Open Access

Common Fixed Point Theorem for Non-Self Mappings Satisfying Generalized Ćirić Type Contraction Condition in Cone Metric Space

  • R Sumitra1,
  • V Rhymend Uthariaraj2,
  • R Hemavathy3Email author and
  • P Vijayaraju4
Fixed Point Theory and Applications20102010:408086

https://doi.org/10.1155/2010/408086

Received: 27 November 2009

Accepted: 9 March 2010

Published: 14 March 2010

Abstract

We prove common fixed point theorem for coincidentally commuting nonself mappings satisfying generalized contraction condition of Ćirić type in cone metric space. Our results generalize and extend all the recent results related to non-self mappings in the setting of cone metric space.

1. Introduction

Recently, Huang and Zhang [1] introduced the concept of cone metric space by replacing the set of real numbers by an ordered Banach space and obtained some fixed point theorems for mappings satisfying different contractive conditions. The category of cone metric spaces is larger than metric spaces and there are different types of cones. Subsequently, many authors like Abbas and Jungck [2], Abbas and Rhoades [3], Ilić and Rakočević [4], Raja andVaezpour[5] have generalized the results of Huang and Zhang [1] and studied the existence of common fixed points of a pair of self mappings satisfying a contractive type condition in the framework of normal cone metric spaces. However, authors like Janković et al. [6], Jungck et al. [7], Kadelburg et al. [8, 9], Radenović and Rhoades [10], Rezapour and Hamlbarani [11] studied the existence of common fixed points of a pair of self and nonself mappings satisfying a contractive type condition in the situation in which the cone does not need be normal.

The study of fixed point theorems for nonself mappings in metrically convex metric spaces was initiated by Assad and Kirk [12]. Utilizing the induction method of Assad and Kirk [12], many authors like Assad [13], Ćirić [14], Hadžić [15], Hadžić and Gajić [16], Imdad and Kumar [17], Rhoades [18, 19] have obtained common fixed point in metrically convex spaces. Recently, Ćirić and Ume [20] defined a wide class of multivalued nonself mappings which satisfy a generalized contraction condition and proved a fixed point theorem which generalize the results of Itoh [21] and Khan [22].

Very recently, Radenović and Rhoades [10] extended the fixed point theorem of Imdad and Kumar [17] for a pair of nonself mappings to nonnormal cone metric spaces. Janković et al. [6] proved new common fixed point results for a pair of nonself mappings defined on a closed subset of metrically convex cone metric space which is not necessarily normal by adapting Assad-Kirk's method.

The aim of this paper is to prove common fixed point theorems for coincidentally commuting nonself mappings satisfying a generalized contraction condition of Ćirić type in the setting of cone metric spaces. Our results generalize mainly results of Ćirić and Ume [20] and all the recent results related to nonself mappings in the setting of cone metric space.

2. Definitions and Preliminaries

We recall some basic definitions and preliminaries that will be needed in the sequel.

Definition 2.1 (see [1]).

Let be a real Banach space. A subset of is called a Cone if and only if

(1) is nonempty, closed and ;

(2) , and ;

(3) .

For a given cone , a partial ordering is defined as on with respect to by , if and only if . It is denoted as to indicate that but , while will stand for , where denotes the interior of .

The cone is called normal, if there is a number such that for all , implies

(2.1)

The least positive number satisfying (2.1) is called the normal constant of . It is clear that . There are nonnormal cones also.

The definition of a cone metric space given by Huang and Zhang [1] is as follows.

Definition 2.2 (see [1]).

Let be a nonempty set. Suppose that is a real Banach space, is a cone with and is a partial ordering with respect to .

If the mapping satisfies the following:

(1) for all and if and only if ;

(2) for all ;

(3) for all ;

then is called a cone metric on and is called a cone metric space.

Example 2.3 (see [1]).

Let , and such that , where is a constant. Then is a cone metric space.

Definition 2.4 (see [1]).

Let be a cone metric space and a sequence in . Then, one has the following.

(1) converges to , if for every with , there is such that for all ,
(2.2)

It is denoted by or , .

(2)If for any , there is a number such that for all
(2.3)

then is called a Cauchy sequence in .

(3) is a complete cone metric space, if every Cauchy sequence in is convergent.

(4)A self mapping is said to be continuous at a point , if implies that for every in .

The following two lemmas of Huang and Zhang [1] will be required in the sequel.

Lemma 2.5 (see [1]).

Let be a cone metric space and a normal cone with normal constant . A sequence in converges to if and only if as .

