, u, u))
limn→∞supζ[G(xn, u, u)G(xn=1 , f u, fu)]
=G(u, f u, fu)
This contradiction shows that G(u, f u, f u) = 0, that is, f u =u. Thus, uis a ﬁxed
point of f.
Example 2.5. Let X= [0,1] and G:X×XRbe deﬁned by G(x, y, z) =
max{|xy|,|yz|,|zx|}. Then, (X, G) is a complete G-Metric space. Deﬁne a
mapping f:XXas f x =x
x+1 for all xX.fis a continuous function but it
is not a Banach contraction. But it is a Z-contraction with respect to ζZ, where
ζ(t, s) = s
s+ 1 tfor all t, s [0,).
Indeed, if x, y X, then by a simple calculation it can be shown that
ζ(G(f x, fy, f y), G(x, y, y)) 0.
Clearly, 0 is the ﬁxed point of f.
Corollary 2.6. Let (X, G) be a complete G-Metric space and f:XXbe
a mapping satisfying the following condition: G(f x, f y, f y)λG(x, y, y) for all
x, y, y X, where λ[0,1]. Then, fhas a unique ﬁxed point in X.
Proof: Deﬁne ζB: [0,)×[0,)Rby ζB(t, s, s) = λstfor all s, t [0,).
Note that, the mapping fis a Z-contraction with respect to ζBZ. Therefore,
the result follows by taking ζ=ζBin Theorem 2.4.
Corollary 2.7. Let (X, G) be a complete G-Metric space and f:XX
be a mapping satisfying the following condition: G(f x, f y, f y)G(x, y, y)
ϕ(G(x, y, y)) for all x, y, y X, where ϕ: [0,)[0,) is lower semi continuous
function and ϕ1(0) = {0}. Then, fhas a unique ﬁxed point in X.
Proof: Deﬁne ζR: [0,)×[0,)Rby ζR(t, s, s) = sϕ(s)tfor all
s, t [0,). Note that, the mapping fis a Z-contraction with respect to ζRZ.
Therefore, the result follows by taking ζ=ζRin Theorem 2.4.
Corollary 2.8. Let Let (X, G) be a complete G-Metric space and f:XX
be a mapping satisfying the following condition: G(f x, fy , fy)ϕ(G(x, y, y))×
118 M. Kumar and R. Sharma
×G(x, y, y) for all x, y, y X, where ϕ: [0,+)[0,1) be a mapping such that
limsuptr+ϕ(t)<1, for all r > 0. Then, fhas a unique ﬁxed point.
Proof: Deﬁne ζR: [0,)×[0,)Rby ζR(t, s, s) = (s)tfor all
s, t [0,). Note that, the mapping fis a Z-contraction with respect to ζRZ.
Therefore, the result follows by taking ζ=ζRin Theorem 2.4.
Corollary 2.9. Let Let (X, G) be a complete G-Metric space and f:XX
be a mapping satisfying the following condition: G(f x, f y, f y)η(G(x, y, y)) for
all x, y, y X, where η: [0,+)[0,+) be an upper semi continuous mapping
such that η(t)< t for all t > 0 and η(0) = 0. Then, fhas a unique ﬁxed point.
Proof: Deﬁne ζBW: [0,)×[0,)Rby ζBW(t, s, s) = (s)tfor all
s, t [0,). Note that, the mapping fis a Z-contraction with respect to ζBWZ.
Therefore, the result follows by taking ζ=ζBWin Theorem 2.4.
Corollary 2.10. Let Let (X, G) be a complete G-Metric space and f:XX
be a mapping satisfying the following condition: RG(f x,f y ,f y
0φ(t)dt G(x, y, y)
for all x, y X, where φ: [0,)[0,) is a function such that Rt
0φ(t)dt exists
and Rc
0φ(t)dt > ǫ, for each ǫ > 0. Then, fhas a unique ﬁxed point.
Proof: Deﬁne ζK: [0,)×[0,)Rby ζK(t, s, s) = sRt
0φ(u)du for
all s, t [0,). Then, ζKZ. Therefore, the result follows by taking ζ=ζKin
Theorem 2.4.
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Manoj Kumar(Corresponding Author),
Rashmi Sharma,
Department of Mathematics,
Lovely Professional University,
Phagwara, Punjab,
India.
... Such family generalized, extended and improved several results that had been obtained in previous years. The simplicity and usefulness of these contractions have inspirited many researchers to diversify it further (see [1,49,50,56,63,[65][66][67]70]). ...
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