Research Article | Open Access

Sunny Chauhan, Huma Sahper, "Common Fixed Point Theorems for Conversely Commuting Mappings Using Implicit Relations", *Journal of Operators*, vol. 2013, Article ID 391474, 5 pages, 2013. https://doi.org/10.1155/2013/391474

# Common Fixed Point Theorems for Conversely Commuting Mappings Using Implicit Relations

**Academic Editor:**Haiyan Wang

#### Abstract

The object of this paper is to utilize the notion of conversely commuting mappings due to Lü (2002) and prove some common fixed point theorems in Menger spaces via implicit relations. We give some examples which demonstrate the validity of the hypotheses and degree of generality of our main results.

#### 1. Introduction

In 1986, Jungck [1] introduced the notion of compatible mappings in metric space. Most of the common fixed point theorems for contraction mappings invariably require a compatibility condition besides continuity of at least one of the mappings. Later on, Jungck and Rhoades [2] studied the notion of weakly compatible mappings and utilized it as a tool to improve commutativity conditions in common fixed point theorems. Many mathematicians proved several fixed point results in Menger spaces (see, e.g., [3–9]). In 2002, Lü [10] presented the concept of the converse commuting mappings as a reverse process of weakly compatible mappings and proved common fixed point theorems for single-valued mappings in metric spaces (also see [11]). Recently, Pathak and Verma [12, 13], Chugh et al. [14], and Chauhan et al. [15] proved some interesting common fixed point theorems for converse commuting mappings.

In this paper, we prove some unique common fixed point theorems for two pairs of converse commuting mappings in Menger spaces by using implicit relations.

#### 2. Preliminaries

*Definition 1 (see [16]). *A -norm is a function satisfying(T1), ;(T2);(T3) for , ;(T4) for all , , in .

Examples of -norms are , , and .

*Definition 2 (see [16]). *A real valued function on the set of real numbers is called a distribution function if it is nondecreasing, left continuous with and .

We shall denote by the set of all distribution functions defined on , while will always denote the specific distribution function defined by

If is a nonempty set, is called a probabilistic distance on and the value of at is represented by .

*Definition 3 (see [17]). *A probabilistic metric space is an ordered pair , where is a nonempty set of elements and is a probabilistic distance satisfying the following conditions: for all and , (1) for all if and only if ;(2);(3);(4)if and , then for all and .

Every metric space can always be realized as a probabilistic metric space by considering defined by for all and . So probabilistic metric spaces offer a wider framework than that of metric spaces and are better suited to cover even wider statistical situations; that is, every metric space can be regarded as a probabilistic metric space of a special kind.

*Definition 4 (see [16]). *A Menger space is a triplet, where is a probabilistic metric space and is a -norm satisfying the following condition:
for all and .

*Definition 5 (see [2]). *A pair of self-mappings defined on a nonempty set is said to be weakly compatible (or coincidentally commuting) if they commute at their coincidence points; that is, if for some , then .

*Definition 6 (see [10]). *A pair of self-mappings defined on a nonempty set is called conversely commuting if, for all , implies .

*Definition 7 (see [10]). *Let and be self-mappings of a nonempty set . A point is called commuting point of and if .

Lemma 8 (see [18]). *Let be a Menger space. If there exists a constant such that
**
for all with fixed , then .*

#### 3. Implicit Relations

In 2005, Singh and Jain [19] studied an implicit function and obtained some fixed point results in framework of fuzzy metric spaces.

Let be the set of all real continuous functions , nondecreasing in first argument and satisfying the following conditions.(*ϕ*-1)For , or implies that .(*ϕ*-2) implies that .

*Example 9. *Define as . Then .

Since then, Imdad and Ali [20] used the following class of implicit functions for the existence of a common fixed point due to Popa [21]. Many authors proved a number of common fixed point theorems using the notion of implicit relation on different spaces (see, e.g., [22–29]).

Let denote the family of all continuous functions satisfying the following conditions.(*ψ*-1)For every with or , we have .(*ψ*-2) for all .

*Example 10. *Define as , where is a continuous function such that for .

*Example 11. *Define as , where .

*Example 12. *Define as , where .

*Example 13. *Define as , where and .

*Example 14. *Define as , where and .

*Example 15. *Define as , where .

In 2011, Gopal et al. [30] showed that the above-mentioned classes of functions and are independent classes.

#### 4. Main Results

First, we prove a unique common fixed point theorem for two pairs of self-mappings satisfying a class of implicit function .

