Common Fixed Points of Generalized Meir-Keeler Type Condition and Nonexpansive Mappings

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 𝑓 and 𝑔 of a metric space (𝑋,𝑑) are called compatible [17] if lim_𝑛𝑑(𝑓𝑔𝑥_𝑛,𝑔𝑓𝑥_𝑛)=0, whenever {𝑥_𝑛} is a sequence in 𝑋 such that lim_𝑛𝑓𝑥_𝑛=lim_𝑛𝑔𝑥_𝑛=𝑡 for some 𝑡 in 𝑋.
The definition of compatibility implies that the mappings 𝑓 and 𝑔 will be noncompatible if there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛𝑓𝑥_𝑛=lim_𝑛𝑔𝑥_𝑛=𝑡 for some 𝑡 in 𝑋 but lim_𝑛𝑑(𝑓𝑔𝑥_𝑛,𝑔𝑓𝑥_𝑛) is either nonzero or nonexistent.

Definition 1.2. Two self-maps 𝑓 and 𝑔 are called pointwise 𝑅-weakly commuting [1] (see also [18, 19]) on 𝑋 if given 𝑥 in 𝑋 there exists 𝑅>0 such that 𝑑(𝑓𝑔𝑥,𝑔𝑓𝑥)≤𝑅𝑑(𝑓𝑥,𝑔𝑥).

Definition 1.3. Two self-maps 𝑓 and 𝑔 are called pointwise 𝑅-weakly commuting of type (𝐴_𝑓) [20] (see also [21]) on 𝑋 if given 𝑥 in 𝑋 there exists 𝑅>0 such that 𝑑(𝑓𝑔𝑥,𝑔𝑔𝑥)≤𝑅𝑑(𝑓𝑥,𝑔𝑥).

Definition 1.4. Two self-maps 𝑓 and 𝑔 are called pointwise 𝑅-weakly commuting of type (𝐴_𝑔) [20] on 𝑋 if given 𝑥 in 𝑋 there exists 𝑅>0 such that 𝑑(𝑓𝑓𝑥,𝑔𝑓𝑥)≤𝑅𝑑(𝑓𝑥,𝑔𝑥).

Definition 1.5. A pair (𝑓,𝑔) of self-mappings defined on a nonempty set 𝑋 is said to be weakly compatible [22] if the pair commutes on the set of coincidence points, that is, 𝑓𝑥=𝑔𝑥(𝑥∈𝑋) implies 𝑓𝑔𝑥=𝑔𝑓𝑥.

It is well known now that pointwise 𝑅-weak commutativity and analogous notions of pointwise 𝑅-weak commutativity of type (𝐴_𝑓) or pointwise 𝑅-weak commutativity of type (𝐴_𝑔) 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 𝑅-weak commutativity of type (𝐴_𝑓) or (𝐴_𝑔) 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 (𝑓,𝑔) of self-mappings defined on a nonempty set 𝑋 is said to be occasionally weakly compatible [23] (in short owc) if there exists a point 𝑥 in 𝑋, which is a coincidence point of 𝑓 and 𝑔 at which 𝑓 and 𝑔 commute.

Definition 1.7. Two self-mappings 𝑓 and 𝑔 of a metric space (𝑋,𝑑) 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 𝑓 and 𝑔(𝑓≠𝑔) be two self-maps of a metric space (𝑋,𝑑), then 𝑓 is called 𝑔-absorbing [25] if there exists some positive real number 𝑅 such that 𝑑(𝑔𝑥,𝑔𝑓𝑥)≤𝑅𝑑(𝑓𝑥,𝑔𝑥) for all 𝑥 in 𝑋. Similarly, 𝑔 will be called 𝑓-absorbing if there exists some positive real number 𝑅 such that 𝑑(𝑓𝑥,𝑓𝑔𝑥)≤𝑅𝑑(𝑓𝑥,𝑔𝑥) for all 𝑥 in 𝑋.
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 𝑓 and 𝑔 of a metric space (𝑋,𝑑) are called reciprocally continuous [1, 2] if and only if 𝑓𝑔𝑥_𝑛→𝑓𝑡 and 𝑔𝑓𝑥_𝑛→𝑔𝑡 whenever {𝑥_𝑛} is a sequence such that lim_𝑛𝑓𝑥_𝑛=lim_𝑛𝑔𝑥_𝑛=𝑡 for some 𝑡 in 𝑋.

If 𝑓 and 𝑔 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 𝑓 and 𝑔 of a metric space (𝑋,𝑑) are called weakly reciprocally continuous [7] iff 𝑓𝑔𝑥_𝑛→𝑓𝑡 or 𝑔𝑓𝑥_𝑛→𝑔𝑡, whenever {𝑥_𝑛} is a sequence in X such that lim_𝑛𝑓𝑥_𝑛=lim_𝑛𝑔𝑥_𝑛=𝑡 for some 𝑡 in 𝑋.

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

Example 1.11. Let 𝑋=[0,1] and 𝑑 be the usual metric on 𝑋. Define 𝑓,𝑔∶𝑋→𝑋 by 𝑥𝑓𝑥=𝑥,∀𝑥,𝑔𝑥=2if𝑥>0,𝑔(0)=1. (1.1) Then it can be verified that 𝑓 and 𝑔 are compatible as well as weakly reciprocally continuous mappings but do not have a common fixed point.

Example 1.12. Let 𝑋=[0,1] and 𝑑 be the usual metric on 𝑋. Define 𝑓,𝑔∶𝑋→𝑋 by 𝑓𝑥=(1−𝑥),∀𝑥,𝑔𝑥=fractionalpartof(1−𝑥). (1.2) It may be noted that 𝑓 and 𝑔 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 𝑓 and 𝑔 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 𝑓and 𝑔 be weakly reciprocally continuous pointwise 𝑅-weakly commuting of type (𝐴_𝑓) self-mappings of a complete metric space (𝑋,𝑑) such that(i)𝑓𝑋⊆𝑔𝑋, (ii)𝑑(𝑓𝑥,𝑓𝑦)≤𝑘𝑑(𝑔𝑥,𝑔𝑦),𝑘∈[0,1).   If 𝑔 is 𝑓-absorbing or 𝑓 is 𝑔-absorbing, then 𝑓 and 𝑔 have a unique common fixed point.

