hardy and rogers type contractive condition and common fixed … · is a cauchy sequence if for...

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


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,


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

.


(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


(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].




(3. 16)

for all where and . Then the family of maps have

unique common fixed point in .



(3. 17)





(3. 18)



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


. 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).

References

1. Gahler, S., 1963, “2-metrische Raume und ihre topologische strukuren”, Math.

Nachr., 26, pp. 115-148.

2. Gahler, S., 1965, “Uberdie Uniformisierbarkeit 2-metricsche Raume”, Math.

Nachr., 28, pp. 235-244.

3. Rhoades, B. E., 1979, “Contraction type mappings on a 2-metric space”, Math.

Nachr., 91, pp. 151-155.

4. Hardy, G. E., and Rogers, T. D., 1973, “A generalization of a fixed point

theorem of Reich”, Canad. Math. Bull, 16(2), pp. 201-206.

5. Edelstein, M., 1962, “On fixed and periodic points under contractive

mappings”, J. London Math. Soc., 37, pp. 74-79.

6. Rhoades, B. E., 1977, “A comparision of various definitions of contractive

mappings”, Trans. Amer. Math. Soc., 226, pp. 257-290.

7. Long-Guang, H., and Xian, Z., 2007, “Cone metric spaces and fixed point

theorems of contractive mappings”, J. Math. Anal. Appl., 332, pp. 1468-1476.

8. Rezapour, Sh., and Hamlbarani, R., 2008 “Some notes on Cone metric spaces

and fixed point theorems of contractive mappings”, J. Math. Anal. Appl., 345,

pp. 719-724.

9. Abbas, M., and Jungck, G., 2008, “Common fixed point results for non-

commuting mappings without continuity in cone metric spaces”, J. Math.

Anal. Appl. 341, pp. 416-420.

10. Asadi, M., and Soleimani, H., 2012, “Examples in cone metric spaces: A

Survey”, Middle-East Journal of Scientific Research., 11(12), pp. 1636-1640.

11. Rudin, W., 1991, “Functinal Analysis”, 2nd

edition., McGraw Hill., New York.

12. Rodrigues, H. M., 2008, “A note on the relationship between spectral radius

and norms of bounded linear operators”, Cadernos de matematica., 09 pp. 61-

65.

13. Liu, H., and Shaoyuan, Xu., 2013, “Cone metric spaces with Banach algebras

and fixed point theorems of generalized Lipschitz mappings”, Fixed point

theory and applications., 2013:320.

14. Shaoyuan, Xu., and Radenovic, S., 2014 “Fixed point theorems of generalized

Lipschitz mappings on cone metric spaces over Banach algebras without

assumption of normality”, Fixed point theory and applications., 2014:102.

15. Singh, B., Jain, S., and Bhagat, P., 2012, “Cone 2-metric space and Fixed

point theorem of contractive mappings”, commentationes matematicae., 52(2),

pp. 143-151.


16. Wang, T., Yin, J., and Yan, Q., 2015, “Fixed point theorems on cone 2-metric

spaces over Banach algebras and an application”, Fixed point theory and

applications., 2015:204.

17. Tiwari, S. K., 2015, “Cone 2-metric spaces and Fixed point theorems for pair

of contractive mappings”, J. Prog. Research. Math., 5(3), pp. 543-552.

18. Tiwari, S. K., 2015, “cone 2-metric spaces and an extension of fixed point

theorems for contractive mappings”, IOSR Journal of mathematics, 11(4), pp.

01-08.

hardy and rogers type contractive condition and common fixed … · is a cauchy sequence if for...

Documents