The existence of common fixed point theorems of generalized contractive mappings in cone metric spaces

Innovative Systems Design and Engineering
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.4, No.13, 2013

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The existence of common fixed point theorems of generalized
contractive mappings in cone metric spaces
Rajesh Shrivastava ,Sudeep Kumar Pandey1 Ramakant Bhardwaj2 R.N.Yadav3
Govt. Science and Commerce College, Benazir, Bhopal (M.P.) India
1. Millennium Group of Institution, Bhopal (M.P.)India
2. Truba Institute of Technology and Science, Bhopal (M.P.)India
3.Patel Group of Institutions, Bhopal (M.P.)India
Abstract
The purpose of this paper is to the study of the existence of common fixed point theorem for a sequence of self
maps satisfying generalized contractive condition for a cone metrice space and obtains some new results in it.
Also the paper contains generalized fixed point theorems of [10, 13, 19] and many others from the current
literature.
1.

Introduction and preliminaries

The well known Banach contraction principal and its several generalizations in the setting of metric spaces play
a central role for solving many problems of non linear analysis. For example, see[2,5,6,15,16,17]
Huang and Zang[8] generalized the concept of the metric spaces by introducing cone metric spaces and proved
some fixed point theorems for mappings satisfying some contractive conditions, subsequentialy, several other
authors [1,9,16,19,21] studied the existence of fixed points and common fixed points of mappings satisfying
contractive type condition on a normal cone metric space.
Recently Razapour and Hamlbarani[19] omitted the assumption of normality in cone metric
space, which is milestone in developing fixed point theory in cone metric space. In[11] the authors introduced
the concept of a compatible pair of self maps in a cone metric space and established a basic result for a non
normal cone metric space with an example which [12] weakly compatible maps have been studied. In this paper
we prove a common fixed point theorem for a sequence of self maps satisfying a generalized contractive
condition for a non normal cone metric space.
Definition:-1.1 [sec [8]] :- Let E be a real Banach space a sub set of p of E is called a cone whenever the
following condition holds.
(c1) P is closed, nonempty and P ≠{0}
(c2) a,bϵR,a,b≥0 and x, y∈P imply ax+by∈P,
(c3)P∩(-P) = {0}
Given a cone P⊂E, we define a partial ordering ≤with respect to P by x≤y if and only if y-x ∈ P. We shall write
x<y to indicate that x≤y but x≠y while x≪y will stand for y-x∈p0 where P0 stands for the interior of P. If
p0≠Ø.then P is called a solid cone(see[20]).
There exist two kinds of cones-normal (with the normal constant k) and non-normal cone [6]. Let E be a
real Banach space,P⊂E a cone and ≤partial ordering defined by P. Then P is called normal it there is a number
k>0 such that for all x,y∈P.
0≤x≤y implies ∥x∥≤k∥y∥
(1.1)
or equivalently if ( ∀ n) xn≤yn≤zn
and

lim xn = lim zn = x imply lim yn = x

n →∞

n →∞

n →∞

(1.2)
The least positive number K satisfying (1.1) is called the normal constant of P.
Example 1.2 (see[20]) let E= C

1

[ 0,1]

with ∥x∥=∥x∥∞+∥x1∥∞ on p={x∈E:x(t)≥0}.This cone is not normal.

Consider for example, xn(t)=tn/n and y(t)=1/n, then 0≤xn≤yn and lim yn = 0 but ∥xn∥=maxt∈[0,1]
n →∞

maxt∈[0,1] t

n − 1

=

tn
n

+

1
+ 1 > 1 ;hence xn does not converge to zero. It follows by (1.2) that P is a nonn

