Unified Fixed Point Theorems via Common Limit Range Property in Modified Intuitionistic Fuzzy Metric Spaces

Abstract

The purpose of this paper is to emphasize the role of “common limit range property” to ascertain the existence of common fixed points in modified intuitionistic fuzzy metric spaces enjoying an implicit function utilized in Tanveer et al. (2012) and Imdad et al. (2012). As an application to our main result, we derive a fixed point theorem for finite families of self-mappings. We also give some examples which demonstrate the validity of the hypotheses and degree of generality of our main results. Our results improve and extend several previously known fixed point theorems of the existing literature.

1. Introduction

The fruitful and productive idea of fuzzy set was initiated by Zadeh [1]. In an attempt to generalize the idea of fuzzy set, Atanassov [2] introduced the notion of intuitionistic fuzzy set. Thereafter, Çoker [3] defined a topology on intuitionistic fuzzy sets, while Mondal and Samanta [4] introduced the idea of intuitionistic gradation of openness. Park [5] introduced the notion of intuitionistic fuzzy metric space (abbreviated by IFMS in the sequel) as a generalization of fuzzy metric space, especially the one due to George and Veeramani [6]. In recent years, many authors proved a multitude of fixed point theorems in IFMS (e.g., see [7–15]).

Later on, Gregori et al. [16] showed that the topology induced by fuzzy metric coincides with the topology induced by intuitionistic fuzzy metric. In an attempt to remove this shortcoming, Saadati and Park [17] proposed the idea of modified IFMS wherein the notions of continuous t-norm and continuous t-conorm are employed besides adopting the notion of compatible mappings (essentially due to Jungck [18]). Jain et al. [19] proved some unique common fixed point theorems for four self-mappings satisfying a new contractive condition in modified IFMS through compatibility of type (𝑃). Saadati and Park [17] extended the notion of weak compatibility (due to Jungck and Rhoades [20]) to modified IFMS. However, the study of common fixed points of noncompatible mappings due to Pant [21] is also equally natural. Tanveer et al. [22] and Imdad et al. [23] utilized the notions of the property (E.A) (due to Aamri and Moutawakil [24]) and the common property (E.A) (due to Liu et al. [25]) to prove some interesting results in modified intuitionistic fuzzy metric spaces. One may notice that the property (E.A) does require the closedness of certain underlying subspaces to ascertain the existence of common fixed point. Sintunavarat and Kumam [26] coined the idea of “common limit range property” which never requires the closedness of any underlying subspace for the existence of common fixed points (also see [27]). Most recently, Chauhan et al. [28, 29] and Sintunavarat et al. [14] proved some interesting fixed point results for mappings defined on modified IFMS via common limit range property. Imdad et al. [30] extended the notion of common limit range property to two pairs of self-mappings and proved some fixed point results in Menger and metric spaces. We cite some recent papers (e.g., [31–37]) which demonstrate the superiority of common limit range property over the property (E.A) in various settings.

In this paper, utilizing an implicit function due to Tanveer et al. [22] (also Imdad et al. [23]), we prove some common fixed point theorems for two pairs of weakly compatible mappings in modified IFMS employing the common limit range property. In process, many known results (especially those contained in Imdad et al. [23]) are enriched and improved. Some related results are also derived besides furnishing illustrative examples.

2. Preliminaries

Lemma 1 (see [38]). Consider the set 𝐿^∗ and operation ≤_𝐿^∗ defined by 𝐿^∗=𝑥₁,𝑥₂∶𝑥₁,𝑥₂∈[0,1]²,𝑥₁+𝑥₂≤1, (1)(𝑥₁,𝑥₂)≤_𝐿^∗(𝑦₁,𝑦₂)⇔𝑥₁≤𝑦₁ and 𝑥₂≥𝑦₂, for every (𝑥₁,𝑥₂),(𝑦₁,𝑦₂)∈𝐿^∗. Then (𝐿^∗,≤_𝐿^∗) is a complete lattice.

Definition 2 (see [2]). An intuitionistic fuzzy set 𝐴_𝜁,𝜂 in a universe 𝒰 is an object 𝐴_𝜁,𝜂={(𝜁_𝐴(𝑢),𝜂_𝐴(𝑢)∣𝑢∈𝒰)}, where, for all 𝑢∈𝒰, 𝜁_𝐴(𝑢)∈[0,1] and 𝜂_𝐴(𝑢)∈[0,1] are, respectively, called the membership degree and the nonmembership degree of 𝑢∈𝐴_𝜁,𝜂 which also satisfy 𝜁_𝐴(𝑢)+𝜂_𝐴(𝑢)≤1.
For every 𝑧_𝑖=(𝑥_𝑖,𝑦_𝑖)∈𝐿^∗, if 𝑐_𝑖∈[0,1] such that ∑^𝑛_𝑗=1𝑐_𝑗=1, then it is easy to see that 𝑐₁𝑥₁,𝑦₁+⋯+𝑐_𝑛𝑥_𝑛,𝑦_𝑛=^𝑛_𝑗=1𝑐_𝑗𝑥_𝑗,𝑦_𝑗=^𝑛_𝑗=1𝑐_𝑗𝑥_𝑗,^𝑛_𝑗=1𝑐_𝑗𝑦_𝑗∈𝐿^∗. (2)

We denote its units by 0_𝐿^∗=(0,1) and 1_𝐿^∗=(1,0). Classically, a triangular norm ∗=T on [0,1] is defined as an increasing, commutative, associative mapping T∶[0,1]²→[0,1] satisfying T(1,𝑥)=1∗𝑥=𝑥, for all 𝑥∈[0,1]. A triangular conorm S=⋄ is defined as an increasing, commutative, associative mapping S∶[0,1]²→[0,1] satisfying S(0,𝑥)=0⋄𝑥=𝑥, for all 𝑥∈[0,1]. Using the lattice (𝐿^∗,≤_𝐿^∗), these definitions can be easily extended.

Definition 3 (see [39]). A triangular norm (t-norm) on 𝐿^∗ is a mapping 𝒯∶(𝐿^∗)²→𝐿^∗ satisfying the following conditions for all 𝑥,𝑦,𝑥^′,𝑦^′∈𝐿^∗: (1)𝒯(𝑥,1_𝐿^∗)=𝑥, (2)𝒯(𝑥,𝑦)=𝒯(𝑦,𝑥), (3)𝒯(𝑥,𝒯(𝑦,𝑧))=𝒯(𝒯(𝑥,𝑦),𝑧), (4)𝑥≤_𝐿^∗𝑥^′ and 𝑦≤_𝐿^∗𝑦^′⇒𝒯(𝑥,𝑦)≤_𝐿^∗𝒯(𝑥^′,𝑦^′).

Definition 4 (see [38, 39]). A continuous t-norm 𝒯 on 𝐿^∗ is called continuous t-representable if and only if there exist a continuous t-norm ∗ and a continuous t-conorm ⋄ on [0,1] such that, for all 𝑥=(𝑥₁,𝑥₂), 𝑦=(𝑦₁,𝑦₂)∈𝐿^∗, 𝒯𝑥,𝑦=𝑥₁∗𝑦₁,𝑥₂⋄𝑦₂. (3) Now, we define a sequence {𝒯_𝑛} recursively by {𝒯¹=𝒯} and 𝒯^𝑛𝑥⁽¹⁾,…,𝑥^(𝑛+1)=𝒯𝒯^𝑛−1𝑥⁽¹⁾,…,𝑥^(𝑛),𝑥^(𝑛+1), (4) for 𝑛≥2 and 𝑥^(𝑖)∈𝐿^∗.

Definition 5 (see [38, 39]). A negator on 𝐿^∗ is any decreasing mapping 𝒩∶𝐿^∗→𝐿^∗ satisfying 𝒩(0_𝐿^∗)=1_𝐿^∗ and 𝒩(1_𝐿^∗)=0_𝐿^∗. If 𝒩(𝒩(𝑥))=𝑥, for all 𝑥∈𝐿^∗, then 𝒩 is called an involutive negator. A negator on [0,1] is a decreasing mapping 𝑁∶[0,1]→[0,1] satisfying 𝑁(0)=1 and 𝑁(1)=0. Notice that 𝑁_𝑠 stands for standard negator on [0,1] defined by (for all 𝑥∈[0,1]) 𝑁_𝑠(𝑥)=1−𝑥.