Proof. Let 𝑥₀ be any point in 𝑋. Define sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 by 𝑦_𝑛=𝑓𝑥_𝑛=𝑔𝑥_𝑛+1. (2.1) We claim that {𝑦_𝑛} is a Cauchy sequence. Using (ii), we obtain 𝑑𝑦_𝑛,𝑦_𝑛+1=𝑑𝑓𝑥_𝑛,𝑓𝑥_𝑛+1≤𝑘𝑑𝑔𝑥_𝑛,𝑔𝑥_𝑛+1=𝑘𝑑𝑦_𝑛−1,𝑦_𝑛≤⋯≤𝑘^𝑛𝑑𝑦₀,𝑦₁. (2.2) Moreover, for every integer 𝑝>0, we get 𝑑𝑦_𝑛,𝑦_𝑛+𝑝≤𝑑𝑦_𝑛,𝑦_𝑛+1+𝑑𝑦_𝑛+1,𝑦_𝑛+2+⋯+𝑑𝑦_{𝑛+𝑝−1},𝑦_𝑛+𝑝≤𝑑𝑦_𝑛,𝑦_𝑛+1+𝑘𝑑𝑦_𝑛,𝑦_𝑛+1+⋯+𝑘^𝑝−1𝑑𝑦_𝑛,𝑦_𝑛+1=1+𝑘+𝑘²+⋯+𝑘^𝑝−1𝑑𝑦_𝑛,𝑦_𝑛+1≤1𝑑𝑦1−𝑘^𝑛,𝑦_𝑛+1≤𝑘^𝑛𝑑𝑦1−𝑘⁰,𝑦₁. (2.3) This means that 𝑑(𝑦_𝑛,𝑦_𝑛+𝑝)→0 as 𝑛→∞. Therefore, {𝑦_𝑛} is a Cauchy sequence. Since X is complete, there exists a point 𝑡 in 𝑋 such that 𝑦_𝑛→𝑡. Moreover, 𝑦_𝑛=𝑓𝑥_𝑛=𝑔𝑥_𝑛+1→𝑡.
Suppose that 𝑔 is 𝑓-absorbing. Now, weak reciprocal continuity of 𝑓 and 𝑔 implies that 𝑓𝑔𝑥_𝑛→𝑓𝑡 or 𝑔𝑓𝑥_𝑛→𝑔𝑡. Let 𝑔𝑓𝑥_𝑛→𝑔𝑡. By virtue of (2.1), this also yields 𝑔𝑔𝑥_𝑛+1=𝑔𝑓𝑥_𝑛→𝑔𝑡. Since 𝑔 is 𝑓-absorbing, 𝑑(𝑓𝑥_𝑛,𝑓𝑔𝑥_𝑛)≤𝑅𝑑(𝑓𝑥_𝑛,𝑔𝑥_𝑛). On letting 𝑛→∞, we obtain 𝑓𝑔𝑥_𝑛→𝑡. Using (ii), we get 𝑑(𝑓𝑡,𝑓𝑔𝑥_𝑛)≤𝑘𝑑(𝑔𝑡,𝑔𝑔𝑥_𝑛). On making 𝑛→∞ we get 𝑓𝑔𝑥_𝑛→𝑓𝑡. Hence 𝑡=𝑓𝑡. Since 𝑓𝑋⊆𝑔𝑋, there exists 𝑢 in 𝑋 such that 𝑡=𝑓𝑡=𝑔𝑢. Now using (ii), we obtain 𝑑(𝑓𝑥_𝑛,𝑓𝑢)≤𝑘𝑑(𝑔𝑥_𝑛,𝑔𝑢). On letting 𝑛→∞, we get 𝑓𝑢=𝑔𝑢. Since 𝑓 and 𝑔 are pointwise 𝑅-weak commutative of type (𝐴_𝑓), we have 𝑑(𝑓𝑔𝑢,𝑔𝑔𝑢)≤𝑅₁𝑑(𝑓𝑢,𝑔𝑢)=0 for some 𝑅₁>0, that is, 𝑓𝑔𝑢=𝑔𝑔𝑢. Thus 𝑓𝑔𝑢=𝑔𝑓𝑢=𝑔𝑔𝑢=𝑓𝑓𝑢. Finally using (ii), we obtain 𝑑(𝑓𝑢,𝑓𝑓𝑢)≤𝑘𝑑(𝑔𝑢,𝑔𝑓𝑢)=𝑘𝑑(𝑓𝑢,𝑓𝑓𝑢), that is, (1−𝑘)𝑑(𝑓𝑢,𝑓𝑓𝑢)=0. Hence 𝑓𝑢=𝑓𝑓𝑢=𝑔𝑓𝑢 and 𝑓𝑢 is a common fixed point of 𝑓 and 𝑔.
Next suppose that 𝑓𝑔𝑥_𝑛→𝑓𝑡. Since 𝑔 is 𝑓-absorbing, 𝑑(𝑓𝑥_𝑛,𝑓𝑔𝑥_𝑛)≤𝑅𝑑(𝑓𝑥_𝑛,𝑔𝑥_𝑛). On letting 𝑛→∞, we get 𝑡=𝑓𝑡. Since 𝑓𝑋⊆𝑔𝑋, there exists 𝑢 in 𝑋 such that 𝑡=𝑓𝑡=𝑔𝑢. Now using (ii), we obtain 𝑑(𝑓𝑥_𝑛,𝑓𝑢)≤𝑘𝑑(𝑔𝑥_𝑛,𝑔𝑢). On letting 𝑛→∞, we get 𝑓𝑢=𝑡. Thus 𝑓𝑢=𝑔𝑢. Since 𝑓 and 𝑔 are pointwise 𝑅-weak commutative of type (𝐴_𝑓), we have 𝑑(𝑓𝑔𝑢,𝑔𝑔𝑢)≤𝑅₁𝑑(𝑓𝑢,𝑔𝑢)=0 for some 𝑅₁>0, that is, 𝑓𝑔𝑢=𝑔𝑔𝑢. Thus 𝑓𝑔𝑢=𝑔𝑓𝑢=𝑔𝑔𝑢=𝑓𝑓𝑢. Finally using (ii), we obtain 𝑑(𝑓𝑢,𝑓𝑓𝑢)≤𝑘𝑑(𝑔𝑢,𝑔𝑓𝑢)=𝑘𝑑(𝑓𝑢,𝑓𝑓𝑢), that is, (1−𝑘)𝑑(𝑓𝑢,𝑓𝑓𝑢)=0. Hence 𝑓𝑢=𝑓𝑓𝑢=𝑔𝑓𝑢 and 𝑓𝑢 is a common fixed point of 𝑓 and 𝑔.
Finally suppose that 𝑓 is 𝑔-absorbing. Now, weak reciprocal continuity of 𝑓 and 𝑔 implies that 𝑓𝑔𝑥_𝑛→𝑓𝑡 or 𝑔𝑓𝑥_𝑛→𝑔𝑡. Let us first assume that 𝑔𝑓𝑥_𝑛→𝑔𝑡. Since 𝑓 is 𝑔-absorbing, 𝑑(𝑔𝑥_𝑛,𝑔𝑓𝑥_𝑛)≤𝑅𝑑(𝑓𝑥_𝑛,𝑔𝑥_𝑛). On making 𝑛→∞, we get 𝑡=𝑔𝑡. Using (ii) we get 𝑑(𝑓𝑥_𝑛,𝑓𝑡)≤𝑘𝑑(𝑔𝑥_𝑛,𝑔𝑡). On letting 𝑛→∞, we get 𝑓𝑥_𝑛→𝑓𝑡. Hence 𝑡=𝑓𝑡=𝑔𝑡 and 𝑡 is a common fixed point of 𝑓 and 𝑔.
Next suppose that 𝑓𝑔𝑥_𝑛→𝑓𝑡. Then 𝑓𝑋⊆𝑔𝑋 implies that 𝑓𝑡=𝑔𝑢 for some 𝑢∈𝑋 and 𝑓𝑔𝑥_𝑛→𝑔𝑢. By virtue of (2.1), this also yields 𝑓𝑓𝑥_𝑛−1→𝑔𝑢. Since 𝑓 is 𝑔-absorbing, 𝑑(𝑔𝑥_𝑛,𝑔𝑓𝑥_𝑛)≤𝑅𝑑(𝑓𝑥_𝑛,𝑔𝑥_𝑛). On letting 𝑛→∞, we get 𝑔𝑓𝑥_𝑛→𝑡. Now, using (ii), we get 𝑑(𝑓𝑥_𝑛,𝑓𝑓𝑥_𝑛)≤𝑘𝑑(𝑔𝑥_𝑛,𝑔𝑓𝑥_𝑛). On making 𝑛→∞, we obtain 𝑡=𝑔𝑢. Again, by virtue of (ii), 𝑑(𝑓𝑥_𝑛,𝑓𝑢)≤𝑘𝑑(𝑔𝑥_𝑛,𝑔𝑢). Making 𝑛→∞ we get 𝑓𝑢=𝑡. Hence 𝑡=𝑓𝑢=𝑔𝑢. Since 𝑓 and 𝑔 are pointwise 𝑅-weak commutative of type (𝐴_𝑓), we have 𝑑(𝑓𝑔𝑢,𝑔𝑔𝑢)≤𝑅₁𝑑(𝑓𝑢,𝑔𝑢)=0 for some 𝑅₁>0, that is, 𝑓𝑔𝑢=𝑔𝑔𝑢. Thus 𝑓𝑔𝑢=𝑔𝑓𝑢=𝑔𝑔𝑢=𝑓𝑓𝑢. Finally using (ii), we obtain 𝑑(𝑓𝑢,𝑓𝑓𝑢)≤𝑘𝑑(𝑔𝑢,𝑔𝑓𝑢)=𝑘𝑑(𝑓𝑢,𝑓𝑓𝑢), that is, (1−𝑘)𝑑(𝑓𝑢,𝑓𝑓𝑢)=0. Hence 𝑓𝑢=𝑓𝑓𝑢=𝑔𝑓𝑢 and 𝑓𝑢 is a common fixed point of 𝑓 and 𝑔.