normal cone.
Definition 1.3 (see [13]): Let P be a cone in a real Banach space E. If for a∈P and a≤ka for same K∈ [0,1],then
a=0.
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Definition1.4 (see [10]): Let P be a cone in a real Banach space E with non-empty interior. If for a∈E and a≪c
for all c∈p0,then a=0.
Remarks 1.5(see [19]) λp0⊆p0 for λ>0 and p0 +p0⊆p0
Definition 1.6(see [8.22]) Let X be a nonempty set suppose that the mapping
d : X×X→E satisfies.
(d1)0≤d(x,y) for all x,y∈X and d(x,y)=0 if amd only if x=y.
(d2) d(x,y)=d(y,x) for all x,y∈X
(d3) d(x, y)≤d(x,z)+d(z,y), x,y,z∈X ……..
Then d is called a cone metric [8] or K_metric [22] on X and (x,d) is called a cone metric[8] or k-metric
space[22] (we shall use the first term).The concept of a cone metric space is more general than that of a metric
space, because each metric space is a cone metric space where E=ℝ and P=[0,+∞].
Example 1.7(see[8])Let E≠ℝ2 p = {(x,y) ∈ℝ2 : x≥0,y≥0}, X=ℝ and d : X×X→E defined by d(x,y)=(|x-y|, a|xy|),where a≥0 is a constant. Then(x,d) is a cone metric space with normal cone P where K=1
Example 1.8(see[18] Let E=l2 ,P={xn}n≥1∈E:xn≥0, for all n, (X, ρ) a metric space ,and d:X×X→E defined, by
d(x,y)={ρ(x,y/2n}n≥1. Then (X,d) is a cone metric space.
Clearly, the above examples show that class of cone metric space contains the class of metric spaces.
Definition 1.9 (sec[8]) let (X,d) be cone metric space. We say that {xn} is
(i)
a Cauchy sequence if for every ε in E with 0≪ε, then there is an N such that for all n,m>N,d(xn,xm) ≪ ε.
(ii)
a convergent sequence if for every ε in E with 0≪ε, then there is an N such that for all n>N, d(xn,x) ≪ ε
for some fixed x in X.
A cone metric space X is said to be complete if every Cauchy sequence in X is convergent in X.
In the following (X,d) will stands for a cone metric space with respect to a cone P with P0≠Ø in a real Banach
space E and ≤ is partial ordering in E with respect to P.
Remarks 1.10 It follows from above definition that if {x2n} is a subspace of a Cauchy sequence {xn} in a cone
metric space (x,d) and x2n→u as n→∞ then xn→u as n→∞.
Definition 1.11 (see[13]) Let (X,d) be a cone metric Space and P be a cone in a real Banach space E. If
u≤v,v≪w, then u≪w.
Lemma 1.12 (see[13]) Let(X,d) be a cone metric space and P be a cone in a real Banach space E and l,l1,l2 >0
are some fixed point real number. If xn→x, yn→y in X and for some a∈p
la≤l1d(xn, x)+l2d(yn, y)
for all n>N, for some integer N then a=0
2. Generalized contraction mapping
Let X be a cone metric space and T:X→X be a mapping then T is called generalized contractive mapping if it
satisfies the following condition:
d(Tx,Ty) ≤ α d(x,y) +β d(x,Tx)+γ(y, Ty)+δ[d(x,Tx)+(y, Ty)]
+η[d(x,Ty)+(y, Ty)+μ[d(x,Ty)+(x, Tx)]
(2.1)
For all x, y ϵ X and α, β, γ, δ, η, μ ϵ[0,1] are constants such that

α + β + γ + 2δ + 2η + 3µ <1

Remarks (2.1):
(i) If (i)δ=η=μ=0 and α, β, γ ∈[0,1], then (2.1) reduce to contraction mapping defined by Banach[3]
(ii) α=β=γ=μ=0 and δ, η∈[0,1/2] then (2.1) reduce to contraction mapping defined by Kannan[14]
(iii) α=β=γ=δ=η=0 and μ∈[0,1/3] then (2.1) reduce to contraction mapping following the condition hold.
3.