Definition 6 (see [17]). Let 𝑀, 𝑁 be fuzzy sets from 𝑋²×(0,∞) to [0,1] such that 𝑀(𝑥,𝑦,𝑡)+𝑁(𝑥,𝑦,𝑡)≤1 for all 𝑥,𝑦∈𝑋 and 𝑡>0. The 3-tuple (𝑋,ℳ_𝑀,𝑁,𝒯) is said to be a modified IFMS if 𝑋 is an arbitrary nonempty set, 𝒯 is a continuous t-representable, and ℳ_𝑀,𝑁 is an intuitionistic fuzzy set from 𝑋²×(0,∞)→𝐿^∗ satisfying the following conditions (for every 𝑥,𝑦,𝑧∈𝑋 and 𝑡,𝑠>0): (1)ℳ_𝑀,𝑁(𝑥,𝑦,𝑡)>_𝐿^∗0_𝐿^∗, (2)ℳ_𝑀,𝑁(𝑥,𝑦,𝑡)=1_𝐿^∗ if and only if 𝑥=𝑦, (3)ℳ_𝑀,𝑁(𝑥,𝑦,𝑡)=ℳ_𝑀,𝑁(𝑦,𝑥,𝑡), (4)ℳ_𝑀,𝑁(𝑥,𝑦,𝑡+𝑠)≥_𝐿^∗𝒯(ℳ_𝑀,𝑁(𝑥,𝑧,𝑡),ℳ_𝑀,𝑁(𝑧,𝑦,𝑠)), (5)ℳ_𝑀,𝑁(𝑥,𝑦,⋅)∶(0,∞)→𝐿^∗ is continuous.In this case, ℳ_𝑀,𝑁 is called a modified intuitionistic fuzzy metric. Here, ℳ_𝑀,𝑁𝑥,𝑦,𝑡=𝑀𝑥,𝑦,𝑡,𝑁𝑥,𝑦,𝑡. (5)

Remark 7. In an intuitionistic fuzzy metric space (𝑋,ℳ_𝑀,𝑁,𝒯), 𝑀(𝑥,𝑦,⋅) is nondecreasing and 𝑁(𝑥,𝑦,⋅) is nonincreasing for all 𝑥,𝑦∈𝑋. Hence (𝑋,ℳ_𝑀,𝑁,𝒯) is nondecreasing function for all 𝑥,𝑦∈𝑋.

Example 8 (see [17]). Let (𝑋,𝑑) be a metric space. Define 𝒯(𝑎,𝑏)=(𝑎₁𝑏₁,min{𝑎₂+𝑏₂,1}) for all 𝑎=(𝑎₁,𝑎₂) and 𝑏=(𝑏₁,𝑏₂)∈𝐿^∗, and let 𝑀 and 𝑁 be fuzzy sets on 𝑋²×(0,∞) defined as follows: ℳ_𝑀,𝑁=𝑀=𝑥,𝑦,𝑡𝑥,𝑦,𝑡,𝑁𝑥,𝑦,𝑡ℎ𝑡^𝑛ℎ𝑡^𝑛,+𝑚𝑑𝑥,𝑦𝑚𝑑𝑥,𝑦ℎ𝑡^𝑛,+𝑚𝑑𝑥,𝑦 (6) for all ℎ,𝑚,𝑛,𝑡∈ℝ⁺. Then (𝑋,ℳ_𝑀,𝑁,𝒯) is a modified IFMS.

Example 9 (see [17]). Let 𝑋=ℕ. Define 𝒯(𝑎,𝑏)=(max{0,𝑎₁+𝑏₁−1},𝑎₂+𝑏₂−𝑎₂𝑏₂) for all 𝑎=(𝑎₁,𝑎₂) and 𝑏=(𝑏₁,𝑏₂)∈𝐿^∗, and let 𝑀 and 𝑁 be fuzzy sets on 𝑋²×(0,∞). Then ℳ_𝑀,𝑁(𝑥,𝑦,𝑡) is defined as (for all 𝑥,𝑦∈𝑋 and 𝑡>0) follows: ℳ_𝑀,𝑁=𝑀=𝑥𝑥,𝑦,𝑡𝑥,𝑦,𝑡,𝑁𝑥,𝑦,𝑡𝑦,𝑦−𝑥𝑦𝑦,if𝑥≤𝑦,𝑥,𝑥−𝑦𝑥,if𝑦≤𝑥. (7) Then (𝑋,ℳ_𝑀,𝑁,𝒯) is a modified IFMS.

Definition 10 (see [17]). Let (𝑋,ℳ_𝑀,𝑁,𝒯) be a modified IFMS. For 𝑡>0, define the open ball ℬ(𝑥,𝑟,𝑡) with center 𝑥∈𝑋 and radius 0<𝑟<1 as ℬ(𝑥,𝑟,𝑡)=𝑦∈𝑋∶ℳ_𝑀,𝑁𝑥,𝑦,𝑡>_𝐿^∗𝑁_𝑠(𝑟),𝑟. (8)
A subset 𝒜⊂𝑋 is called open if for each 𝑥∈𝒜 there exist 𝑡>0 and 0<𝑟<1 such that ℬ(𝑥,𝑟,𝑡)⊆𝒜. If 𝜏_ℳ𝑀,𝑁 denotes the family of all open subsets of 𝑋, then 𝜏_ℳ𝑀,𝑁 is called the topology induced by intuitionistic fuzzy metric ℳ_𝑀,𝑁. Notice that this topology is Hausdorff (see [5], Remark 3.3, Theorem 3.5).

Definition 11 (see [17]). A sequence {𝑥_𝑛} in a modified IFMS (𝑋,ℳ_𝑀,𝑁,𝒯) is called a Cauchy sequence if for each 0<𝜖<1 and 𝑡>0 there exists 𝑛₀∈ℕ such that ℳ_𝑀,𝑁𝑥_𝑛,𝑦_𝑚,𝑡>_𝐿^∗𝑁_𝑠(𝜖),𝜖, (9) and for each 𝑛,𝑚≥𝑛₀, where 𝑁_𝑠 is a standard negator. The sequence {𝑥_𝑛} is said to be convergent to 𝑥∈𝑋 in the modified IFMS (𝑋,ℳ_𝑀,𝑁,𝒯) and is generally denoted by 𝑥_𝑛→^ℳ𝑀,𝑁𝑥 if ℳ_𝑀,𝑁(𝑥_𝑛,𝑥,𝑡)→1_𝐿^∗ whenever 𝑛→∞ for every 𝑡>0. A modified IFMS is said to be complete if and only if every Cauchy sequence is convergent.

Lemma 12 (see [17]). Let ℳ_𝑀,𝑁 be an intuitionistic fuzzy metric. Then, for any 𝑡>0, ℳ_𝑀,𝑁(𝑥,𝑦,𝑡) is nondecreasing with respect to 𝑡 in (𝐿^∗,≤_𝐿^∗), for all 𝑥,𝑦∈𝑋.

Definition 13 (see [17]). Let (𝑋,ℳ_𝑀,𝑁,𝒯) be a modified IFMS. Then ℳ_𝑀,𝑁 is said to be continuous on 𝑋×𝑋×(0,∞), if lim_𝑛→∞ℳ_𝑀,𝑁𝑥_𝑛,𝑦_𝑛,𝑡_𝑛=ℳ_𝑀,𝑁𝑥,𝑦,𝑡, (10) whenever a sequence {(𝑥_𝑛,𝑦_𝑛,𝑡_𝑛)} in 𝑋×𝑋×(0,∞) converges to a point {(𝑥,𝑦,𝑡)}∈𝑋×𝑋×(0,∞); that is, lim_𝑛→∞ℳ_𝑀,𝑁𝑥_𝑛,𝑥,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝑦_𝑛,𝑦,𝑡=1_𝐿^∗,lim_𝑛→∞ℳ_𝑀,𝑁𝑥,𝑦,𝑡_𝑛=ℳ_𝑀,𝑁𝑥,𝑦,𝑡. (11)

Lemma 14 (see [17]). Let (𝑋,ℳ_𝑀,𝑁,𝒯) be a modified IFMS. Then, ℳ_𝑀,𝑁 is continuous function on 𝑋×𝑋×(0,∞).