Lemma 2.6 (see [1]).

Let be a cone metric space and a normal cone with normal constant . A sequence in is a Cauchy sequence if and only if as .

The following Corollary of Rezapour [23] will be needed in the sequel.

Corollary 2.7 (see [23]).

Let , the real Banach space.

(i)If and , then .

(ii)If and , then .

(iii)If for each , then .

The following remarks of Radenović and Rhoades [10] will be needed in the sequel.

Remark 2.8 (see [10]).

If , and , then there exists such that for all , it follows that .

Remark 2.9 (see [10]).

If and , then , where is a sequence and is a given point in .

Remark 2.10 (see [10]).

If and , , then for each cone .

Remark 2.11 (see [10]).

If is a real Banach space with a cone and if , where and , then .

3. Main Results

In the following, we suppose that is a Banach space, is a cone in with and is partial ordering with respect to .

Theorem 3.1.

Let be a complete cone metric space and a nonempty closed subset of such that for each and there exists a point such that
(3.1)
Suppose that are two nonself mappings satisfying for all with ,
(3.2)
and are nonnegative real numbers such that
(3.3)

Also assume that

(i) ;

(ii) ;

(iii) is closed in ;

Then there exists a coincidence point of and in . Moreover, if and are coincidentally commuting, then and have a unique common fixed point in .

Proof.

Two sequences and are constructed in the following way. Let . As , by (i) there exists a point such that . Since , from (ii) it follows that . Let be such that . Since , there exists such that .

If , then which implies that there exists a point such that . Otherwise, if , then there exists a point such that

(3.4)
Since , there exists a point such that and thus
(3.5)

Assume that .

Thus repeating the arguments, two sequences and are obtained such that

(i) ;

(ii) ;

(iii) whenever , then there exists such that

(3.6)
Next, we claim that is a Cauchy sequence in . The following are derived. Let us denote
(3.7)

Obviously, two consecutive terms cannot lie in . Note that, if , then and belong to . Now, three cases are distinguished.

Case 1.

If , then . Now from (3.2),
(3.8)
where
(3.9)
Thus
(3.10)

Now foursubcasesarise.

Subcase 1.1.

If and , then (3.8) becomes
(3.11)

Subcase 1.2.

If and , then (3.8) becomes
(3.12)

Subcase 1.3.

If and , then (3.8) becomes
(3.13)

Subcase 1.4.

If and , then (3.8) becomes
(3.14)
Combining allSubcases1.1, 1.2, 1.3, and 1.4, it follows that
(3.15)
where . Hence
(3.16)

Case 2.

If , then . Now,
(3.17)
Proceeding as in Case 1,
(3.18)

Case 3.

If , then , and . Now,
(3.19)
Thus
(3.20)
where
(3.21)
Thus
(3.22)

Again foursubcasesarise.

Subcase 3.1.

If , then (3.20) becomes
(3.23)
Thus
(3.24)
where using Case 2,
(3.25)
Then (3.24) becomes
(3.26)

Subcase 3.2.

If , then (3.20) becomes
(3.27)
Proceeding as in Subcase 3.1, it follows that
(3.28)

Subcase 3.3.

If , then (3.20) becomes
(3.29)
Thus
(3.30)
where using Case 2,
(3.31)
Then (3.30) becomes
(3.32)

Subcase 3.4.

If , then (3.20) becomes
(3.33)
Proceeding as in Subcase 3.1, it follows that
(3.34)
Combining all fourSubcases3.1, 3.2, 3.3, and 3.4, we have
(3.35)
where by (3.3). Hence
(3.36)
Now, combining main Cases 1, 2, and 3, it follows that
(3.37)
where
(3.38)
Following the procedure of Assad and Kirk [12], it can be easily shown by induction that for ,
(3.39)
By triangle inequality, for , it follows that
(3.40)
From Remark 2.9, , which implies by Definition 2.4(2) that is a Cauchy sequence in which is a closed subset of the complete cone metric space and hence is complete. Then there exists a point such that as . Thus
(3.41)
Since , there exists a point such that . By the construction of , it was seen that there exists a subsequence such that
(3.42)
We will prove that . Consider
(3.43)
where
(3.44)
Thus
(3.45)

Now again four cases arise.

Case 1.

If , then (3.43) becomes
(3.46)

Case 2.

If , then (3.43) becomes
(3.47)

Case 3.

If , then (3.43) becomes
(3.48)

Case 4.