Main Results
In this section we shall prove some fixed point theorems of generalized contractive mapping.
Theorem 3.1: let (X,d) be a complete cone metric space with respect to a cone p contained in real Banach
space E. let {Tn}be a sequence of self maps on x satisfying generalized contractive condition (2.1) with for
some α, β, γ, δ, η,μ ϵ[0,1] for x0 ∈X, let xn =Tnxn-1 for all n . then the sequence {xn} converges in X and its
limit v is a common fixed point of all the maps of the sequence {Tn}. This common fixed point is unique if
α+2η+µ<1
Proof:- taking x=xn-1 , y= xn T= Tn and T= Tn+1 in (2.1) we have

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d(Tnxn-1, Tn+1xn)≤αd(xn-1, xn)+βd(xn-1, Tnxn-1)
+γd(xn, Tn+1xn)+δ[d(xn-1, Tnxn-1)+d(xn, Tn+1xn)]
+η[d(xn-1, Tn+1xn)+d(xn, Tnxn-1)]
+μ[d(xn-1, Tn+1xn)+d(xn-1, Tnxn)]
As xn= Tnxn-1, we have
d(xn, xn+1) ≤αd(xn-1, xn)+βd(xn-1, xn)
+γd(xn, xn+1)+δ[d(xn-1, xn)+d(xn, xn+1)]
+η[d(xn-1, xn+1)+d(xn, xn)]+μ[d(xn-1, xn+1)+d(xn-1, xn)]
≤αd(xn-1, xn)+βd(xn-1, xn)+γd(xn, xn+1)
+δ[d(xn-1, xn)+d(xn, xn+1)]+η[d(xn-1, xn)+d(xn, xn+1)]
+μ[d(xn-1, xn)+d(xn, xn+1)+d(xn-1, xn)]
Writing d(xn, xn+1)=ρn we have
ρ n ≤ (α + β + δ + η + 2 µ ) ρ n −1 + (γ + δ + η + µ ) ρ n

( 1 − γ − δ − η − µ ) ρ n ≤ (α + β + δ + η + 2 µ ) ρ n − 1

This implies that

ρ n ≤ t ρ n −1

t=

Where

α + β + δ +η + 2µ
1 − γ − δ −η − µ

As ( α
Now

Where

+ β + 2δ + γ + 2η + 3 µ < 1 ) ,

we obtain that t<1

ρn ≤ t ρn −1 ≤ t ρn −2 ≤ ........... ≤ t n ρ0
ρ0 = d(x0 , x1 ) also for n>m we have
2

d (x n , x m ) ≤ d (x n , x n − 1 ) + d (x n − 1 , x n − 2 ) + ........... + d (x m + 1 , x m )
≤ (t n − 1 + t n − 2 + ........ + t m ) d(x1 , x0 ) ≤
=

tm
d(x1 , x0 )
1−t

tm
ρ0
1−t

As t<1 and p is closed, thus we obtain that

d(xn ,xm ) ≤

tm
ρ0
1−t

(3.2)

Now for εϵp0 ,there exists r>0 such that ε-yϵ p0, if ∥y∥<r. choose a positive integer Nℇ such that for all n≥Nℇ
m

t
tm
ρ0 ∈p0 and
ρ0 y∥<r which implies ℇ −
∥
1−t
1−t
tm
ρ0 − d(xn,xm)∈p by using (3.2).
1−t
0

So we have ℇ‒d(xn,xm)∈p for all n>Nℇ, and for all m by definition(1.11). this implies d(xn,xm)<<ℇ. for all n>Nℇ
and for all m hence {xn} is a Cauchy sequence in X. by the completeness of X, there exists, Z∈x such that xn→z
as n→∞. for an arbitrary fixed m we show that Tmz=z. now
d(Tmz,z) ≤d(Tmz,Tn xn-1)+d(Tnxn-1,z)
=d(xn,z)+ d(Tmz,Tn xn-1)
Using (2.1) we have
d(Tmz,z)≤ d(Tmz,Tn xn-1)+ d(Tn,xn-1 z)

