Semicompatibility and Fixed Point Theorems for Reciprocally Continuous Maps in a Fuzzy Metric Space

Abstract

The aim of this paper is to prove a common fixed point theorem for six mappings on fuzzy metric space using notion of semicompatibility and reciprocal continuity of maps satisfying an implicit relation. We proposed to reanalysis the theorems of Imdad et al. (2002), Popa (2001), Popa (2002) and Singh and Jain (2005).

1. Introduction

The fuzzy theory has become an area of active research for the last forty years. It has a wide range of applications in the field of science and engineering, for example, population dynamics, computer programming, nonlinear dynamical systems, medicine and so forth. The concept of fuzzy sets was introduced initially by Zadeh [1] in 1965. Since then, to use this concept in topology and analysis, many authors have expansively developed the theory of fuzzy sets and application. Following the concept of fuzzy sets, fuzzy metric spaces have been introduced by Kramosil and Michalek [2], and George and Veeramani [3] modified the notion of fuzzy metric spaces with the help of continuous 𝑡-norms. Later, many authors, for example, [4–6] proved common fixed point theorems in fuzzy metric spaces. Grabiec’s [7] followed Kramosil and Michalek and obtained the fuzzy version of Banach contraction principle. Vasuki [8] obtained the fuzzy version of common fixed point theorem which had extra conditions. Pant [9] introduced the notion of reciprocal continuity of mappings in metric spaces. Pant and Jha [10] proved an analogue of the result given by Balasubramaniam et al. [11]. Popa [12] proved theorem for weakly compatible noncontinuous mapping using implicit relation. Recently Singh and Jain [13], and Singh and Chauhan [14] have introduced semicompatible, compatible, and weak compatible maps in fuzzy metric space.

The purpose of this paper is to prove a common fixed point theorem in fuzzy metric spaces using weak compatibility, semicompatibility, an implicit relation, and reciprocal continuity. Here, we generalize the result of [15] by(1)increasing the number of self maps from four to six,(2)using the notion of reciprocal continuity.

2. Preliminaries

Definition 2.1. A binary operation ∗∶[0,1]²→[0,1] is called a continuous 𝑡-norm if ([0,1],∗) is an abelian topological monoid with unit 1 such that 𝑎∗𝑏≤𝑐∗𝑑 whenever 𝑎≤𝑐 and 𝑏≤𝑑 for all 𝑎,𝑏,𝑐 and 𝑑∈[0,1].

Examples of 𝑡-norm are 𝑎∗𝑏=𝑎𝑏 and 𝑎∗𝑏=min{𝑎,𝑏}.

Definition 2.2. The 3-tuple (𝑋,𝑀,∗) is called a fuzzy metric space if 𝑋 is an arbitrary set, ∗ is a continuous 𝑡-norm, and 𝑀 is a fuzzy set in 𝑋²×[0,∞) satisfying the following conditions for all 𝑥,𝑦,𝑧∈𝑋 and 𝑠,𝑡>0:(FMS-1)  𝑀(𝑥,𝑦,0)=0,(FMS-2)  𝑀(𝑥,𝑦,𝑡)=1, for all 𝑡>0 if and only if 𝑥=𝑦,(FMS-3)  𝑀(𝑥,𝑦,𝑡)=𝑀(𝑦,𝑥,𝑡),(FMS-4)  𝑀(𝑥,𝑦,𝑡)∗𝑀(𝑦,𝑧,𝑠)≥𝑀(𝑥,𝑧,𝑡+𝑠),(FMS-5)  𝑀(𝑥,𝑦,⋅)∶[0,∞)→[0,1] is left continuous.
Note that 𝑀(𝑥,𝑦,𝑡) can be thought of as the degree of nearness between 𝑥 and 𝑦 with respect to 𝑡.

Remark 2.3. Every metric space (𝑋,𝑑) induces a fuzzy metric space (𝑋,𝑀,∗), where 𝑎∗𝑏=min{𝑎,𝑏} and for all 𝑎,𝑏∈𝑋, 𝑀(𝑥,𝑦,𝑡)=𝑡/(𝑡+𝑑(𝑥,𝑦)), for all 𝑡>0, 𝑀(𝑥,𝑦,0)=0, which is called the fuzzy metric space induced by the metric 𝑑.

Lemma 2.4. For all 𝑥,𝑦∈𝑋, let 𝑀(𝑥,𝑦,⋅) be a nondecreasing function.

Definition 2.5. Let (𝑋,𝑀,∗) be a fuzzy metric space.(a)The sequence {𝑥_𝑛} in 𝑋 is said to be convergent to a point 𝑥∈𝑋 if lim_𝑛→∞𝑀𝑥_𝑛,𝑥,𝑡=1∀𝑡>0. (2.1)(b)The sequence {𝑥_𝑛} in 𝑋 is said to be a Cauchy sequence in 𝑋 if lim_𝑛→∞𝑀𝑥_𝑛,𝑥_𝑛+𝑝,𝑡=1∀𝑡>0,𝑝>0. (2.2)(c)The space is said to be complete if every Cauchy sequence in it converges to a point of it.

