Journal of Applied Mathematics

Volume 2013 (2013), Article ID 510691, 7 pages

http://dx.doi.org/10.1155/2013/510691

## A Common Fixed Point Theorem in Metric Space under General Contractive Condition

^{1}R.H. Government Postgraduate College, Udham Singh Nagar, Kashipur, Uttarakhand 244713, India^{2}Near Nehru Training Centre, H. No. 274, Nai Basti B-14, Bijnor, Uttar Pradesh 246701, India^{3}Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Beograd, Serbia^{4}Department of Mathematics, Jazan University, Saudi Arabia

Received 30 April 2013; Revised 12 July 2013; Accepted 4 August 2013

Academic Editor: E. Karapinar

Copyright © 2013 Sunny Chauhan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We prove a common fixed point theorem for two pairs of compatible and subsequentially continuous (alternately subcompatible and reciprocally continuous) mappings satisfying a general contractive condition in a metric space. Some illustrative examples are furnished to highlight the realized improvements. Our result improves the main result of Sedghi and Shobe (2007).

#### 1. Introduction

Fixed point theory is one of the most fruitful and effective tools in mathematics which has enormous applications within as well as outside mathematics. Starting from the celebrated Banach contraction principle [1], many authors have obtained its several generalizations in different ways (see, e.g., [2–4]).

A metrical common fixed point theorem generally involves conditions on commutativity, continuity, and contraction of the given mappings, as well as completeness (or closedness) of the underlying space (or subspaces), along with conditions on suitable containment amongst the ranges of involved mappings. Hence, in order to prove a new metrical common fixed point theorem, one is always required to weaken one or more of these conditions. In order to weaken commutativity conditions in common fixed point theorems, Sessa [5] introduced the concept of weakly commuting mappings. Jungck [6] defined the notion of compatible mappings in order to generalize the concept of weak commutativity and showed that weakly commuting mappings are compatible, but the converse is not true. Afterwards Jungck and Rhoades [7] introduced the concept of weak compatibility to the setting of single-valued and multivalued mappings which is more general than compatibility. However, the study of common fixed points of noncompatible mappings is also equally interesting, and it was initiated by Pant [8] in metric spaces. Researchers of this domain introduced several definitions of weak commutativity such as compatible mappings, compatibility of type , , , and , and several others, whose systematic comparisons and illustrations are available in Murthy [9] and Singh and Tomar [10].

In 2002, Aamri and El Moutawakil [11] introduced the notion of property (E.A). Further, Al-Thagafi and Shahzad [12] introduced the notion of occasionally weakly compatible mappings. It was used further by many authors (see, e.g., [13]). However, in [14], Đorić et al. observed that the condition of occasionally weak compatibility reduces to weak compatibility in the presence of a unique point of coincidence of the given pair of mappings. Thus, no generalization can be obtained by replacing weak compatibility with occasionally weak compatibility (see also [15–17]).

In 2009, Bouhadjera and Godet-Thobie [18] further enlarged the class of compatible (reciprocally continuous) pairs by introducing the concept of subcompatibility (subsequential continuity) of pairs of mappings, which is substantially weaker than compatibility (reciprocal continuity). Since then, Imdad et al. [19] improved the results of Bouhadjera and Godet-Thobie and showed that these results can easily be recovered by replacing subcompatibility with compatibility or subsequential continuity with reciprocal continuity. Very recently, Chauhan et al. [20] obtained some results of this kind using integral type contractive conditions. Many authors established a number of other fixed point results in metric and related spaces (see, e.g., [21–31]).

In 2007, Sedghi and Shobe [32] proved a common fixed point theorem for weakly compatible mappings satisfying a new general contractive type condition. The aim of this paper is to prove a common fixed point theorem for two pairs of self-mappings by using the notions of compatibility and subsequential continuity (alternately subcompatibility and reciprocal continuity) satisfying general contractive condition in a metric space. Some examples are furnished which demonstrate the validity of our result.

