Employing Common Limit Range Property to Prove Unified Metrical Common Fixed Point Theorems

Abstract

The purpose of this paper is to emphasize the role of “common limit range property” to ascertain the existence of common fixed point in metric spaces satisfying an implicit function essentially due to the paper of Ali and Imdad (2008). As an application to our main result, we derive a fixed point theorem for four finite families of self-mappings which can be utilized to derive common fixed point theorems involving any finite number of mappings. Our results improve and extend a host of previously known results including the ones contained in the paper of Ali and Imdad (2008). We also furnish some illustrative examples to support our main results.

1. Introduction and Preliminaries

The well-known Banach fixed point theorem is often referred to as Banach contraction principle which appeared in its explicit form in the thesis of Banach in 1922 [1]. Owing to its simplicity and usefulness, it became a very popular and powerful tool in solving existence problems in pure and applied sciences which include biology, medicine, physics, and computer science (see [2, 3]). This natural theorem asserts that every contraction mapping defined on a complete metric space has a unique fixed point and that fixed point can be explicitly obtained as limit of repeated iteration of the mapping at any point of the underlying space. Evidently, every contraction mapping is a continuous but not conversely.

A result on the existence and uniqueness of common fixed point in metric spaces generally involves conditions on commutativity, continuity, and contraction along with a suitable condition on the containment of range of one mapping into the range of other. Hence, one is always required to improve one or more of these conditions to prove a new fixed point theorem. In the recent past, several authors have contributed to the vigorous development of metric fixed point theory (e.g., [4–18]).

In metric fixed point theory, recently implicit relations are utilized to cover several contraction conditions in one go rather than proving a separate theorem for each contraction condition. The first ever attempt to coin an implicit relation can be traced back to Popa [19]. In 2008, Ali and Imdad [20] introduced a new class of implicit functions which covers several classes of contraction conditions such as Ćirić quasi-contractions, generalized contractions, 𝜓-type contractions, rational inequalities, and among others. They (i.e., Ali and Imdad [20]) proved some fixed point theorems for weakly compatible mappings satisfying common property (E.A). Thereafter, many researchers utilized various implicit relations to prove a number of fixed point theorems in different settings (e.g., [21–27]).

In this paper, utilizing the implicit function of Ali and Imdad [20], we prove some common fixed point theorems for two pairs of weakly compatible mappings employing common limit range property. In process, many known results (especially the ones contained in Ali and Imdad [20]) are enriched and improved. Some related results are also derived besides furnishing illustrative examples.

Definition 1. Let 𝐴,𝑆∶𝑋→𝑋 be two self-mappings of a metric space (𝑋,𝑑). Then the pair (𝐴,𝑆) is said to be (1)commuting if 𝐴𝑆𝑥=𝑆𝐴𝑥, for all 𝑥∈𝑋, (2)weakly commuting (cf. [28]) if 𝑑(𝐴𝑆𝑥,𝑆𝐴𝑥)≤𝑑(𝐴𝑥,𝑆𝑥), for all 𝑥∈𝑋, (3)compatible (cf. [29]) if lim_𝑛→∞𝑑(𝐴𝑆𝑥_𝑛,𝑆𝐴𝑥_𝑛)=0 for each sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛, (4)noncompatible (cf. [30]) if there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛 but lim_𝑛→∞𝑑(𝐴𝑆𝑥_𝑛,𝑆𝐴𝑥_𝑛) is either nonzero or nonexistent, (5)weakly compatible (cf. [31]) if they commute at their coincidence points, that is, 𝐴𝑆𝑢=𝑆𝐴𝑢 whenever 𝐴𝑢=𝑆𝑢, for some 𝑢∈𝑋.
For more details on systematic comparisons and illustrations of earlier described notions, we refer to Murthy [32].

Definition 2 (cf. [33]). A pair (𝐴,𝑆) of self-mappings of a metric space (𝑋,𝑑) is said to satisfy the property (E.A) if there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=𝑡, (1) for some 𝑡∈𝑋.

Hence any pair of compatible as well as noncompatible self-mappings of a metric space (𝑋,𝑑) satisfy the property (E.A) but a pair of mappings satisfying the property (E.A) need not be noncompatible (see [34, Example 1]).

In 2005, Liu et al. [35] defined the notion of common property (E.A) for hybrid pairs of mappings which contains the property (E.A).

Definition 3. Two pairs (𝐴,𝑆) and (𝐵,𝑇) of self-mappings of a metric space (𝑋,𝑑) are said to satisfy the common property (E.A), if there exist two sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=lim_𝑛→∞𝐵𝑦_𝑛=lim_𝑛→∞𝑇𝑦_𝑛=𝑡, (2) for some 𝑡∈𝑋.

Motivated by the observation that the majority of the fixed point results always require the closedness of the underlying subspaces for the existence of common fixed point under the property (E.A) and common property (E.A), Sintunavarat and Kumam [36] coined the idea of “common limit range property” (also see [37]). Recently, Imdad et al. [38] extended the notion of common limit range property to two pairs of self-mappings which relaxes the requirements on closedness of the underlying subspaces.

Definition 4 (cf. [36]). A pair (𝐴,𝑆) of self-mappings of a metric space (𝑋,𝑑) is said to satisfy the common limit range property with respect to 𝑆, denoted by (CLR_𝑆), if there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=𝑡, (3) where 𝑡∈𝑆(𝑋).

Thus, one can infer that a pair (𝐴,𝑆) satisfying the property (E.A) along with closedness of the subspace 𝑆(𝑋) always enjoys the (CLR_𝑆) property with respect to the mapping 𝑆 (see [38, Examples 2.16-2.17]).

Definition 5. Two pairs (𝐴,𝑆) and (𝐵,𝑇) of self-mappings of a metric space (𝑋,𝑑) are said to satisfy the common limit range property with respect to mappings 𝑆 and 𝑇, denoted by (CLR_𝑆𝑇), if there exist two sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=lim_𝑛→∞𝐵𝑦_𝑛=lim_𝑛→∞𝑇𝑦_𝑛=𝑡, (4) where 𝑡∈𝑆(𝑋)∩𝑇(𝑋).

Definition 6 (cf. [39]). Two families of self-mappings {𝐴_𝑖}^𝑚_𝑖=1 and {𝑆_𝑘}^𝑛_𝑘=1 are said to be pairwise commuting if (1)𝐴_𝑖𝐴_𝑗=𝐴_𝑗𝐴_𝑖 for all 𝑖,𝑗∈{1,2,…,𝑚}, (2)𝑆_𝑘𝑆_𝑙=𝑆_𝑙𝑆_𝑘 for all 𝑘,𝑙∈{1,2,…,𝑛}, (3)𝐴_𝑖𝑆_𝑘=𝑆_𝑘𝐴_𝑖 for all 𝑖∈{1,2,…,𝑚} and 𝑘∈{1,2,…,𝑛}.