Remark 2.6. Since ∗ is continuous, it follows from (FMS-4) that the limit of a sequence in a fuzzy metric space is unique.
In this paper, (𝑋,𝑀,∗) is considered to be the fuzzy metric space with condition
(FMS-6)    lim_𝑛→∞𝑀𝑥,𝑦,𝑡=1∀𝑥,𝑦∈𝑋. (2.3)

Lemma 2.7. Let {𝑦_𝑛} be a sequence in a fuzzy metric space (𝑋,𝑀,∗) with the condition (FMS-6). If there exists a number 𝑘∈(0,1) such that 𝑀𝑦_𝑛+2,𝑦_𝑛+1,𝑘𝑡≥𝑀𝑦_𝑛+1,𝑦_𝑛,𝑡 (2.4) for all 𝑡>0, then {𝑦_𝑛} is a Cauchy sequence in 𝑋.

Lemma 2.8. Let 𝐴 and 𝐵 be two self-maps on a complete fuzzy metric space (𝑋,𝑀,∗) such that for some 𝑘∈(0,1), for all 𝑥,𝑦∈𝑋 and 𝑡>0, 𝑀𝐴𝑥,𝐵𝑦,𝑘𝑡≥min𝑀𝑥,𝑦,𝑡,𝑀(𝐴𝑥,𝑥,𝑡). (2.5) Then, 𝐴 and 𝐵 have a unique common fixed point in 𝑋.

Definition 2.9. Let 𝐴 and 𝑆 be mappings from a fuzzy metric space (𝑋,𝑀,∗) into itself. Then, the mappings are said to be weak compatible if they commute at their coincidence points, that is, 𝐴𝑥=𝑆𝑥 implies that 𝐴𝑆𝑥=𝑆𝐴𝑥.

Definition 2.10. Let 𝐴 and 𝑆 be mappings from a fuzzy metric space (𝑋,𝑀,∗) into itself. Then, the mappings are said to be compatible if lim_𝑛→∞𝑀𝐴𝑆𝑥_𝑛,𝑆𝐴𝑥_𝑛,𝑡=1,∀𝑡>0 (2.6) whenever {𝑥_𝑛} is a sequence in 𝑋 such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=𝑥∈𝑋. (2.7)

Remark 2.11. Compatibility implies weak compatibility. The converse is not true.

Definition 2.12. Let 𝐴 and 𝑆 be mappings from a fuzzy metric space (𝑋,𝑀,∗) into itself. Then the mappings are said to be semicompatible if lim_𝑛→∞𝑀𝐴𝑆𝑥_𝑛,𝑆𝑥,𝑡=1,∀𝑡>0. (2.8) whenever {𝑥_𝑛} is a sequence in 𝑋 such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=𝑥∈𝑋. (2.9)

Remark 2.13. Semicompatibility implies weak compatibility. The converse is not true.

Proposition 2.14. Let 𝐴 and 𝑆 be self-maps on a fuzzy metric space (𝑋,𝑀,∗). If 𝑆 is continuous, then (𝐴,𝑆) is semicompatible if and only if (𝐴,𝑆) is compatible.

Remark 2.15. Semicompatibility of the pair (𝐴,𝑆) does not imply the semicompatibility of (𝑆,𝐴) as seen in Example 2.16.

Example 2.16. Let 𝑋=[0,1] and let (𝑋,𝑀,∗) be the fuzzy metric space with 𝑀(𝑥,𝑦,𝑡)=[exp|𝑥−𝑦|/𝑡]⁻¹, for all 𝑥,𝑦∈𝑋, 𝑡>0. Define self-map 𝑆 as follows: 1𝐒𝐱=𝑥if0≤𝑥<3,11if𝑥≥3. (2.10) Let 𝐼 be the identity map on 𝑋 and 𝑥_𝑛=1/3−1/𝑛. Then, {𝐼𝑥_𝑛}={𝑥_𝑛}→1/3 and{𝑆𝑥_𝑛}={𝑥_𝑛}→1/3. Thus, {𝐼𝑆𝑥_𝑛}={𝑆𝑥_𝑛}→1/3≠𝑆(1/3). Hence, (𝐼,𝑆) is not semicompatible. Again, as (𝐼,𝑆) is commuting, it is compatible. Further, for any sequence {𝑥_𝑛} in 𝑋 such that {𝑥_𝑛}→𝑥 and {𝑆𝑥_𝑛}→𝑥, we have {𝑆𝐼𝑥_𝑛}={𝑆𝑥_𝑛}→𝑥=𝐼𝑥. Hence, (𝑆,𝐼) is always semicompatible.

