Common Fixed Point Theorems Using the E.A. and CLR Properties in 2-Menger Spaces

Abstract

First, we prove a common fixed point theorem using weakly compatible maps in 2-Menger space with t-norm of Hadzic type. Second, we prove a common fixed point theorem using the E.A. property along with weakly compatible maps. Further, we obtained a common fixed point theorem using the CLR property along with weakly compatible maps. At the end, we provide an application of our main theorem for four finite families of mappings.

1. Introduction

The theory of probabilistic metric spaces is an important part of stochastic analysis, and so it is of interest to develop the fixed point theory in such spaces. The first result from the fixed point theory in probabilistic metric spaces was obtained by Sehgal and Bharucha-Reid [1]. Since then many fixed points theorems for single valued and multivalued mappings in probabilistic metric spaces have been proved in [2–8]. The study of 2-metric spaces was initiated by Gähler [9] and some fixed point theorems in 2-metric spaces were proved in [2, 8, 10–13]. In 1987, Zeng [14] gave the generalization of 2-metric to probabilistic 2-metric as follows.

A probabilistic 2-metric space is an ordered pair (𝑋,𝐹), where 𝑋 is an arbitrary set and 𝐹 is a mapping from 𝑋³ into the set of distribution functions. The distribution function 𝐹_{𝑥,𝑦,𝑧}(𝑡) will denote the value of 𝐹_{𝑥,𝑦,𝑧} at the positive real number 𝑡. The function 𝐹_{𝑥,𝑦,𝑧} is assumed to satisfy the following conditions:(i)𝐹_{𝑥,𝑦,𝑧}(0)=0 for all 𝑥,𝑦,𝑧∈𝑋;(ii)𝐹_{𝑥,𝑦,𝑧}(𝑡)=1 for all 𝑡>0 if and only if at least two of the three points 𝑥,𝑦,𝑧 are equal;(iii)for distinct points 𝑥,𝑦∈𝑋, there exists a point 𝑧∈𝑋 such that 𝐹_{𝑥,𝑦,𝑧}(𝑡)≠1 for 𝑡>0;(iv)𝐹_{𝑥,𝑦,𝑧}(𝑡)=𝐹_{𝑦,𝑧,𝑥}(𝑡)=𝐹_{𝑧,𝑥,𝑦}(𝑡)=⋯ for all 𝑥,𝑦,𝑧∈𝑋 and 𝑡>0;(v)if 𝐹_{𝑥,𝑦,𝑤}(𝑡₁)=1, 𝐹_{𝑥,𝑤,𝑧}(𝑡₂)=1, and 𝐹_{𝑤,𝑦,𝑧}(𝑡₃)=1, then 𝐹_{𝑥,𝑦,𝑧}(𝑡₁+𝑡₂+𝑡₃)=1 for all 𝑥,𝑦,𝑧,𝑤∈𝑋 and 𝑡₁,𝑡₂,𝑡₃>0.

In 2003, Shi et al. [15] gave the notion of 𝑛th order 𝑡-norm as follows.

