Gen. Math. Notes, Vol. 19, No. 2, December, 2013, pp. 1-9
ISSN 2219-7184; Copyright © ICSRS Publication, 2013
www.i-csrs.org
Available free online at http://www.geman.in
Cone Metric Space and Some Fixed Point
Results for Pair of Expansive Mappings
S.K. Tiwari1, R.P. Dubey2 and A.K. Dubey3
1, 2
3
Department of Mathematics, Dr. C.V. Raman University, Bilaspur
Department of Mathematics, Bhilai Institute of Technology, Durg
1
E-mail: sk10tiwari@gmail.com
2
E-mail: ravidubey_1963@yahho.co.in
3
E-mail: anilkumardby@rediffmail.com
(Received: 3-10-13/ Accepted: 9-11-13)
Abstract
The purpose of this work is to extend and generalize some fixed point theorems
for Expansive type mappings in complete cone metric spaces. Our theorems
improve and generalize of the results [1] and [3].
Keywords: Complete cone metric space, common fixed point, expansive type
contractive mapping, Non-Normal cones.
1
Introduction
Very recently, Huang and Zhang [1] introduce the concept of cone metric spaces.
They have proved some fixed point Theorems for contractive mappings using
normality of the cone. The results in [1] were generalized by Sh. Rezapour and
Hamlbarani [2] omitted the assumption of normality on the cone, which is a
milestone in cone metric space. Many authors have studied fixed point theorem in
such spaces, see for instance [4]. [5], [8]. [11] and [13]. In sequel, the authors [7],
[8] and [12] introduced a new class of multifunction and obtained a unique fixed
2
S.K. Tiwari et al.
point. These results are also generalized by [15] with normal constant K = 1.Also
we have observed the recent work of fixed point s for non-explosive map in cone
metric spaces see ([6], [10]). Recently, Azam introduced cone rectangular metric
spaces in [14].
The purpose of this paper is to analyse the existence and uniqueness of fixed
points for pair of expansive mappings defined on a complete metric space. we
generalize the results of [3].
2
Preliminary Notes
First we recall the definition of cone metric spaces and some properties of theirs
[1].
Definition: 2.1 [1]: Let E be a real Banach space and P a subset of E. Then P is
called a cone if and only if:
(i) P is closed, non-empty and P ≠ {0};
(ii) , ϵ , , ≥ 0 , ϵ ⇒ + ϵ ;
(iii)
ϵ P and – ϵ P => = 0.
For given a cone ⊂ E , we define a Partial ordering ≤ on E with respect to P by
≤ if and only if - ϵ . We shall write x ≪ y to denote ≤
but ≠
0
to denote - ϵ p , where stands for the interior of P.
> 0 such that for all , ∈
The cone P is called normal if there is a number
,0 ≤ ≤
implies ‖ ‖ ≤ ‖ ‖ .The least positive number satisfying the
above is called the normal constant of P. The least positive number satisfying the
above is called the normal constant P. The cone P is called regular if every
increasing sequence which is bounded from above is convergent .that is , if
!" ≥ 1 $% sequence such that x1 ≤ 2 ≤ ….. n ≤ …..≤ for some ∈ ,
then there is ∈ such that ‖ &' ‖ ⟶0 (" ⟶∞). Equivalently the cone p is
regular if and only if every decreasing sequence which is bounded from below is
convergent.
Lemma 2.2[2]:
(i) Every regular cone is normal
(ii) For each k > 1, there is a normal cone with normal constant K > k.
In following we always suppose E is a Banach space. P is a cone in E with
$") ≠ *and ≤ is partial ordering with respect to P.
Definition 2.3[1]: Let X be a non – empty set. Suppose the mapping +: - ⨉ - →
satisfies the following condition:
Cone Metric Space and Some Fixed Point…
3
(i) 0 < d ( , ) for all , ∊ X and + ( , ) = 0 if and only if x = ;
(ii) d ( , ) = d ( , ) for all , ∊ X;
(iii) d ( , ) ≤ d ( , z) + d (z, ) for 11 , , 2 ∊ -.
