#### Abstract

The aim of the present paper is to obtain common fixed point theorems by employing the recently introduced notion of weak reciprocal continuity. The new notion is a proper generalization of reciprocal continuity and is applicable to compatible mappings as well as noncompatible mappings. We demonstrate that weak reciprocal continuity ensures the existence of common fixed points under contractive conditions, which otherwise do not ensure the existence of fixed points. Our results generalize and extend Banach contraction principle and Meir-Keeler-type fixed point theorem.

#### 1. Introduction

In his earlier works, Pant [1, 2] introduced the notion of reciprocal continuity and obtained the first results that established a situation in which a collection of mappings has a fixed point, which is a point of discontinuity for all the mappings. These papers are the genesis of a large number of papers (e.g., [3–16]) that employ or deal with reciprocal continuity to study fixed points of discontinuous mappings in various settings. Imdad and Ali [4] used this concept in the setting of non-self-mappings. Singh et al. [9, 10] have obtained applications of reciprocal continuity for hybrid pair of mappings. Balasubramaniam et al. [14] (see also [15]) extended the study of reciprocal continuity to fuzzy metric spaces. Kumar and Pant [16] studied this concept in the setting of probabilistic metric space. Muralisankar and Kalpana [11] established a common fixed point theorem in an intuitionistic fuzzy metric space using contractive condition of integral type.

In 1986, Jungck [17] generalized the notion of weakly commuting maps by introducing the concept of compatible maps.

*Definition 1.1. *Two self-maps and of a metric space are called compatible [17] if , whenever is a sequence in such that for some in .

The definition of compatibility implies that the mappings and will be noncompatible if there exists a sequence in such that for some in but is either nonzero or nonexistent.

*Definition 1.2. *Two self-maps and are called pointwise -weakly commuting [1] (see also [18, 19]) on if given in there exists such that .

*Definition 1.3. *Two self-maps and are called pointwise -weakly commuting of type [20] (see also [21]) on if given in there exists such that .

*Definition 1.4. *Two self-maps and are called pointwise -weakly commuting of type [20] on if given in there exists such that .

*Definition 1.5. *A pair of self-mappings defined on a nonempty set is said to be weakly compatible [22] if the pair commutes on the set of coincidence points, that is, implies .

It is well known now that pointwise -weak commutativity and analogous notions of pointwise -weak commutativity of type or pointwise -weak commutativity of type are equivalent to commutativity at coincidence points and in the setting of metric spaces these notions are equivalent to weak compatibility. On the other hand, pointwise R-weak commutativity and analogous notions of pointwise -weak commutativity of type or are more useful in establishing common fixed point theorems since they not only imply commutativity at coincidence points but may also help in the determination of coincidence points [19, 21].

In a recent work, Al-Thagafi and Shahzad [23] generalized the notion of nontrivial weakly compatible maps by introducing the notion of occasionally weakly compatible mappings.

*Definition 1.6. *A pair of self-mappings defined on a nonempty set is said to be occasionally weakly compatible [23] (in short owc) if there exists a point in , which is a coincidence point of and at which and commute.

*Definition 1.7. *Two self-mappings and of a metric space are called conditionally commuting [24] if they commute on a nonempty subset of the set of coincidence points whenever the set of their coincidences is nonempty.

From the definition itself, it is clear that if two maps are weakly compatible or owc then they are necessarily conditionally commuting; however, the conditionally commuting mappings are not necessarily weakly compatible or owc [24].

*Definition 1.8. *Let and be two self-maps of a metric space , then is called -absorbing [25] if there exists some positive real number such that for all in . Similarly, will be called -absorbing if there exists some positive real number such that for all in .

It is well known that the absorbing maps are neither a subclass of compatible maps nor a subclass of noncompatible maps [25].

*Definition 1.9. *Two self-mappings and of a metric space are called reciprocally continuous [1, 2] if and only if and whenever is a sequence such that for some in .