Definition 2.17. Let 𝐴 and 𝑆 be mappings from a fuzzy metric space (𝑋,𝑀,∗) into itself. Then, the mappings are said to be reciprocally continuous if lim_𝑛→∞𝐴𝑆𝑥_𝑛=𝐴𝑥,lim_𝑛→∞𝑆𝐴𝑥_𝑛=𝑆𝑥 (2.11) whenever {𝑥_𝑛} is a sequence in 𝑋 such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=𝑥∈𝑋. (2.12)

If 𝐴 and 𝑆 are both continuous, then they are obviously reciprocally continuous, but the converse is not true.

2.1. A Class of Implicit Relations

Let 𝐾₄ be the set of all real continuous functions

𝐹∶𝑅⁴₊→𝑅, nondecreasing in first argument and satisfying the following conditions:(i)for 𝑢,𝑣≥0, 𝐹(𝑢,𝑣,𝑣,𝑢)≥0 or 𝐹(𝑢,𝑣,𝑢,𝑣)≥0 implies that 𝑢≥𝑣;(ii)𝐹(𝑢,𝑢,1,1)≥0 implies that 𝑢≥1.

3. Main Result

Theorem 3.1. Let 𝐴,𝐵,𝑆,𝑇,𝐼, and 𝐽 be self-mappings of a complete fuzzy metric space (𝑋,𝑀,∗) such that(a)𝐴𝐵(𝑋)⊂𝐽(𝑋) and 𝑆𝑇(𝑋)⊂𝐼(𝑋);(b)the pair (𝐴𝐵,𝐼) is semicompatible and (𝑆𝑇,𝐽) is weak compatible;(c)the pair (𝐴𝐵,𝐼) is reciprocally continuous.

For some 𝐹∈𝐾₄, there exists 𝑘∈(0,1) such that for all 𝑥,𝑦∈𝑋 and 𝑡>0,𝐹𝑀()𝐹𝑀()𝐴𝐵𝑥,𝑆𝑇𝑦,𝑘𝑡,𝑀𝐼𝑥,𝐽𝑦,𝑡,𝑀𝐴𝐵𝑥,𝐼𝑥,𝑡,𝑀𝑆𝑇𝑦,𝐽𝑦,𝑘𝑡≥0,𝐴𝐵𝑥,𝑆𝑇𝑦,𝑘𝑡,𝑀𝐼𝑥,𝐽𝑦,𝑡,𝑀𝐴𝐵𝑥,𝐼𝑥,𝑘𝑡,𝑀𝑆𝑇𝑦,𝐽𝑦,𝑡≥0. (3.1)(3.2) Then 𝐴𝐵,𝑆𝑇,𝐼, and 𝐽 have a unique common fixed point. Furthermore, if the pairs (𝐴,𝐵),(𝐴,𝐼),(𝐵,𝐼),(𝑆,𝑇),(𝑆,𝐽), and (𝑇,𝐽) are commuting mapping then, 𝐴,𝐵,𝑆,𝑇,𝐼, and 𝐽 have a unique common fixed point.