Theorem 16. *Let , , , and be four self-mappings of a Menger space , where is a continuous -norm and the pairs and are conversely commuting, respectively, and satisfy the following conditions:
**
for all , , and . If and have a commuting point and and have a commuting point, then , and have a unique common fixed point in .*

*Proof. *Let be the commuting point of and . Then . And let be the commuting point of and . Then . Since and are conversely commuting, we have . Since and are conversely commuting, we have . Hence and .(i)We claim that . On using (4) with , , we get
or, equivalently,
Hence, for for all , we have . Thus .(ii)Now, we show that is a fixed point of mapping . In order to establish this, using (4) with , , we have
and so
Hence, for for all , we get . Similarly we show that . On using (4) with , , we have
or, equivalently,
Thus, for all and we obtain = . Since , we have = = = which shows that is a fixed point of the mapping . On the other hand, = = = = and = = = . Hence is a common fixed point of , , , and .(iii)For the uniqueness of common fixed point, we use (4) with and , where is another common fixed point of the mappings , , , and . Now we have
and so
Hence, we get . Therefore, is a unique common fixed point of the mappings , , , and .

Now, we give an example which illustrates Theorem 16.

*Example 17. *Let with the metric defined by and for each , define
for all . Define . Clearly is a Menger space. Let as with for . Define the self-mappings , and by

Hence the pairs and are conversely commuting and 1 is a unique common fixed point of the mappings , and .

Corollary 18. *The conclusions of Theorem 16 remain true if condition (4) is replaced by one of the following conditions: for all **
where is a continuous function such that for all ;
**
where ;
**
where ;
**
where and ;
**
where and ;
**
where .*

*Proof. *The proof of each inequality (15)–(20) easily follows from Theorem 16 in view of Examples 10–15.

Now we state a unique common fixed point theorem satisfying a class of implicit function .

Theorem 19. *Let , , , and be four self-mappings on a Menger space , where is a continuous t-norm, the pairs and are conversely commuting, respectively, and satisfying
*

*for all , , and . If and have a commuting point and and have a commuting point, then , , , and have a unique common fixed point in .*

* Proof. *The proof of this theorem can be completed on the lines of the proof of Theorem 16 (in view of Lemma 8); hence details are omitted.

*Example 20. *In the setting of Example 17, define as besides retaining the rest. Therefore, all the conditions of Theorem 19 are satisfied for some fixed and 1 is a unique common fixed point of the mappings , and .

By choosing , , , and suitably, we can deduce corollaries involving two as well as three self-mappings. For the sake of naturality, we only derive the following corollary (due to Theorem 16) involving a pair of self-mappings.

Corollary 21. *Let and be two self-mappings of a Menger space , where is a continuous t-norm and the mappings and are conversely commuting satisfying
*

*for all , , and . If and have a commuting point, then and have a unique common fixed point in .*

#### Conflict of Interests

The authors declare that they have no conflict of interests.

