Advances in Environmental Biology M.Reza Heidari Tavani

Advances in Environmental Biology, 8(7) May 2014, Pages: 3387-3392
AENSI Journals
Advances in Environmental Biology
ISSN-1995-0756
EISSN-1998-1066
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Some Fixed Point and Common Fixed Point Theorems in G-Cone Metric Spaces
M.Reza Heidari Tavani
Department of Mathematics, Ramhormoz Branch, Islamic Azad University, Ramhormoz, Iran.
ARTICLE INFO
Article history:
Received 25 March 2014
Received in revised form 20 April
2014
Accepted 15 May 2014
Available online 10 June 2014
ABSTRACT
Background: In this paper the concept of G-cone metric is introduced Objective: Also
some results of common fixed point is expressed for mappings which satisfies in
weakly compatible condition. Results: Are presented the results were extension of
some existing results.
Keywords:
G-cone metric; expansive mapping;
weakly compatible; common fixed
point
© 2014 AENSI Publisher All rights reserved.
To Cite This Article: M.Reza Heidari Tavani., Some Fixed Point and Common Fixed Point Theorems in G-Cone Metric Spaces. Adv.
Environ. Biol., 8(7), 3387-3392, 2014
INTRODUCTION
Fixed point theory plays a major role in mathematics and applied sciences ,such that optimization, economy
and medicine. In the past two decades ,fixed point theory concepts quickly expanded. For example in 2007
Huang and Zhang in [4] introduced the concept of cone metric spaces and proved some fixed point theorems for
contractive mappings. Then in 2010 Beg and Abbas and Nazir in [1] presented G-cone metric spaces that were
generalized of G-metric spaces. More results are obtained for fixed point in this spaces.(see for example
,[1,5,6,7,8,9,11]). The purpose of this paper is applying suitable conditions for two mappings to obtained
common fixed point on G-cone metric spaces. To obtain these results some preliminary results and definitions
were referred. However for more details see also [2,4,6,9,12,13].
2.Preliminaries:
Definition 2.1([4]). Let E be a real Banach space and P a subset of E. P is called a cone if and only if :
(i) P is nonempty , closed , and
,
(ii) If
,
and ,
, then
,
(iii) If both
and
then
.
Given a cone
, it will be defined a partial ordering which respect to P by
written
to indicate that
but
, while
will stand for
interior of P. The cone P is said to be normal if there exists a real number
if
. It will be
, where Int P denotes the
such that for all
,
.
The least positive number K satisfying the above statement is called the normal constant of P . The cone P
is regular if every increasing sequence which is bounded from above is convergent that is , if
is a
sequence such that
, for some
, then there is
such that
as
.
Equivalently , the cone P is regular if and only if every decreasing sequence which is bounded from below
is convergent.
Lemma 2.1([8]). Every regular cone is a normal cone.
Lemma 2.2 Let E be a real Banach space with a cone P. Then:
(i) If
and
, then
[13] ,
(ii) If
and
, then
[13] ,
Corresponding Author: M.Reza Heidari Tavani, Department of Mathematics, Ramhormoz Branch, Islamic Azad
University, Ramhormoz, Iran.
E-mail: [email protected]
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M.Reza Heidari Tavani, 2014
Advances in Environmental Biology, 8(7) May 2014, Pages: 3387-3392
(iii) If
(iv) If
for each
, and
and
, then
[7] ,
(v) P is normal if and only if
Lemma 2.3. Let
,
, then
and
,
imply
and ,
.If
Proof . According to property of P ,
, then
.
is achieved
is said to be continuous at
is continuous if
Definition 2.3. Let X be a nonempty set and suppose
(G1)
if
,
whenever
(G3)
(G4)
,
whenever
if for any sequence
is continuous at all
In the following it is suppose that E is a Banach space , P is a cone in E and
respect to P.
(G2)
. Now
and finally from condition of (iii) in definition 2.1
be a function .
in P ,
[3] .