If , then (3.43) becomes
(3.49)
Combining Cases 1, 2, 3, and 4, it follows that
(3.50)
Thus
(3.51)

where .

Let be given with . From as and Definition 2.4(1),

(3.52)
From as and by Definition 2.4 (1),
(3.53)
From the definition of convergence in cone metric space and by (3.52) and (3.53), inequality (3.43) becomes
(3.54)

Therefore, for each . Then by (iii) of Corollary 2.7, we have , that is, which implies that is the coincidence point of and .

Since and are coincidentally commuting, for which implies . Consider

(3.55)
where
(3.56)

Thus and . Two cases arise.

Case 1.

If and , then (3.55) becomes
(3.57)

Case 2.

If and , then (3.55) becomes
(3.58)
Combining Cases 1 and 2, it follows that
(3.59)

Since by (3.3), it follows from Remark 2.11 that which implies that . Thus .

Uniqueness: if is another common fixed point of and in , then . Now by (3.2), it follows that

(3.60)
where
(3.61)

Thus and . Two cases arise.

Case 1.

If and , then (3.60) becomes . Since , by Remark 2.11 we have which implies that is the unique common fixed point of and .

Case 2.

If and , then (3.60) becomes
(3.62)

Since , by Remark 2.11 we have which implies is the unique common fixed point of and . Hence is the unique common fixed point of and in .

The following example illustrates Theorem 3.1.

Example 3.2.

Let , , , and . Define two nonself mappings as and for all .

Now let us see that conditions (i)–(iii) in Theorem 3.1 are satisfied.

It may be seen that and . Then and . Also, as . Moreover is closed in .

Next, we shall see that inequality (3.2) is satisfied by taking and . It is easy to see that .

Now, LHS of inequality (3.2) is . Taking and , it follows that .

Next, RHS of inequality (3.2) is , where , and . Then RHS of inequality (3.2) is if and . Thus LHS of inequality (3.2) RHS of inequality (3.2). Similarly, LHS of inequality (3.2) RHS of inequality (3.2) for all possible cases of and . Thus all the conditions of Theorem 3.1 are satisfied. Hence "0" is the unique common fixed point of and in .

Corollary 3.3.

Let be a complete cone metric space and a nonempty closed subset of such that for each and there exists a point such that
(3.63)
Suppose that is a nonself mapping satisfying for all with ,
(3.64)

and are nonnegative real numbers such that . Also assume that . Then there exists a unique fixed point of in .

Proof.

The proof of this corollary follows by taking , the identity mapping of in Theorem 3.1.

Remark 3.4.

Our results generalize the results of Radenović and Rhoades [10] and Janković et al. [6] and extend the results of Ćirić and Ume [20] to cone metric space for single valued mappings.

Declarations

Acknowledgment

The authors would like to thank the referees for their valuable suggestions which lead to the improvement of the presentation of the paper.

Authors’ Affiliations

(1)
Department of Mathematics, SMK Fomra Institute of Technology
(2)
Ramanujan Computing Centre, Anna University Chennai
(3)
Department of Mathematics, Easwari Engineering College, Ramapuram
(4)
Department of Mathematics, Anna University Chennai