#### 2. Preliminaries

*Definition 1. *Let be two self-mappings on a metric space . The mappings and are said to be (1)weakly commuting if for all [5],(2)compatible if for each sequence in such that [6],(3)noncompatible if there exists a sequence in such that , but is either nonzero or nonexistent [8], (4)weakly compatible if they commute at their coincidence points; that is, whenever , for some [7],(5)occasionally weakly compatible if there is a point which is a coincidence point of and at which and commute [12],(6)with the property (E.A) if there exists a sequence in and some such that [11].

It can be noticed that arbitrary noncompatible self-mappings satisfy the property (E.A) but two mappings satisfying the property (E.A), need not be noncompatible (see [33, Example 1]). Also, weak compatibility and property (E.A) are independent of each other (see [31, Examples 2.1 and 2.2]).

*Definition 2 (see [34]). * A pair of self-mappings on a metric space is called reciprocally continuous if for a sequence in , and , whenever , for some .

It is easy to see that if two self-mappings are continuous, then they are obviously reciprocally continuous, but the converse is not true. Moreover, in the setting of common fixed point theorems for compatible pairs of self-mappings satisfying contractive conditions, continuity of one of the mappings implies their reciprocal continuity but not conversely (see [35]).

*Definition 3 (see [18]). *A pair of self-mappings on a metric space is said to be subcompatible if there exists a sequence such that , for some and .

A pair of subcompatible mappings satisfies the property (E.A). Obviously, compatible mappings which satisfy the property (E.A) are subcompatible, but the converse statement does not hold in general (see [36, Example 2.3]). Two occasionally weakly compatible mappings are subcompatible; however, the converse is not true in general (see [18, Example 1.2]).

*Definition 4 (see [18]). *A pair of self-mappings on a metric space is called subsequentially continuous if there exists a sequence in such that , for some such that and .

One can easily check that if two self-mappings and are both continuous, hence also reciprocally continuous mappings but and are not subsequentially continuous (see [35, Example 1]).

*Definition 5 (see [32]). *By , a binary operation will be denoted, satisfying the following conditions: (1) is associative and commutative, (2) is continuous.

Some typical examples of are , , , , and , for each .

*Definition 6 (see [32]). *The binary operation is said to satisfy -property if there exists a positive real number such that
for all .

*Example 7 (see [32]). * (1) If , for each , then for , we have .

(2) If , for each , then for , we have .

#### 3. Main Results

In 2007, Sedghi and Shobe [32] proved the following result.

Theorem 8 (see [32, Theorem ]). *Let be a complete metric space such that satisfies the -property with . Let , , , and be self-mappings on satisfying the following conditions: *(1)*, , and or is a closed subset of , *(2)* the pairs and are weakly compatible, *(3)* for all ,
* *where and . ** Then, , , , and have a unique common fixed point in . *

Now we prove our main result.

Theorem 9. *Let , , , and be four self-mappings on a metric space , and let the operation satisfy the -property with . Suppose that the pairs and are compatible and subsequentially continuous (alternately subcompatible and reciprocally continuous), satisfying inequality (2) of Theorem 8. Then , , , and have a unique common fixed point in .*

*Proof. *If the pair of mappings is subsequentially continuous and compatible, there exists a sequence in such that
for some , and
that is, . Similarly, with respect to the pair , there exists a sequence in such that
for some , and
that is, . Hence is a coincidence point of the pair whereas is a coincidence point of the pair .

Now we assert that . If then using inequality (2) with and , we have
Letting , we get
Since satisfies the -property, we obtain
which is a contradiction. Hence . Now we prove that . If we suppose that , then from inequality (2) with and , we have
Taking the limit as , we get
that is,
Then, simplifying, we obtain
a contradiction. Hence, . Therefore, . Now we show that . If then using (2) with and , we have
As , we get
Then, simplifying, we obtain
a contradiction. Hence, . Therefore, ; that is, is a common fixed point of , , , and . The uniqueness of common fixed point is an easy consequence of inequality (2).