Definition 15. Let 𝐴 and 𝑆 be two mappings from a modified IFM space (𝑋,ℳ_𝑀,𝑁,𝒯) into itself. Then this pair of mappings is said to be (1)commuting if 𝐴𝑆𝑥=𝑆𝐴𝑥, for all 𝑥∈𝑋;(2)weakly commuting [17] if ℳ_𝑀,𝑁(𝐴𝑆𝑥,𝑆𝐴𝑥,𝑡)≥_𝐿^∗ℳ_𝑀,𝑁(𝐴𝑥,𝑆𝑥,𝑡), (12) for all 𝑥∈𝑋 and 𝑡>0;(3)compatible [17] if lim_𝑛→∞ℳ_𝑀,𝑁𝐴𝑆𝑥_𝑛,𝑆𝐴𝑥_𝑛,𝑡=1_𝐿^∗, (13) for all 𝑡>0 whenever {𝑥_𝑛} is a sequence in 𝑋 such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=𝑥∈𝑋; (14)(4)noncompatible [22] if there exists at least one sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=𝑥∈𝑋, (15) but lim_𝑛→∞ℳ_𝑀,𝑁(𝐴𝑆𝑥_𝑛,𝑆𝐴𝑥_𝑛,𝑡)≠1_𝐿^∗ or nonexistent for at least one 𝑡>0.

Definition 16 (see [40]). 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_𝑎𝐴_𝑏=𝐴_𝑏𝐴_𝑎, for all 𝑎,𝑏∈{1,2,…,𝑚}; (2)𝑆_𝑐𝑆_𝑑=𝑆_𝑑𝑆_𝑐, for all 𝑐,𝑑∈{1,2,…,𝑛}; (3)𝐴_𝑎𝑆_𝑐=𝑆_𝑐𝐴_𝑎, for all 𝑎∈{1,2,…,𝑚} and 𝑐∈{1,2,…,𝑛}.

Definition 17 (see [12]). Let 𝐴 and 𝑆 be two mappings from a modified IFMS (𝑋,ℳ_𝑀,𝑁,𝒯) into itself. Then this pair of mappings is said to satisfy the property (E.A) if there exists a sequence {𝑥_𝑛} in 𝑋 such that for all 𝑡>0lim_𝑛→∞ℳ_𝑀,𝑁𝐴𝑥_𝑛,𝑧,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝑆𝑥_𝑛,𝑧,𝑡=1_𝐿^∗, (16) for some 𝑧∈𝑋.

Definition 18 (see [22]). Two pairs (𝐴,𝑆) and (𝐵,𝑇) of self-mappings of a modified IFMS (𝑋,ℳ_𝑀,𝑁,𝒯) are said to satisfy the common property (E.A) if there exist sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that lim_𝑛→∞ℳ_𝑀,𝑁𝐴𝑥_𝑛,𝑧,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝑆𝑥_𝑛,𝑧,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝐵𝑦_𝑛,𝑧,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝑇𝑦_𝑛,𝑧,𝑡=1_𝐿^∗, (17) for some 𝑧∈𝑋 and 𝑡>0.

Definition 19 (see [14]). A pair (𝐴,𝑆) of self-mappings of a modified IFMS (𝑋,ℳ_𝑀,𝑁,𝒯) 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 for all 𝑡>0lim_𝑛→∞ℳ_𝑀,𝑁𝐴𝑥_𝑛,𝑧,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝑆𝑥_𝑛,𝑧,𝑡=1_𝐿^∗, (18) where 𝑧∈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 [14, 29]).

Now, we extend common limit range property for two pairs of self-mappings in the framework of modified IFMS (𝑋,ℳ_𝑀,𝑁,𝒯) as follows.

Definition 20. Two pairs (𝐴,𝑆) and (𝐵,𝑇) of self-mappings of a modified IFMS (𝑋,ℳ_𝑀,𝑁,𝒯) 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→∞ℳ_𝑀,𝑁𝐴𝑥_𝑛,𝑧,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝑆𝑥_𝑛,𝑧,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝐵𝑦_𝑛,𝑧,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝑇𝑦_𝑛,𝑧,𝑡=1_𝐿^∗, (19) where 𝑧∈𝑆(𝑋)∩𝑇(𝑋) and 𝑡>0.

By setting 𝐴=𝐵 and 𝑆=𝑇 in Definition 20 implies Definition 19 (due to Sintunavarat et al. [14]), whereas Definition 20 implies Definition 18, but the converse implications are not true in general. The following example substantiates this fact.

Example 21. Let (𝑋,ℳ_𝑀,𝑁,𝒯) be a modified IFMS, where 𝑋=[3,20] and ℳ_𝑀,𝑁(𝑥,𝑦,𝑡)=(𝑡/(𝑡+|𝑥−𝑦|),|𝑥−𝑦|/(𝑡+|𝑥−𝑦|)) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Define four self-mappings 𝐴, 𝐵, 𝑆, and 𝑇 on 𝑋 as 𝐴(𝑥)=7,if𝑥=3;5,if3<𝑥≤14;𝑥+15𝐵(𝑥)=,if𝑥>14,4,if𝑥=3;4𝑥+35𝑆(𝑥)=(),if3<𝑥≤14;13,if𝑥>14,5,if𝑥=3;15,if3<𝑥≤14;2𝑥−19𝑇(𝑥)=,if𝑥>14,6,if𝑥=3;𝑥+32,if3<𝑥≤14;17,if𝑥>14. (20)
If we choose two sequences as {𝑥_𝑛}={14+1/𝑛}_𝑛∈ℕ and {𝑦_𝑛}={3+1/𝑛}_𝑛∈ℕ), then the pairs (𝐴,𝑆) and (𝐵,𝑇) enjoy the common property (E.A) for all 𝑡>0: lim_𝑛→∞ℳ_𝑀,𝑁𝐴𝑥_𝑛,3,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝑆𝑥_𝑛,3,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝐵𝑦_𝑛,3,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝑇𝑦_𝑛,3,𝑡=1_𝐿^∗, (21) where 3∈𝑋. Here it is noticed that 3∉𝑆(𝑋)∩𝑇(𝑋). Therefore, the pairs (𝐴,𝑆) and (𝐵,𝑇) do not satisfy the common limit range property with respect to mappings 𝑆 and 𝑇.

In view of Example 21, the following proposition is predictable.

Proposition 22. If the pairs (𝐴,𝑆) and (𝐵,𝑇) share the common property (E.A) and 𝑆(𝑋) as well as 𝑇(𝑋) are closed subsets of 𝑋, then the pairs also enjoy the (𝐶𝐿𝑅_𝑆𝑇) property.

3. Implicit Relations

On the lines of Imdad et al. [23], we adopt an implicit function which covers a multitude of contraction conditions in one go as exhibited by demonstrative examples.

Let Ψ be the set of all upper continuous functions 𝜑(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶𝐿^∗6→𝐿^∗, satisfying the following conditions (for all 𝑢,0,1∈𝐿^∗, where 𝑢=(𝑢₁,𝑢₂), 𝟎=0_𝐿^∗=(0,0), and 𝟏=1_𝐿^∗=(1,0)):(𝜑₁)𝜑(𝑢,𝟏,𝑢,𝟏,𝟏,𝑢)<_𝐿^∗𝟎, for all 𝑢>_𝐿^∗𝟎;(𝜑₂)𝜑(𝑢,𝟏,𝟏,𝑢,𝑢,𝟏)<_𝐿^∗𝟎, for all 𝑢>_𝐿^∗𝟎;(𝜑₃)𝜑(𝑢,𝑢,𝟏,𝟏,𝑢,𝑢)<_𝐿^∗𝟎, for all 𝑢>_𝐿^∗𝟎.

Example 23. Define 𝜑(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶𝐿^∗6→𝐿^∗ as 𝜑𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝛼min𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆, (22) where 𝛼>1.

Example 24. Define 𝜑(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶𝐿^∗6→𝐿^∗ as 𝜑𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡²₁−𝛼₁min𝑡²₂,𝑡²₃,𝑡²₄−𝛼₂min𝑡₃𝑡₆,𝑡₄𝑡₅, (23) where 𝛼₁,𝛼₂>0, 𝛼₁+𝛼₂>1, and 𝛼₁≥1.

Example 25. Define 𝜑(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶𝐿^∗6→𝐿^∗ as 𝜑𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡³₁−𝛼min𝑡²₁𝑡₂,𝑡₁𝑡₃𝑡₄,𝑡²₅𝑡₆,𝑡₅𝑡²₆, (24) where 𝛼>1.

Example 26. Define 𝜑(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶𝐿^∗6→𝐿^∗ as 𝜑𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡³₁𝑡−𝛼²³𝑡²₄+𝑡²₅𝑡²₆𝑡₂+𝑡₃+𝑡₄, (25) where 𝛼≥3/2.

