Global Journal of Pure and Applied Mathematics.
ISSN 0973-1768 Volume 12, Number 3 (2016), pp. 2375-2383
© Research India Publications
http://www.ripublication.com/gjpam.htm
Hardy and Rogers type Contractive condition and
common fixed point theorem in Cone 2-metric space
for a family of self-maps
M. Rangamma1 and P. Rama Bhadra Murthy
2*
Author 1: Professor of Mathematics, Department of Mathematics, Osmania University, Telangana, India.
Author2: Research Scholar, Department of mathematics, Osmania University, Telangana, India.
Abstract
In this paper, Hardy and Rogers type common fixed point theorem for family
of self-maps in cone 2-metric spaces over Banach algebras is proved. Our
result generalizes one of the result of Wang et. al. Fixed Point Theory and
Applications(2015) 2015:204. An example supports our result.
Keywords: Cone 2-metric, Banach algebras, common fixed point theorems,
family of self-maps, spectral radius.
1. Introduction The concept of 2-metric space was introduced by S. Gahler in 1960’s by generalizing
the metric space and proved several fixed point theorems in such spaces See[1-2]. In
1979, B. E. Rhoades obtained several fixed point theorems in cone 2-metric spaces.
See[3-6]. In 2007, L. G. Huang and X. Zhang introduced metric spaces by replacing
the real numbers set with an ordered Banach space in codomain and obtained some
fixed point theorems. Thereafter, several authors worked on cone metric spaces and
obtained many fixed and common fixed point results. See[7-10]. Hao Liu and
Shaoyuan Xu worked on cone metric spaces over Banach Algebras. See[11-14].
Recently, B. Singh et. al introduced cone 2-metric spaces generalizing both 2-metric
and cone metric spaces and proved some fixed point theorems for a self-maps
satisfying certain contractive conditions. See[15-18]. Here, we have obtained Hardy
and Rogers[4] type common fixed point theorem for family of self-maps in cone 2-
metric spaces over Banach algebras, in which one of the result of Wang et. al [16] is
obtained as corollary. An example is also given to support our result.
2376 M. Rangamma and P. Rama Bhadra Murthy
2. Preliminaries
Let always represents a real Banach Algebra. Then we have the following
properties: For all and :
1.
2. and
3.
4.
Throughout this paper, we shall assume that the Banach Algebra has a unit
(multiplicative identity) such that for all . An element is
said to be invertible if there is an element such that . The inverse
of is denoted by .
Let be a Banach algebra and be a subset of . is called a cone if:
1. is closed, nonempty and .
2. .
3. and
For a cone , define a partial ordering ≼ w. r. t as follows:
1. iff .
2. will stand for and .
3. means , where denotes interior of .
Definition 2.1: ([16]) Let be a non-empty set. Suppose the mapping
satisfies:
1. for every pair of distinct points , there exists a point such that
.
2. , for all and if and only if at least two
of are equal.
3. for all and for all permutations
of
4. for all permutations
Then is called a cone 2-metric on , and is called a cone 2-metric space.
If is non-empty, then is a solid cone. In the following we always assume that
is a solid cone.
Definition 2.2: ([16]) Let be a cone 2-metric space. Let and be a
sequence in . Then the convergence and Cauchyness are defined as follows:
1. whenever for every with , there is a natural number
such that .
2. is a Cauchy Sequence if for every with , there is a natural
number such that and
Hardy and Rogers type Contractive condition and common fixed point theorem 2377
3. is a complete cone 2-metric space if every Cauchy sequence in is
convergent in .
Lemma 2.1: ([14]) Let If we have,
1. with and then .
2. for each , then
3. then for any there is a natural number such that
for all
Preposition 2.1([11]) Let be a Banach Algebra with a unit and If the
spectral radius 1,
then is invertible and .
Lemma 2.2: ([14]) For a Banach Algebra with a unit ,
1. for all .
2. If where , we have .
3. If , then as .
Lemma 2.3: ([14]) Let If commute, then
1.
2.
3.
Lemma 2.4: ([14]) Let be a Banach Algebra and let be a vector in . If
, then we have .
Lemma 2.5: ([16]) If is a real Banach space with a solid cone and if
as , then for any , there exists such that for all we have
.