2. Implicit Functions

In an attempt to cover Ćirić’s quasi-contractions, Ali and Imdad [20] introduced a new class of implicit functions which includes a variety of contraction classes such as Ćirić quasi-contractions, generalized contractions, 𝜓-type contractions, rational inequalities, and among others.

Let Φ be the family of lower semicontinuous functions 𝜙∶ℝ⁶→ℝ satisfying the following conditions:(𝜙₁)∶𝜙(𝑡,0,𝑡,0,0,𝑡)>0, for all 𝑡>0,(𝜙₂)∶𝜙(𝑡,0,0,𝑡,𝑡,0)>0, for all 𝑡>0,(𝜙₃)∶𝜙(𝑡,𝑡,0,0,𝑡,𝑡)>0, for all 𝑡>0.

The following examples furnished in Ali and Imdad’s [20] establishes the utility of the preceeding definition.

Example 7. Define 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶ℝ⁶→ℝ as 𝜙𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝑘max𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆,where𝑘∈[0,1). (5)

Example 8. Define 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶ℝ⁶→ℝ as 𝜙𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝑘max𝑡₂,𝑡₃,𝑡₃𝑡₅,𝑡₄𝑡₆,where𝑘∈[0,1). (6)

Example 9. Define 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶ℝ⁶→ℝ as 𝜙𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝑘max𝑡²₂,𝑡₃𝑡₄,𝑡₅𝑡₆,𝑡₃𝑡₅,𝑡₄𝑡₆^1/2,where𝑘∈[0,1). (7)

Example 10. Define 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶ℝ⁶→ℝ as 𝜙𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝛼𝛽max𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆+1−𝛽max𝑡²₂,𝑡₃𝑡₄,𝑡₅𝑡₆,𝑡₃𝑡₆,𝑡₄𝑡₅^1/2, (8) where 𝛼∈[0,1) and 𝛽≥0.

Example 11. Define 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶ℝ⁶→ℝ as 𝜙𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡²₁−𝛼max𝑡²₂,𝑡²₃,𝑡²₄−𝛽max𝑡₃𝑡₅,𝑡₄𝑡₆−𝛾𝑡₅𝑡₆, (9) where 𝛼,𝛽,𝛾≥0 and 𝛼+𝛾<1.

Example 12. Define 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶ℝ⁶→ℝ as 𝜙𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=1+𝛼𝑡₂𝑡₁−𝛼max𝑡₃𝑡₄,𝑡₅𝑡₆−𝛽max𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆, (10) where 𝛼≥0 and 𝛽∈[0,1).

Example 13. Define 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶ℝ⁶→ℝ as 𝜙𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝛼𝑡₂−𝛽max𝑡₃,𝑡₄−𝛾max𝑡₃+𝑡₄,𝑡₅+𝑡₆, (11) where 𝛼,𝛽,𝛾≥0 and 𝛼+𝛽+2𝛾<1.

Example 14. Define 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶ℝ⁶→ℝ as 𝜙𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝜓max𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆, (12) where 𝜓∶ℝ→ℝ is an upper semicontinuous function such that 𝜓(0)=0 and 𝜓(𝑡)<𝑡 for all 𝑡>0.

Example 15. Define 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶ℝ⁶→ℝ as 𝜙𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝜓𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆, (13) where 𝜓∶ℝ⁵→ℝ is an upper semicontinuous function such that max{𝜓(0,𝑡,0,0,𝑡), 𝜓(0,0,𝑡,𝑡,0), 𝜓(𝑡,0,0,𝑡,𝑡)}<𝑡 for each 𝑡>0.

Example 16. Define 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶ℝ⁶→ℝ as 𝜙𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡²₁−𝜓𝑡²₂,𝑡₃𝑡₄,𝑡₅𝑡₆,𝑡₃𝑡₆,𝑡₄𝑡₅, (14) where 𝜓∶ℝ⁵→ℝ is an upper semicontinuous function such that max{𝜓(0,0,0,𝑡,0), 𝜓(0,0,0,0,𝑡), 𝜓(𝑡,0,𝑡,0,0)}<𝑡 for each 𝑡>0.

Example 17. Define 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶ℝ⁶→ℝ as 𝜙𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝛼𝑡₂𝑡−𝛽²³+𝑡²₄𝑡₃+𝑡₄−𝛾𝑡₅+𝑡₆,if𝑡₃+𝑡₄𝑡≠0;₁,if𝑡₃+𝑡₄=0, (15) where 𝛼,𝛽,𝛾≥0 and 𝛽+𝛾<1.

Example 18. Define 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶ℝ⁶→ℝ as 𝜙𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡^𝑝₁−𝑘𝑡^𝑝₂−𝑡₃𝑡^𝑝₄+𝑡₅𝑡^𝑝₆𝑡₃+𝑡₄,if𝑡₃+𝑡₄𝑡≠0;₁,if𝑡₃+𝑡₄=0, (16) where 𝑝≥1 and 0≤𝑘<∞.

Example 19. Define 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶ℝ⁶→ℝ as 𝜙𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝛼𝑡₂𝑡−𝛽²⁵+𝑡²₆𝑡₅+𝑡₆−𝛾𝑡₃+𝑡₄,if𝑡₅+𝑡₆𝑡≠0;₁,if𝑡₅+𝑡₆=0, (17) where 𝛼,𝛽,𝛾≥0 and 𝛽+𝛾<1.

Example 20. Define 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶ℝ⁶→ℝ as 𝜙𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝑘𝑡₂−𝑡₃𝑡₄+𝑡₅𝑡₆𝑡₅+𝑡₆,if𝑡₅+𝑡₆𝑡≠0;₁,if𝑡₅+𝑡₆=0, (18) where 0≤𝑘<∞.

Example 21. Define 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶ℝ⁶→ℝ as 𝜙𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝑘𝑡₂−𝑡₃𝑡₄+𝑡₅𝑡₆𝑡₃+𝑡₄−𝑡₃𝑡₅+𝑡₄𝑡₆𝑡₅+𝑡₆,if𝑡₃+𝑡₄≠0,𝑡₅+𝑡₆𝑡≠0;₁,if𝑡₃+𝑡₄=0,𝑡₅+𝑡₆=0, (19) where 0≤𝑘<∞.