If and are both continuous, then they are obviously reciprocally continuous but the converse is not true [1, 2]. The notion of reciprocal continuity is mainly applicable to compatible mapping satisfying contractive conditions [7]. To widen the scope of the study of fixed points from the class of compatible mappings satisfying contractive conditions to a wider class including compatible as well as noncompatible mappings satisfying contractive, nonexpansive, or Lipschitz-type condition Pant et al. [7] generalized the notion of reciprocal continuity by introducing the new concept of weak reciprocal continuity as follows.

*Definition 1.10. *Two self-mappings and of a metric space are called weakly reciprocally continuous [7] iff or , whenever is a sequence in *X* such that for some in .

We now give examples of compatible and weakly reciprocally continuous mappings with or without common fixed points.

*Example 1.11. *Let and be the usual metric on . Define by
Then it can be verified that and are compatible as well as weakly reciprocally continuous mappings but do not have a common fixed point.

*Example 1.12. *Let and be the usual metric on . Define by
It may be noted that and are compatible as well as weakly reciprocally continuous mappings and have infinitely many common fixed points. Examples of noncompatible weakly reciprocally continuous mappings are given on the following pages.

If and are reciprocally continuous, then they are obviously weakly reciprocally continuous but, as shown in Example 2.2 below, the converse is not true. As an application of weak reciprocal continuity we prove common fixed point theorems under contractive conditions that extend the scope of the study of common fixed point theorems from the class of compatible continuous mappings to a wider class of mappings, which also includes noncompatible and discontinuous mappings. Our results also demonstrate the usefulness of the notion of the absorbing maps in fixed point considerations.

#### 2. Main Results

Theorem 2.1. *Let and be weakly reciprocally continuous pointwise -weakly commuting of type self-mappings of a complete metric space such that*(i)*,
*(ii)*.
* * If is -absorbing or is -absorbing, then and have a unique common fixed point.*

*Proof. *Let be any point in . Define sequences and in by
We claim that is a Cauchy sequence. Using (ii), we obtain
Moreover, for every integer , we get
This means that as . Therefore, is a Cauchy sequence. Since is complete, there exists a point in such that . Moreover, .

Suppose that is -absorbing. Now, weak reciprocal continuity of and implies that or . Let . By virtue of (2.1), this also yields . Since is -absorbing, . On letting , we obtain . Using (ii), we get . On making we get Hence . Since , there exists in such that . Now using (ii), we obtain . On letting , we get . Since and are pointwise -weak commutative of type , we have for some , that is, . Thus . Finally using (ii), we obtain , that is, . Hence and is a common fixed point of and .

Next suppose that . Since is -absorbing, . On letting , we get Since , there exists in such that . Now using (ii), we obtain . On letting , we get . Thus Since and are pointwise -weak commutative of type , we have for some , that is, . Thus . Finally using (ii), we obtain , that is, . Hence and is a common fixed point of and .

Finally suppose that is -absorbing. Now, weak reciprocal continuity of and implies that or . Let us first assume that . Since is -absorbing, . On making , we get Using (ii) we get . On letting , we get Hence and is a common fixed point of and .

Next suppose that . Then implies that for some and . By virtue of (2.1), this also yields . Since is -absorbing, . On letting , we get . Now, using (ii), we get . On making , we obtain . Again, by virtue of (ii), . Making we get Hence . Since and are pointwise -weak commutative of type , we have for some , that is, . Thus . Finally using (ii), we obtain , that is, . Hence and is a common fixed point of and .

Uniqueness of the common fixed point theorem follows easily in each of the two cases.

We now give an example to illustrate the above theorem.

*Example 2.2. *Let and be the usual metric on . Define as follows:

Then and satisfy all the conditions of Theorem 2.1 and have a unique common fixed point at . It can be verified in this example that and satisfy the contraction condition (ii) for . The mappings and are pointwise -weakly commuting of type maps as they commute at their only coincidence point . Furthermore, is -absorbing with . It can also be noted that and are weakly reciprocally continuous. To see this, let be a sequence in such that for some . Then and either for each from some place onwards or , where as . If for each from some place onwards, and . If , then , and . Thus but . Hence and are weakly reciprocally continuous. It is also obvious that and are not reciprocally continuous mappings.

*Remark 2.3. * Putting equal to identity map, we get the famous Banach fixed point theorem as a particular case of the above theorem.