Preposition 2.2 Let . Then
1. A sequence is a -sequence if for each there exists
such that for all
2. If , and are -sequences in then for all ;
is a -sequence
3. For any , we have whenever
Preposition 2.3:([16]) Let P be a solid cone in the Banach algebra A. Let If
and for any , then for any
2378 M. Rangamma and P. Rama Bhadra Murthy
3. MAIN RESULT
Theorem 3.1: ([4])Let be a complete metric space and be a self-mapping of
satisfying the condition for all
(3. 1)
Where are non-negative and Then has a unique
fixed point in .
Theorem 3.2: Let (X, d) be a complete cone 2-metric space in a Banach algebra
and be the underlying cone. Let be family of self-maps on satisfying
(3. 2)
for all where If commute and
Then the family of maps have unique common fixed point in .
Proof: Write for
Then (3. 2) becomes
(3. 3)
Choose arbitrary and define the sequence by for
Now we show that is a Cauchy sequence in .
where
(3. 4)
Because of symmetry of the cone 2-metric , we have
Or where
(3. 5)
Then we claim that, either or .
Suppose if and Then we get,
Hardy and Rogers type Contractive condition and common fixed point theorem 2379
which implies
(3. 6)
which implies
(3. 7)
Adding (3.6) and (3.7), we get , a
contradiction.
Thus, our claim is true.
Take
Then clearly and
(3. 8)
Thus for all where
(3. 9)
Suppose Then
……..
Thus, for all , we have .
(3. 10)
For , using (3. 10) in the below steps, we have
(3. 11)
Hence from Lemma 2. 5 and the fact that
we have for any with , there exists such that for all
we have
, showing that is a Cauchy
sequence in Since is complete, there exists such that
Now we show that this is the unique common fixed point of the family of maps
.
2380 M. Rangamma and P. Rama Bhadra Murthy
(3. 12)
Keeping fixed and using the Preposition 2. 2(2), the R. H. S of the above inequality
(3. 12) is a c-sequence in , say
Therefore, for any with , we have
for all
Since by Lemma 2. 2(2), we have is invertible and
hence This is true for any and for any with Thus we
have, for all and
Therefore, for all which implies is common fixed point of
,
(3. 13)
Suppose be any other common fixed point of ,
, Then using (3. 2)
which implies for all .
Therefore is the unique common fixed point of the family of maps .
Thus we have,
(3. 14)
Also,
which implies is also fixed point of .
But is the unique fixed point of by (3. 14), which means
Suppose is any other fixed point of
Then and hence
Since fixed point of is unique, which is we get , proving the uniqueness.
The following list of corollaries can be obtained from our main theorem.
Corollary 3.1: Let (X, d) be a complete cone 2-metric space in a Banach algebra
and be the underlying cone. Let be family of self-maps on satisfying
(3. 15)
for all where If commute and Then
the family of maps have unique common fixed point in .
The above fixed point theorem was proved by Wang et. al in [16].
Corollary 3.2: Let (X, d) be a complete cone 2-metric space in a Banach algebra
and be the underlying cone. Let be family of self-maps on satisfying
Hardy and Rogers type Contractive condition and common fixed point theorem 2381
(3. 16)
for all where and . Then the family of maps have
unique common fixed point in .
Corollary 3.3: Let (X, d) be a complete cone 2-metric space in a Banach algebra
and be the underlying cone. Let be family of self-maps on satisfying
(3. 17)
for all where and . Then the family of maps have
unique common fixed point in .
Corollary 3.4: Let (X, d) be a complete cone 2-metric space in a Banach algebra
and be the underlying cone. Let be family of self-maps on satisfying
(3. 18)
for all where and . Then the family of maps have
unique common fixed point in .
EXAMPLE 3.1: Let for each . The
multiplication is defined by . Then is a
Banach algebra with unit Let then is a
cone in .
Let .
The metric is defined by where ,
are such that
and
Then is a complete cone 2-metric space over the Banach Algebra .
Now define by
And
Then we have and
Also satisfies the contractive condition of the main Theorem 3. 2 with
. Also
2382 M. Rangamma and P. Rama Bhadra Murthy
. Hence by our main Theorem 3. 2, has a unique fixed point
which is for all .
Acknowledgement:
The Research was supported by Council of Scientific and Industrial Research (CSIR).
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Hardy and Rogers type Contractive condition and common fixed point theorem 2383
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