Example 22. Define 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶ℝ⁶→ℝ as 𝜙𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝑡₃𝑡₄+𝑡₅𝑡₆1+𝑡₂. (20)

Example 23. Define 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶ℝ⁶→ℝ as 𝜙𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝛼𝑡₂𝑡−𝛽³+𝑡₄1+𝑡₅𝑡₆, (21) where 𝛼,𝛽∈[0,1).

Example 24. Define 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶ℝ⁶→ℝ as 𝜙𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡²₁−𝛼𝑡²₂𝑡−𝛽⁵𝑡₆1+𝑡²₃+𝑡²₄, (22) where 𝛼,𝛽≥0 and 𝛼+𝛽<1.

Example 25. Define 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶ℝ⁶→ℝ as 𝜙𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡³₁−𝑡²₃𝑡²₄+𝑡²₅𝑡²₆1+𝑡₂. (23)

Example 26. Define 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶ℝ⁶→ℝ as 𝜙𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡³₁−𝛼𝑡²₁𝑡₂−𝛽𝑡₁𝑡₃𝑡₄−𝛾𝑡²₅𝑡₆−𝜂𝑡₅𝑡²₆, (24) where 𝛼,𝛽,𝛾,𝜂≥0 and 𝛼+𝛾+𝜂<1.

Apart from earlier stated definitions, still there are many contractive definitions which meet the requirements (𝜙₁), (𝜙₂) and (𝜙₃) but due to paucity of the space we have not opted to include more examples.

3. Main Results

Firstly, we observe the following.

Lemma 27. Let 𝐴,𝐵,𝑆, and 𝑇 be self-mappings of a metric space (𝑋,𝑑). Suppose the following: (1)the pair (𝐴,𝑆) satisfies the (CLR_𝑆) property (or (𝐵,𝑇) satisfies the (CLR_𝑇) property), (2)𝐴(𝑋)⊂𝑇(𝑋)  (or 𝐵(𝑋)⊂𝑆(𝑋)), (3)𝑇(𝑋)  (or 𝑆(𝑋)) is a closed subset of 𝑋, (4)there exists 𝜙∈Φ such that (for all 𝑥,𝑦∈𝑋)𝜙𝑑(),𝑑𝐴𝑥,𝐵𝑦,𝑑𝑆𝑥,𝑇𝑦,𝑑𝐴𝑥,𝑆𝑥,𝑑𝐵𝑦,𝑇𝑦𝑆𝑥,𝐵𝑦,𝑑𝑇𝑦,𝐴𝑥≤0. (25)
Then the pairs (𝐴,𝑆) and (𝐵,𝑇) share the (CLR_𝑆𝑇) property.

Proof. If the pair (𝐴,𝑆) enjoys the (CLR_𝑆) property with respect to mapping 𝑆, then there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=𝑡, (26) where 𝑡∈𝑆(𝑋). Since 𝐴(𝑋)⊂𝑇(𝑋), for each sequence {𝑥_𝑛} there exists a sequence {𝑦_𝑛} in 𝑋 such that 𝐴𝑥_𝑛=𝑇𝑦_𝑛. Therefore, due to closedness of 𝑇(𝑋), lim_𝑛→∞𝑇𝑦_𝑛=lim_𝑛→∞𝐴𝑥_𝑛=𝑡, (27) where 𝑡∈𝑆(𝑋)∩𝑇(𝑋). Thus in all, we have 𝐴𝑥_𝑛→𝑡, 𝑆𝑥_𝑛→𝑡 and 𝑇𝑦_𝑛→𝑡 as 𝑛→∞. Now, we show that 𝐵𝑦_𝑛→𝑡 as 𝑛→∞. On using inequality (25) with 𝑥=𝑥_𝑛, 𝑦=𝑦_𝑛, we have 𝜙𝑑𝐴𝑥_𝑛,𝐵𝑦_𝑛,𝑑𝑆𝑥_𝑛,𝑇𝑦_𝑛,𝑑𝐴𝑥_𝑛,𝑆𝑥_𝑛,𝑑𝐵𝑦_𝑛,𝑇𝑦_𝑛,𝑑𝑆𝑥_𝑛,𝐵𝑦_𝑛,𝑑𝑇𝑦_𝑛,𝐴𝑥_𝑛≤0. (28)
Let on contrary that 𝐵𝑦_𝑛→𝑙(≠𝑡) as 𝑛→∞. Then, on taking limit as 𝑛→∞ in the preceeding inequality, we get 𝜙(𝑑(𝑡,𝑙),𝑑(𝑡,𝑡),𝑑(𝑡,𝑡),𝑑(𝑙,𝑡),𝑑(𝑡,𝑙),𝑑(𝑡,𝑡))≤0, (29) or 𝜙(𝑑(𝑡,𝑙),0,0,𝑑(𝑙,𝑡),𝑑(𝑡,𝑙),0)≤0, (30) which is a contradiction to (𝜙₂). Hence, 𝐵𝑦_𝑛→𝑡 which shows that the pairs (𝐴,𝑆) and (𝐵,𝑇) share the (CLR_𝑆𝑇) property. This concludes the proof.

Remark 28. In general, the converse of Lemma 27 is not true (see [38, Example 3.5]).

Now, we state and prove our main result for two pairs of weakly compatible mappings satisfying the (CLR_𝑆𝑇) property.

Theorem 29. Let 𝐴,𝐵,𝑆, and 𝑇 be self-mappings of a metric space (𝑋,𝑑) satisfying the inequality (25) of Lemma 27. If the pairs (𝐴,𝑆) and (𝐵,𝑇) share the (CLR_𝑆𝑇) property, then (𝐴,𝑆) and (𝐵,𝑇) have a coincidence point each. Moreover, 𝐴,𝐵,𝑆 and 𝑇 have a unique common fixed point provided that both pairs (𝐴,𝑆) and (𝐵,𝑇) are weakly compatible.