.
and also from
from condition of (ii) in definition 2.1 ,
,
, which implies
.
Definition 2.2. Let
[13] ,
.
is partial ordering with
satisfies :
,
,
, (symmetry in all three variables) ,
(G5)
,
.
Then G is called a generalized cone metric on X, and (X,G) is called a generalized cone metric space or a
G-cone metric space.
Corollary 2.1. if (X,G) be a G-cone metric space then
,
.
Proof . It is sufficient be replaced z with y and a with z in (G5) from definition 2.3.
Definition 2.4. A G-cone metric space is said to be symmetric if :
,
Following is example of non symmetric G-cone metric space.
Example 2.1([10]). Let
,
,
.Define
by
,
,
.
Here X is non symmetric G-cone metric space since
Definition 2.5. Let (X,G) be a G-cone metric space and let
(1) a G- Cauchy sequence if ,for every
,
, with
.
be a sequence in X .It is said that
, there is
such that for
is
,
(2) a G- convergent sequence to
if for every
, with
, there is
such that for
,
. Here
is called the limit of the sequence
.
A G-cone metric space (X,G) is said to be G-complete if every G-Cauchy sequence in X is G-convergent in X.
The following results are proved in [1]
Lemma 2.4. If (X,G) is a G-cone metric space , then the following are equivalent:
(i)
converges to x.
(ii)
as
.
(iii)
as
.
(iv)
as
.
Lemma 2.5. Let
be a sequence in X. if for every
, with
there is
such that for
,
then
is a G-Cauchy sequence in X.
Proof . Suppose
that for
obtained
is an arbitrarily element .According to the condition of lemma there is
,
and ,
such
. Using the property (G5) of Definition 2.3 will be
and therefore,
is a G-Cauchy sequence in X.
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M.Reza Heidari Tavani, 2014
Advances in Environmental Biology, 8(7) May 2014, Pages: 3387-3392
Lemma 2.6. Let (X,G) be a G-cone metric space and let
such that
(2.1)
, n= 1,2,…
be a sequence in X. If there exists the number
Then
is a G- Cauchy sequence in X .
Proof . using the condition (2.1) to be achieved
So for
according to corollary 2.1
+
(
Let
.
be given. Choose,
such that
P , where
. Also choose a natural number
such that
for all
Thus
. therefore
, for all
and hence according to the lemma 2.5 ,
is a
G- Cauchy sequence in X .
Definition 3.1. Let (X,G) be a G-cone metric space and let T be an onto self mapping on X. Then T is
called an expansive mapping if there exist a constant
such that
for all
.
Definition 3.2. Let T and S be self mappings of a set X. If
for some x in X , then x is called a
coincidence point of T and S and w is called a point of coincidence of T and S.
Definition 3.3. The mappings
are weakly compatible , if for every
, the following holds :
whenever
.
There are many theorems that prove existence of a fixed point for a mapping satisfies in expansive
conditions. For example see two next theorems.
Theorem 2.1([10]). Let (X,G) be a complete G-cone metric space . If there exist a constant
and an
onto self mapping T on X satisfying
for all
, then T has a unique fixed point.
Theorem 2.2([10]). Let (X,G) be a complete G-cone metric space and let
be an onto mapping
satisfying
Where
For all
,
, and constant
,then T has a unique fixed point.
3. Main results:
In this section will be presented some fixed point theorems for expansive mappings.
Theorem 3.1 Let (X,G) be a complete symmetric G-cone metric space and let
be an onto mapping
satisfying
(3.1)
Where
with
for all
, then T has a fixed point in X.
Proof. Since
is onto then T is injective and invertible .Suppose that H be the inverse mapping
of T . Let
, so there exists
such that
. Continuing in this way can be obtained a sequence
in X where
.
