Abstract

It is shown that occasionally operators as well as occasionally weakly biased mappings reduce to weakly compatible mappings in the presence of a unique point of coincidence (and a unique common fixed point) of the given maps.

1. Introduction and Preliminaries

The study of finding a common fixed point of pair of commuting mappings was initiated by Jungck [1]. Later, on this condition was weakened in various ways. Sessa [2] introduced the notion of weakly commuting maps. Jungck [3] gave the notion of compatible mappings in order to generalize the concept of weak commutativity. One of the conditions that was used most often was the weak compatibility, introduced by Jungck in [4] fixed point results for various classes of mappings on a metric space, utilizing these concepts. Jungck and Pathak [5] defined the concept of a weakly biased maps in order to generalize the concept of weak compatibility. In the paper [6], published in 2008, Al-Thagafi and Shahzad introduced an even weaker condition which they called occasionally weak compatibility (see also [7]). Many authors (see, e.g., [8–13]) used this condition to obtain common fixed point results, sometimes trying to generalize results that were known to use (formally stronger) condition of weak compatibility. Recently, Hussain et al. [14] have introduced two new and different classes of noncommuting self-maps: -operators and occasionally weakly biased mappings. These classes contain the occasionally weakly compatible and weakly biased self-maps as proper subclasses. For these new classes, authors have proved common fixed point results on the space which is more general than metric space. We will show in this short note that in the presence of a unique point of coincidence (and a unique common fixed point) of the given mappings, occasionally -operators as well as occasionally weakly biased mappings reduce to weakly compatible mappings (and so occasionally weakly compatible mappings). For more details on the subject, we refer the reader to [15–18].

Let be a nonempty set and let and be two self-mappings on . The set of fixed points of (resp., ) is denoted by (resp., ). A point is called a coincidence point of the pair if . The point is then called a point of coincidence for . The set of coincidence points of will be denoted as . Let represent the set of points of coincidence of the pair . A point is a common fixed point of and if . The self-maps and on are called(1)commuting if for all ;(2)weakly compatible (WC) if they commute at their coincidence points, that is, if whenever [4];(3)occasionally weakly compatible (OWC) if for some with [6].

Let be symmetric on . Then and are called(4)-operators if there is a point such that and , where [19];(5)-operators if there is a point in such that [14];(6)weakly -biased, if whenever [5];(7)occasionally weakly -biased, if there exists some such that and [14].

Let be a mapping such that if and only if . Then and are called(8)-operators if there is a point in such that and , where

2. Results

We begin with the following results.

Lemma 2.1 (see [20]). If a WC pair of self-maps on has a unique POC, then it has a unique common fixed point.

The following lemma is according to Jungck and Rhoades [12].

Lemma 2.2 (see [12]). If an OWC pair of self-maps on has a unique POC, then it has a unique common fixed point.

Proof. Since is an OWC, there exists such that and . Hence, is also a POC for , and since it must be unique, we have that is a common fixed point for . If is any common fixed point for (i.e., , then, again by the uniqueness of POC, it must be .

The following result is due to Ðorić et al. [21]. It shows that the results of Jungck and Rhoades are not generalizations of results obtained from Lemma 2.1.

Proposition 2.3 (see [21]). Let a pair of mappings have a unique POC. Then it is WC if and only if it is OWC.

Proof. In this case, we have only to prove that OWC implies WC. Let be the given POC, and let be OWC. Let . We have to prove that . Now is a POC for the pair . By the assumption, . Since, by Lemma 2.2, is a unique common fixed point of the pair , it follows that and , hence . The pair is WC.

Proposition 2.4. Let be a mapping such that if and only if . Let a pair of mappings have a unique POC. If it is a pair of -operators, then it is WC.

Proof. Let be a pair of -operators. Then there is a point in such that and . Clearly is a singleton. If not, then is a POC for the pair . By the assumption, . As a result, we have , which implies , that is, . Consequently, we have and thus is OWC. By Proposition 2.3, is WC.

It is worth mentioning that if is the identity mapping, then the pair is always WC, but it is a pair of -operators if and only if has a fixed point.

Proposition 2.5. Let be a mapping such that if and only if . Suppose is a pair of -operators satisfying for each , where are real numbers such that .Then is WC.

Proof. By hypothesis, there exists some such that . It remains to show that has a unique POC. Suppose there exists another point with . Then, we have which is a contradiction since . Thus has a unique POC. By Proposition 2.4, is WC.

Proposition 2.6. Let be symmetric on . Let a pair of mappings have a unique POC which belongs to . If it is a pair of occasionally weakly -biased mappings, then it is WC.

Proof. Let be a pair of occasionally weakly -biased mappings. Then there exists some such that and . Since belong to , then . Thus and thus . Hence is OWC. By Proposition 2.3, is WC.

Let be a nondecreasing function satisfying the condition for each .

Proposition 2.7. Let be self-maps of symmetric space and let the pair be occasionally weakly -biased. If for the control function , we have for each , then is WC.

Proof. It remains to show that has a unique POC which belongs to . Since are occasionally weakly -biased mappings, there exists some such that and . If and , then which is a contradiction. Also if , we have which is a contradiction. By Proposition 2.6, is WC.

Remark 2.8. According to Propositions 2.4, 2.5, 2.6, and 2.7, it follows that results from [14]: (Theorems 2.8, 2.9, 2.10, 2.11, 2.12, 3.7, 3.9 and Corollary 3.8) are not generalizations (extensions) of some common fixed point theorems due to Bhatt et al. [9], Jungck and Rhoades [12, 13], and Imdad and Soliman [11]. Moreover, all mappings in these results are WC.

Proposition 2.9. Let be symmetric on , and let a pair of mappings have a unique , that is, is a singleton. If is -operator pair, then it is WC.

Proof. According to (4), there is a point such that and . Hence, is a unique POC of pair and since is OWC. By Proposition 2.3 it is WC.

Remark 2.10. By Proposition 2.9, it follows that Theorem 2.1 from [19] is not a generalization result of [6], the main result of Jungck [1], and other results.

Proposition 2.11. Let be symmetric on , and let a pair of mappings have a unique POC. Then it is weakly -biased if and only if it is occasionally weakly -biased.

Proof. In this case, we have only to prove that (7) implies (6). Let be the given POC. Let . We have to prove that . Now is a POC for the pair . By the assumption, , that is, . Further, we have and , which implies that , that is, the pair satisfies (6).

The following example shows that the assumption about the uniqueness of POC in Propositions 2.3, 2.4, 2.6, and 2.11 cannot be removed.

Example 2.12. Let (see [21]). It is obvious that , the pair is occasionally weakly -biased, but it is not weakly -biased. Also, is occasionally weakly compatible, but it is not weakly compatible. However, the pair has not the unique POC.

Acknowledgments

The authors are very grateful to the anonymous referees for their valuable comments and suggestions. The second author is thankful to the Ministry of Science and Technological Development of Serbia.