Proof. Since the pairs (𝐴,𝑆) and (𝐵,𝑇) satisfy the (CLR_𝑆𝑇) property, there exist two sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=lim_𝑛→∞𝐵𝑦_𝑛=lim_𝑛→∞𝑇𝑦_𝑛=𝑡, (31) where 𝑡∈𝑆(𝑋)∩𝑇(𝑋). Since 𝑡∈𝑆(𝑋), there exists a point 𝑢∈𝑋 such that 𝑆𝑢=𝑡. We show that 𝐴𝑢=𝑆𝑢. If not, then using inequality (25) with 𝑥=𝑢, 𝑦=𝑦_𝑛, we get 𝜙𝑑𝐴𝑢,𝐵𝑦_𝑛,𝑑𝑆𝑢,𝑇𝑦_𝑛,𝑑(𝐴𝑢,𝑆𝑢),𝑑𝐵𝑦_𝑛,𝑇𝑦_𝑛,𝑑𝑆𝑢,𝐵𝑦_𝑛,𝑑𝑇𝑦_𝑛,𝐴𝑢≤0, (32) which on making 𝑛→∞ reduces to 𝜙(𝑑(𝐴𝑢,𝑡),𝑑(𝑡,𝑡),𝑑(𝐴𝑢,𝑡),𝑑(𝑡,𝑡),𝑑(𝑡,𝑡),𝑑(𝑡,𝐴𝑢))≤0, (33) or 𝜙(𝑑(𝐴𝑢,𝑡),0,𝑑(𝐴𝑢,𝑡),0,0,𝑑(𝑡,𝐴𝑢))≤0, (34) a contradiction to (𝜙₁). Hence, 𝐴𝑢=𝑆𝑢=𝑡. Therefore, 𝑢 is a coincidence point of the pair (𝐴,𝑆).
Also 𝑡∈𝑇(𝑋), there exists a point 𝑣∈𝑋 such that 𝑇𝑣=𝑡. We assert that 𝐵𝑣=𝑇𝑣. If not, then using inequality (25) with 𝑥=𝑢, 𝑦=𝑣, we get 𝜙(𝑑()()()(),𝑑()(𝐴𝑢,𝐵𝑣,𝑑𝑆𝑢,𝑇𝑣,𝑑𝐴𝑢,𝑆𝑢,𝑑𝐵𝑣,𝑇𝑣𝑆𝑢,𝐵𝑣,𝑑𝑇𝑣,𝐴𝑢))≤0, (35) which reduces to 𝜙(𝑑(𝑡,𝐵𝑣),𝑑(𝑡,𝑡),𝑑(𝑡,𝑡),𝑑(𝐵𝑣,𝑡),𝑑(𝑡,𝐵𝑣),𝑑(𝑡,𝑡))≤0, (36) or 𝜙(𝑑(𝑡,𝐵𝑣),0,0,𝑑(𝐵𝑣,𝑡),𝑑(𝑡,𝐵𝑣),0)≤0, (37) a contradiction to (𝜙₂). Hence, 𝐵𝑣=𝑇𝑣=𝑡, which shows that 𝑣 is a coincidence point of the pair (𝐵,𝑇).
Since the pair (𝐴,𝑆) is weakly compatible and 𝐴𝑢=𝑆𝑢, hence 𝐴𝑡=𝐴𝑆𝑢=𝑆𝐴𝑢=𝑆𝑡. Now, we show that 𝑡 is a common fixed point of the pair (𝐴,𝑆). Suppose that 𝐴𝑡≠𝑡, using inequality (25) with 𝑥=𝑡, 𝑦=𝑣, we have 𝜙(𝑑()()()(),𝑑()(𝐴𝑡,𝐵𝑣,𝑑𝑆𝑡,𝑇𝑣,𝑑𝐴𝑡,𝑆𝑡,𝑑𝐵𝑣,𝑇𝑣𝑆𝑡,𝐵𝑣,𝑑𝑇𝑣,𝐴𝑡))≤0, (38) or 𝜙(𝑑(𝐴𝑡,𝑡),𝑑(𝐴𝑡,𝑡),0,0,𝑑(𝐴𝑡,𝑡),𝑑(𝑡,𝐴𝑡))≤0, (39) a contradiction to (𝜙₃); we get 𝐴𝑡=𝑡=𝑆𝑡 which shows that 𝑡 is a common fixed point of the pair (𝐴,𝑆).
Also the pair (𝐵,𝑇) is weakly compatible and 𝐵𝑣=𝑇𝑣, then 𝐵𝑡=𝐵𝑇𝑣=𝑇𝐵𝑣=𝑇𝑡. Suppose that 𝐵𝑡≠𝑡, then using inequality (25) with 𝑥=𝑢, 𝑦=𝑡, we have 𝜙(𝑑()()()(),𝑑()(𝐴𝑢,𝐵𝑡,𝑑𝑆𝑢,𝑇𝑡,𝑑𝐴𝑢,𝑆𝑢,𝑑𝐵𝑡,𝑇𝑡𝑆𝑢,𝐵𝑡,𝑑𝑇𝑡,𝐴𝑢))≤0, (40) or 𝜙(𝑑(𝑡,𝐵𝑡),𝑑(𝑡,𝐵𝑡),0,0,𝑑(𝑡,𝐵𝑡),𝑑(𝐵𝑡,𝑡))≤0, (41) a contradiction to (𝜙₃). Therefore, 𝐵𝑡=𝑡=𝑇𝑡 which shows that 𝑡 is a common fixed point of the pair (𝐵,𝑇) and in all 𝑡 is a common fixed point of both pairs (𝐴,𝑆) and (𝐵,𝑇). Uniqueness of common fixed point is an easy consequence of the inequality (25) in view of condition (𝜙₃). This concludes the proof.

Remark 30. Theorem 29 improves the corresponding results contained in Ali and Imdad’s [20, Theorem 3.1] as closedness of the underlying subspaces is not required.

Now, we present an example which demonstrates the validity of the hypotheses and degree of generality of our main result over comparable ones from the existing literature.