#### References

- 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 Site | Google Scholar | 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 | Zentralblatt MATH | MathSciNet - S. Chauhan and B. D. Pant, “Common fixed point theorem for weakly compatible mappings in Menger space,”
*Journal of Advanced Research in Pure Mathematics*, vol. 3, no. 2, pp. 107–119, 2011. View at: Publisher Site | Google Scholar | MathSciNet - M. Imdad, J. Ali, and M. Tanveer, “Coincidence and common fixed point theorems for nonlinear contractions in Menger PM spaces,”
*Chaos, Solitons & Fractals*, vol. 42, no. 5, pp. 3121–3129, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. Miheţ, “A note on a common fixed point theorem in probabilistic metric spaces,”
*Acta Mathematica Hungarica*, vol. 125, no. 1-2, pp. 127–130, 2009. View at: Publisher Site | Google Scholar | MathSciNet - B. D. Pant and S. Chauhan, “A contraction theorem in Menger space,”
*Tamkang Journal of Mathematics*, vol. 42, no. 1, pp. 59–68, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. D. Pant and S. Chauhan, “Common fixed point theorems for two pairs of weakly compatible mappings in Menger spaces and fuzzy metric spaces,”
*Scientific Studies and Research*, vol. 21, no. 2, pp. 81–96, 2011. View at: Google Scholar | Zentralblatt MATH | MathSciNet - B. D. Pant, S. Chauhan, and Q. Alam, “Common fixed point theorem in probabilistic metric space,”
*Kragujevac Journal of Mathematics*, vol. 35, no. 3, pp. 463–470, 2011. View at: Google Scholar | Zentralblatt MATH | MathSciNet - R. Saadati, D. O'Regan, S. M. Vaezpour, and J. K. Kim, “Generalized distance and common fixed point theorems in Menger probabilistic metric spaces,”
*Iranian Mathematical Society*, vol. 35, no. 2, pp. 97–117, 2009. View at: Google Scholar | MathSciNet - Z. X. Lü, “Common fixed points for converse commuting selfmaps on a metric space,”
*Acta Analysis Functionalis Applicata.*, vol. 4, no. 3, pp. 226–228, 2002 (Chinese). View at: Google Scholar | Zentralblatt MATH | MathSciNet - Q. K. Liu and X. Q. Hu, “Some new common fixed point theorems for converse commuting multi-valued mappings in symmetric spaces with applications,”
*Nonlinear Analysis Forum*, vol. 10, no. 1, pp. 97–104, 2005. View at: Google Scholar | Zentralblatt MATH | MathSciNet - H. K. Pathak and R. K. Verma, “Integral type contractive condition for converse commuting mappings,”
*International Journal of Mathematical Analysis*, vol. 3, no. 21–24, pp. 1183–1190, 2009. View at: Google Scholar | Zentralblatt MATH | MathSciNet - H. K. Pathak and R. K. Verma, “An integral type implicit relation for converse commuting mappings,”
*International Journal of Mathematical Analysis*, vol. 3, no. 21–24, pp. 1191–1198, 2009. View at: Google Scholar | Zentralblatt MATH | MathSciNet - R. Chugh, Sumitra, and M. Alamgir Khan, “Common fixed point theorems for converse commuting maps in fuzzy metric spaces,”
*Journal for Theory and Applications*, vol. 6, no. 37–40, pp. 1845–1851, 2011. View at: Google Scholar | Zentralblatt MATH | MathSciNet - S. Chauhan, M. A. Khan, and W. Sintunavarat, “Fixed points of converse commuting mappings using an implicit relation,”
*Honam Mathematical Journal*, vol. 35, no. 2, pp. 109–117, 2013. View at: Google Scholar - B. Schweizer and A. Sklar, “Statistical metric spaces,”
*Pacific Journal of Mathematics*, vol. 10, pp. 313–334, 1960. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. Menger, “Statistical metrics,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 28, pp. 535–537, 1942. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. N. Mishra, “Common fixed points of compatible mappings in PM-spaces,”
*Mathematica Japonica*, vol. 36, no. 2, pp. 283–289, 1991. View at: Google Scholar | Zentralblatt MATH | MathSciNet - B. Singh and S. Jain, “Semicompatibility and fixed point theorems in fuzzy metric space using implicit relation,”
*International Journal of Mathematics and Mathematical Sciences*, no. 16, pp. 2617–2629, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Imdad and J. Ali, “A general fixed point theorems in fuzzy metric spaces via implicit function,”
*Journal of Applied Mathematics & Informatics*, vol. 26, no. 3-4, pp. 591–603, 2008. View at: Google Scholar - V. Popa, “A fixed point theorem for mapping in d-complete topological spaces,”
*Mathematica Moravica*, vol. 3, pp. 43–48, 1999. View at: Google Scholar | Zentralblatt MATH | MathSciNet - S. Chauhan, M. Imdad, and C. Vetro, “Unified metrical common fixed point theorems in 2-metric spaces via an implicit relation,”
*Journal of Operators*, vol. 2013, Article ID 186910, 11 pages, 2013. View at: Publisher Site | Google Scholar - S. Chauhan, M. A. Khan, and S. Kumar, “Unified fixed point theorems in fuzzy metric spaces via common limit range property,”
*Journal of Inequalities and Applications*, vol. 2013, article 182, 17 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - S. Chauhan and B. D. Pant, “Fixed points of weakly compatible mappings using common (E.A) like property,”
*Le Matematiche*, vol. 68, no. 1, pp. 99–116, 2013. View at: Google Scholar | 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Kumar and S. Chauhan, “Common fixed point theorems using implicit relation and property (E.A) in fuzzy metric spaces,”
*Annals of Fuzzy Mathematics and Informatics*, vol. 5, no. 1, pp. 107–114, 2013. View at: Google Scholar | MathSciNet - S. Kumar and B. D. Pant, “Common fixed point theorems in probabilistic metric spaces using implicit relation and property (E.A),”
*Bulletin of the Allahabad Mathematical Society*, vol. 25, no. 2, pp. 223–235, 2010. View at: Google Scholar | Zentralblatt MATH | MathSciNet - V. Popa and D. Turkoǧlu, “Some fixed point theorems for hybrid contractions satisfying an implicit relation,”
*Studii şi Cercetări Ştiinţifice*, no. 8, pp. 75–86, 1998. View at: Google Scholar | Zentralblatt MATH | MathSciNet - S. Sharma and B. Deshpande, “On compatible mappings satisfying an implicit relation in common fixed point consideration,”
*Tamkang Journal of Mathematics*, vol. 33, no. 3, pp. 245–252, 2002. View at: Google Scholar | Zentralblatt MATH | MathSciNet - D. Gopal, M. Imdad, and C. Vetro, “Impact of common property (E.A.) on fixed point theorems in fuzzy metric spaces,”
*Fixed Point Theory and Applications*, vol. 2011, Article ID 297360, 14 pages, 2011. View at: Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2013 Sunny Chauhan and Huma Sahper. 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.