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 𝑋=[2,20] and 𝑑 be the usual metric on 𝑋. Define 𝑓,𝑔∶𝑋→𝑋 as follows: ()𝑓𝑥=2if𝑥=2or𝑥>5,𝑓𝑥=6if2<𝑥≤5,𝑔2=2,𝑔𝑥=12if2<𝑥≤5,𝑔𝑥=𝑥+13if𝑥>5. (2.4)
Then 𝑓 and 𝑔 satisfy all the conditions of Theorem 2.1 and have a unique common fixed point at 𝑥=2. It can be verified in this example that 𝑓 and 𝑔 satisfy the contraction condition (ii) for 𝑘=4/5. The mappings 𝑓 and 𝑔 are pointwise 𝑅-weakly commuting of type (𝐴_𝑓) maps as they commute at their only coincidence point 𝑥=2. Furthermore, 𝑓 is 𝑔-absorbing with 𝑅=29/18. It can also be noted that 𝑓 and 𝑔 are weakly reciprocally continuous. To see this, let {𝑥_𝑛} be a sequence in 𝑋 such that 𝑓𝑥_𝑛→𝑡,𝑔𝑥_𝑛→𝑡 for some 𝑡. Then 𝑡=2 and either 𝑥_𝑛=2 for each 𝑛 from some place onwards or 𝑥_𝑛=5+𝜀_𝑛, where 𝜀_𝑛→0 as 𝑛→∞. If 𝑥_𝑛=2 for each 𝑛 from some place onwards, 𝑓𝑔𝑥_𝑛→2=𝑓2 and 𝑔𝑓𝑥_𝑛→2=𝑔2. If 𝑥_𝑛=5+𝜀_𝑛, then 𝑓𝑥_𝑛→2,𝑔𝑥_𝑛=(2+∈_𝑛/3)→2,𝑓𝑔𝑥_𝑛=𝑓(2+∈_𝑛/3)→6≠𝑓2, and 𝑔𝑓𝑥_𝑛→𝑔2=2. Thus lim_𝑛→∞𝑔𝑓𝑥_𝑛=𝑔2 but lim_𝑛→∞𝑓𝑔𝑥_𝑛≠𝑓2. Hence 𝑓 and 𝑔 are weakly reciprocally continuous. It is also obvious that 𝑓 and 𝑔 are not reciprocally continuous mappings.