We now establish a common fixed point theorem for a pair of mappings satisfying an type contractive condition. It is now well known (e.g., Example 2.4 below) that an contractive condition does not ensure the existence of a fixed point.

*Example 2.4 (see [26]). *Let and be the Euclidean metric on . Define by

Then satisfies the contractive condition
with for and for but does not have a fixed point.

In view of the above example, the next theorem demonstrates the usefulness of weak reciprocal continuity and shows that the new notion ensures the existence of a common fixed point under an contractive condition.

Theorem 2.5. *Let and be weakly reciprocally continuous pointwise R-weakly commuting of type self-mappings of a complete metric space such that*(i)*;*(ii)* whenever ;*(iii)*given there exists such that
**If is -absorbing or is -absorbing, then and have a unique common fixed point.*

*Proof. *Let be any point in . Define sequences and in by
We claim that is a Cauchy sequence. Using (ii), we obtain
Thus is a strictly decreasing sequence of positive real numbers and, therefore, tends to a limit , that is, . We assert that . For, if not, suppose that . Then given , no matter small may be, there exists a positive integer such that for each , we have
that is,
Selecting in (2.11) in accordance with (iii), for each , we get , that is, , a contradiction to (2.11). Therefore, . We now show that is a Cauchy sequence. Suppose it is not. Then there exist an and a subsequence of such that . Select in (iii) so that . Since , there exists an integer such that whenever .

Let . Then, there exist integers satisfying such that . If not, then
a contradiction. Let be the smallest integer such that . Then and
that is, . In view of (iii), this yields . But then
which contradicts (2.13). Hence is a Cauchy sequence. Since is complete, there exists a point in such that . Moreover, .

Suppose that is -absorbing. Now, weak reciprocal continuity of and implies that or . Let . By virtue of (2.8), this also yields . Since is -absorbing, . On letting , we get . Using (ii), we get . On making , we get Hence . Since , there exists in such that . Now using (ii), we obtain . On letting , we get . Thus . Since and are pointwise -weak commutative of type , we have for some , that is, . Thus . If , then using (ii) we get , a contradiction. Hence and is a common fixed point of and .

Next suppose that . Since is -absorbing, . On letting , we get . Since , there exists in such that . Now using (ii), we obtain . On letting , we get . Thus . Since and are pointwise -weak commutative of type , we have for some , that is, . Thus . If then using (ii) we get , a contradiction. Hence and is a common fixed point of and .

When is assumed -absorbing, the proof follows on similar lines as in the corresponding part of Theorem 2.1.

We now give an example to illustrate Theorem 2.5.

*Example 2.6. *Let and be the usual metric on . Define as follows:
Then and satisfy all the conditions of Theorem 2.5 and have a unique common fixed point at . It can be seen in this example that and satisfy the condition (ii) and the condition
with for and for . Furthermore, is -absorbing with . It can also be noted that and are weakly reciprocally continuous. To see this, let be a sequence in such that for some . Then and either for each from some place onwards or where as . If for each from some place onwards, and . If , then , and 2. Thus but . Hence and are weakly reciprocally continuous. It is also obvious that and are not reciprocally continuous mappings. Further, and are pointwise -weakly commuting of type maps as they commute at their only coincidence point .

*Remark 2.7. *Theorem 2.5 generalizes the well-known fixed point theorem of Meir and Keeler [27].

It may be observed that the mappings and in Examples 2.2 and 2.6 are noncompatible mappings. However, in the case of noncompatible mappings there is an alternative method of proving the existence of fixed points [6, 7, 11, 19, 24, 26, 28–32]. This alternative method was introduced by Pant [19, 26, 28–30] and is also applicable under strictly contractive [19, 26, 31–33], nonexpansive [7], and Lipschitz-type conditions [6, 24, 30]. The existence of such a method is important since there is no general method for studying the fixed points of nonexpansive or Lipschitz-type mapping pairs in ordinary metric spaces.

In the area of fixed point theory, Lipschitz type mappings constitute a very important class of mappings and include contraction mappings, contractive mappings and, nonexpansive mappings as subclasses. The next theorem provides a good illustration of the applicability of recently introduced notions of conditional commutativity and weak reciprocal continuity to establish a situation in which a pair of mappings may possess common fixed points as well as coincidence points, which may not be common fixed points.