Now suppose that the mappings (as well as ) are subcompatible and reciprocally continuous. Then there exists a sequence in such that
for some , and
whereas in respect of the pair , there exists a sequence in with
for some , and
Therefore, and ; that is, is a coincidence point of the pair whereas is a coincidence point of the pair . The rest of the proof can be completed easily.

*Example 10. * Let , and let be the usual metric on . Define self-mappings , , , and by
Consider the sequence in . Then
Next,
Consider another sequence in . Then
However,
but . Thus, the pair is compatible as well as subsequentially continuous but not reciprocally continuous (the same for the pair ). It is easy to check that condition (2) is satisfied with , , and . Therefore, all the conditions of Theorem 9 are satisfied. Here, is a coincidence as well as the unique common fixed point of the mappings , , , and .

It can be noted that this example cannot be covered by those fixed point theorems which assume both compatibility and reciprocal continuity or by involving conditions of closedness of respective ranges. Indeed, in this example and ; hence, neither of and is closed.

*Example 11. *Let (set of real numbers), and let be the usual metric on . Define self-mappings , , , and by
Consider the sequence in . Then
Also,
Consider another sequence in . Then
Next,
and . Thus, the pair is reciprocally continuous as well as subcompatible but not compatible (the same for the pair ). It is easy to check that condition (2) is satisfied with , , and . Therefore, all the conditions of Theorem 9 are satisfied. Thus, is a coincidence as well as the unique common fixed point of the pair .

It can be noted that this example cannot be covered by those fixed point theorems which involve both compatibility and reciprocal continuity. Again, is not closed. Note also that the mappings and have two points of coincidence ( and ), which are occasionally weakly compatible but not weakly compatible.

In the next example (taken from [18, Example 1.4]), we demonstrate the situation when conditions of Theorem 9 are not satisfied, and the given pairs have no common fixed points.

*Example 12. *Let with the standard metric , and let be given by
Then, as it was shown in [18, Example 1.4], the pairs and are subsequentially continuous and subcompatible. However, they are neither reciprocally continuous nor compatible (not even occasionally weakly compatible). We note that has no common fixed points, although it has a unique point of coincidence .

We present an example of different kind, inspired by [20, Example 3].

*Example 13. *Let , and define a metric on by
Consider the mappings given by
Take (which satisfies -condition with ). Then,(1)the pair (as well as ) is compatible and subsequentially continuous, (2)condition (2) is satisfied with . Indeed, in order to prove (1), take for all but finitely many (which is the only possibility to obtain the same limit for and ). Then and ; also and . Hence, the pair is compatible and subsequentially continuous.

In order to prove (2), suppose that , (the case is trivial). Since we have , , and , condition (2) is symmetric in ; hence, without loss of generality, we can suppose that . Consider the following possible cases.*Case ** 1*. One has and . Then , , and (2) is satisfied.*Case ** 2*. One has and . Then , , . The right-hand side of (2) becomes ()
*Case ** 3*. One has and . Then
*Case ** 4*. One has . Then and
*Case ** 5*. One has . Then (2) again reduces to .

All the conditions of Theorem 9 are satisfied, and , , , and have a unique common fixed point (which is ).

By choosing , , , and suitably in Theorem 9, we can deduce corollaries for two or three self-mappings. As a sample, we deduce the following corollary for two self-mappings.

Corollary 14. *Let and be two self-mappings on a metric space such that satisfies the -property with . If the pair is compatible and subsequentially continuous (alternately subcompatible and reciprocally continuous) satisfying
**
for all , where and , then and have a unique common fixed point in .*

*Remark 15. *The conclusion of Theorem 9 remains true if we replace inequality (2) by the following:
for all , where and .

Similarly, other variants of contractive condition can be obtained by specifying operation .

*Remark 16. *Similar results can be obtained if condition (2) is replaced by the following one:
for a suitable function .