Example 27. Define 𝜑(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶𝐿^∗6→𝐿^∗ as 𝜑𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=1+𝛼𝑡₂𝑡₁−𝛼min𝑡₃𝑡₄,𝑡₅𝑡₆−𝜓min𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆, (26) where 𝛼≥0 and 𝜓∶𝐿^∗→𝐿^∗ is a continuous function such that 𝜓(𝑠)>_𝐿^∗𝑠 for all 𝑠∈𝐿^∗⧵{𝟎,𝟏}.

Example 28. Define 𝜑(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶𝐿^∗6→𝐿^∗ as 𝜑𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡²₁𝑡−𝛼²²+𝑡²₃+𝑡²₄𝑡₅+𝑡₆, (27) where 𝛼≥2.

Example 29. Define 𝜑(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶𝐿^∗6→𝐿^∗ as 𝜑𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝜓min𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆, (28) where 𝜓∶𝐿^∗→𝐿^∗ is a continuous function such that 𝜓(𝑠)>_𝐿^∗𝑠 for all 𝑠∈𝐿^∗⧵{𝟎,𝟏}.

Example 30. Define 𝜑(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶𝐿^∗6→𝐿^∗ as 𝜑𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡³₁𝑡−𝛼²³𝑡²₄𝑡₂+𝑡₅+𝑡₆, (29) where 𝛼≥3.

Example 31. Define 𝜑(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶𝐿^∗6→𝐿^∗ as 𝜑𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡²₁−𝛼₁min𝑡²₂,𝑡²₃,𝑡²₄−𝛼₂𝑡₅𝑡₅+𝑡₆, (30) where 𝛼₁≥1 and 𝛼₂>0.

Example 32. Define 𝜑(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶𝐿^∗6→𝐿^∗ as 𝜑𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡²₁−𝛼₁min𝑡²₂,𝑡²₅,𝑡²₆−𝛼₂𝑡₃𝑡₃+𝑡₄, (31) where 𝛼₁≥1 and 𝛼₂>0.

Example 33. Define 𝜑(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶𝐿^∗6→𝐿^∗ as 𝜑𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝛼₁𝑡₂−𝛼₂𝑡₃−𝛼₃𝑡₄−𝛼₄𝑡₅−𝛼₅𝑡₆, (32) where 𝛼₁,𝛼₂,𝛼₃,𝛼₄,𝛼₅>0, 𝛼₂+𝛼₅≥1, 𝛼₃+𝛼₄≥1, and 𝛼₁+𝛼₄+𝛼₅≥1.

On the lines of Tanveer et al. [22], let Φ be the set of all continuous functions 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶𝐿^∗6→𝐿^∗, satisfying (for all 𝑢,𝑣,𝟏∈𝐿^∗, where 𝑢=(𝑢₁,𝑢₂), 𝑣=(𝑣₁,𝑣₂), and 𝟏=1_𝐿^∗=(1,0)) the following:(𝜙₁) for all 𝑢,𝑣>_𝐿^∗𝟎, 𝜙(𝑢,𝑣,𝑢,𝑣,𝑣,𝑢)≥_𝐿^∗𝟎 or 𝜙(𝑢,𝑣,𝑣,𝑢,𝑢,𝑣)≥_𝐿^∗𝟎 implies that 𝑢≥_𝐿^∗𝑣;(𝜙₂)𝜙(𝑢,𝑢,𝟏,𝟏,𝑢,𝑢)≥_𝐿^∗𝟎 implies that 𝑢≥_𝐿^∗𝟏.

Example 34. Define 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)=18𝑡₁−16𝑡₂+8𝑡₃−10𝑡₄+𝑡₅−𝑡₆. Then 𝜙∈Φ.

Example 35. Define 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)=𝑡₁−(1/2)𝑡₂−(5/6)𝑡₃+(1/3)𝑡₄+𝑡₅−𝑡₆. Then 𝜙∈Φ.

Here, it can be pointed out that the abovementioned classes of functions, namely, Ψ and Φ, are independent to each other as the implicit function 𝜑(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)=𝑡₁−𝛼min{𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆} (where 𝛼>1 and 𝜑∈Ψ) does not belong to Φ as 𝜑(𝑢,𝑢,𝟏,𝟏,𝑢,𝑢)<_𝐿^∗𝟎, for all 𝑢>_𝐿^∗𝟎, while the implicit function 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)=15𝑡₁−13𝑡₂+5𝑡₃−7𝑡₄+𝑡₅−𝑡₆ (where 𝜙∈Φ) does not belong to Ψ as 𝜙(𝑢,𝑣,𝑢,𝑣,𝑣,𝑢)=0 implies 𝑢=𝑣 instead of 𝑢>_𝐿^∗𝑣.

For an extensive collection of implicit relations on different settings, we refer to [41–45].

4. Results

Before proving our main results, we observe the following.

Lemma 36. Let 𝐴, 𝐵, 𝑆, and 𝑇 be self-mappings of a modified IFMS (𝑋,ℳ_𝑀,𝑁,𝒯). Suppose that (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){𝐵𝑦_𝑛} converges for every sequence {𝑦_𝑛} in 𝑋 whenever {𝑇𝑦_𝑛} converges (or {𝐴𝑥_𝑛} converges for every sequence {𝑥_𝑛} in 𝑋 whenever {𝑆𝑥_𝑛} converges), (5)for all 𝑥,𝑦∈𝑋 and 𝜑∈Ψ𝜑ℳ_𝑀,𝑁𝐴𝑥,𝐵𝑦,𝑡,ℳ_𝑀,𝑁𝑆𝑥,𝑇𝑦,𝑡,ℳ_𝑀,𝑁(),ℳ𝐴𝑥,𝑆𝑥,𝑡_𝑀,𝑁𝐵𝑦,𝑇𝑦,𝑡,ℳ_𝑀,𝑁𝑆𝑥,𝐵𝑦,𝑡,ℳ_𝑀,𝑁≥𝐴𝑥,𝑇𝑦,𝑡^𝐿∗𝟎. (33)
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→∞ℳ_𝑀,𝑁𝐴𝑥_𝑛,𝑧,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝑆𝑥_𝑛,𝑧,𝑡=1_𝐿^∗, (34) where 𝑧∈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→∞ℳ_𝑀,𝑁𝑇𝑦_𝑛,𝑧,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝐴𝑥_𝑛,𝑧,𝑡=1_𝐿^∗, (35) where 𝑧∈𝑆(𝑋)∩𝑇(𝑋). Thus, in all, we have 𝐴𝑥_𝑛→𝑧, 𝑆𝑥_𝑛→𝑧, and 𝑇𝑦_𝑛→𝑧 as 𝑛→∞. Moreover, in view of (4), {𝐵𝑦_𝑛} converges. Now, we show that 𝐵𝑦_𝑛→𝑧 as 𝑛→∞. On using inequality (33) with 𝑥=𝑥_𝑛, 𝑦=𝑦_𝑛, we have𝜑ℳ_𝑀,𝑁𝐴𝑥_𝑛,𝐵𝑦_𝑛,𝑡,ℳ_𝑀,𝑁𝑆𝑥_𝑛,𝑇𝑦_𝑛,𝑡,ℳ_𝑀,𝑁𝐴𝑥_𝑛,𝑆𝑥_𝑛,ℳ,𝑡_𝑀,𝑁𝐵𝑦_𝑛,𝑇𝑦_𝑛,𝑡,ℳ_𝑀,𝑁𝑆𝑥_𝑛,𝐵𝑦_𝑛,𝑡,ℳ_𝑀,𝑁𝐴𝑥_𝑛,𝑇𝑦_𝑛≥,𝑡^𝐿∗𝟎. (36)
Let, on contrary, 𝐵𝑦_𝑛→𝑙(≠𝑧) as 𝑛→∞. Then, on making 𝑛→∞, we get 𝜑ℳ_𝑀,𝑁(𝑧,𝑙,𝑡),ℳ_𝑀,𝑁(𝑧,𝑧,𝑡),ℳ_𝑀,𝑁(),ℳ𝑧,𝑧,𝑡_𝑀,𝑁(𝑙,𝑧,𝑡),ℳ_𝑀,𝑁(𝑧,𝑙,𝑡),ℳ_𝑀,𝑁()≥𝑧,𝑧,𝑡^𝐿∗𝟎, (37) or, equivalently, 𝜑ℳ_𝑀,𝑁(𝑧,𝑙,𝑡),𝟏,𝟏,ℳ_𝑀,𝑁(𝑙,𝑧,𝑡),ℳ_𝑀,𝑁(𝑧,𝑙,𝑡),𝟏≥_𝐿^∗𝟎, (38) which is a contradiction to (𝜑₂). Hence ℳ_𝑀,𝑁(𝑧,𝑙,𝑡)=𝟏, that is, 𝑧=𝑙, which shows that 𝐵𝑦_𝑛→𝑧 as 𝑛→∞. Hence both the pairs (𝐴,𝑆) and (𝐵,𝑇) share the (CLR_𝑆𝑇) property. This concludes the proof.