Remark 2.3. Putting 𝑔 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 (∈,𝛿) type contractive condition. It is now well known (e.g., Example 2.4 below) that an (∈,𝛿) contractive condition does not ensure the existence of a fixed point.

Example 2.4 (see [26]). Let 𝑋=[0,2] and 𝑑 be the Euclidean metric on 𝑋. Define 𝑓∶𝑋→𝑋 by ()𝑓𝑥=1+𝑥2if𝑥<1,𝑓𝑥=0if𝑥≥1. (2.5)
Then 𝑓 satisfies the contractive condition 𝜀≤max𝑑𝑥,𝑦,𝑑𝑥,𝑓𝑥,𝑑𝑦,𝑓𝑦<𝜀+𝛿⟹𝑑𝑓𝑥,𝑓𝑦<𝜀 (2.6) with 𝛿(𝜀)=1 for 𝜀≥1 and 𝛿(𝜀)=1−𝜀 for 𝜀<1 but 𝑓 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 (𝜀,𝛿) contractive condition.

Theorem 2.5. Let 𝑓 and 𝑔 be weakly reciprocally continuous pointwise R-weakly commuting of type (𝐴_𝑓) self-mappings of a complete metric space (𝑋,𝑑) such that(i)𝑓𝑋⊆𝑔𝑋;(ii)𝑑(𝑓𝑥,𝑓𝑦)<𝑑(𝑔𝑥,𝑔𝑦) whenever 𝑔𝑥≠𝑔𝑦;(iii)given 𝜀>0 there exists 𝛿>0 such that 𝜀<𝑑𝑔𝑥,𝑔𝑦<𝜀+𝛿⟹𝑑𝑓𝑥,𝑓𝑦≤𝜀. (2.7)If 𝑔 is 𝑓-absorbing or 𝑓 is 𝑔-absorbing, then 𝑓 and 𝑔 have a unique common fixed point.