Example 31. Consider 𝑋=[5,21) equipped with the usual metric. Define four self-mappings 𝐴,𝐵,𝑆, and 𝑇 by 𝐴(𝑥)={5}∪();(],𝐵(𝑥)={5}∪();(],𝑆(𝑥)=(];5,if𝑥∈9,2120,if𝑥∈5,95,if𝑥∈9,2113,if𝑥∈5,95,if𝑥=5;10,if𝑥∈5,9𝑥+12(),𝑇(𝑥)=(];().,if𝑥∈9,215,if𝑥=5;18,if𝑥∈5,9𝑥−4,if𝑥∈9,21 (42)
Also, define implicit function 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶(ℝ⁺)⁶→ℝ as 𝜙𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝑘max𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆ (43) or 𝑑𝑑(),𝑑,𝐴𝑥,𝐵𝑦≤𝑘max𝑆𝑥,𝑇𝑦,𝑑𝐴𝑥,𝑆𝑥,𝑑𝐵𝑦,𝑇𝑦𝑆𝑥,𝐵𝑦,𝑑𝑇𝑦,𝐴𝑥 (44) for all 𝑥,𝑦∈𝑋 and 𝑘∈[0,1). By a routine calculation, one can verify the inequality (44) with 𝑘=1/2. If we choose two sequences as {𝑥_𝑛}={9+(1/𝑛)}_𝑛∈ℕ, {𝑦_𝑛}={5} (or {𝑥_𝑛}={5}, {𝑦_𝑛}={9+(1/𝑛)}_𝑛∈ℕ), then the pairs (𝐴,𝑆) and (𝐵,𝑇) satisfy the (CLR_𝑆𝑇) property: lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=lim_𝑛→∞𝐵𝑦_𝑛=lim_𝑛→∞𝑇𝑦_𝑛=5∈𝑆(𝑋)∩𝑇(𝑋). (45)
Also, 𝐴(𝑋)={5,20}⊈[5,17)∪{18}=𝑇(𝑋) and 𝐵(𝑋)={5,13}⊈[5,11)=𝑆(𝑋). Thus all the conditions of Theorem 29 are satisfied and 5 is a unique common fixed point of the pairs (𝐴,𝑆) and (𝐵,𝑇) which also remains a point of coincidence as well. Here, one may notice that all the involved mappings are discontinuous at their unique common fixed point 5.
Notice that the subspaces 𝑆(𝑋) and 𝑇(𝑋) are not closed subspaces of 𝑋; therefore, Theorem 29 of Ali and Imdad [20] cannot be used in the context of this example which establishes the genuineness of our extension.

Theorem 32. Let 𝐴,𝐵,𝑆, and 𝑇 be self-mappings of a metric space (𝑋,𝑑) satisfying all the hypotheses of Lemma 27. Then 𝐴,𝐵,𝑆, and 𝑇 have a unique common fixed point provided that both pairs (𝐴,𝑆) and (𝐵,𝑇) are weakly compatible.

Proof. In view of Lemma 27, the pairs (𝐴,𝑆) and (𝐵,𝑇) share the (CLR_𝑆𝑇) property so that there exist two sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=lim_𝑛→∞𝑇𝑦_𝑛=lim_𝑛→∞𝐵𝑦_𝑛=𝑡, (46) where 𝑡∈𝑆(𝑋)∩𝑇(𝑋). The rest of the proof can be completed on the lines of the proof of Theorem 29. This completes the proof.

The following example demonstrates the utility of Theorem 32 over Theorem 29.

Example 33. In the setting of Example 31, replace the self-mappings 𝐴 and 𝐵 by the following besides retaining the rest (i.e., 𝑆 and 𝑇): 𝐴(𝑥)={5}∪();(],𝐵(𝑥)={5}∪();(],𝑆(𝑥)=(];5,if𝑥∈9,2115,if𝑥∈5,95,if𝑥∈9,2110,if𝑥∈5,95,if𝑥=5;11,if𝑥∈5,9𝑥+12(),𝑇(𝑥)=(];().,if𝑥∈9,215,if𝑥=5;17,if𝑥∈5,9𝑥−4,if𝑥∈9,21 (47)
Then, like earlier example, the pairs (𝐴,𝑆) and (𝐵,𝑇) satisfy the (CLR_𝑆𝑇) property. Consider the implicit function 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶(ℝ⁺)⁶→ℝ as 𝜙𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝑘max𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆, (48) which implies 𝑑𝑑(),𝑑,𝐴𝑥,𝐵𝑦≤𝑘max𝑆𝑥,𝑇𝑦,𝑑𝐴𝑥,𝑆𝑥,𝑑𝐵𝑦,𝑇𝑦𝑆𝑥,𝐵𝑦,𝑑𝑇𝑦,𝐴𝑥 (49) for all 𝑥,𝑦∈𝑋 and 𝑘∈[0,1). By a routine calculation, one can verify inequality (49) with 𝑘=1/2. Also, 𝐴(𝑋)={5,15}⊂[5,17]=𝑇(𝑋) and 𝐵(𝑋)={5,10}⊂[5,11]=𝑆(𝑋). The pairs (𝐴,𝑆) and (𝐵,𝑇) commute at 5 which is also their common coincidence point as well. Thus all the conditions of Theorems 32 are satisfied and 5 is a unique common fixed point of the involved mappings 𝐴,𝐵,𝑆, and 𝑇.
Here, it can be pointed out that Theorem 29 is not applicable to this example as both 𝑆(𝑋) and 𝑇(𝑋) are closed subsets of 𝑋 which demonstrates the situational utility of Theorem 32 over Theorem 29.