If for a positive integer n ,
then
is a fixed point of T and proof is complete. So suppose
that
for all n = 1,2,…
It follows that from condition (3.1)
(3.2)
Since G-cone metric space is assumed symmetric then from (3.2) is obtained
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M.Reza Heidari Tavani, 2014
Advances in Environmental Biology, 8(7) May 2014, Pages: 3387-3392
(3.3)
If
, then
or
and hence The inequality (3.3) implies that
.
also according the cone property ,
and thus
, that is contradiction . Hence
and
or
. Therefore
(3.4)
Since
, by lemma 2.6
is a G-Cauchy sequence in X . Because (X,G) is complete and therefore
the sequence
is converges to a point
. Now from condition (3.1) ,
Since T is onto mapping so there exists
, such that
Which implies that as
or
. Hence
and therefore
T and hence proof is complete .
Theorem 3.2. Let (X,G) be a complete G-cone metric space and let
For all
, and constant
This gives that w is a fixed point of
□
be a onto
mapping satisfying
(3.5)
, then T has a fixed point.
Proof. Similar to the proof of Theorem 2.1, we can obtain a sequence
positive integer n ,
then
for all n = 1,2,
It follows that from condition (3.5)
such that
. If for a
is a fixed point of T and proof is complete. So suppose that
or
(3.6)
Where
. By lemma 2.6
is a G-Cauchy sequence in X and since (X,G) is complete therefore
the sequence
is G- converges to a point
. Now from condition (3.5) ,
Since T is onto mapping so there exists
, such that
.
Taking ,
Since
, in above inequality .
and
as
, by lemma 2.3
and
,(
), which implies,
as
.hence w = u . But
, so ,
. Therefor, w is a fixed point of T.
□
Now , is presented a common fixed point theorem of two weakly compatible mappings in G-cone metric
spaces.
Theorem 3.3. Let (X,G) be a G- cone metric space. Let S and T be weakly compatible self –mapping of X and
. Suppose that there exists
such that
(3.7) , for all
.
If one of the subspaces T(X) or S(X) is complete , then S and T have a unique common fixed point in X.
Proof . let
. Since
, choose
such that
. In general for every positive integer
, n , choose
such that
. Then from (3.7) ,
.
Thus by lemma 2.6
G- converges to a point
Thus ,
point
such that
is a G- Cauchy sequence in
and hence is
. (Because Under the assumption T(X) or S(X) is complete )
. Since T(X) or S(X) is complete and
.Now from (3.7)
, there exists a
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Advances in Environmental Biology, 8(7) May 2014, Pages: 3387-3392
(3.8) .
Taking ,
in (3.8) .therefore ,
, which implies that,
and
hence ,
. Since S and T are weakly compatible ,therefore ,
or
. Now it
can be shown that u is a fixed point of S and T .
For this purpose from (3.7)
(3.9)
Taking ,
in (3.8) .therefore ,
, which implies that
.
Thus
.To prove uniqueness suppose that, y, is also another common fixed point of S and T
(
. Now from (3.7)
or
, which implies that
and hence,
.
This completes the proof.
□
Corollary 3.1. Let (X,G) be a complete G-cone metric space and
be a surjection and
be an
injective .If
S
and T are commutative , and there is constant
such that ,
, for all
.
Then S and T have a unique common fixed point in X.
Proof . clearly S and T are weakly compatible and since S is surjection then
and therefore S(X)
is G-complete .Also
.Thus all conditions of theorem 3.3 is satisfied and hence S and T have a
unique common fixed point in X.
Now , an example is provided to illustrate the corollary 3.1.
Example 3.1. let
,
and
be a cone in E .
Define
by
for all
.Then X is a complete G-cone metric space. Define
by
and
(1). S and T are weakly compatible .
(2). S is surjection ( range of S is
(3).
is complete .
(4).
Let
. Now for all
for all
.Then following conditions is satisfied.
) and T is injective and
and
.
.
the following is satisfied .
Thus according to corollary 3.1, S and T have a unique common fixed point in X .
, is unique common fixed point of , S and T, (
.
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