#### 4. Conclusion

Theorem 9 is proved for two pairs of compatible and subsequentially continuous (alternately subcompatible and reciprocally continuous) mappings satisfying a general contractive condition. Theorem 9 improves the main result of Sedghi and Shobe [32, Theorem 2.1] as we do not require any condition on the containment of ranges of involved mappings and completeness (or closedness) of the whole space (or any subspace). A natural result is defined in the form of a corollary (see Corollary 14). On the other hand, Remark 15 is developed for a particular case, , which also improves the result of Sedghi and Shobe [32, Corollary 2.2].

#### Acknowledgments

The authors would like to thank Professor Mohammad Imdad for the preprint of paper [36] and anonymous referees for their valuable comments that helped them to improve the exposition, in particular for Remark 16. The second author is thankful to the Ministry of Education, Science and Technological Development of Serbia.

#### References

- S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales,”
*Fundamenta Mathematicae*, vol. 3, no. 1, pp. 133–181, 1922. View at Google Scholar - L. B. Ćirić, “On a family of contractive maps and fixed points,”
*Publications de l'Institut Mathématique*, vol. 17, no. 31, pp. 45–51, 1974. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Lj. B. \'Cirić, “A generalization of Banach's contraction principle,”
*Proceedings of the American Mathematical Society*, vol. 45, pp. 267–273, 1974. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. E. Rhoades, “A comparison of various definitions of contractive mappings,”
*Transactions of the American Mathematical Society*, vol. 226, pp. 257–290, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Sessa, “On a weak commutativity condition of mappings in fixed point considerations,”
*Publications de l'Institut Mathématique*, vol. 32, no. 46, pp. 149–153, 1982. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Jungck, “Compatible mappings and common fixed points,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 9, no. 4, pp. 771–779, 1986. View at Publisher · View at Google Scholar · View at MathSciNet - G. Jungck and B. E. Rhoades, “Fixed points for set valued functions without continuity,”
*Indian Journal of Pure and Applied Mathematics*, vol. 29, no. 3, pp. 227–238, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. P. Pant, “Noncompatible mappings and common fixed points,”
*Soochow Journal of Mathematics*, vol. 26, no. 1, pp. 29–35, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. P. Murthy, “Important tools and possible applications of metric fixed point theory,”
*Nonlinear Analysis*, vol. 47, no. 5, pp. 3479–3490, 2001. View at Publisher · View at Google Scholar · View at MathSciNet - S. L. Singh and A. Tomar, “Weaker forms of commuting maps and existence of fixed points,”
*Journal of the Korean Society of Mathematical Education B*, vol. 10, no. 3, pp. 145–161, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Aamri and D. El Moutawakil, “Some new common fixed point theorems under strict contractive conditions,”
*Journal of Mathematical Analysis and Applications*, vol. 270, no. 1, pp. 181–188, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Al-Thagafi and N. Shahzad, “Generalized $I$-nonexpansive selfmaps and invariant approximations,”
*Acta Mathematica Sinica*, vol. 24, no. 5, pp. 867–876, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - G. Jungck and B. E. Rhoades, “Fixed point theorems for occasionally weakly compatible mappings,”
*Fixed Point Theory*, vol. 7, no. 2, pp. 287–296, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Đorić, Z. Kadelburg, and S. Radenović, “A note on occasionally weakly compatible mappings and common fixed points,”
*Fixed Point Theory*, vol. 13, no. 2, pp. 475–479, 2012. View at Google Scholar · View at MathSciNet - M. A. Alghamdi, S. Radenović, and N. Shahzad, “On some generalizations of commuting mappings,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 952052, 6 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - R. K. Bisht and R. P. Pant, “A critical remark on “Fixed point theorems for occasionally weakly compatible mappings”,”
*Journal of the Egyptian Mathematical Society*, 2013. View at Publisher · View at Google Scholar - R. K. Bisht and N. Shahzad, “Faintly compatible mappings and common fixed points,”