Corollary 34. The conclusions of Lemma 27, Theorems 29, and 32 remain true if inequality (25) is replaced by one of the following contraction conditions. For all 𝑥,𝑦∈𝑋, 𝑑𝑑(),𝑑,𝐴𝑥,𝐵𝑦≤𝑘max𝑆𝑥,𝑇𝑦,𝑑𝐴𝑥,𝑆𝑥,𝑑𝐵𝑦,𝑇𝑦𝑆𝑥,𝐵𝑦,𝑑𝑇𝑦,𝐴𝑥 (50) where 𝑘∈[0,1).
Consider 𝑑𝑑()()𝑑,𝑑𝑑,𝐴𝑥,𝐵𝑦≤𝑘max𝑆𝑥,𝑇𝑦,𝑑𝐴𝑥,𝑆𝑥,𝑑𝐴𝑥,𝑆𝑥𝑆𝑥,𝐵𝑦𝐵𝑦,𝑇𝑦𝑇𝑦,𝐴𝑥 (51) where 𝑘∈[0,1).
Consider 𝑑𝑑𝐴𝑥,𝐵𝑦≤𝑘max²()𝑑,𝑑𝑑()𝑑,𝑑𝑑𝑆𝑥,𝑇𝑦,𝑑𝐴𝑥,𝑆𝑥𝐵𝑦,𝑇𝑦𝑆𝑥,𝐵𝑦𝑇𝑦,𝐴𝑥,𝑑𝐴𝑥,𝑆𝑥𝑆𝑥,𝐵𝑦𝐵𝑦,𝑇𝑦𝑇𝑦,𝐴𝑥^1/2, (52) where 𝑘∈[0,1).
Consider 𝑑𝑑(),𝑑+𝑑𝐴𝑥,𝐵𝑦≤𝛼𝛽max𝑆𝑥,𝑇𝑦,𝑑𝐴𝑥,𝑆𝑥,𝑑𝐵𝑦,𝑇𝑦𝑆𝑥,𝐵𝑦,𝑑𝑇𝑦,𝐴𝑥1−𝛽max²()𝑑,𝑑𝑑,𝑑()𝑑,𝑆𝑥,𝑇𝑦,𝑑𝐴𝑥,𝑆𝑥𝐵𝑦,𝑇𝑦𝑆𝑥,𝐵𝑦𝑇𝑦,𝐴𝑥𝐴𝑥,𝑆𝑥𝑇𝑦,𝐴𝑥𝑑(𝐵𝑦,𝑇𝑦)𝑑(𝑆𝑥,𝐵𝑦)^1/2, (53) where 𝛼∈[0,1) and 𝛽≥0.
Consider 𝑑²𝑑𝐴𝑥,𝐵𝑦≤𝛼max²𝑆𝑥,𝑇𝑦,𝑑²(𝐴𝑥,𝑆𝑥),𝑑²𝑑()𝑑𝑑𝑑,𝐵𝑦,𝑇𝑦+𝛽max𝐴𝑥,𝑆𝑥𝑆𝑥,𝐵𝑦,𝑑𝐵𝑦,𝑇𝑦𝑇𝑦,𝐴𝑥+𝛾𝑑𝑆𝑥,𝐵𝑦𝑇𝑦,𝐴𝑥 (54) where 𝛼,𝛽,𝛾≥0 and 𝛼+𝛾<1.
Consider 𝑑𝑑()𝑑𝑑𝑑(),𝑑,1+𝛼𝑑𝑆𝑥,𝑇𝑦𝐴𝑥,𝐵𝑦≤𝛼max𝐴𝑥,𝑆𝑥𝐵𝑦,𝑇𝑦,𝑑𝑆𝑥,𝐵𝑦𝑇𝑦,𝐴𝑥+𝛽max𝑆𝑥,𝑇𝑦,𝑑𝐴𝑥,𝑆𝑥,𝑑𝐵𝑦,𝑇𝑦𝑆𝑥,𝐵𝑦,𝑑𝑇𝑦,𝐴𝑥 (55) where 𝛼≥0 and 𝛽∈[0,1).
Consider 𝑑𝑑()𝑑(),𝑑,𝐴𝑥,𝐵𝑦≤𝛼𝑑𝑆𝑥,𝑇𝑦+𝛽max𝐴𝑥,𝑆𝑥,𝑑𝐵𝑦,𝑇𝑦+𝛾max𝐴𝑥,𝑆𝑥+𝑑𝐵𝑦,𝑇𝑦𝑆𝑥,𝐵𝑦+𝑑𝑇𝑦,𝐴𝑥 (56) where 𝛼,𝛽,𝛾≥0 and 𝛼+𝛽+2𝛾<1.
Consider 𝑑𝑑(),𝑑,𝐴𝑥,𝐵𝑦≤𝜓max𝑆𝑥,𝑇𝑦,𝑑𝐴𝑥,𝑆𝑥,𝑑𝐵𝑦,𝑇𝑦𝑆𝑥,𝐵𝑦,𝑑𝑇𝑦,𝐴𝑥 (57) where 𝜓∶ℝ⁺→ℝ is an upper semicontinuous function such that 𝜓(0)=0 and 𝜓(𝑡)<𝑡 for all 𝑡>0.
Consider 𝑑𝑑(),𝑑,𝐴𝑥,𝐵𝑦≤𝜓𝑆𝑥,𝑇𝑦,𝑑𝐴𝑥,𝑆𝑥,𝑑𝐵𝑦,𝑇𝑦𝑆𝑥,𝐵𝑦,𝑑𝑇𝑦,𝐴𝑥 (58) where 𝜓∶(ℝ⁺)⁵→ℝ is an upper semicontinuous function such that max{𝜓(0,𝑡,0,0,𝑡), 𝜓(0,0,𝑡,𝑡,0),𝜓(𝑡,0,0,𝑡,𝑡)}<𝑡 for each 𝑡>0.
Consider 𝑑²𝑑𝐴𝑥,𝐵𝑦≤𝜓²()𝑑,𝑑𝑑,𝑑()𝑑𝑑,𝑆𝑥,𝑇𝑦,𝑑𝐴𝑥,𝑆𝑥𝐵𝑦,𝑇𝑦𝑆𝑥,𝐵𝑦𝑇𝑦,𝐴𝑥𝐴𝑥,𝑆𝑥𝑇𝑦,𝐴𝑥,𝑑𝐵𝑦,𝑇𝑦𝑆𝑥,𝐵𝑦 (59) where 𝜓∶(ℝ⁺)⁵→ℝ is an upper semicontinuous function such that max{𝜓(0,0,0,𝑡,0), 𝜓(0,0,0,0,𝑡),𝜓(𝑡,0,𝑡,0,0)}<𝑡 for each 𝑡>0.
In the following contraction conditions, one denotes 𝐷=𝑑(𝐴𝑥,𝑆𝑥)+𝑑(𝐵𝑦,𝑇𝑦) and 𝐷₁=𝑑(𝑆𝑥,𝐵𝑦)+𝑑(𝑇𝑦,𝐴𝑥): 𝑑≤𝑑𝐴𝑥,𝐵𝑦𝛼𝑑𝑆𝑥,𝑇𝑦+𝛽²(𝐴𝑥,𝑆𝑥)+𝑑²𝐵𝑦,𝑇𝑦𝑑()𝑑𝐴𝑥,𝑆𝑥+𝑑𝐵𝑦,𝑇𝑦+𝛾𝑆𝑥,𝐵𝑦+𝑑𝑇𝑦,𝐴𝑥,𝑖𝑓𝐷≠0;0,𝑖𝑓𝐷=0, (60) where 𝛼,𝛽,𝛾≥0 and 𝛽+𝛾<1, 𝑑^𝑝≤𝐴𝑥,𝐵𝑦𝑘𝑑^𝑝+𝑑()𝑑𝑆𝑥,𝑇𝑦𝐴𝑥,𝑆𝑥^𝑝𝐵𝑦,𝑇𝑦+𝑑𝑆𝑥,𝐵𝑦𝑑^𝑝𝑇𝑦,𝐴𝑥𝑑(𝐴𝑥,𝑆𝑥)+𝑑𝐵𝑦,𝑇𝑦,𝑖𝑓𝐷≠0;0,𝑖𝑓𝐷=0, (61) where 𝑝≥1 and 0≤𝑘<∞, 𝑑≤𝑑𝐴𝑥,𝐵𝑦𝛼𝑑𝑆𝑥,𝑇𝑦+𝛽²𝑆𝑥,𝐵𝑦+𝑑²𝑇𝑦,𝐴𝑥𝑑𝑑()𝑆𝑥,𝐵𝑦+𝑑𝑇𝑦,𝐴𝑥+𝛾𝐴𝑥,𝑆𝑥+𝑑𝐵𝑦,𝑇𝑦,𝑖𝑓𝐷₁≠0;0,𝑖𝑓𝐷₁=0, (62) where 𝛼,𝛽,𝛾≥0 and 𝛽+𝛾<1, 𝑑≤+𝑑()𝑑𝑑𝐴𝑥,𝐵𝑦𝑘𝑑𝑆𝑥,𝑇𝑦𝐴𝑥,𝑆𝑥𝐵𝑦,𝑇𝑦+𝑑𝑆𝑥,𝐵𝑦𝑇𝑦,𝐴𝑥𝑑𝑆𝑥,𝐵𝑦+𝑑𝑇𝑦,𝐴𝑥,𝑖𝑓𝐷₁≠0;0,𝑖𝑓𝐷₁=0, (63) where 0≤𝑘<∞, 𝑑≤+𝑑()𝑑𝑑𝐴𝑥,𝐵𝑦𝑘𝑑𝑆𝑥,𝑇𝑦𝐴𝑥,𝑆𝑥𝐵𝑦,𝑇𝑦+𝑑𝑆𝑥,𝐵𝑦𝑇𝑦,𝐴𝑥𝑑()+𝑑()𝑑𝑑𝐴𝑥,𝑆𝑥+𝑑𝐵𝑦,𝑇𝑦𝐴𝑥,𝑆𝑥𝑆𝑥,𝐵𝑦+𝑑𝐵𝑦,𝑇𝑦𝑇𝑦,𝐴𝑥𝑑𝑆𝑥,𝐵𝑦+𝑑𝑇𝑦,𝐴𝑥,𝑖𝑓𝐷≠0,𝐷₁≠0;0,𝑖𝑓𝐷=0,𝐷₁=0, (64) where 0≤𝑘<∞𝑑≤𝑑()𝑑𝑑𝐴𝑥,𝐵𝑦𝐴𝑥,𝑆𝑥𝐵𝑦,𝑇𝑦+𝑑𝑆𝑥,𝐵𝑦𝑇𝑦,𝐴𝑥,𝑑𝑑()1+𝑑𝑆𝑥,𝑇𝑦𝐴𝑥,𝐵𝑦≤𝛼𝑑𝑆𝑥,𝑇𝑦+𝛽𝐴𝑥,𝑆𝑥+𝑑𝐵𝑦,𝑇𝑦𝑑,1+𝑑𝑆𝑥,𝐵𝑦𝑇𝑦,𝐴𝑥 (65) where 𝛼,𝛽∈[0,1), 𝑑²𝐴𝑥,𝐵𝑦≤𝛼𝑑²𝑑𝑑𝑆𝑥,𝑇𝑦+𝛽𝑆𝑥,𝐵𝑦𝑇𝑦,𝐴𝑥1+𝑑²(𝐴𝑥,𝑆𝑥)+𝑑²,𝐵𝑦,𝑇𝑦 (66) where 𝛼,𝛽≥0 and 𝛼+𝛽<1, 𝑑³≤𝑑𝐴𝑥,𝐵𝑦₂(𝐴𝑥,𝑆𝑥)𝑑²𝐵𝑦,𝑇𝑦+𝑑²𝑆𝑥,𝐵𝑦𝑑²𝑇𝑦,𝐴𝑥,𝑑1+𝑑𝑆𝑥,𝑇𝑦₃𝐴𝑥,𝐵𝑦≤𝛼𝑑²𝑑𝑑()𝑑𝐴𝑥,𝐵𝑦𝑆𝑥,𝑇𝑦+𝛽𝑑𝐴𝑥,𝐵𝑦𝐴𝑥,𝑆𝑥𝐵𝑦,𝑇𝑦+𝛾𝑑²𝑆𝑥,𝐵𝑦𝑑𝑇𝑦,𝐴𝑥+𝜂𝑑𝑆𝑥,𝐵𝑦𝑑²𝑇𝑦,𝐴𝑥, (67) where 𝛼,𝛾,𝜂,𝛽≥0 and 𝛼+𝛾+𝜂<1.