Remark 37. In general, the converse of Lemma 36 is not true. For a counter example, one can see Example 42.

Now, we state and prove our first main result as follows.

Theorem 38. Let 𝐴, 𝐵, 𝑆, and 𝑇 be self-mappings of a modified IFMS (𝑋,ℳ_𝑀,𝑁,𝒯) satisfying inequality (33) of Lemma 36. 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 both the 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→∞ℳ_𝑀,𝑁𝐴𝑥_𝑛,𝑧,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝑆𝑥_𝑛,𝑧,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝐵𝑦_𝑛,𝑧,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝑇𝑦_𝑛,𝑧,𝑡=1_𝐿^∗, (39) where 𝑧∈𝑆(𝑋)∩𝑇(𝑋). Since 𝑧∈𝑆(𝑋), there exists a point 𝑤∈𝑋 such that 𝑆𝑤=𝑧. We show that 𝐴𝑤=𝑆𝑤. If not, then using inequality (33) with 𝑥=𝑤, 𝑦=𝑦_𝑛, we get𝜑ℳ_𝑀,𝑁𝐴𝑤,𝐵𝑦_𝑛,𝑡,ℳ_𝑀,𝑁𝑆𝑤,𝑇𝑦_𝑛,𝑡,ℳ_𝑀,𝑁(),ℳ𝐴𝑤,𝑆𝑤,𝑡_𝑀,𝑁𝐵𝑦_𝑛,𝑇𝑦_𝑛,𝑡,ℳ_𝑀,𝑁𝑆𝑤,𝐵𝑦_𝑛,𝑡,ℳ_𝑀,𝑁𝐴𝑤,𝑇𝑦_𝑛≥,𝑡^𝐿∗𝟎, (40)
which, on making 𝑛→∞, reduces to𝜑ℳ_𝑀,𝑁(𝐴𝑤,𝑧,𝑡),ℳ_𝑀,𝑁(𝑧,𝑧,𝑡),ℳ_𝑀,𝑁(),ℳ𝐴𝑤,𝑧,𝑡_𝑀,𝑁(𝑧,𝑧,𝑡),ℳ_𝑀,𝑁(𝑧,𝑧,𝑡),ℳ_𝑀,𝑁()≥𝐴𝑤,𝑧,𝑡^𝐿∗𝟎 (41)
so that𝜑ℳ_𝑀,𝑁(𝐴𝑤,𝑧,𝑡),𝟏,ℳ_𝑀,𝑁(𝐴𝑤,𝑧,𝑡),𝟏,𝟏,ℳ_𝑀,𝑁(𝐴𝑤,𝑧,𝑡)≥_𝐿^∗𝟎, (42)
a contradiction to (𝜑₁). Hence ℳ_𝑀,𝑁(𝐴𝑤,𝑧,𝑡)=𝟏; that is, 𝐴𝑤=𝑆𝑤=𝑧. Therefore, 𝑤 is a coincidence point of the pair (𝐴,𝑆).
Also 𝑧∈𝑇(𝑋); there exists a point 𝑣∈𝑋 such that 𝑇𝑣=𝑧. We assert that 𝐵𝑣=𝑇𝑣. If not, then using inequality (33) with 𝑥=𝑤, 𝑦=𝑣, we get𝜑ℳ_𝑀,𝑁(𝐴𝑤,𝐵𝑣,𝑡),ℳ_𝑀,𝑁(𝑆𝑤,𝑇𝑣,𝑡),ℳ_𝑀,𝑁(),ℳ𝐴𝑤,𝑆𝑤,𝑡_𝑀,𝑁(𝐵𝑣,𝑇𝑣,𝑡),ℳ_𝑀,𝑁(𝑆𝑤,𝐵𝑣,𝑡),ℳ_𝑀,𝑁()≥𝐴𝑤,𝑇𝑣,𝑡^𝐿∗𝟎 (43)
so that𝜑ℳ_𝑀,𝑁(𝑧,𝐵𝑣,𝑡),ℳ_𝑀,𝑁(𝑧,𝑧,𝑡),ℳ_𝑀,𝑁(),ℳ𝑧,𝑧,𝑡_𝑀,𝑁(𝐵𝑣,𝑧,𝑡),ℳ_𝑀,𝑁(𝑧,𝐵𝑣,𝑡),ℳ_𝑀,𝑁()≥𝑧,𝑧,𝑡^𝐿∗𝟎 (44)
or𝜑ℳ_𝑀,𝑁(𝑧,𝐵𝑣,𝑡),𝟏,𝟏,ℳ_𝑀,𝑁(𝐵𝑣,𝑧,𝑡),ℳ_𝑀,𝑁(𝑧,𝐵𝑣,𝑡),𝟏≥_𝐿^∗𝟎, (45)
a contradiction to (𝜑₂). Hence ℳ_𝑀,𝑁(𝑧,𝐵𝑣,𝑡)=𝟏, and so 𝐵𝑣=𝑇𝑣=𝑧, which shows that 𝑣 is a coincidence point of the pair (𝐵,𝑇).
Since the pair (𝐴,𝑆) is weakly compatible and 𝐴𝑤=𝑆𝑤, hence 𝐴𝑧=𝐴𝑆𝑤=𝑆𝐴𝑤=𝑆𝑧. Now, we show that 𝑧 is a common fixed point of the pair (𝐴,𝑆). Suppose that 𝐴𝑧≠𝑧; using inequality (33) with 𝑥=𝑧, 𝑦=𝑣, we have𝜑ℳ_𝑀,𝑁(𝐴𝑧,𝐵𝑣,𝑡),ℳ_𝑀,𝑁(𝑆𝑧,𝑇𝑣,𝑡),ℳ_𝑀,𝑁(),ℳ𝐴𝑧,𝑆𝑧,𝑡_𝑀,𝑁(𝐵𝑣,𝑇𝑣,𝑡),ℳ_𝑀,𝑁(𝑆𝑧,𝐵𝑣,𝑡),ℳ_𝑀,𝑁()≥𝐴𝑧,𝑇𝑣,𝑡^𝐿∗𝟎 (46)
so that𝜑ℳ_𝑀,𝑁(𝐴𝑧,𝑧,𝑡),ℳ_𝑀,𝑁(𝐴𝑧,𝑧,𝑡),ℳ_𝑀,𝑁(),ℳ𝐴𝑧,𝐴𝑧,𝑡_𝑀,𝑁(𝑧,𝑧,𝑡),ℳ_𝑀,𝑁(𝐴𝑧,𝑧,𝑡),ℳ_𝑀,𝑁()≥𝐴𝑧,𝑧,𝑡^𝐿∗𝟎 (47)
or 𝜑ℳ_𝑀,𝑁(𝐴𝑧,𝑧,𝑡),ℳ_𝑀,𝑁()ℳ𝐴𝑧,𝑧,𝑡,𝟏,𝟏,_𝑀,𝑁(𝐴𝑧,𝑧,𝑡),ℳ_𝑀,𝑁()≥𝐴𝑧,𝑧,𝑡^𝐿∗𝟎, (48)
a contradiction to (𝜑₃) yielding thereby 𝐴𝑧=𝑧=𝑆𝑧 which shows that 𝑧 is a common fixed point of the pair (𝐴,𝑆).
Also the pair (𝐵,𝑇) is weakly compatible, and 𝐵𝑣=𝑇𝑣; therefore, 𝐵𝑧=𝐵𝑇𝑣=𝑇𝐵𝑣=𝑇𝑧. Suppose that 𝐵𝑧≠𝑧; then using inequality (33) with 𝑥=𝑤, 𝑦=𝑧, we have𝜑ℳ_𝑀,𝑁(𝐴𝑤,𝐵𝑧,𝑡),ℳ_𝑀,𝑁(𝑆𝑤,𝑇𝑧,𝑡),ℳ_𝑀,𝑁(),ℳ𝐴𝑤,𝑆𝑤,𝑡_𝑀,𝑁(𝐵𝑧,𝑇𝑧,𝑡),ℳ_𝑀,𝑁(𝑆𝑤,𝐵𝑧,𝑡),ℳ_𝑀,𝑁()≥𝐴𝑤,𝑇𝑧,𝑡^𝐿∗𝟎, (49)
so that𝜑ℳ_𝑀,𝑁(𝑧,𝐵𝑧,𝑡),ℳ_𝑀,𝑁(𝑧,𝐵𝑧,𝑡),ℳ_𝑀,𝑁(),ℳ𝑧,𝑧,𝑡_𝑀,𝑁(𝐵𝑧,𝐵𝑧,𝑡),ℳ_𝑀,𝑁(𝑧,𝐵𝑧,𝑡),ℳ_𝑀,𝑁()≥𝑧,𝐵𝑧,𝑡^𝐿∗𝟎 (50)
or 𝜑ℳ_𝑀,𝑁(𝑧,𝐵𝑧,𝑡),ℳ_𝑀,𝑁()ℳ𝑧,𝐵𝑧,𝑡,𝟏,𝟏,_𝑀,𝑁(𝑧,𝐵𝑧,𝑡),ℳ_𝑀,𝑁()≥𝑧,𝐵𝑧,𝑡^𝐿∗𝟎, (51)
a contradiction to (𝜑₃). Therefore, 𝐵𝑧=𝑧=𝑇𝑧 which shows that 𝑧 is a common fixed point of the pair (𝐵,𝑇). Hence 𝑧 is a common fixed point of both the pairs (𝐴,𝑆) and (𝐵,𝑇). Uniqueness of common fixed point is an easy consequence of inequality (33) (owing to condition (𝜑₃)). This completes the proof.