Proof. Let 𝑥₀ be an arbitrary point in 𝑋. Since 𝐴𝐵(𝑋)⊂𝐽(𝑋) and 𝑆𝑇(𝑋)⊂𝐼(𝑋), there exist 𝑥₁,𝑥₂∈𝑋 such that 𝐴𝐵𝑥₀=𝐽𝑥₁ and 𝑆𝑇𝑥₁=𝐼𝑥₂. Inductively, we construct the sequences {𝑦_𝑛} and {𝑥_𝑛} in 𝑋 such that 𝑦_2𝑛+1=𝐴𝐵𝑥_2𝑛=𝐽𝑥_2𝑛+1,𝑦_2𝑛+2=𝑆𝑇𝑥_2𝑛+1=𝐼𝑥_2𝑛+2 (3.3) for 𝑛=0,1,2,…. Now putting in (3.1) 𝑥=𝑥_2𝑛, 𝑦=𝑥_2𝑛+1, we obtain 𝐹𝑀𝐴𝐵𝑥_2𝑛,𝑆𝑇𝑥_2𝑛+1,𝑘𝑡,𝑀𝐼𝑥_2𝑛,𝐽𝑥_2𝑛+1,𝑡,𝑀𝐴𝐵𝑥_2𝑛,𝐼𝑥_2𝑛,𝑡,𝑀𝑆𝑇𝑥_2𝑛+1,𝐽𝑥_2𝑛+1,𝑘𝑡≥0 (3.4) that is, 𝐹𝑀𝑦_2𝑛+1,𝑦_2𝑛+2,𝑘𝑡,𝑀𝑦_2𝑛,𝑦_2𝑛+1,𝑡,𝑀𝑦_2𝑛+1,𝑦_2𝑛,𝑡,𝑀𝑦_2𝑛+2,𝑦_2𝑛+1,𝑘𝑡≥0. (3.5) Using (i), we get 𝑀𝑦_2𝑛+2,𝑦_2𝑛+1,𝑘𝑡≥𝑀𝑦_2𝑛+1,𝑦_2𝑛,𝑡. (3.6) Analogously, putting 𝑥=𝑥_2𝑛+2, 𝑦=𝑥_2𝑛+1 in (3.2), we have 𝐹𝑀𝑦_2𝑛+3,𝑦_2𝑛+2,𝑘𝑡,𝑀𝑦_2𝑛+1,𝑦_2𝑛+2,𝑡,𝑀𝑦_2𝑛+3,𝑦_2𝑛+2,𝑘𝑡,𝑀𝑦_2𝑛+1,𝑦_2𝑛+2,𝑡≥0. (3.7) Using (i), we get 𝑀𝑦_2𝑛+3,𝑦_2𝑛+2,𝑘𝑡≥𝑀𝑦_2𝑛+1,𝑦_2𝑛+2,𝑡. (3.8) Thus, from (3.6) and (3.8), for 𝑛 and 𝑡, we have 𝑀𝑦_𝑛,𝑦_𝑛+1,𝑘𝑡≥𝑀𝑦_𝑛−1,𝑦_𝑛,𝑡. (3.9) Hence, by Lemma 2.7, {𝑦_𝑛} is a Cauchy sequence in 𝑋, which is complete. Therefore, {𝑦_𝑛} converges to 𝑝∈𝑋. The sequences {𝐴𝐵𝑥_2𝑛}, {𝑆𝑇𝑥_2𝑛+1}, {𝐼𝑥_2𝑛}, and {𝐽𝑥_2𝑛+1}, being subsequences of {𝑦_𝑛}, also converge to 𝑝, that is, 𝐴𝐵𝑥_2𝑛⟶𝑝,𝑆𝑇𝑥_2𝑛+1⟶𝑝,𝐼𝑥_2𝑛⟶𝑝,𝐽𝑥_2𝑛+1⟶𝑝. (3.10) The reciprocal continuity of the pair (𝐴𝐵,𝐼) gives 𝐴𝐵𝐼𝑥_2𝑛⟶𝐴𝐵𝑝,𝐼𝐴𝐵𝑥_2𝑛⟶𝐼𝑝. (3.11) The semicompatibility of the pair (𝐴𝐵,𝐼) gives lim_𝑛→∞𝐴𝐵𝐼𝑥_2𝑛=𝐼𝑝. (3.12) From the uniqueness of the limit in a fuzzy metric space, we obtain that 𝐴𝐵𝑝=𝐼𝑝. (3.13)
Step 1. By putting 𝑥=𝑝, 𝑦=𝑥_2𝑛+1 in (3.1), we obtain 𝐹𝑀𝐴𝐵𝑝,𝑆𝑇𝑥_2𝑛+1,𝑘𝑡,𝑀𝐼𝑝,𝐽𝑥_2𝑛+1,𝑡,𝑀𝐴𝐵𝑝,𝐼𝑝,𝑡,𝑀𝑆𝑇𝑥_2𝑛+1,𝐽𝑥_2𝑛+1,𝑘𝑡≥0. (3.14) Letting 𝑛 tends to infinity and using (3.10) and (3.13), we get 𝐹𝑀𝐼𝑝,𝑝,𝑘𝑡,𝑀𝐼𝑝,𝑝,𝑡,𝑀𝐼𝑝,𝐼𝑝,𝑡,𝑀𝑝,𝑝,𝑘𝑡≥0. (3.15) As 𝐹 is nondecreasing in first argument, we have 𝐹𝑀𝐼𝑝,𝑝,𝑡,𝑀𝐼𝑝,𝑝,𝑡,1,1≥0. (3.16) Using (ii), we have 𝑀(𝐼𝑝,𝑝,𝑡)≥1 for all 𝑡>0, which gives 𝑀(𝐼𝑝,𝑝,𝑡)=1, that is, 𝐼𝑝=𝑝=𝐴𝐵𝑝. (3.17)
Step 2. As 𝐴𝐵(𝑋)⊂𝐽(𝑋), there exists 𝑢∈𝑋 such that 𝐴𝐵𝑝=𝐼𝑝=𝑝=𝐽𝑢.
Putting 𝑥=𝑥_2𝑛, 𝑦=𝑢 in (3.1), we obtain that 𝐹𝑀𝐴𝐵𝑥_2𝑛,𝑆𝑇𝑢,𝑘𝑡,𝑀𝐼𝑥_2𝑛,𝐽𝑢,𝑡,𝑀𝐴𝐵𝑥_2𝑛,𝐼𝑥_2𝑛,𝑡,𝑀(𝑆𝑇𝑢,𝐽𝑢,𝑘𝑡)≥0. (3.18) Letting 𝑛 tends to infinity and using (3.10), we get 𝐹𝑀𝑝,𝑆𝑇𝑢,𝑘𝑡,1,1,𝑀𝑆𝑇𝑢,𝑝,𝑘𝑡≥0. (3.19) Using (i), we have 𝑀(𝑝,𝑆𝑇𝑢,𝑘𝑡)≥1 for all 𝑡>0, which gives 𝑀(𝑝,𝑆𝑇𝑢,𝑘𝑡)=1.
Thus, 𝑝=𝑆𝑇𝑢. Therefore, 𝑆𝑇𝑢=𝐽𝑢=𝑝. Since (𝑆𝑇,𝐽) is weak compatible, we get 𝐽𝑆𝑇𝑢=𝑆𝑇𝐽𝑢, that is, 𝑆𝑇𝑝=𝐽𝑝. (3.20)