Then d is called a cone metric on X, and (X, d) is called a cone metric space. It is
obvious that cone metric spaces generalize metric space.
Example 2.3 [1]: Let
- → ,
=
2
,
On defined by +( , ) = (| −
(-, +) is a cone metric space.
= {( , ) ∈
≥ 0}, - =
!" ≥ 1 ∈ :
≥ 0, :;< 11 "=
(ii)
(iii)
?(',@)
, defined by +( , ) = >
Definition 2.5 [1]: Let (X, d) be a cone metric space,
sequence in X. Then,
(i)
and +: - ×
|,∝ | − |) where ∝ ≥ 0 is a constant. Then
Example: 2.4: Let E= 1 1, P = 9
metric space and +: - × - →
(-, +) is a cone metric space.
: ,
AB
(-, +) a
C " ≥ 1.Then
∊ X and { n}n
≥ 1
a
{ n} n ≥ 1 converges to x whenever for every D ∊ with0 ≪ D, there is a
natural number N. such that d ( n, ) ≪c for all n ≥ N. We denote this by
lim n→∞ n = or n → , ( n → ∞).
{ n}n ≥ 1 is said to be a Cauchy sequence if for every c ∊ E with o ≪ c,
there is a natural number N such that d ( n, m) ≪ c for all n, m ≥ N.
(X, d) is called a complete cone metric space if every Cauchy sequence in
X is convergent.
Definition 2.6[1]: Let (-, +) be a cone metric space, P be a cone in real Banach
space, if
(i) ∈ and ≪ D for some E ∈ F0, 1G, then = 0.
(ii) H ≤ I, I ≪ J , then H ≪ J.
3
Main Results
In this section we shall prove some fixed point theorems for pair of expansive
type contractive mappings by using omitting the assumption of normality of the
theorems 2.1, 2.3, 2.5, 2.6 of [3].
Theorem 3.1: Let (-, +) be a cone metric space and suppose :K, :A : - → - be
any two onto mapping satisfies the contractive condition
+(:K , :A ) ≥ +( , )
(1)
4
S.K. Tiwari et al.
for all , ∈ -,where
common fixed point in X.
> 1 is a constant. Then :K and :A have a unique
Proof: If :K = :A then
0 ≥ +( , ) ⇒ 0 = + ( , ) ⇒
= .
Thus, :K is one to one. Define VK W :K &K
+X :K &K , :K &K Y
+( , ) ≥
= +(VK , VK ).
So, +(VK , VK ) ≤ ℎ+( , ) where ℎ =
unique fixed point
Therefore,
= :A .
Hence ∗ =:K
:K and:A .
∗
∗
in X.$. ^. VK
∗
=
K
∗
[
< 1. By theorem 2.3 in [2]. VK Has a
⇒ :K &K
∗
=
∗
⇒
∗
= :K .
is a fixed point of :K . Similarly it can be established that
=:A . Thus
∗
∗
is the common fixed point of pair maps
Corollary 3.2: Let (-, +) be a cone metric space and suppose :K, :A : - → - be
any two onto mapping satisfying the condition
+X:K A
kK
, :A A
for all , ∈ -,where
point in X.
kA
Y ≥ +( , )
(2)
> 1 is a constant. Then :K and :A have a common fixed
Theorem 3.1: Let (-, +) be a cone metric space and suppose :K, :A : - → - be
any two continuous and onto mapping satisfying the condition
+(:K , :A ) ≥ F+(:K ,
K
for all , ∈ -,where A <
common fixed point in X.
) + +(:A , )G
(3)
≤ 1 is a constant. Then :K and :A have a unique
Proof: Let
be an arbitrary point in-. Since :K and :A be onto (surjective), there
exist ∈ - and K ∈ - such that
:K ( K ) =
and :A ( A ) =
K.