Remark 39. Theorem 38 improves the corresponding results contained in Imdad et al. [23] 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 40. Let (𝑋,ℳ_𝑀,𝑁,𝒯) be a modified IFMS, wherein 𝑋=[5,21), 𝒯(𝑎,𝑏)=(𝑎₁𝑏₁,min{𝑎₂+𝑏₂,1}) for all 𝑎=(𝑎₁,𝑎₂) and 𝑏=(𝑏₁,𝑏₂)∈𝐿^∗ with ℳ_𝑀,𝑁(𝑥,𝑦,𝑡)=(𝑡/(𝑡+|𝑥−𝑦|),|𝑥−𝑦|/(𝑡+|𝑥−𝑦|)) for all 𝑥,𝑦∈𝑋 and 𝑡>0. 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 (52) Define an implicit function 𝜑(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶𝐿^∗6→𝐿^∗ by 𝜑𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝜓min𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆, (53) where 𝜓∶𝐿^∗→𝐿^∗ is a continuous function such that 𝜓(𝑠)>_𝐿^∗𝑠 (that is, 𝜓(𝑠)=√𝑠) for all 𝑠∈𝐿^∗⧵{𝟎,𝟏} and 𝜑∈Ψ. Hence (53) impliesℳ_𝑀,𝑁≥𝐴𝑥,𝐵𝑦,𝑡_𝐿∗ℳmin^𝑀,𝑁𝑆𝑥,𝑇𝑦,𝑡,ℳ_𝑀,𝑁(𝐴𝑥,𝑆𝑥,𝑡),ℳ_𝑀,𝑁,ℳ𝐵𝑦,𝑇𝑦,𝑡_𝑀,𝑁𝑆𝑥,𝐵𝑦,𝑡,ℳ_𝑀,𝑁,𝐴𝑥,𝑇𝑦,𝑡 (54)
for all 𝑥,𝑦∈𝑋 and 𝑡>0. With two sequences {𝑥_𝑛}={9+1/𝑛}_𝑛∈ℕ and {𝑦_𝑛}={5} (or {𝑥_𝑛}={5}, {𝑦_𝑛}={9+1/𝑛}_𝑛∈ℕ), the pairs (𝐴,𝑆) and (𝐵,𝑇) satisfy the (CLR_𝑆𝑇) property: lim_𝑛→∞ℳ_𝑀,𝑁𝐴𝑥_𝑛,5,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝑆𝑥_𝑛,5,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝐵𝑦_𝑛,5,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝑇𝑦_𝑛,5,𝑡=1_𝐿^∗, (55) where 5∈𝑆(𝑋)∩𝑇(𝑋). Also, 𝐴(𝑋)={5,20}⊆[̸5,17)∪{18}=𝑇(𝑋) and 𝐵(𝑋)={5,13}⊆[̸5,11)=𝑆(𝑋). By a routine calculation, one can easily verify the inequality (54) for all 𝑥,𝑦∈𝑋. Thus all the conditions of Theorem 38 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 even at their unique common fixed point 5.
Notice that the subspaces 𝑆(𝑋) and 𝑇(𝑋) are not closed subspaces of 𝑋; therefore, the main result contained in Imdad et al. [23] can not be used in the context of this example which establishes the genuineness of our extension.

In the proof of our next theorem, Lemma 36 is utilized.

Theorem 41. Let 𝐴, 𝐵, 𝑆, and 𝑇 be self-mappings of a modified IFMS (𝑋,ℳ_𝑀,𝑁,𝒯) satisfying all the hypotheses of Lemma 36. Then 𝐴, 𝐵, 𝑆, and 𝑇 have a unique common fixed point provided both the pairs (𝐴,𝑆) and (𝐵,𝑇) are weakly compatible.

Proof. In view of Lemma 36, 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→∞ℳ_𝑀,𝑁𝐴𝑥_𝑛,𝑧,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝑆𝑥_𝑛,𝑧,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝐵𝑦_𝑛,𝑧,𝑡=lim_𝑛→∞ℳ_𝑀,𝑁𝑇𝑦_𝑛,𝑧,𝑡=1_𝐿^∗, (56) where 𝑧∈𝑆(𝑋)∩𝑇(𝑋). The rest of the proof can be completed on the lines of the proof of Theorem 38. This completes the proof.

The following example demonstrates the utility of Theorem 41 over Theorem 38.

Example 42. In the setting of Example 40, replace the self-mappings 𝐴, 𝐵, 𝑆, and 𝑇 by the following, besides retaining the rest: 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 (57) Then, like the earlier example, it is easy to see that both the pairs (𝐴,𝑆) and (𝐵,𝑇) satisfy the (𝐶𝐿𝑅_𝑆𝑇) property. Consider an implicit function described by Example 40. 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. Thus all the conditions of Theorems 41 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 38 is not applicable to this example as both 𝑆(𝑋), 𝑇(𝑋) are closed subsets of 𝑋 which demonstrates the situational utility of Theorem 41 over Theorem 38.

In view of the earlier demonstrative examples, one can outline the following corollary.