Proof. Let 𝑥₀ be any point in 𝑋. Define sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 by 𝑦_𝑛=𝑓𝑥_𝑛=𝑔𝑥_𝑛+1. (2.8) We claim that {𝑦_𝑛} is a Cauchy sequence. Using (ii), we obtain 𝑑𝑦_𝑛,𝑦_𝑛+1=𝑑𝑓𝑥_𝑛,𝑓𝑥_𝑛+1<𝑑𝑔𝑥_𝑛,𝑔𝑥_𝑛+1=𝑑𝑦_𝑛−1,𝑦_𝑛. (2.9) Thus {𝑑(𝑦_𝑛,𝑦_𝑛+1)} is a strictly decreasing sequence of positive real numbers and, therefore, tends to a limit 𝑟≥0, that is, lim_𝑛→∞𝑑(𝑦_𝑛,𝑦_𝑛+1)=𝑟,𝑟≥0. We assert that 𝑟=0. For, if not, suppose that 𝑟>0. Then given 𝛿>0, no matter small 𝛿 may be, there exists a positive integer 𝑁 such that for each 𝑛≥𝑁, we have 𝑟<𝑑𝑦_𝑛,𝑦_𝑛+1=𝑑𝑓𝑥_𝑛,𝑓𝑥_𝑛+1<𝑟+𝛿, (2.10) that is, 𝑟<𝑑𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+2<𝑟+𝛿. (2.11) Selecting 𝛿 in (2.11) in accordance with (iii), for each 𝑛≥𝑁, we get 𝑑(𝑓𝑥_𝑛+1,𝑓𝑥_𝑛+2)≤𝑟, that is, 𝑑(𝑦_𝑛+1,𝑦_𝑛+2)≤𝑟, a contradiction to (2.11). Therefore, lim_𝑛→∞𝑑(𝑦_𝑛,𝑦_𝑛+1)=0. We now show that {𝑦_𝑛} is a Cauchy sequence. Suppose it is not. Then there exist an 𝜀>0 and a subsequence {𝑦_𝑛𝑖} of {𝑦_𝑛} such that 𝑑(𝑦_𝑛𝑖,𝑦_𝑛𝑖+1)≥2𝜀. Select 𝛿 in (iii) so that 0<𝛿≤𝜀. Since lim_𝑛→∞𝑑(𝑦_𝑛,𝑦_𝑛+1)=0, there exists an integer 𝑁 such that 𝑑(𝑦_𝑛,𝑦_𝑛+1)<𝛿/6 whenever 𝑛≥𝑁.
Let 𝑛_𝑖≥𝑁. Then, there exist integers 𝑚_𝑖 satisfying 𝑛_𝑖<𝑚_𝑖<𝑛_𝑖+1 such that 𝑑(𝑦_𝑛𝑖,𝑦_𝑚𝑖)≥𝜀+(𝛿/3). If not, then 𝑑𝑦_𝑛𝑖,𝑦_𝑛𝑖+1≤𝑑𝑦_𝑛𝑖,𝑦_{𝑛𝑖+1−1}+𝑑𝑦_{𝑛𝑖+1−1},𝑦_𝑛𝑖+1𝛿<𝜀+3+𝛿6<2𝜀, (2.12) a contradiction. Let 𝑚_𝑖 be the smallest integer such that 𝑑(𝑦_𝑛𝑖,𝑦_𝑚𝑖)≥𝜀+(𝛿/3). Then 𝑑(𝑦_𝑛𝑖,𝑦_𝑚𝑖−2)<𝜀+(𝛿/3) and 𝛿𝜀+3≤𝑑𝑦_𝑛𝑖,𝑦_𝑚𝑖≤𝑑𝑦_𝑛𝑖,𝑦_𝑚𝑖−2+𝑑𝑦_𝑚𝑖−2,𝑦_𝑚𝑖−1+𝑑𝑦_𝑚𝑖−1,𝑦_𝑚𝑖𝛿<𝜀+3+𝛿6+𝛿6<𝜀+2𝛿3, (2.13) that is, 𝜀<𝜀+(𝛿/3)≤𝑑(𝑔𝑥_𝑛𝑖+1,𝑔𝑥_𝑚𝑖+1)<𝜀+(2/3)𝛿. In view of (iii), this yields 𝑑(𝑦_𝑛𝑖+1,𝑦_𝑚𝑖+1)≤𝜀. But then 𝑑𝑦_𝑛𝑖,𝑦_𝑚𝑖≤𝑑𝑦_𝑛𝑖,𝑦_𝑛𝑖+1+𝑑𝑦_𝑛𝑖+1,𝑦_𝑚𝑖+1+𝑑𝑦_𝑚𝑖+1,𝑦_𝑚𝑖<𝛿6𝛿+𝜀+6𝛿=𝜀+3, (2.14) which contradicts (2.13). Hence {𝑦_𝑛} is a Cauchy sequence. Since 𝑋 is complete, there exists a point 𝑡 in 𝑋 such that 𝑦_𝑛→𝑡. Moreover, 𝑦_𝑛=𝑓𝑥_𝑛=𝑔𝑥_𝑛+1→𝑡.
Suppose that 𝑔 is 𝑓-absorbing. Now, weak reciprocal continuity of 𝑓 and 𝑔 implies that 𝑓𝑔𝑥_𝑛→𝑓𝑡 or 𝑔𝑓𝑥_𝑛→𝑔𝑡. Let 𝑔𝑓𝑥_𝑛→𝑔𝑡. By virtue of (2.8), this also yields 𝑔𝑔𝑥_𝑛+1=𝑔𝑓𝑥_𝑛→𝑔𝑡. Since 𝑔 is 𝑓-absorbing, 𝑑(𝑓𝑥_𝑛,𝑓𝑔𝑥_𝑛)≤𝑅𝑑(𝑓𝑥_𝑛,𝑔𝑥_𝑛). On letting 𝑛→∞, we get 𝑓𝑔𝑥_𝑛→𝑡. Using (ii), we get 𝑑(𝑓𝑡,𝑓𝑔𝑥_𝑛)<𝑑(𝑔𝑡,𝑔𝑔𝑥_𝑛). On making 𝑛→∞, we get 𝑓𝑔𝑥_𝑛→𝑓𝑡. Hence 