Theorem 2.8. *Let and be weakly reciprocally continuous noncompatible self-mappings of a metric space satisfying*(i)*,*(ii)*.
**If and are conditionally commuting and is -absorbing or is -absorbing, then and have a common fixed point.*

*Proof. *Since and are noncompatible maps, there exists a sequence in such that and for some in but either or the limit does not exist. Since , for each there exists in such that . Thus and as . By virtue of this and using (ii), we obtain . Therefore, we have
Suppose that is -absorbing. Then and . On letting , these inequalities yield
Weak reciprocal continuity of and implies that or . Let . By virtue of (ii), we get . On letting , we get In view of (2.18), this yields . Since , there exists in such that . Now using (ii), we obtain . On letting , we get . Thus . Conditional commutativity of and implies that and commute at , or there exists a coincidence point of and at which and commute. Suppose and commute at the coincidence point . Then and . Also . Since is -absorbing . This yields . Hence and is a common fixed point of and .

Next suppose that . In view of (2.18), we get . Since , there exists in such that . Now using (ii), we obtain . On letting , we get . Thus . This, in view of conditional commutativity and -absorbing property of , implies that and have a common fixed point.

Now suppose that is -absorbing. Then and . On letting , these inequalities yield
Weak reciprocal continuity of and implies that or . Let us first assume that . In view of (2.19), this yields . Using (ii) we get . On letting , we obtain . Hence and is a common fixed point of and .

Next suppose that . Then implies that for some . Therefore, . Using (ii) and in view of (2.19), we get . On letting , we get . Again, by virtue of (ii), we obtain . Making , we get Hence . Conditional commutativity of and implies that and commute at or there exists a coincidence point of and at which and commute. Suppose and commute at the coincidence point . Then and . Also . Since is -absorbing, . This yields . Hence and is a common fixed point of and . This completes the proof of the theorem.

We now give examples to illustrate Theorem 2.8.

*Example 2.9. *Let and be the usual metric on . Define by ,

Then and satisfy all the conditions of the above theorem and have two coincidence points and a common fixed point . It may be verified in this example that and . Also that and are noncompatible but conditionally commuting maps. Furthermore, and are conditionally commuting since they commute at their coincidence point . To see that and are noncompatible, let us consider the sequence given by . Then , and . Hence and are noncompatible. It may also be verified that and are not pointwise *R*-weakly commuting of type as they do not commute at the coincidence point , since and . It is also easy to verify that and satisfy the Lipschitz-type condition together with -absorbing condition for all . It can also be noted that and are weakly reciprocally continuous since both and are continuous.

In Example 2.9, and are not pointwise -weakly commuting of type as they do not commute at the coincidence point . We now give an example of pointwise -weakly commuting of type maps satisfying Theorem 2.8.

*Example 2.10. *Let and be the usual metric on . Define as follows:
Then and satisfy all the conditions of the above theorem and have three coincidence points and two common fixed point . It may be verified in this example that and . Also, and are pointwise -weakly commuting of type maps, hence also conditionally commuting, since they commute at each of their coincidence points, namely, . To see that and are noncompatible, let us consider the sequence given by . Then , and . Hence and are noncompatible. It is also easy to verify that and satisfy the Lipschitz-type condition . The mapping is -absorbing since for all . It can also be noted that and are weakly reciprocally continuous. To see this, let be a sequence in such that for some . Then and either for each or . If for each , then , and . If , then , and . Thus but . Hence and are weakly reciprocally continuous.

Putting in Theorem 2.8, we get a common fixed point theorem for a non-expansive-type mapping pair.

Corollary 2.11. *Let and be weakly reciprocally continuous noncompatible self-mappings of a metric space satisfying*(i)*,*(ii)*.
**If and are conditionally commuting and is -absorbing or is -absorbing, then and have a common fixed point.*

#### Acknowledgment

The author is thankful to the learned referee for his deep observations and pertinent suggestions, which improved the exposition of the paper.