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= d( xn, z)+ d(Tmz,Tn xn-1)
≤ d( xn ,z)+α d(z, xn-1)+ d(z,Tmz,)+ d(xn-1,Tn xn-1)
+ δ[d(z,Tmz)+ d(xn-1,Tn xn-1)]+η[ d(z,Tn xn-1)+ d(xn-1,Tmz )]
+μ[ d(z,Tn xn-1)+ d(z,Tmz,)]
= d(xn, z)+αd(z,xn-1)+βd(z,Tmz)+ γd(xn-1,xn)+δ[ d(z,Tmz,)+ d(xn-1, xn)]
+η[ d(z,xn)+ d(xn-1,Tmz)]+μ[ d(z, xn)+ d(z,Tmz)]
≤ d(xn,z)+ αd(z,xn-1)+ βd(z,Tmz)+ γd( xn-1,z)
+δ[ d(z,Tmz)+ d(xn-1,z)+ d(z,xn)]+η[ d(z, xn)+ d(xn-1,z)+ d(z,Tmz)]
+μ[ d(z,xn)+ d(xn-1,z)+ d(z,Tmz)]
=(1+ +η+µ )d(xn,z)+(α+ + +η+µ) d( z,xn-1)+( + +µ+η) d(Tmz,z)
So we have
(1- ‒ ‒µ‒η)d(Tmz,z)≤(1+ +η+µ) d(xn, z)+(α+ + + +η)d(z,xn-1)
As xn→z, xn-1→z(n→∞) and (1‒ ‒ ‒η‒µ)>0, Using lemma 1.12 we have d(Tmz,z)=0 and we get Tmz=z, thus z
is a common fixed point of all the maps of sequence{Tn}.
Uniqueness:Let Tnv=v for all n be another common fixed point of all the maps of the sequence {Tn}. Now d(v,z)= d(Tnv,Tn
z)
≤ αd(v,z)+ βd(v,Tn v)+γ d(z,Tn z)+δ[ d(v,Tn v)+ d(z,Tn z)]
+η[ d(v,Tn z)+ d(z,Tn v)]+μ[ d(v,Tn z)+ d(v,Tn v)]
Which gives
d(v,z)≤(α+2η+µ)d(v,z)
as α+2η+µ<1 using definition 1.3 we have d(v,z)=0, i.e. v=z. thus v is the unique common fixed point of all the
maps of the sequence {Tn}.
Theorem 3.2:- let (X,d) be a compact cone metric space with respect to a cone p contained in a real Banach
space E. Let {Sn}be a sequence of self maps in X satisfying for some αn, βn, n n, n, n ∈[0,1] with αn+
βn,+ n +2 n,+2 n,+3 n<1 and αn+2 n+ n<1 there exists positive integer mi for each i such that for all
x,y∈X.

d(simi x,smi y) ≤αnd(x,y)+ βnd(x,simi x)+γnd(y,simi y)
j
+δn [d(x,simi x)+ d(y,smi y)] +ηn [d(x,smi y)+ d(x,simi x)] (3.3)
j
j
+ µn [d(x,smi y)+ d(x,simi x)]
j
Then all the maps of the sequence {sn} have a unique common fixed point in X.
mi
Proof: - from theorem 3.1 all the maps of the sequence { i },have a unique common fixed point , say z.

s

hence

simi z=z

For all i. now

simi

z=z implies

simi

s1z=s1z. taking x=s1z, y=z, i=1 and j=2 in (3.3), we have s1z = z.

continuing in similar way it follows that siz = z for all i . thus z is a common fixed point of all the maps of the
mi
sequence {si}. Its uniqueness follows from the fact that siz = z implies i z=z for all i.

s

In theorem 3.1 taking T1= T2= T3=……………………..= Tn=………= T, we get the following general form of
Banach contraction principle in a cone metric space which is not necessarily normal.
Theorem 3.3:- let(X,d) be a complete cone metric space with respect to a cone p contained in real Banach space
E. Let T be a self map in X satisfying generalized contractive condition (2.1) with α+ β+ +2 +2 +3 <1 and
n
for some α,β, , , , ∈[0,1]then for each x∈X sequence
converges in X and its limit u is a fixed
x
point T. This fixed point is unique if α+2 + <1.