Cone Metric Space and Some Fixed Point…
5
In this way, we define the sequence% A ! and A kK ! by
A = :K A kK for n = 0, 1, 2, 3……… and
A kK = :A A kA for n =o, 1, 2, 3……………
Note that, if
=
A
A kK ,
=
Now putting
A kK
A kA ,
= +(:K
+X
A ,
A kK Y
≥ F+(:K
K&[
A kK Y
A ,
A kK Y
A ,
is fixed point of :K and :A .
we have
A kK )
= F+(
A
≥ K +(
A kK ,
K
,
A kK ,
≤ ℎ+(
,
A
A kK )++(:A A kA ,
A kK ,
≥ K&[ +(
A kA Y
A kK,
A
A kK , :A A kA )
A kK Y
⇒+X
In general
=
A ,
+X
[
and
+X
(1 − )+X
Where ℎ =
for some " ≥ 1, then
++(
A kA )G
A kA )G
A kK ,
A kA )
A kA )
A kK )
.
(4)
, 0 ≤ ℎ ≤ 1.
+X
A kK Y ≤
A ,
So for " < l, we have
+X A ,
Am Y
ℎ +(
≤ +X
A &K , A
A kK Y +
A ,
≤(ℎA + ℎA
opB
≤
K&o
) ≤ ……… ≤ ℎA +X
+( ,
kK
……….+ +X
K Y.
,
Am Y
Am&K,
+ … … … + ℎAm&K ) +( ,
K)
K)
(5)
Let 0 ≤ D be given, choose a natural number qK such that
opB
K&o
K)
+( ,
≤ D, for
all " ≥ qK . Thus +X A , Am Y ≤ D, for " < l. Therefore A ! is a Cauchy
sequence in(-, +). since(-, +) is a complete cone metric space, there exist ∗ ∈ such that A → ∗ % " → ∞. If :K is a continuous, then
+(:K
Since
∗
,
A
∗)
≤ +X:K
→
∗
and :K
This implies that :K
:K
A kK,
∗
=
∗
A kK,
∗
Y + +X:K
→ :K
. hence
∗
∗
A kK,
∗
Y → 0 % " → ∞.
% " → ∞. Therefore +(:K
is a fixed point of:K .
∗
,
∗)
= 0.
6
S.K. Tiwari et al.
∗
Similarly, it can be established that :A
Thus
proof.
∗
=
∗
, Therefore :K
∗
=
∗
= :A
∗
.
is the common fixed point of pair of maps :K and :A .This completes the
Theorem 3.2: Let (-, +) be a cone metric space and suppose :K, :A : - → - be
any two continuous and onto mapping satisfying the condition
+(:K , :A ) ≥ +( ,
for all , ∈ -, where s ≥ 0,
common fixed point in X.
) + s+(:A , )
(6)
> 1 is a constant. Then :K and :A have a
Proof: Let
be an arbitrary point in-. Since :K and :A be onto (surjective), there
exist ∈ - and K ∈ - such that
:K ( K ) =
and :A ( A ) =
In this way, we define the sequence%
Note that, if
=
A
=
Now putting
+X
⇒ +(
A
! and
A kK !
by
A
= :K
A kK
for n = 0, 1, 2, 3……… and
A kK
= :A
A kA
for n =o, 1, 2, 3……………
A kK ,
for some " ≥ 1, then
A kK
and
A kK Y
A ,
K.
A kK , A kA )
=
A kA ,
A
is fixed point of :K and :A .
we have
= +(:K
A kK , :A A kA )
≥ +(
A kK , A kA )
+ s+(:A
= +(
A kK , A kA )
+ s+(
≥ +(
A kK , A kA )
≤
K
[
+X
A ,
A kA , A kK )
A kK , A kK )
A kK Y,
(7)
K
where ℎ = [ , 0 ≤ ℎ ≤ 1
So for " < l, we have
+X
A ,
Am Y
≤ +X
A ,
A kK Y +
…….................+ +X
Am&K,
Am Y
Cone Metric Space and Some Fixed Point…
≤ (ℎA + ℎA
≤
opB
K&o
kK
7
+ … … … … … + ℎAm&K ) +( ,
K)
+( ,
K)
(8)
opB
+( , K ) ≤ D, for
all " ≥ qK . Thus +X A , Am Y ≤ D, for " < l. Therefore A ! is a Cauchy
sequence in(-, +). since(-, +) is a complete cone metric space, there exist ∗ ∈ such that A → ∗ % " → ∞. If :K is a continuous, then
Let 0 ≤ D be given, choose a natural number qK such that
+(:K ∗ ,
Since
A
∗
→
∗)
and :K
This implies that :K
∗
=
≤ ++X:K
A kK,
∗
→ :K
∗
. hence
Similarly, it can be established that :A
Thus
proof.