References

  1. Huang L-G, Zhang X: Cone metric spaces and fixed point theorems of contractive mappings. Journal of Mathematical Analysis and Applications 2007,332(2):1468–1476. 10.1016/j.jmaa.2005.03.087MathSciNetView ArticleMATHGoogle Scholar
  2. Abbas M, Jungck G: Common fixed point results for noncommuting mappings without continuity in cone metric spaces. Journal of Mathematical Analysis and Applications 2008,341(1):416–420. 10.1016/j.jmaa.2007.09.070MathSciNetView ArticleMATHGoogle Scholar
  3. Abbas M, Rhoades BE: Fixed and periodic point results in cone metric spaces. Applied Mathematics Letters 2009,22(4):511–515. 10.1016/j.aml.2008.07.001MathSciNetView ArticleMATHGoogle Scholar
  4. Ilić D, Rakočević V: Common fixed points for maps on cone metric space. Journal of Mathematical Analysis and Applications 2008,341(2):876–882. 10.1016/j.jmaa.2007.10.065MathSciNetView ArticleMATHGoogle Scholar
  5. Raja P, Vaezpour SM: Some extensions of Banach's contraction principle in complete cone metric spaces. Fixed Point Theory and Applications 2008, Article ID 768294 2008:-11 Pages.Google Scholar
  6. Janković S, Kadelburg Z, Radenović S, Rhoades BE: Assad-Kirk-type fixed point theorems for a pair of nonself mappings on cone metric spaces. Fixed Point Theory and Applications 2009, Article ID 761086 2009:-16 Pages.Google Scholar
  7. Jungck G, Radenović S, Radojević S, Rakočević V: Common fixed point theorems for weakly compatible pairs on cone metric spaces. Fixed Point Theory and Applications 2009, Article ID 643840 2009:-13 Pages.Google Scholar
  8. Kadelburg Z, Radenović S, Rakočević V: Remarks on "Quasi-contraction on a cone metric space". Applied Mathematics Letters 2009,22(11):1674–1679. 10.1016/j.aml.2009.06.003MathSciNetView ArticleMATHGoogle Scholar
  9. Kadelburg Z, Radenović S, Rosić B: Strict contractive conditions and common fixed point theorems in cone metric spaces. Fixed Point Theory and Applications 2009, Article ID 173838 2009:-14 Pages.Google Scholar
  10. Radenović S, Rhoades BE: Fixed point theorem for two non-self mappings in cone metric spaces. Computers & Mathematics with Applications 2009,57(10):1701–1707. 10.1016/j.camwa.2009.03.058MathSciNetView ArticleMATHGoogle Scholar
  11. Rezapour Sh, Hamlbarani R: Some notes on the paper: "Cone metric spaces and fixed point theorems of contractive mappings". Journal of Mathematical Analysis and Applications 2008,345(2):719–724. 10.1016/j.jmaa.2008.04.049MathSciNetView ArticleMATHGoogle Scholar
  12. Assad NA, Kirk WA: Fixed point theorems for set-valued mappings of contractive type. Pacific Journal of Mathematics 1972,43(3):553–562.MathSciNetView ArticleMATHGoogle Scholar
  13. Assad NA: On a fixed point theorem of Kannan in Banach spaces. Tamkang Journal of Mathematics 1976,7(1):91–94.MathSciNetMATHGoogle Scholar
  14. Ćirić L: Non-self mappings satisfying non-linear contractive condition with applications. Nonlinear Analysis: Theory, Methods & Applications 2009,71(7–8):2927–2935. 10.1016/j.na.2009.01.174View ArticleMathSciNetMATHGoogle Scholar
  15. Hadžić O: On coincidence points in convex metric spaces. Univerzitet u Novom Sadu. Zbornik Radova Prirodno-Matematičkog Fakulteta. Serija za Matematiku 1989,19(2):233–240.MathSciNetMATHGoogle Scholar
  16. Hadžić O, Gajić L: Coincidence points for set-valued mappings in convex metric spaces. Univerzitet u Novom Sadu. Zbornik Radova Prirodno-Matematičkog Fakulteta. Serija za Matematiku 1986,16(1):13–25.MathSciNetMATHGoogle Scholar
  17. Imdad M, Kumar S: Rhoades-type fixed-point theorems for a pair of nonself mappings. Computers & Mathematics with Applications 2003,46(5–6):919–927. 10.1016/S0898-1221(03)90153-2MathSciNetView ArticleMATHGoogle Scholar
  18. Rhoades BE: A fixed point theorem for some non-self-mappings. Mathematica Japonica 1978,23(4):457–459.MathSciNetMATHGoogle Scholar
  19. Rhoades BE: A fixed point theorem for non-self set-valued mappings. International Journal of Mathematics and Mathematical Sciences 1997,20(1):9–12. 10.1155/S0161171297000021MathSciNetView ArticleMATHGoogle Scholar
  20. Ćirić LB, Ume JS: Multi-valued non-self-mappings on convex metric spaces. Nonlinear Analysis: Theory, Methods & Applications 2005,60(6):1053–1063. 10.1016/j.na.2004.09.057MathSciNetView ArticleMATHGoogle Scholar
  21. Itoh S: Multivalued generalized contractions and fixed point theorems. Commentationes Mathematicae Universitatis Carolinae 1977,18(2):247–258.MathSciNetMATHGoogle Scholar
  22. Khan MS: Common fixed point theorems for multivalued mappings. Pacific Journal of Mathematics 1981,95(2):337–347.MathSciNetView ArticleMATHGoogle Scholar
  23. Rezapour Sh: A review on topological properties of cone metric spaces. Proceedings of the Conference on Analysis, Topology and Applications (ATA '08), May-June 2008, Vrnjacka Banja, SerbiaGoogle Scholar

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© R. Sumitra et al. 2010

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