Step 3. By putting 𝑥=𝑝, 𝑦=𝑝 in (3.1) and using (3.17) and (3.20), we obtain 𝐹𝑀𝐴𝐵𝑝,𝑆𝑇𝑝,𝑘𝑡,𝑀𝐼𝑝,𝐽𝑝,𝑡,𝑀𝐴𝐵𝑝,𝐼𝑝,𝑡,𝑀𝑆𝑇𝑝,𝐽𝑝,𝑘𝑡≥0 (3.21) that is, 𝐹(𝑀(𝐴𝐵𝑝,𝑆𝑇𝑝,𝑘𝑡),𝑀(𝐴𝐵𝑝,𝑆𝑇𝑝,𝑡),1,1)≥0.
As 𝐹 is nondecreasing in first argument, we have 𝐹𝑀𝐴𝐵𝑝,𝑆𝑇𝑝,𝑡,𝑀𝐴𝐵𝑝,𝑆𝑇𝑝,𝑡,1,1≥0. (3.22) Using (ii), we have 𝑀(𝐴𝐵𝑝,𝑆𝑇𝑝,𝑡)≥1 for all 𝑡>0, which gives 𝑀(𝐴𝐵𝑝,𝑆𝑇𝑝,𝑡)=1.
Thus, 𝐴𝐵𝑝=𝑆𝑇𝑝. (3.23) Therefore, 𝑝=𝐴𝐵𝑝=𝑆𝑇𝑝=𝐼𝑝=𝐽𝑝, that is, 𝑝 is a common fixed point of 𝐴𝐵,𝑆𝑇,𝐼, and 𝐽.
Uniqueness. Let 𝑞 be another common fixed point of 𝐴𝐵,𝑆𝑇,𝐼, and 𝐽. Then, 𝑞=𝐴𝐵𝑞=𝑆𝑇𝑞=𝐼𝑞=𝐽𝑞.
By putting 𝑥=𝑝 and 𝑦=𝑞 in (3.1), we get 𝐹𝑀𝐴𝐵𝑝,𝑆𝑇𝑞,𝑘𝑡,𝑀𝐼𝑝,𝐽𝑞,𝑡,𝑀𝐴𝐵𝑝,𝐼𝑝,𝑡,𝑀𝑆𝑇𝑞,𝐽𝑞,𝑘𝑡≥0 (3.24) that is, 𝐹(𝑀(𝑝,𝑞,𝑘𝑡),𝑀(𝑝,𝑞,𝑡),1,1)≥0.
As 𝐹 is nondecreasing in first argument, we have 𝐹𝑀𝑝,𝑞,𝑡,𝑀𝑝,𝑞,𝑡,1,1≥0. (3.25) Using (ii), we have 𝑀(𝑝,𝑞,𝑡)≥1 for all 𝑡>0, which gives 𝑀(𝑝,𝑞,𝑡)=1, that is, 𝑝=𝑞. Therefore, 𝑝 is the unique common fixed point of the self-maps 𝐴𝐵,𝑆𝑇,𝐼, and 𝐽.
Finally, we need to show that 𝑝 is also a common fixed point of 𝐴,𝐵,𝑆,𝑇,𝐼, and 𝐽. For this, let 𝑝 be the unique common fixed point of both the pairs (𝐴𝐵,𝐼) and (𝑆𝑇,𝐽). Then, by using commutativity of the pair (𝐴,𝐵), (𝐴,𝐼), and (𝐵,𝐼), we obtain ,𝐴,𝐴𝑝=𝐴𝐴𝐵𝑝=𝐴𝐵𝐴𝑝=𝐴𝐵𝐴𝑝,𝐴𝑝=𝐴𝐼𝑝=𝐼𝐴𝑝𝐵𝑝=𝐵𝐴𝐵𝑝=𝐵𝐵𝑝=𝐵𝐴𝐵𝑝=𝐴𝐵𝐵𝑝,𝐵𝑝=𝐵𝐼𝑝=𝐼𝐵𝑝 (3.26) which shows that 𝐴𝑝 and 𝐵𝑝 are common fixed point of (𝐴𝐵,𝐼), yielding thereby 𝐴𝑝=𝑝=𝐵𝑝=𝐼𝑝=𝐴𝐵𝑝 (3.27) in the view of uniqueness of the common fixed point of the pair (𝐴𝐵,𝐼). Similarly, using the commutativity of (𝑆,𝑇), (𝑆,𝐽), (𝑇,𝐽), it can be shown that 𝑆𝑝=𝑇𝑝=𝐽𝑝=𝑆𝑇𝑝=𝑝. (3.28) Now, we need to show that 𝐴𝑝=𝑆𝑝(𝐵𝑝=𝑇𝑝) also remains a common fixed point of both the pairs (𝐴𝐵,𝐼) and (𝑆𝑇,𝐽). For this, put 𝑥=𝑝 and 𝑦=𝑝 in (3.1), and using (3.27) and (3.28), we get 𝐹𝑀𝐴𝐵𝑝,𝑆𝑇𝑝,𝑘𝑡,𝑀𝐼𝑝,𝐽𝑝,𝑡,𝑀𝐴𝐵𝑝,𝐼𝑝,𝑡,𝑀𝑆𝑇𝑝,𝐽𝑝,𝑘𝑡≥0, (3.29) that is, 𝐹𝑀𝐴𝑝,𝑆𝑝,𝑘𝑡,𝑀𝐴𝑝,𝑆𝑝,𝑡,𝑀𝐴𝑝,𝐴𝑝,𝑡,𝑀𝑆𝑝,𝑆𝑝,𝑘𝑡≥0. (3.30) As 𝐹 is nondecreasing in first argument, we have 𝐹𝑀𝐴𝑝,𝑆𝑝,𝑡,𝑀𝐴𝑝,𝑆𝑝,𝑡,1,1≥0. (3.31) Using (ii), we obtain 𝑀𝐴𝑝,𝑆𝑝,𝑡≥1∀𝑡>0, (3.32) which gives 𝑀(𝐴𝑝,𝑆𝑝,𝑡)=1, that is, 𝐴𝑝=𝑆𝑝. Similarly, it can be shown that 𝐵𝑝=𝑇𝑝. Thus, 𝑝 is the unique common fixed point of 𝐴,𝐵,𝑆,𝑇,𝐼, and 𝐽. This completes the proof of our theorem.