*Fixed Point Theory and Applications*, vol. 2013, article 156, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - H. Bouhadjera and C. Godet-Thobie, “Common fixed point theorems for pairs of subcompatible maps,” old version, 2009, http://arxiv.org/abs/0906.3159.
- M. Imdad, J. Ali, and M. Tanveer, “Remarks on some recent metrical common fixed point theorems,”
*Applied Mathematics Letters*, vol. 24, no. 7, pp. 1165–1169, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - S. Chauhan, H. Aydi, W. Shatanawi, and C. Vetro, “Some integral type fixed point theorems and an application to system of functional equations,”
*Vietnam Journal of Mathematics*. In press. View at Publisher · View at Google Scholar - M. A. Ahmed, “Common fixed point theorems for weakly compatible mappings,”
*The Rocky Mountain Journal of Mathematics*, vol. 33, no. 4, pp. 1189–1203, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Akkouchi, “Common fixed points for three mappings using G-functions and the property (E.A),”
*Acta Universitatis Apulensis*, no. 23, pp. 223–231, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Akkouchi, “A common fixed point theorem for four maps using a Lipschitz type condition,”
*Mathematical Journal of the Ovidius University of Constantza*, vol. 19, no. 3, pp. 17–26, 2011. View at Google Scholar · View at MathSciNet - H. Aydi and E. Karapinar, “A Meir-Keeler common type fixed point theorem on partial metric spaces,”
*Fixed Point Theory and Applications*, vol. 2012, article 26, 10 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - I. Beg and M. Abbas, “Coincidence and common fixed points of noncompatiblemaps,”
*Journal of Applied Mathematics & Informatics*, vol. 29, no. 3-4, pp. 743–752, 2011. View at Google Scholar · View at MathSciNet - R. Chugh and S. Kumar, “Common fixed points for weakly compatible maps,”
*Proceedings of the Indian Academy of Sciences*, vol. 111, no. 2, pp. 241–247, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. S. Ding, Z. Kadelburg, E. Karapınar, and S. Radenović, “Common fixed points of weak contractions in cone metric spaces,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 793862, 18 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Imdad and S. Chauhan, “Employing common limit range property to prove unified metrical common fixed
point theorems,”
*International Journal of Analysis*, vol. 2013, Article ID 763261, 10 pages, 2013. View at Publisher · View at Google Scholar - M. Imdad, S. Chauhan, and Z. Kadelburg, “Fixed point theorems for mappings with common limit range property satisfying generalized (
*ψ*,*ϕ*)-weak contractive conditions,”*Mathematical Sciences*, vol. 7, article 16, 2013. View at Publisher · View at Google Scholar - E. Karapinar, N. Shobkolaei, S. Sedghi, and S. M. Vaezpour, “A common fixed point theorem for cyclic operators on partial metric spaces,”
*Filomat*, vol. 26, no. 2, pp. 407–414, 2012. View at Publisher · View at Google Scholar - H. K. Pathak, R. Rodríguez-López, and R. K. Verma, “A common fixed point theorem using implicit relation and property (E.A) in metric spaces,”
*Filomat*, vol. 21, no. 2, pp. 211–234, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Sedghi and N. Shobe, “Common fixed point theorems for four mappings in complete metric spaces,”
*Bulletin of the Iranian Mathematical Society*, vol. 33, no. 2, pp. 37–47, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. X. Fang and Y. Gao, “Common fixed point theorems under strict contractive conditions in Menger spaces,”
*Nonlinear Analysis*, vol. 70, no. 1, pp. 184–193, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. P. Pant, “Common fixed points of Lipschitz type mapping pairs,”
*Journal of Mathematical Analysis and Applications*, vol. 240, no. 1, pp. 280–283, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. P. Pant and R. K. Bisht, “Common fixed point theorems under a new continuity condition,”
*Annali dell'Universitá di Ferrara VII*, vol. 58, no. 1, pp. 127–141, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - F. Rouzkard, M. Imdad, and H. K. Nashine, “New common fixed point theorems and invariant approximation in convex metric spaces,”
*Bulletin of the Belgian Mathematical Society*, vol. 19, no. 2, pp. 311–328, 2012. View at Google Scholar · View at MathSciNet