Proof. The proof of each inequality (50)–(67) easily follows from Theorem 29 in view of Examples 7–26.

Remark 35. Corollary 34 improves and generalizes a multitude of well-known results especially those contained in [12, 14, 29, 30, 35, 40–42].
By choosing 𝐴,𝐵,𝑆, and 𝑇 suitably, we can deduce corollaries involving two as well as three self-mappings. For the sake of naturality, we only derive the following corollary involving a pair of self-mappings.

Corollary 36. Let 𝐴 and 𝑆 be self-mappings of a metric space (𝑋,𝑑). Suppose the following: (1)the pair (𝐴,𝑆) satisfies the (CLR_𝑆) property, (2)there exists 𝜙∈Φ such that 𝜙𝑑(),𝑑𝐴𝑥,𝐴𝑦,𝑑𝑆𝑥,𝑆𝑦,𝑑𝐴𝑥,𝑆𝑥,𝑑𝐴𝑦,𝑆𝑦𝑆𝑥,𝐴𝑦,𝑑𝑆𝑦,𝐴𝑥≤0, (68) for all 𝑥,𝑦∈𝑋.
Then (𝐴,𝑆) has a coincidence point. Moreover, if the pair (𝐴,𝑆) is weakly compatible, then the pair has a unique common fixed point in 𝑋.

As an application of Theorem 29, we have the following result involving four finite families of self-mappings.

Theorem 37. Let {𝐴_𝑖}^𝑚_𝑖=1, {𝐵_𝑗}^𝑛_𝑟=1, {𝑆_𝑘}^𝑝_𝑘=1, and {𝑇_𝑙}^𝑞_𝑙=1 be four finite families of self-mappings of a metric space (𝑋,𝑑) with 𝐴=𝐴₁𝐴₂⋯𝐴_𝑚, 𝐵=𝐵₁𝐵₂⋯𝐵_𝑛, 𝑆=𝑆₁𝑆₂⋯𝑆_𝑝 and 𝑇=𝑇₁𝑇₂⋯𝑇_𝑞 satisfying the condition (25). Suppose that the pairs (𝐴,𝑆) and (𝐵,𝑇) enjoy the (CLR_𝑆𝑇) property, then (𝐴,𝑆) and (𝐵,𝑇) have a point of coincidence each.
Moreover, {𝐴_𝑖}^𝑚_𝑖=1,{𝐵_𝑗}^𝑛_𝑗=1,{𝑆_𝑘}^𝑝_𝑘=1, and {𝑇_𝑙}^𝑞_𝑙=1 have a unique common fixed point if the families ({𝐴_𝑖},{𝑆_𝑘}) and ({𝐵_𝑟},{𝑇_ℎ}) commute pairwise wherein 𝑖∈{1,2,…,𝑚}, 𝑘∈{1,2,…,𝑝}, 𝑗∈{1,2,…,𝑛}, and 𝑙∈{1,2,…,𝑞}.