𝑡=𝑓𝑡. Since 𝑓𝑋⊆𝑔𝑋, there exists 𝑢 in 𝑋 such that 𝑡=𝑓𝑡=𝑔𝑢. Now using (ii), we obtain 𝑑(𝑓𝑥_𝑛,𝑓𝑢)<𝑑(𝑔𝑥_𝑛,𝑔𝑢). On letting 𝑛→∞, we get 𝑓𝑢=𝑡. Thus 𝑓𝑢=𝑔𝑢. Since 𝑓 and 𝑔 are pointwise 𝑅-weak commutative of type (𝐴_𝑓), we have 𝑑(𝑓𝑔𝑢,𝑔𝑔𝑢)≤𝑅₁𝑑(𝑓𝑢,𝑔𝑢)=0 for some 𝑅₁>0, that is, 𝑓𝑔𝑢=𝑔𝑔𝑢. Thus 𝑓𝑔𝑢=𝑔𝑓𝑢=𝑔𝑔𝑢=𝑓𝑓𝑢. If 𝑓𝑢≠𝑓𝑓𝑢, then using (ii) we get 𝑑(𝑓𝑢,𝑓𝑓𝑢)<𝑑(𝑔𝑢,𝑔𝑓𝑢)=𝑑(𝑓𝑢,𝑓𝑓𝑢), a contradiction. Hence 𝑓𝑢=𝑓𝑓𝑢=𝑔𝑓𝑢 and 𝑓𝑢 is a common fixed point of 𝑓 and 𝑔.
Next suppose that 𝑓𝑔𝑥_𝑛→𝑓𝑡. Since 𝑔 is 𝑓-absorbing, 𝑑(𝑓𝑥_𝑛,𝑓𝑔𝑥_𝑛)≤𝑅𝑑(𝑓𝑥_𝑛,𝑔𝑥_𝑛). On letting 𝑛→∞, we get 𝑡=𝑓𝑡. Since 𝑓𝑋⊆𝑔𝑋, there exists 𝑢 in 𝑋 such that 𝑡=𝑓𝑡=𝑔𝑢. Now using (ii), we obtain 𝑑(𝑓𝑥_𝑛,𝑓𝑢)<𝑑(𝑔𝑥_𝑛,𝑔𝑢). On letting 𝑛→∞, we get 𝑓𝑢=𝑡. Thus 𝑓𝑢=𝑔𝑢. Since 𝑓 and 𝑔 are pointwise 𝑅-weak commutative of type (𝐴_𝑓), we have 𝑑(𝑓𝑔𝑢,𝑔𝑔𝑢)≤𝑅₁𝑑(𝑓𝑢,𝑔𝑢)=0 for some 𝑅₁>0, that is, 𝑓𝑔𝑢=𝑔𝑔𝑢. Thus 𝑓𝑔𝑢=𝑔𝑓𝑢=𝑔𝑔𝑢=𝑓𝑓𝑢. If 𝑓𝑢≠𝑓𝑓𝑢 then using (ii) we get 𝑑(𝑓𝑢,𝑓𝑓𝑢)<𝑑(𝑔𝑢,𝑔𝑓𝑢)=𝑑(𝑓𝑢,𝑓𝑓𝑢), a contradiction. Hence 𝑓𝑢=𝑓𝑓𝑢=𝑔𝑓𝑢 and 𝑓𝑢 is a common fixed point of 𝑓 and 𝑔.
When 𝑓 is assumed 𝑔-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 𝑋=[2,20] and 𝑑 be the usual metric on 𝑋. Define 𝑓,𝑔∶𝑋→𝑋 as follows: ()𝑓𝑥=2if𝑥=2or𝑥>5,𝑓𝑥=6if2<𝑥≤5,𝑔2=2,𝑔𝑥=𝑥+313()if2<𝑥≤5,𝑔𝑥=𝑥+13if𝑥>5. (2.15) Then 𝑓 and 𝑔 satisfy all the conditions of Theorem 2.5 and have a unique common fixed point at 𝑥=2. It can be seen in this example that 𝑓 and 𝑔 satisfy the condition (ii) and the condition 𝜀<𝑑𝑔𝑥,𝑔𝑦<𝜀+𝛿⟹𝑑𝑓𝑥,𝑓𝑦≤𝜀 (2.16) with 𝛿(𝜀)=1 for 𝜀≥4 and 𝛿(𝜀)=4−𝜀 for 𝜀<4. Furthermore, 𝑓 is 𝑔-absorbing with 𝑅=2. It can also be noted that 𝑓 and 𝑔 are weakly reciprocally continuous. To see this, let {𝑥_𝑛} be a sequence in 𝑋 such that 𝑓𝑥_𝑛→𝑡,𝑔𝑥_𝑛→𝑡 for some 𝑡. Then 𝑡=2 and either 𝑥_𝑛=2 for each 𝑛 from some place onwards or 𝑥_𝑛=5+𝜀_𝑛 where 𝜀_𝑛→0 as 𝑛→∞. If 𝑥_𝑛=2 for each 𝑛 from some place onwards, 𝑓𝑔𝑥_𝑛→2=𝑓2 and 𝑔𝑓𝑥_𝑛→2=𝑔2. If 𝑥_𝑛=5+𝜀_𝑛, then 𝑓𝑥_𝑛→2,𝑔𝑥_𝑛=(2+∈_𝑛/3)→2,𝑓𝑔𝑥_𝑛=𝑓(2+∈_𝑛/3)→6≠𝑓2, and 𝑔𝑓𝑥_𝑛→𝑔2=2. Thus lim_𝑛→∞𝑔𝑓𝑥_𝑛=𝑔2 but lim_𝑛→∞𝑓𝑔𝑥_𝑛≠𝑓2. Hence 𝑓 and 𝑔 are weakly reciprocally continuous. It is also obvious that 𝑓 and 𝑔 are not reciprocally continuous mappings. Further, 𝑓 and 𝑔 are pointwise 𝑅-weakly commuting of type (𝐴_𝑓) maps as they commute at their only coincidence point 𝑥=2.