Corollary 43. The conclusions of Lemma 36, Theorem 38, and Theorem 41 remain true if inequality (33) is replaced by one of the following contraction conditions. For all 𝑥,𝑦∈𝑋 and 𝑡>0,ℳ_𝑀,𝑁≥𝐴𝑥,𝐵𝑦,𝑡_𝐿∗ℳ𝛼min^𝑀,𝑁𝑆𝑥,𝑇𝑦,𝑡,ℳ_𝑀,𝑁(𝐴𝑥,𝑆𝑥,𝑡),ℳ_𝑀,𝑁,ℳ𝐵𝑦,𝑇𝑦,𝑡_𝑀,𝑁𝑆𝑥,𝐵𝑦,𝑡,ℳ_𝑀,𝑁,𝐴𝑥,𝑇𝑦,𝑡 (58)
where 𝛼>1,ℳ²_𝑀,𝑁≥𝐴𝑥,𝐵𝑦,𝑡_𝐿∗𝛼₁minℳ²_𝑀,𝑁𝑆𝑥,𝑇𝑦,𝑡,ℳ²_𝑀,𝑁(𝐴𝑥,𝑆𝑥,𝑡),ℳ²_𝑀,𝑁𝐵𝑦,𝑇𝑦,𝑡+𝛼₂ℳmin^𝑀,𝑁(𝐴𝑥,𝑆𝑥,𝑡)ℳ_𝑀,𝑁,ℳ𝐴𝑥,𝑇𝑦,𝑡_𝑀,𝑁𝐵𝑦,𝑇𝑦,𝑡ℳ_𝑀,𝑁,𝑆𝑥,𝐵𝑦,𝑡 (59)
where 𝛼₁,𝛼₂>0, 𝛼₁+𝛼₂>1 and 𝛼₁≥1,ℳ³_𝑀,𝑁≥𝐴𝑥,𝐵𝑦,𝑡_𝐿∗ℳ𝛼min^2𝑀,𝑁𝐴𝑥,𝐵𝑦,𝑡ℳ_𝑀,𝑁,ℳ𝑆𝑥,𝑇𝑦,𝑡_𝑀,𝑁𝐴𝑥,𝐵𝑦,𝑡ℳ_𝑀,𝑁(𝐴𝑥,𝑆𝑥,𝑡)ℳ_𝑀,𝑁,ℳ𝐵𝑦,𝑇𝑦,𝑡_2𝑀,𝑁𝑆𝑥,𝐵𝑦,𝑡ℳ_𝑀,𝑁𝐴𝑥,𝑇𝑦,𝑡,ℳ_𝑀,𝑁𝑆𝑥,𝐵𝑦,𝑡ℳ²_𝑀,𝑁,𝐴𝑥,𝑇𝑦,𝑡 (60)
where 𝛼>1, ℳ³_𝑀,𝑁𝐴𝑥,𝐵𝑦,𝑡≥_𝐿^∗𝛼ℳ²_𝑀,𝑁(𝐴𝑥,𝑆𝑥,𝑡)ℳ²_𝑀,𝑁𝐵𝑦,𝑇𝑦,𝑡+ℳ²_𝑀,𝑁𝑆𝑥,𝐵𝑦,𝑡,ℳ²_𝑀,𝑁𝐴𝑥,𝑇𝑦,𝑡ℳ_𝑀,𝑁𝑆𝑥,𝑇𝑦,𝑡+ℳ_𝑀,𝑁(𝐴𝑥,𝑆𝑥,𝑡)+ℳ_𝑀,𝑁,𝐵𝑦,𝑇𝑦,𝑡 (61)
where 𝛼≥3/2,1+𝛼ℳ_𝑀,𝑁𝑆𝑥,𝑇𝑦,𝑡ℳ_𝑀,𝑁≥𝐴𝑥,𝐵𝑦,𝑡_𝐿∗ℳ𝛼min^𝑀,𝑁(𝐴𝑥,𝑆𝑥,𝑡)ℳ_𝑀,𝑁,ℳ𝐵𝑦,𝑇𝑦,𝑡_𝑀,𝑁𝑆𝑥,𝐵𝑦,𝑡ℳ_𝑀,𝑁ℳ𝐴𝑥,𝑇𝑦,𝑡−𝜓min^𝑀,𝑁𝑆𝑥,𝑇𝑦,𝑡,ℳ_𝑀,𝑁(),ℳ𝐴𝑥,𝑆𝑥,𝑡_𝑀,𝑁𝐵𝑦,𝑇𝑦,𝑡,ℳ_𝑀,𝑁,ℳ𝑆𝑥,𝐵𝑦,𝑡_𝑀,𝑁,𝐴𝑥,𝑇𝑦,𝑡 (62)
where 𝛼≥0 and 𝜓∶𝐿^∗→𝐿^∗ is a continuous function such that 𝜓(𝑠)>_𝐿^∗𝑠 for all 𝑠∈𝐿^∗⧵{𝟎,𝟏},ℳ²_𝑀,𝑁≥𝐴𝑥,𝐵𝑦,𝑡_𝐿∗𝛼ℳ²_𝑀,𝑁𝑆𝑥,𝑇𝑦,𝑡+ℳ²_𝑀,𝑁(𝐴𝑥,𝑆𝑥,𝑡)+ℳ²_𝑀,𝑁𝐵𝑦,𝑇𝑦,𝑡ℳ_𝑀,𝑁𝑆𝑥,𝐵𝑦,𝑡+ℳ_𝑀,𝑁,𝐴𝑥,𝑇𝑦,𝑡 (63)
where 𝛼≥2, ℳ_𝑀,𝑁≥𝐴𝑥,𝐵𝑦,𝑡_𝐿∗𝜓ℳmin^𝑀,𝑁𝑆𝑥,𝑇𝑦,𝑡,ℳ_𝑀,𝑁(𝐴𝑥,𝑆𝑥,𝑡),ℳ_𝑀,𝑁,ℳ𝐵𝑦,𝑇𝑦,𝑡_𝑀,𝑁𝑆𝑥,𝐵𝑦,𝑡,ℳ_𝑀,𝑁,𝐴𝑥,𝑇𝑦,𝑡 (64)
where 𝜓∶𝐿^∗→𝐿^∗ is a continuous function such that 𝜓(𝑠)>_𝐿^∗𝑠 for all 𝑠∈𝐿^∗⧵{𝟎,𝟏},ℳ³_𝑀,𝑁≥𝐴𝑥,𝐵𝑦,𝑡_𝐿∗𝛼ℳ²_𝑀,𝑁(𝐴𝑥,𝑆𝑥,𝑡)ℳ²_𝑀,𝑁𝐵𝑦,𝑇𝑦,𝑡ℳ_𝑀,𝑁𝑆𝑥,𝑇𝑦,𝑡+ℳ_𝑀,𝑁𝑆𝑥,𝐵𝑦,𝑡+ℳ_𝑀,𝑁,𝐴𝑥,𝑇𝑦,𝑡 (65)
where 𝛼≥3,ℳ²_𝑀,𝑁≥𝐴𝑥,𝐵𝑦,𝑡_𝐿∗𝛼₁minℳ²_𝑀,𝑁𝑆𝑥,𝑇𝑦,𝑡,ℳ²_𝑀,𝑁(𝐴𝑥,𝑆𝑥,𝑡),ℳ²_𝑀,𝑁𝐵𝑦,𝑇𝑦,𝑡+𝛼₂ℳ_𝑀,𝑁𝑆𝑥,𝐵𝑦,𝑡ℳ_𝑀,𝑁𝑆𝑥,𝐵𝑦,𝑡+ℳ_𝑀,𝑁,𝐴𝑥,𝑇𝑦,𝑡 (66)
where 𝛼₁≥1 and 𝛼₂>0,ℳ²_𝑀,𝑁≥𝐴𝑥,𝐵𝑦,𝑡_𝐿∗𝛼₁minℳ²_𝑀,𝑁𝑆𝑥,𝑇𝑦,𝑡,ℳ²_𝑀,𝑁𝑆𝑥,𝐵𝑦,𝑡,ℳ²_𝑀,𝑁𝐴𝑥,𝑇𝑦,𝑡+𝛼₂ℳ_𝑀,𝑁(𝐴𝑥,𝑆𝑥,𝑡)ℳ_𝑀,𝑁(𝐴𝑥,𝑆𝑥,𝑡)+ℳ_𝑀,𝑁,𝐵𝑦,𝑇𝑦,𝑡 (67)
where 𝛼₁≥1 and 𝛼₂>0,ℳ_𝑀,𝑁≥𝐴𝑥,𝐵𝑦,𝑡_𝐿∗𝛼₁ℳ_𝑀,𝑁𝑆𝑥,𝑇𝑦,𝑡+𝛼₂ℳ_𝑀,𝑁(𝐴𝑥,𝑆𝑥,𝑡)+𝛼₃ℳ_𝑀,𝑁𝐵𝑦,𝑇𝑦,𝑡+𝛼₄ℳ_𝑀,𝑁𝑆𝑥,𝐵𝑦,𝑡+𝛼₅ℳ_𝑀,𝑁𝐴𝑥,𝑇𝑦,𝑡, (68)
where 𝛼₁,𝛼₂,𝛼₃,𝛼₄,𝛼₅>0, 𝛼₂+𝛼₅≥1, 𝛼₃+𝛼₄≥1, and 𝛼₁+𝛼₄+𝛼₅≥1.

Proof. The proof for each inequality (58)–(68) easily follows from Theorem 38 in view of Examples 23–33.

Remark 44. Corollary 43 improves and generalizes a multitude of well-known results especially those contained in [8, 10, 11, 17, 19, 22, 23].

Now we state the next theorem for another independent class of implicit functions 𝜙∈Φ utilized in Tanveer et al. [22].