Definition 1. A mapping Δ∶∏^𝑛_𝑖=1[0,1]→[0,1] is called an 𝑛th order 𝑡-norm if the following conditions are satisfied:(i)Δ(0,0,…,0)=0, Δ(𝑎,1,1,…,1)=𝑎 for all 𝑎∈[0,1];(ii)Δ(𝑎₁,𝑎₂,𝑎₃,…,𝑎_𝑛)=Δ(𝑎₂,𝑎₁,𝑎₃,…,𝑎_𝑛)=Δ(𝑎₂,𝑎₃,𝑎₁,…,𝑎_𝑛)=⋯=Δ(𝑎₂,𝑎₃,𝑎₄,…,𝑎_𝑛,𝑎₁); (iii)𝑎_𝑖≥𝑏_𝑖,𝑖=1,2,3…,𝑛, implies Δ(𝑎₁,𝑎₂,𝑎₃,…,𝑎_𝑛)≥Δ(𝑏₁,𝑏₂,𝑏₃,…,𝑏_𝑛);(iv)Δ(Δ(𝑎₁,𝑎₂,𝑎₃,…,𝑎_𝑛),𝑏₂,𝑏₃,…,𝑏_𝑛) =Δ(𝑎₁,Δ(𝑎₂,𝑎₃,…,𝑎_𝑛,𝑏₂),𝑏₃,…𝑏_𝑛) =Δ(𝑎₁,𝑎₂,Δ(𝑎₃,𝑎₄,….,𝑎_𝑛,𝑏₂,𝑏₃),𝑏₄,…,𝑏_𝑛) =⋯=Δ(𝑎₁,𝑎₂,𝑎₃,…,𝑎_𝑛−1,Δ(𝑎_𝑛,𝑏₂,𝑏₃,…,𝑏_𝑛)).
For 𝑛=2, we have a binary 𝑡-norm, which is commonly known as 𝑡-norm.
Basic examples of 𝑡-norm are the Lukasiewicz 𝑡-norm Δ_𝐿,Δ_𝐿(𝑎,𝑏)=max(𝑎+𝑏−1,0),𝑡-norm Δ_𝑃, Δ_𝑃(𝑎,𝑏)=𝑎𝑏, and 𝑡-norm Δ_𝑀, Δ_𝑀(𝑎,𝑏)=min{𝑎,𝑏}.

Definition 2 (see [6]). A special class of t-norms (called a Hadzic-type 𝑡-norm) is introduced as follows.
Let Δ be a 𝑡-norm and let Δ_𝑛∶[0,1]→[0,1](𝑛∈𝑁) be defined in the following way: Δ₁(𝑥)=Δ(𝑥,𝑥),Δ_𝑛+1(𝑥)=ΔΔ_𝑛(𝑥)([.,𝑥𝑛∈𝑁,𝑥∈0,1]) (1)
We say that the 𝑡-norm Δ is of 𝐻 type if Δ is continuous and the family {Δ_𝑛(𝑥),𝑛∈𝑁} is equicontinuous at 𝑥=1.
The family {Δ_𝑛(𝑥),𝑛∈𝑁} is equicontinuous at 𝑥=1 if for every 𝜆∈(0,1) there exists 𝛿(𝜆)∈(0,1) such that the following implication holds: 𝑥>1−𝛿(𝜆)impliesΔ_𝑛(𝑥)>1−𝜆∀𝑛∈𝑁. (2)
A trivial example of t-norm of 𝐻 type is Δ=Δ_𝑀.

Remark 3. Every 𝑡-norm Δ_𝑀 is of Hadzic type but the converse need not be true; see [7].
There is a nice characterization of continuous 𝑡-norm.(i)If there exists a strictly increasing sequence {𝑏_𝑛}_𝑛∈𝑁∈[0,1] such that lim_𝑛→∞𝑏_𝑛=1and Δ(𝑏_𝑛,𝑏_𝑛)=𝑏_𝑛 for all 𝑛∈𝑁, then Δ is of Hadzic type.(ii)If Δ is continuous and Δ is of Hadzic type, then there exists a sequence {𝑏_𝑛}_𝑛∈𝑁 as in (i).

Definition 4 (see [7]). If Δ is a 𝑡-norm and (𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛)∈[0,1]^𝑛(𝑛∈𝑁), then Δ^𝑛_𝑖=1𝑥_𝑖 is defined recurrely by 1, if 𝑛=1 and Δ^𝑛_𝑖=1𝑥_𝑖=Δ(Δ^𝑛−1_𝑖=1𝑥_𝑖,𝑥_𝑛) for all 𝑛≥2. If {𝑏_𝑛}_𝑛∈𝑁 is a sequence of numbers from [0,1], then Δ^∞_𝑖=1𝑥_𝑖 is defined as lim_𝑛→∞Δ^𝑛_𝑖=1𝑥_𝑖 (this limit always exists) and Δ^∞_𝑖=1𝑥_𝑖 as Δ^∞_𝑖=1𝑥_𝑛+𝑖.