Since semicompatibility implies weak compatibility, we have the following.

Corollary 3.2. Let 𝐴,𝐵,𝑆,𝑇,𝐼, and 𝐽 be self-maps of a complete fuzzy metric space (𝑋,𝑀,∗) satisfying the conditions (a), (3.1), and (3.2) of the above theorem, and the pairs (𝐴𝐵,𝐼) and (𝑆𝑇,𝐽) are semicompatible and one of the pair, (𝐴𝐵,𝐼) or (𝑆𝑇,𝐽), is reciprocally continuous. Then, 𝐴𝐵,𝑆𝑇,𝐼, and 𝐽 have a unique common fixed point. Furthermore, if the pairs (𝐴,𝐵),(𝐴,𝐼),(𝐵,𝐼),(𝑆,𝑇),(𝑆,𝐽), and (𝑇,𝐽) are commuting mapping, then 𝐴,𝐵,𝑆,𝑇,𝐼, and 𝐽 have a unique common fixed point.

If we take 𝐵 = 𝑇 = identity map in Theorem 3.1, then we have the following.

Corollary 3.3. Let 𝐴,𝑆,𝐼, and 𝐽 be self-mappings of a complete fuzzy metric space (𝑋,𝑀,∗) such that(a)𝐴(𝑋)⊂𝐽(𝑋) and 𝑆(𝑋)⊂𝐼(𝑋);(b)(𝐴,𝐼), (𝑆,𝐽) are semicompatible commuting pair of maps;(c)the pair (𝐴,𝐼) or (𝑆,𝐽) is reciprocally continuous.

For some 𝐹∈𝐾₄, there exists 𝑘∈(0,1) such that for all 𝑥,𝑦∈𝑋 and 𝑡>0,𝐹𝑀()𝐹𝑀()𝐴𝑥,𝑆𝑦,𝑘𝑡,𝑀𝐼𝑥,𝐽𝑦,𝑡,𝑀𝐴𝑥,𝐼𝑥,𝑡,𝑀𝑆𝑦,𝐽𝑦,𝑘𝑡≥0,𝐴𝑥,𝑆𝑦,𝑘𝑡,𝑀𝐼𝑥,𝐽𝑦,𝑡,𝑀𝐴𝑥,𝐼𝑥,𝑘𝑡,𝑀𝑆𝑦,𝐽𝑦,𝑡≥0. (3.33) Then, 𝐴,𝑆,𝐼, and 𝐽 have a unique common fixed point.

This theorem proves that the theorem of Singh holds even when the pairs are reciprocally continuous.

On taking 𝐼 = identity mapping in Theorem 3.1, we have the following result for 5 self-maps.

Corollary 3.4. Let 𝐴,𝐵,𝑆,𝑇, and 𝐽 be self-mappings of a complete fuzzy metric space (𝑋,𝑀,∗) such that(a)𝐴𝐵(𝑋)∩𝑆𝑇(𝑋)⊂𝐽(𝑋),(b)(𝑆𝑇,𝐽) is weak compatible.