Theorem 45. Let 𝐴, 𝐵, 𝑆, and 𝑇 be self-mappings of a modified IFMS (𝑋,ℳ_𝑀,𝑁,𝒯) satisfying𝜙ℳ_𝑀,𝑁𝐴𝑥,𝐵𝑦,𝑡,ℳ_𝑀,𝑁𝑆𝑥,𝑇𝑦,𝑡,ℳ_𝑀,𝑁(),ℳ𝐴𝑥,𝑆𝑥,𝑡_𝑀,𝑁𝐵𝑦,𝑇𝑦,𝑡,ℳ_𝑀,𝑁𝑆𝑥,𝐵𝑦,𝑡,ℳ_𝑀,𝑁≥𝐴𝑥,𝑇𝑦,𝑡^𝐿∗𝟎, (69)
for all 𝑥,𝑦∈𝑋 and 𝜙∈Φ. 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 both the pairs (𝐴,𝑆) and (𝐵,𝑇) are weakly compatible.

Proof. The proof of this theorem can be completed on the lines of the proof of Theorem 38; hence we skip the details.

Example 46. In the setting of Example 40, one can define an implicit function 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)∶𝐿^∗6→𝐿^∗ by 𝜙(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)=18𝑡₁−16𝑡₂+8𝑡₃−10𝑡₄+𝑡₅−𝑡₆, where 𝜙∈Φ (besides retaining the rest). Hence the pairs (𝐴,𝑆) and (𝐵,𝑇) enjoy the (CLR_𝑆𝑇) property. Thus all the conditions of Theorem 45 are satisfied, and 5 is a unique common fixed point of the pairs (𝐴,𝑆) and (𝐵,𝑇).

By choosing 𝐴, 𝐵, 𝑆, and 𝑇 suitably, we can deduce corollaries involving two as well as three self-mappings. For the sake naturality, we only derive the following corollary involving a pair of self-mappings.

Corollary 47. Let 𝐴 and 𝑆 be self-mappings of a modified IFMS (𝑋,ℳ_𝑀,𝑁,𝒯). Suppose that (1)the pair (𝐴,𝑆) satisfies the (𝐶𝐿𝑅_𝑆) property, (2)for all 𝑥,𝑦∈𝑋 and 𝜑∈Ψ,𝜑ℳ_𝑀,𝑁𝐴𝑥,𝐴𝑦,𝑡,ℳ_𝑀,𝑁𝑆𝑥,𝑆𝑦,𝑡,ℳ_𝑀,𝑁(),ℳ𝐴𝑥,𝑆𝑥,𝑡_𝑀,𝑁𝐴𝑦,𝑆𝑦,𝑡,ℳ_𝑀,𝑁𝑆𝑥,𝐴𝑦,𝑡,ℳ_𝑀,𝑁≥𝐴𝑥,𝑆𝑦,𝑡^𝐿∗𝟎. (70)
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 38, we have the following result involving four finite families of self-mappings.

Theorem 48. 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 modified IFMS (𝑋,ℳ_𝑀,𝑁,𝒯) with 𝐴=𝐴₁𝐴₂⋯𝐴_𝑚, 𝐵=𝐵₁𝐵₂⋯𝐵_𝑛, 𝑆=𝑆₁𝑆₂⋯𝑆_𝑝, and 𝑇=𝑇₁𝑇₂⋯𝑇_𝑞 satisfying the condition (33). 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 the proof of a similar theorem contained in Imdad et al. [40].

Remark 49. A result similar to Theorem 48 can be outlined in respect of Theorem 38. Notice that Theorem 48 generalizes certain results of Sharma and Deshpande [13].

Now, we indicate that Theorem 48 can be utilized to derive common fixed point theorems for any finite number of mappings. As a sample, we derive a theorem involving five mappings by setting one family of two members while the remaining three of single members.

Corollary 50. Let 𝐴, 𝐵, 𝑅, 𝑆, and 𝑇 be self-mappings of a modified IFMS (𝑋,ℳ_𝑀,𝑁,𝒯). Suppose that (1)the pairs (𝐴,𝑆𝑅) and (𝐵,𝑇) share the (𝐶𝐿𝑅_{(𝑆𝑅)(𝑇)}) property, (2)for all 𝑥,𝑦∈𝑋 and 𝜑∈Ψ,𝜑ℳ_𝑀,𝑁𝐴𝑥,𝐵𝑦,𝑡,ℳ_𝑀,𝑁𝑆𝑅𝑥,𝑇𝑦,𝑡,ℳ_𝑀,𝑁(),ℳ𝐴𝑥,𝑆𝑅𝑥,𝑡_𝑀,𝑁𝐵𝑦,𝑇𝑦,𝑡,ℳ_𝑀,𝑁𝑆𝑅𝑥,𝐵𝑦,𝑡,ℳ_𝑀,𝑁≥𝐴𝑥,𝑇𝑦,𝑡^𝐿∗𝟎. (71)
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 both the pairs (𝐴,𝑆𝑅) and (𝐵,𝑇) commute pairwise; that is, 𝐴𝑆=𝑆𝐴, 𝐴𝑅=𝑅𝐴, 𝑆𝑅=𝑅𝑆, and 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 51. Let 𝐴, 𝐵, 𝐻, 𝑅, 𝑆, and 𝑇 be self-mappings of a modified IFMS (𝑋,ℳ_𝑀,𝑁,𝒯). Suppose that (1)the pairs (𝐴,𝑆𝑅) and (𝐵,𝑇𝐻) share the (𝐶𝐿𝑅_{(𝑆𝑅)(𝑇𝐻)}) property, (2)for all 𝑥,𝑦∈𝑋 and 𝜑∈Ψ,𝜑ℳ_𝑀,𝑁𝐴𝑥,𝐵𝑦,𝑡,ℳ_𝑀,𝑁𝑆𝑅𝑥,𝑇𝐻𝑦,𝑡,ℳ_𝑀,𝑁(),ℳ𝐴𝑥,𝑆𝑅𝑥,𝑡_𝑀,𝑁𝐵𝑦,𝑇𝐻𝑦,𝑡,ℳ_𝑀,𝑁𝑆𝑅𝑥,𝐵𝑦,𝑡,ℳ_𝑀,𝑁≥𝐴𝑥,𝑇𝐻𝑦,𝑡^𝐿∗𝟎. (72)
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 both the 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 48, we deduce the following.

Corollary 52. Let 𝐴, 𝐵, 𝑆, and 𝑇 be self-mappings of a modified IFMS (𝑋,ℳ_𝑀,𝑁,𝒯). Suppose that (1)the pairs (𝐴^𝑚,𝑆^𝑝) and (𝐵^𝑛,𝑇^𝑞) share the (𝐶𝐿𝑅_𝑆^𝑝_𝑇^𝑞) property, (2)for all 𝑥,𝑦∈𝑋 and 𝜑∈Ψ,𝜑ℳ_𝑀,𝑁𝐴^𝑚𝑥,𝐵^𝑛𝑦,𝑡,ℳ_𝑀,𝑁𝑆^𝑝𝑥,𝑇^𝑞𝑦,𝑡,ℳ_𝑀,𝑁𝐴^𝑚𝑥,𝑆^𝑝,ℳ𝑥,𝑡_𝑀,𝑁𝐵^𝑛𝑦,𝑇^𝑞𝑦,𝑡,ℳ_𝑀,𝑁𝑆^𝑝𝑥,𝐵^𝑛𝑦,𝑡,ℳ_𝑀,𝑁𝐴^𝑚𝑥,𝑇^𝑞≥𝑦,𝑡^𝐿∗𝟎, (73)
where 𝑚, 𝑛, 𝑝, and 𝑞 are fixed positive integers.
Then 𝐴, 𝐵, 𝑆, and 𝑇 have a unique common fixed point provided 𝐴𝑆=𝑆𝐴 and 𝐵𝑇=𝑇𝐵.

Remark 53. Corollary 52 is a slight but partial generalization of Theorem 38 as the commutativity requirements (that is, 𝐴𝑆=𝑆𝐴 and 𝐵𝑇=𝑇𝐵) in this corollary are relatively stronger as compared to weak compatibility in Theorem 38.

Remark 54. Results similar to Corollary 52 can be derived from Theorem 38 and Corollary 43.

Remark 55. It is noticed that Lemma 36, Theorems 38–48, and Corollaries 43–52 can be also proved for the implicit function Φ, but due to paucity of the space we have opted not to include the entire details.

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgments

The authors are thankful to all anonymous referees for their valuable suggestions and critical remarks towards the improvement of this paper. The authors are also very grateful to Professor Erdal Karapnar (Guest Editor of Abstract and Applied Analysis) for his encouragements, especially in the course of two revisions.