Definition 5. Let 𝑋 be any nonempty set and 𝐷 the set of all left-continuous distribution functions. A triplet (𝑋,𝐹,Δ) is said to be a 2-Menger space if the probabilistic 2-metric space (𝑋,𝐹) satisfies the following condition: 𝐹_{𝑥,𝑦,𝑧}(𝑡)≥Δ(𝐹_{𝑥,𝑦,𝑤}(𝑡₁), 𝐹_{𝑥,𝑤,𝑧}(𝑡₂), 𝐹_{𝑤,𝑦,𝑧}(𝑡₃)), where 𝑡₁,𝑡₂,𝑡₃>0, 𝑡₁+𝑡₂+𝑡₃=𝑡; 𝑥,𝑦,𝑧,𝑤∈𝑋; and Δ is the 3rd order 𝑡-norm.

Definition 6. A sequence {𝑥_𝑛} in a 2-Menger space (𝑋,𝐹,Δ) is said to be(i)converge with limit 𝑥 if lim_𝑛→∞𝐹_{𝑥𝑛,𝑥,𝑎}(𝑡)=1 for all 𝑡>0 and for every 𝑎∈𝑋,(ii)Cauchy sequence in 𝑋, if given 𝜖>0, 𝜆>0, there exists a positive integer 𝑁_𝜖,𝜆 such that 𝐹_{𝑥𝑛,𝑥𝑚,𝑎}(𝜖)>1−𝜆∀𝑚,𝑛>𝑁_𝜖,𝜆,forevery𝑎∈𝑋, (3)(iii)complete if every Cauchy sequence in 𝑋 is convergent in 𝑋.In 1996, Jungck’s [16] introduced the notion of weakly compatible as follows.

Definition 7. Two maps 𝑓 and 𝑔 are said to be weakly compatible if they commute at their coincidence points.

In 2002, Aamri and Moutawakil [17] generalized the notion of noncompatible mapping to the E.A. property. It was pointed out in [17] that the property E.A. buys containment of ranges without any continuity requirements besides minimizing the commutativity conditions of the maps to the commutativity at their points of coincidence. Moreover, the E.A. property allows replacing the completeness requirement of the space with a more natural condition of closeness of the range. Recently, some common fixed point theorems in probabilistic metric spaces/fuzzy metric spaces by the E.A. property under weak compatibility have been obtained in [18–20].

Definition 8 (see [17]). Let 𝑓 and 𝑔 be two self-maps of a metric (𝑋, 𝑑). The maps 𝑓 and 𝑔 are said to satisfy the E.A. property if there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝑢forsome𝑢∈𝑋. (4)
Now in a similar mode, we can state the E.A. property in 2-Menger space as follows.

Definition 9. A pair of self-mappings (𝑓,𝑔) of 2-Menger spaces (𝑋,𝐹,Δ) is said to hold the E.A. property if there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝐹_{𝑓𝑥𝑛,𝑔𝑥𝑛,𝑝}(𝑡)=1∀𝑡>0,𝑝∈𝑋. (5)

Example 10. Let 𝑋=[0,∞) be the usual metric space. Define 𝑓,𝑔∶𝑋→𝑋 by 𝑓𝑥=𝑥/4 and 𝑔𝑥=3𝑥/4 for all 𝑥∈𝑋. Consider the sequence {𝑥_𝑛}=1/𝑛. Since lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=0, then 𝑓 and 𝑔 satisfy the E.A. property.
Although E.A property is generalization of the concept of noncompatible maps, yet it requires either completeness of the whole space or any of the range spaces or continuity of maps. But on the contrary, the new notion of the CLR property (common limit range property) recently given by Sintunavarat and Kumam [21] does not impose such conditions. The importance of the CLR property ensures that one does not require the closeness of range subspaces.

Definition 11 (see [21]). Two maps 𝑓 and 𝑔 on 2-Menger spaces 𝑋 satisfy the common limit in the range of 𝑔 (CLRg) property if lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝑔𝑥 for some 𝑥∈𝑋.