For some 𝐹∈𝐾₄, there exists 𝑘∈(0,1) such that for all 𝑥,𝑦∈𝑋 and 𝑡>0, 𝐹𝑀()𝐹𝑀()𝐴𝐵𝑥,𝑆𝑇𝑦,𝑘𝑡,𝑀𝑥,𝐽𝑦,𝑡,𝑀𝐴𝐵𝑥,𝑥,𝑡,𝑀𝑆𝑇𝑦,𝐽𝑦,𝑘𝑡≥0𝐴𝐵𝑥,𝑆𝑇𝑦,𝑘𝑡,𝑀𝑥,𝐽𝑦,𝑡,𝑀𝐴𝐵𝑥,𝑥,𝑘𝑡,𝑀𝑆𝑇𝑦,𝐽𝑦,𝑡≥0. (3.34) Then, 𝐴𝐵,𝑆𝑇, and 𝐽 have a unique common fixed point. Furthermore, if the pairs (𝐴,𝐵), (𝑆,𝑇), (𝑆,𝐽), and (𝑇,𝐽) are commuting mapping then 𝐴,𝐵,𝑆,𝑇, and 𝐽 have a unique common fixed point.

If we take 𝐵=𝑇=𝐼=𝐽= identity mapping in Theorem 3.1, then the conditions (a), (b), and (c) are satisfied trivially, and we get the following result.

Corollary 3.5. Let 𝐴 and 𝑆 be self-mappings of a complete fuzzy metric space (𝑋,𝑀,∗) such that for some 𝐹∈𝐾₄, there exists 𝑘∈(0,1) such that for all 𝑥,𝑦∈𝑋 and 𝑡>0, 𝐹𝑀()𝐹𝑀()𝐴𝑥,𝑆𝑦,𝑘𝑡,𝑀𝑥,𝑦,𝑡,𝑀𝐴𝑥,𝑥,𝑡,𝑀𝑆𝑦,𝑦,𝑘𝑡≥0,𝐴𝑥,𝑆𝑦,𝑘𝑡,𝑀𝑥,𝑦,𝑡,𝑀𝐴𝑥,𝑥,𝑘𝑡,𝑀𝑆𝑦,𝑦,𝑡≥0. (3.35) Then, 𝐴 and 𝑆 have a unique common fixed point in 𝑋.

Theorem 3.6. Let 𝐴,𝐵,𝑆,𝑇,𝐼, and 𝐽 be self-maps on a complete fuzzy metric space (𝑋,𝑀,∗) satisfying conditions (a), (c), (3.1), and (3.2) of Theorem 3.1, and (𝐴𝐵,𝐼) is compatible, and (𝑆𝑇,𝐽) is weak compatible. Then, 𝐴𝐵,𝑆𝑇,𝐼, and 𝐽 have a unique common fixed point. Furthermore, if the pairs (𝐴,𝐵), (𝐴,𝐼), (𝐵,𝐼), (𝑆,𝑇), (𝑆,𝐽), and (𝑇,𝐽) are commuting mapping then 𝐴,𝐵,𝑆,𝑇,𝐼, and 𝐽 have a unique common fixed point.