∗
:K ∗ Y + +X:K
A kK,
∗
K&o
A kK,
∗
Y → 0 % " → ∞.
% " → ∞. Therefore +(:K
∗
,
∗)
= 0.
is a fixed point of :K .
∗
=
∗
.Therefore:K
∗
=
∗
= :A
∗
.
is the common fixed point of pair of maps :K and :A .This completes the
Theorem 3.3: Let (-, +) be a cone metric space and suppose :K, :A : - → - be
any two continuous and onto mapping satisfying the condition
+(:K , :A ) ≥ +( ,
) + s+( , :K ) +M +( , :A )
for all , ∈ -,where ≥ −1, s ≥ 1 "+ M < <^ constant, with
u > 1. Then :K and :A have a common fixed point in X.
(9)
+s+
Proof: Let
be an arbitrary point in-. Since :K and :A be onto (surjective), there
exist ∈ - and K ∈ - such that
:K ( K ) =
and :A ( A ) =
In this way, we define the sequence%
A
! and
A kK !
Now putting
A
=
=
+X
A
and
=
A kK Y
= +(:K
A kK
A ,
by
= :K A kK for n = 0, 1, 2, 3……… and
A kK = :A A kA for n =o, 1, 2, 3…………
kK , for some " ≥ 1, then A is fixed point of :K and :A .
A
Note that, if
K.
A kA ,
we have
A kK , :A A kA )
8
S.K. Tiwari et al.
+X
+( A kK , A kA ) + s+(
+ u+( A kA , :A A kA )
A kK Y ≥
A ,
A kK , A kA )
≥ +(
+ u+(
A kA ,
= s+(
(1 − s)+X
A ,
A kK Y
≥ (K+ u) +(
+X
A ,
A kK Y
≥
⇒ +X
A kK,
A kA Y
≤
⇒ +X
A kK,
A kA Y
≤ ℎ +(
[kv
+(
K&w
K&w
[kv
A ,
A kK )
)
A kK , A kA )
A kA )
A kA )
A kK Y
A kK )
,
:K
A kK , A
+ ( + u)+(
A kK ,
A kK ,
+X
A
A
A kK )
,
A
+ s+(
kK )
A kK ,
.
K&w
Where ℎ = [kv , 0 ≤ ℎ ≤ 1.
In general
+X
A kK Y ≤
A ,
So for " < l, we have
+X A ,
Am Y
ℎ +(
≤ +X
A &K , A
A kK Y
A ,
≤(ℎA + ℎA
≤
opB
K&o
+( ,
kK
) ≤ ……… ≤ ℎA +X
+ ……….+ +X
,
Am&K,
K Y.
Am Y
+ … … … … + ℎAm&K ) +( ,
K)
Let 0 ≤ D be given, choose a natural number qK such that
K)
(10)
opB
K&o
+( ,
K)
≤ D, for
all " ≥ qK . Thus +X A , Am Y ≤ D, for " < l. Therefore A ! is a Cauchy
sequence in(-, +). since(-, +) is a complete cone metric space, there exist ∗ ∈ such that A → ∗ % " → ∞. If :K is a continuous, then
Since
A
+(:K ∗ ,
→ ∗ and :K
This implies that :K
∗
∗)
A
=
≤ +X:K A kK, :K ∗ Y + +X:K A kK, ∗ Y → 0 % " → ∞.
∗
% " → ∞. Therefore +(:K ∗ , ∗ ) = 0.
kK, → :K
∗
., Hence
∗
is a fixed point of:K .
Similarly, it can be established that :A ∗ = ∗ . Therefore :K ∗ = ∗ = :A ∗ .
Thus ∗ is the common fixed point of pair of maps :K and :A . This completes the
proof.
Cone Metric Space and Some Fixed Point…
9
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