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 𝑓 and 𝑔 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 𝑓 and 𝑔 be weakly reciprocally continuous noncompatible self-mappings of a metric space (𝑋,𝑑) satisfying(i)𝑓𝑋⊆𝑔𝑋,(ii)𝑑(𝑓𝑥,𝑓𝑦)≤𝑘𝑑(𝑔𝑥,𝑔𝑦),𝑘≥0.
If 𝑓 and 𝑔 are conditionally commuting and 𝑔 is 𝑓-absorbing or 𝑓 is 𝑔-absorbing, then 𝑓 and 𝑔 have a common fixed point.

Proof. Since 𝑓 and 𝑔 are noncompatible maps, there exists a sequence {𝑥_𝑛} in 𝑋 such that 𝑓𝑥_𝑛→𝑡 and 𝑔𝑥_𝑛→𝑡 for some 𝑡 in 𝑋 but either lim_𝑛𝑑(𝑓𝑔𝑥_𝑛,𝑔𝑓𝑥_𝑛)≠0 or the limit does not exist. Since 𝑓𝑋⊆𝑔𝑋, for each 𝑥_𝑛 there exists 𝑦_𝑛 in 𝑋 such that 𝑓𝑥_𝑛=𝑔𝑦_𝑛. Thus 𝑓𝑥_𝑛→𝑡,𝑔𝑥_𝑛→𝑡 and 𝑔𝑦_𝑛→𝑡 as 𝑛→∞. By virtue of this and using (ii), we obtain 𝑓𝑦_𝑛→𝑡. Therefore, we have 𝑓𝑥_𝑛=𝑔𝑦_𝑛⟶𝑡,𝑔𝑥_𝑛⟶𝑡,𝑓𝑦_𝑛⟶𝑡. (2.17) Suppose that 𝑔 is 𝑓-absorbing. Then 𝑑(𝑓𝑥_𝑛,𝑓𝑔𝑥_𝑛)≤𝑅𝑑(𝑓𝑥_𝑛,𝑔𝑥_𝑛) and 𝑑(𝑓𝑦_𝑛,𝑓𝑔𝑦_𝑛)≤𝑅𝑑(𝑓𝑦_𝑛,𝑔𝑦_𝑛). On letting 𝑛→∞, these inequalities yield 𝑓𝑔𝑥_𝑛⟶𝑡,𝑓𝑔𝑦_𝑛=𝑓𝑓𝑥_𝑛⟶𝑡. (2.18) Weak reciprocal continuity of 𝑓 and 𝑔 implies that 𝑓𝑔𝑥_𝑛→𝑓𝑡 or 𝑔𝑓𝑥_𝑛→𝑔𝑡. Let 𝑔𝑓𝑥_𝑛→𝑔𝑡. By virtue of (ii), we get 𝑑(𝑓𝑓𝑥_𝑛,𝑓𝑡)≤𝑘𝑑(𝑔𝑓𝑥_𝑛,𝑔𝑡). On letting 𝑛→∞, we get 𝑓𝑓𝑥_𝑛→𝑓𝑡. In view of (2.18), this yields 𝑡=𝑓𝑡. Since 𝑓𝑋⊆𝑔𝑋, there exists 𝑢 in 𝑋 such that 𝑡=𝑓𝑡=𝑔𝑢. Now using (ii), we obtain 𝑑(𝑓𝑥_𝑛,𝑓𝑢)≤𝑘𝑑(𝑔𝑥_𝑛,𝑔𝑢). On letting 𝑛→∞, we get 𝑓𝑢=𝑡. Thus 𝑓𝑢=𝑔𝑢. Conditional commutativity of 𝑓 and 𝑔 implies that 𝑓 and 𝑔 commute at 𝑢, or there exists a coincidence point 𝑣 of 𝑓 and 𝑔 at which 𝑓 and 𝑔 commute. Suppose 𝑓 and 𝑔 commute at the coincidence point 𝑣. Then 𝑓𝑣=𝑔𝑣 and 𝑓𝑔𝑣=𝑔𝑓𝑣. Also 𝑓𝑓𝑣=𝑓𝑔𝑣=𝑔𝑓𝑣=𝑔𝑔𝑣. Since 𝑔 is 𝑓-absorbing 𝑑(𝑓𝑣,𝑓𝑔𝑣)≤𝑅𝑑(𝑓𝑣,𝑔𝑣). This yields 𝑓𝑣=𝑓𝑔𝑣. Hence 𝑓𝑣=𝑓𝑓𝑣=𝑔𝑓𝑣 and 𝑓𝑣 is a common fixed point of 𝑓 and 𝑔.
Next suppose that 𝑓𝑔𝑥_𝑛→𝑓𝑡. In view of (2.18), we get 𝑡=𝑓𝑡. Since 𝑓𝑋⊆𝑔𝑋, there exists 𝑢 in 𝑋 such that 𝑡=𝑓𝑡=𝑔𝑢. Now using (ii), we obtain 𝑑(𝑓𝑥_𝑛,𝑓𝑢)≤𝑘𝑑(𝑔𝑥_𝑛,𝑔𝑢). On letting 𝑛→∞, we get 𝑓𝑢=𝑡. Thus 𝑓𝑢=𝑔𝑢. This, in view of conditional commutativity and 𝑓-absorbing property of 𝑔, implies that 𝑓 and 𝑔 have a common fixed point.
Now suppose that 𝑓 is 𝑔-absorbing. Then 𝑑(𝑔𝑥_𝑛,𝑔𝑓𝑥_𝑛)≤𝑅𝑑(𝑓𝑥_𝑛,𝑔𝑥_𝑛) and 𝑑(𝑔𝑦_𝑛,𝑔𝑓𝑦_𝑛)≤𝑅𝑑(𝑓𝑦_𝑛,𝑔𝑦_𝑛). On letting 𝑛→∞, these inequalities yield 𝑔𝑓𝑥_𝑛=𝑔𝑔𝑦_𝑛⟶𝑡,𝑔𝑓𝑦_𝑛⟶𝑡. (2.19) Weak reciprocal continuity of 𝑓 and 𝑔 implies that 𝑓𝑔𝑦_𝑛→𝑓𝑡 or 𝑔𝑓𝑦_𝑛→𝑔𝑡. Let us first assume that 𝑔𝑓𝑦_𝑛→𝑔𝑡. In view of (2.19), this yields 𝑡=𝑔𝑡. Using (ii) we get 𝑑(𝑓𝑥_𝑛,𝑓𝑡)≤𝑘𝑑(𝑔𝑥_𝑛,𝑔𝑡). On letting 𝑛→∞, we obtain 𝑡=𝑓𝑡. Hence 𝑡=𝑓𝑡=𝑔𝑡 and 𝑡 is a common fixed point of 𝑓 and 𝑔.
Next suppose that 𝑓𝑔𝑦_𝑛→𝑓𝑡. Then 𝑓𝑋⊆𝑔𝑋 implies that 𝑓𝑡=𝑔𝑢 for some 𝑢∈𝑋. Therefore, 𝑓𝑔𝑦_𝑛→𝑓𝑡=𝑔𝑢. Using (ii) and in view of (2.19), we get 𝑑(𝑓𝑦_𝑛,𝑓𝑔𝑦_𝑛)≤𝑘𝑑(𝑔𝑦_𝑛,𝑔𝑔𝑦_𝑛). On letting 𝑛→∞, we get 𝑡=𝑔𝑢. Again, by virtue of (ii), we obtain 𝑑(𝑓𝑦_𝑛,𝑓𝑢)≤𝑘𝑑(𝑔𝑦_𝑛,𝑔𝑢). Making 𝑛→∞, we get 𝑡=𝑓𝑢. Hence 𝑓𝑢=𝑔𝑢. Conditional commutativity of 𝑓 and 𝑔 implies that 𝑓 and 𝑔 commute at 𝑢 or there exists a coincidence point 𝑣 of 𝑓 and 𝑔 at which 𝑓 and 𝑔 commute. Suppose 𝑓 and 𝑔 commute at the coincidence point 𝑣. Then 𝑓𝑣=𝑔𝑣 and 𝑓𝑔𝑣=𝑔𝑓𝑣. Also 𝑓𝑓𝑣=𝑓𝑔𝑣=𝑔𝑓𝑣=𝑔𝑔𝑣. Since 𝑓 is 𝑔-absorbing, 𝑑(𝑔𝑣,𝑔𝑓𝑣)≤𝑅𝑑(𝑓𝑣,𝑔𝑣). This yields 𝑔𝑣=𝑔𝑓𝑣. Hence 𝑓𝑣=𝑓𝑓𝑣=𝑔𝑓𝑣 and 𝑓𝑣 is a common fixed point of 𝑓 and 𝑔. This completes the proof of the theorem.