Proof. As in the proof of Theorem 3.1, the sequence {𝑦_𝑛} converges to 𝑝∈𝑋, and (3.10) is satisfied.
The reciprocal continuity of the pair (𝐴𝐵,𝐼) gives 𝐴𝐵𝐼𝑥_2𝑛⟶𝐴𝐵𝑝,𝐼𝐴𝐵𝑥_2𝑛⟶𝐼𝑝. (3.36) The compatibility of the pair (𝐴𝐵,𝐼) gives lim_𝑛→∞𝐴𝐵𝐼𝑥_2𝑛=𝐴𝐵𝑝=lim_𝑛→∞𝐼𝐴𝐵𝑥₂𝑛. (3.37) From the uniqueness of the limit in a fuzzy metric space, we obtain that 𝐴𝐵𝑝=𝐼𝑝.
Step 1. By putting 𝑥=𝐴𝐵𝑥_2𝑛, 𝑦=𝑥_2𝑛+1 in (3.1), we obtain 𝐹𝑀𝐴𝐵𝐴𝐵𝑥_2𝑛,𝑆𝑇𝑥_2𝑛+1,𝑘𝑡,𝑀𝐼𝐴𝐵𝑥_2𝑛,𝐽𝑥_2𝑛+1,𝑀,𝑡𝐴𝐵𝐴𝐵𝑥_2𝑛,𝐼𝐴𝐵𝑥_2𝑛,𝑡,𝑀𝑆𝑇𝑥_2𝑛+1,𝐽𝑥_2𝑛+1,𝑘𝑡≥0. (3.38) Letting 𝑛 tends to infinity and using (3.10) and (3.13), we get 𝐹𝑀𝐴𝐵𝑝,𝑝,𝑘𝑡,𝑀𝐴𝐵𝑝,𝑝,𝑡,𝑀𝐴𝐵𝑝,𝐴𝐵𝑝,𝑡,𝑀𝑝,𝑝,𝑘𝑡≥0. (3.39) As 𝐹 is nondecreasing in first argument, we have 𝐹𝑀𝐴𝐵𝑝,𝑝,𝑡,𝑀𝐴𝐵𝑝,𝑝,𝑡,1,1≥0. (3.40) Using (ii), we have 𝑀(𝐴𝐵𝑝,𝑝,𝑡)≥1 for all 𝑡>0, which gives 𝑀(𝐴𝐵𝑝,𝑝,𝑡)=1, that is, 𝐼𝑝=𝑝=𝐴𝐵𝑝. (3.41)
Step 2. As 𝐴𝐵(𝑋)⊂𝐽(𝑋), there exists 𝑢∈𝑋 such that 𝐴𝐵𝑝=𝐼𝑝=𝑝=𝐽𝑢.
Putting 𝑥=𝑥_2𝑛, 𝑦=𝑢 in (3.1), we obtain that 𝐹𝑀𝐴𝐵𝑥_2𝑛,𝑆𝑇𝑢,𝑘𝑡,𝑀𝐼𝑥_2𝑛,𝐽𝑢,𝑡,𝑀𝐴𝐵𝑥_2𝑛,𝐼𝑥_2𝑛,𝑡,𝑀(𝑆𝑇𝑢,𝐽𝑢,𝑘𝑡)≥0. (3.42) Letting 𝑛 tends to infinity and using (3.10), we get 𝐹𝑀𝑝,𝑆𝑇𝑢,𝑘𝑡,1,1,𝑀𝑆𝑇𝑢,𝑝,𝑘𝑡≥0. (3.43) Using (i), we have 𝑀(𝑝,𝑆𝑇𝑢,𝑘𝑡)≥1 for all 𝑡>0, which gives 𝑀(𝑝,𝑆𝑇𝑢,𝑘𝑡)=1. Thus, 𝑝=𝑆𝑇𝑢. Therefore, 𝑆𝑇𝑢=𝐽𝑢=𝑝. Since (𝑆𝑇,𝐽) is weak compatible, we get  𝐽𝑆𝑇𝑢=𝑆𝑇𝐽𝑢, that is, 𝑆𝑇𝑝=𝐽𝑝. (3.44)
Step 3. By putting 𝑥=𝑝, 𝑦=𝑝 in (3.1) and using (3.17) and (3.20), we obtain 𝐹𝑀𝐴𝐵𝑝,𝑆𝑇𝑝,𝑘𝑡,𝑀𝐼𝑝,𝐽𝑝,𝑡,𝑀𝐴𝐵𝑝,𝐼𝑝,𝑡,𝑀𝑆𝑇𝑝,𝐽𝑝,𝑘𝑡≥0, (3.45) that is, 𝐹(𝑀(𝐴𝐵𝑝,𝑆𝑇𝑝,𝑘𝑡),𝑀(𝐴𝐵𝑝,𝑆𝑇𝑝,𝑡),1,1)≥0.
As 𝐹 is nondecreasing in first argument, we have 𝐹𝑀𝐴𝐵𝑝,𝑆𝑇𝑝,𝑡,𝑀𝐴𝐵𝑝,𝑆𝑇𝑝,𝑡,1,1≥0. (3.46) Using (ii), we have 𝑀(𝐴𝐵𝑝,𝑆𝑇𝑝,𝑡)≥1 for all 𝑡>0, which gives 𝑀(𝐴𝐵𝑝,𝑆𝑇𝑝,𝑡)=1. Thus, 𝐴𝐵𝑝=𝑆𝑇𝑝. (3.47)
Therefore, 𝑝=𝐴𝐵𝑝=𝑆𝑇𝑝=𝐼𝑝=𝐽𝑝, that is, 𝑝 is a common fixed point of 𝐴𝐵,𝑆𝑇,𝐼, and 𝐽.
The rest of the proof is the same as in Theorem 3.1.

Since compatibility implies weak compatibility, we have the following.

Corollary 3.7. Let 𝐴,𝐵,𝑆,𝑇,𝐼, and 𝐽 be self-maps on a complete fuzzy metric space (𝑋,𝑀,∗) satisfying conditions (a), (3.1), and (3.2) of Theorem 3.1, and the pairs (𝐴𝐵,𝐼) and (𝑆𝑇,𝐽) are compatible, and one of the pair, (𝐴𝐵,𝐼) or (𝑆𝑇,𝐽), is reciprocally continuous. Then, 𝐴𝐵,𝑆𝑇,𝐼, and 𝐽 have a unique common fixed point in 𝑋. Furthermore, if the pairs (𝐴,𝐵), (𝐴,𝐼), (𝐵,𝐼), (𝑆,𝑇), (𝑆,𝐽), and (𝑇,𝐽) are commuting mapping, then 𝐴,𝐵,𝑆,𝑇,𝐼, and 𝐽 have a unique common fixed point.