Proof. The proof of this theorem can be completed on the lines of Theorem 2.2 of Imdad et al. [10, Theorem 2.2].

Remark 38. (1) A result similar to Theorem 37 can be outlined in respect of Theorem 29.
(2) Theorem 37 improves and extends the result of Ćirić et al. [5] and Pathak et al. [25].
Now, we indicate that Theorem 37 can be utilized to derive common fixed point theorems for any finite number of mappings. As a sample, for five mappings, we can derive the following by setting one family of two members while the remaining three of single members.

Corollary 39. Let 𝐴,𝐵,𝑅,𝑆, and 𝑇 be self-mappings of a metric space (𝑋,𝑑). Suppose the following: (1)the pairs (𝐴,𝑆𝑅) and (𝐵,𝑇) share the (CLR_{(𝑆𝑅)(𝑇)}) property, (2)there exists 𝜙∈Φ such that 𝜙𝑑(),𝑑𝐴𝑥,𝐵𝑦,𝑑𝑆𝑅𝑥,𝑇𝑦,𝑑𝐴𝑥,𝑆𝑅𝑥,𝑑𝐵𝑦,𝑇𝑦𝑆𝑅𝑥,𝐵𝑦,𝑑𝑇𝑦,𝐴𝑥≤0, (69) for all 𝑥,𝑦∈𝑋.
Then (𝐴,𝑆𝑅) and (𝐵,𝑇) have a coincidence point each. Moreover, 𝐴,𝐵,𝑅,𝑆, and 𝑇 have a unique common fixed point provided that both pairs (𝐴,𝑆𝑅) and (𝐵,𝑇) commute pairwise, that is, 𝐴𝑆=𝑆𝐴, 𝐴𝑅=𝑅𝐴, 𝑆𝑅=𝑅𝑆, 𝐵𝑇=𝑇𝐵.

Similarly, we can derive a common fixed point theorem for six mappings by setting two families of two members while the rest two of single members.

Corollary 40. Let 𝐴,𝐵,𝐻,𝑅,𝑆, and 𝑇 be self-mappings of a metric space (𝑋,𝑑). Suppose that (1)the pairs (𝐴,𝑆𝑅) and (𝐵,𝑇𝐻) share the (CLR_{(𝑆𝑅)(𝑇𝐻)}) property, (2)there exists 𝜙∈Φ such that 𝜙𝑑(),𝑑𝐴𝑥,𝐵𝑦,𝑑𝑆𝑅𝑥,𝑇𝐻𝑦,𝑑𝐴𝑥,𝑆𝑅𝑥,𝑑𝐵𝑦,𝑇𝐻𝑦𝑆𝑅𝑥,𝐵𝑦,𝑑𝑇𝐻𝑦,𝐴𝑥≤0, (70) for all 𝑥,𝑦∈𝑋.
Then (𝐴,𝑆𝑅) and (𝐵,𝑇𝐻) have a coincidence point each. Moreover, 𝐴,𝐵,𝐻,𝑅,𝑆, and 𝑇 have a unique common fixed point provided that both pairs (𝐴,𝑆𝑅) and (𝐵,𝑇𝐻) commute pairwise, that is, 𝐴𝑆=𝑆𝐴, 𝐴𝑅=𝑅𝐴, 𝑆𝑅=𝑅𝑆, 𝐵𝑇=𝑇𝐵, 𝐵𝐻=𝐻𝐵, and 𝑇𝐻=𝐻𝑇.

By setting 𝐴₁=𝐴₂=⋯=𝐴_𝑚=𝐴, 𝐵₁=𝐵₂=⋯=𝐵_𝑛=𝐵, 𝑆₁=𝑆₂=⋯=𝑆_𝑝=𝑆 and 𝑇₁=𝑇₂=⋯=𝑇_𝑞=𝑇 in Theorem 37, we deduce the following.

Corollary 41. Let 𝐴,𝐵,𝑆, and 𝑇 be self-mappings of a metric space (𝑋,𝑑). Suppose the following: (1)the pairs (𝐴^𝑚,𝑆^𝑝) and (𝐵^𝑛,𝑇^𝑞) share the (CLR_𝑆^𝑝_,𝑇^𝑞) property, (2)there exists 𝜙∈Φ such that 𝜙𝑑𝐴^𝑚𝑥,𝐵^𝑛𝑦,𝑑𝑆^𝑝𝑥,𝑇^𝑞𝑦,𝑑𝐴^𝑚𝑥,𝑆^𝑝𝑥,𝑑𝐵^𝑛𝑦,𝑇^𝑞𝑦,𝑑𝑆^𝑝𝑥,𝐵^𝑛𝑦,𝑑𝑇^𝑞𝑦,𝐴^𝑚𝑥≤0, (71) for all 𝑥,𝑦∈𝑋 where 𝑚,𝑛,𝑝,𝑞 are fixed positive integers.
Then 𝐴,𝐵,𝑆, and 𝑇 have a unique common fixed point provided that 𝐴𝑆=𝑆𝐴 and 𝐵𝑇=𝑇𝐵.

Remark 42. Corollary 41 is a slight but partial generalization of Theorem 29 as the commutativity requirements (i.e., 𝐴𝑆=𝑆𝐴 and 𝐵𝑇=𝑇𝐵) in this corollary are relatively stronger as compared to weak compatibility in Theorem 29.

Remark 43. Results similar to Corollary 41 can be derived from Theorem 29 and Corollary 34.

Remark 44. It may be noticed that the earlier proved results, namely, Theorems 29–37 (also Corollaries 36–41) remain valid in symmetric space (𝑋,𝑑) whenever 𝑑 is continuous.

Acknowledgment

The authors wish to dedicate this work to Professor Mohammad Saeed Khan for his crucial role in their initial development in general and his fruitful observations on the first draft of this paper in particular.