We now give examples to illustrate Theorem 2.8.

Example 2.9. Let 𝑋=[0,1] and 𝑑 be the usual metric on 𝑋. Define 𝑓,𝑔∶𝑋→𝑋 by 𝑓𝑥=(1/2)−|𝑥−(1/2)|, 2𝑔𝑥=3(1−𝑥). (2.20)
Then 𝑓 and 𝑔 satisfy all the conditions of the above theorem and have two coincidence points 𝑥=1,2/5 and a common fixed point 𝑥=2/5. It may be verified in this example that 𝑓(𝑋)=[0,1/2],𝑔(𝑋)=[0,2/3] and 𝑓𝑋⊆𝑔𝑋. Also that 𝑓 and 𝑔 are noncompatible but conditionally commuting maps. Furthermore, 𝑓 and 𝑔 are conditionally commuting since they commute at their coincidence point 2/5. To see that 𝑓 and 𝑔 are noncompatible, let us consider the sequence {𝑥_𝑛} given by 𝑥_𝑛=1−(1/𝑛). Then 𝑓𝑥_𝑛→0,𝑔𝑥_𝑛→0,𝑓𝑔𝑥_𝑛→0, and 𝑔𝑓𝑥_𝑛→2/3. Hence 𝑓 and 𝑔 are noncompatible. It may also be verified that 𝑓 and 𝑔 are not pointwise R-weakly commuting of type (𝐴_𝑓) as they do not commute at the coincidence point 𝑥=1, since 𝑓(𝑔(1))=0 and 𝑔(𝑓(1))=2/3. It is also easy to verify that 𝑓 and 𝑔 satisfy the Lipschitz-type condition 𝑑(𝑓𝑥,𝑓𝑦)≤(3/2)𝑑(𝑔𝑥,𝑔𝑦) together with 𝑓-absorbing condition 𝑑(𝑓𝑥,𝑓𝑔𝑥)≤𝑑(𝑓𝑥,𝑔𝑥) for all 𝑥. It can also be noted that 𝑓 and 𝑔 are weakly reciprocally continuous since both 𝑓 and 𝑔 are continuous.

In Example 2.9, 𝑓 and 𝑔 are not pointwise 𝑅-weakly commuting of type (𝐴_𝑓) as they do not commute at the coincidence point 𝑥=1. We now give an example of pointwise 𝑅-weakly commuting of type (𝐴_𝑓) maps satisfying Theorem 2.8.

Example 2.10. Let 𝑋=[0,1] and 𝑑 be the usual metric on 𝑋. Define 𝑓,𝑔∶𝑋→𝑋 as follows: 1𝑓𝑥=2−1𝑥−2,2𝑔𝑥=3fractionalpartof(1−𝑥). (2.21) Then 𝑓 and 𝑔 satisfy all the conditions of the above theorem and have three coincidence points 𝑥=0,2/5,1 and two common fixed point 𝑥=0,2/5. It may be verified in this example that 𝑓(𝑋)=[0,1/2],𝑔(𝑋)=[0,2/3) and 𝑓𝑋⊆𝑔𝑋. Also, 𝑓 and 𝑔 are pointwise 𝑅-weakly commuting of type (𝐴_𝑓) maps, hence also conditionally commuting, since they commute at each of their coincidence points, namely, 𝑥=0,2/5,1. To see that 𝑓 and 𝑔 are noncompatible, let us consider the sequence {𝑥_𝑛} given by 𝑥_𝑛=1−1/𝑛. Then 𝑓𝑥_𝑛→0,𝑔𝑥_𝑛→0,𝑓𝑔𝑥_𝑛→0, and 𝑔𝑓𝑥_𝑛→2/3. Hence 𝑓 and 𝑔 are noncompatible. It is also easy to verify that 𝑓 and 𝑔 satisfy the Lipschitz-type condition 𝑑(𝑓𝑥,𝑓𝑦)≤(3/2)𝑑(𝑔𝑥,𝑔𝑦). The mapping 𝑔 is 𝑓-absorbing since 𝑑(𝑓𝑥,𝑓𝑔𝑥)≤𝑑(𝑓𝑥,𝑔𝑥) for all 𝑥. It can also be noted that 𝑓 and 𝑔 are weakly reciprocally continuous. To see this, let {𝑥_𝑛} be a sequence in 𝑋 such that 𝑓𝑥_𝑛→𝑡,𝑔𝑥_𝑛→𝑡 for some 𝑡. Then 𝑡=0 and either 𝑥_𝑛=0 for each 𝑛 or 𝑥_𝑛→1. If 𝑥_𝑛=0 for each 𝑛, then 𝑓𝑥_𝑛→0,𝑔𝑥_𝑛→0,𝑓𝑔𝑥_𝑛→0=𝑓(0), and 𝑔𝑓𝑥_𝑛→0=𝑔(0). If 𝑥_𝑛→1, then 𝑓𝑥_𝑛→0,𝑔𝑥_𝑛→0,𝑓𝑔𝑥_𝑛→0=𝑓(0), and 𝑔𝑓𝑥_𝑛→2/3≠𝑔(0). Thus lim_𝑛→∞𝑓𝑔𝑥_𝑛=𝑓(0) but lim_𝑛→∞𝑔𝑓𝑥_𝑛≠𝑔(0). Hence 𝑓 and 𝑔 are weakly reciprocally continuous.
Putting 𝑘=1 in Theorem 2.8, we get a common fixed point theorem for a non-expansive-type mapping pair.

Corollary 2.11. Let 𝑓 and 𝑔 be weakly reciprocally continuous noncompatible self-mappings of a metric space (𝑋,𝑑) satisfying(i)𝑓𝑋⊆𝑔𝑋,(ii)𝑑(𝑓𝑥,𝑓𝑦)≤𝑑(𝑔𝑥,𝑔𝑦).
If 𝑓 and 𝑔 are conditionally commuting and 𝑔 is 𝑓-absorbing or 𝑓 is 𝑔-absorbing, then 𝑓 and 𝑔 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.