#### 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  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  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., ). In 2002, Lü  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 ). Recently, Pathak and Verma [12, 13], Chugh et al. , and Chauhan et al.  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 ). A  -norm is a function satisfying(T1),  ;(T2);(T3) for ,  ;(T4) for all ,  ,   in .
Examples of -norms are , , and .

Definition 2 (see ). 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 ). 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 ). 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 ). 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 ). A pair of self-mappings defined on a nonempty set is called conversely commuting if, for all , implies .

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

Lemma 8 (see ). 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  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  used the following class of implicit functions for the existence of a common fixed point due to Popa . Many authors proved a number of common fixed point theorems using the notion of implicit relation on different spaces (see, e.g., ).

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.  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 1015.

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.