{T }

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Theorem 3.4:- let (X,d)be a complete cone metric space with respect to a cone p contained in a real Banach
space E. suppose the mapping T:X→X satisfying for some positive integer n

d(Tn x,Tn y) ≤αnd(x,y) + βnd( x,Tn x) +γnd(y,Tn y)
+δn [d(x,Tn x) + d(y,Tn y)] +ηn [d(x,Tn y) + d(y,Tn x)]
+ µn [d(x,Tn y) + d(x,Tn x)]
For all x,y ∈ X and αn, βn, n, n, n, n ∈[0,1] are constants such that αn+ βn,+ n +2 n,+2 n,+3 n<1 then T has a
unique fixed point in X.
Proof :- from theorem 3.3 Tn has a unique fixed point u. but Tn(Tu)=T(Tnu)=Tu, so Tu is also a fixed point of Tn
hence Tu=u, u is a fixed of T. since the fixed point of T is unique.
Corollary 3.5:- Let (X,d) be a complete cone metric space with respect to a cone p contained in real Banach
space E . Suppose the mapping T:X→X satisfies for some positive integer m, n.

d(Txm x,Tyn y) ≤ α n d(x, y) + βn d( x,T m x) + γ n d(y,T n y)
+ δ n [d(x,T m x ) + d(y,T n y)] + ηn [d(x,T n y) + d(y,T m x )]
+ µn [d(x,T n y) + d(x,T m y)]
For all x,y ∈ X and αn, βn, n, n, n, n ∈[0,1] are constants such that αn+ βn,+ n +2 n,+2 n,+3 n<1 and
then T has a unique fixed point in X.
Proof:- by theorem 3.4 we get x ∈ X such that Tmx=Tny=x. the result then follows from the fact that
d(T x,x) = d(TT m x,T n y) = d(T mTx,T n x)

n,= n

≤ αn d(Tx,x) + βnd(Tx,T mTx) + γ n d(x,T n x)
+ δ n [d(Tx,T mTx) + d(x,T n x)] +ηn [d(Tx,T n x) + d(x,T m Tx )]
+ µn [d(Tx,T n x) + d(x,T m Tx )]
≤ αn d(Tx,x) + βnd(Tx,Tx) + γ nd(x,x)
+ δ n [d(Tx,Tx) + d(x,x)] +ηn [d(Tx,x) + d(x,Tx )]
+ µn [d(Tx,x) + d(Tx,Tx )]
=( αn + 2ηn + µn )d(Tx,x)
Which implies Tx=x.
Theorem 1[8] and theorem 2.3 [20]:-. Let (X,d) be a complete cone metric space. Suppose the mapping T:
X→X satisfies the contractive condition
d(Tx,Ty)<=kd(x,y)
For all x,y ϵ X where k∈[0,1], is a constant. Then T has a unique fixed point in X. And for any x∈X, the
iterative sequence {Tnx} converges to the fixed point.
Theorem 3[8] and theorem 2.6[20]:- Let (X,d) be a complete cone metric space. Suppose the mapping T:X→X
satisfies the contractive condition
d(Tx,Ty)<=k[d(x,Tx)+d(y,Ty)]
For all x,y e X where k∈[0,1/2], is a constant. Then T has a unique fixed point in X. And for any x∈X, the
Theorem 4[8] and Theorem 2.7[20]:- Theorem 1[8] and theorem 2.3 [20]. Let (X,d) be a complete cone metric
space. Suppose the mapping T:X→X satisfies the contractive condition
d(Tx,Ty)<=k[d(y,Tx)+d(x,Ty)]
For all x,y e X where k∈[0,1], is a constant. Then T has a unique fixed point in X. And for any x∈X, the
Remark 3.8:- Above theorems os [8] and [20] follows Theorem 3.3 of this paper by taking
(i) β, , , , and α=k
(ii) α, , , , and β=k
(iii) α ,β, , , , and =k
(iv) α, β , , , and =k
(v) α, β, , ,μ and η = k

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(vi) α ,β , , , and =k
Precisely, Theorem 3.3 synthesize and generalizes all the results of [9] and [20] for a non normal cone metric
space. Theorem 3.2 is a generalized form of Banach contraction principle in a complete cone metric space which
is not necessarily normal
Acknowledgement: One of the authors (R.B.) is thank full to MPCST Bhopal for project no. 2556
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The existence of common fixed point theorems of generalized contractive mappings in cone metric spaces