Example 12. Let 𝑋=[0,∞) be the usual metric space. Define 𝑓,𝑔∶𝑋→𝑋 by 𝑓𝑥=𝑥+1 and 𝑔𝑥=2𝑥 for all 𝑥∈𝑋. Consider the sequence {𝑥_𝑛}=1+(1/𝑛). Since lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=2=𝑔1, 𝑓 and 𝑔 satisfy the CLRg property.

Now we state a lemma which is useful in our study.

Lemma 13 (see [22]). Let (𝑋,𝐹,Δ) be a 2-Menger space. If there exists 𝑞∈(0,1) such that 𝐹(𝑥,𝑦,𝑧,𝑞𝑡)≥𝐹(𝑥,𝑦,𝑧,𝑡) for all 𝑥,𝑦,𝑧∈𝑋 with 𝑧≠𝑥,𝑧≠𝑦, and 𝑡>0, then 𝑥=𝑦.

2. Weakly Compatible Maps

Theorem 14. Let (𝑋,𝐹,Δ) be a complete 2-Menger space with continuous t-norm Δ of 𝐻 type. Let 𝑆 and 𝑇 be self-mappings of 𝑋. Then 𝑆 and 𝑇 have a unique common fixed point in 𝑋 if and only if there exist two self-mappings 𝐴,𝐵 of 𝑋 satisfying the following:(2.1)𝐴(𝑋)⊂𝑇(𝑋) and 𝐵(𝑋)⊂𝑆(𝑋);(2.2) the pairs (𝐴,𝑆) and (𝐵,𝑇) are weakly compatible;(2.3) there exists 𝑞∈(0,1) such that for every 𝑥,𝑦,𝑝∈𝑋 and 𝑡>0, 𝐹(𝐴𝑥,𝐵𝑦,𝑝,𝑞𝑡)≥min{𝐹(𝑆𝑥,𝑇𝑦,𝑝,𝑡),𝐹(𝐴𝑥,𝑆𝑥,𝑝,𝑡), 𝐹(𝐵𝑦,𝑇𝑦,𝑝,𝑡),𝐹(𝐴𝑥,𝑇𝑦,𝑝,𝑡)};(2.4) one of the subsets 𝐴(𝑋), 𝐵(𝑋), 𝑆(𝑋), or 𝑇(𝑋) is a closed subset of 𝑋. Indeed, 𝐴,𝐵,𝑆, and 𝑇 have a unique common fixed point in 𝑋.

Proof. Suppose that 𝑆 and 𝑇 have a unique common fixed point, say 𝑧∈𝑋.
Define 𝐴∶𝑋→𝑋 by 𝐴𝑥=𝑧 for all 𝑥∈𝑋 and 𝐵∶𝑋→𝑋 by 𝐵𝑥=𝑧 for all 𝑥∈𝑋.
Then one can see that (2.1)–(2.4) are satisfied.
Conversely, assume that there exist two self-mappings 𝐴,𝐵 of 𝑋 satisfying conditions (2.1)–(2.4). From condition (2.1) we can construct two sequences {𝑥_𝑛} and {𝑦_𝑛} of 𝑋 such that 𝑦_2𝑛−1=𝑇𝑥_2𝑛−1=𝐴𝑥_2𝑛−2,𝑦_2𝑛=𝑆𝑥_2𝑛=𝐵𝑥_2𝑛−1for𝑛=1,2,…. (6) Putting 𝑥=𝑥_2𝑛 and 𝑦=𝑥_2𝑛+1 in (2.3), we have that for all 𝑝∈𝑋 and 𝑡>0𝐹𝑦_2𝑛+1,𝑦_2𝑛+2,𝑝,𝑞𝑡=𝐹𝐴𝑥_2𝑛,𝐵𝑥_2𝑛+1𝐹,𝑝,𝑞𝑡≥min𝑆𝑥_2𝑛,𝑇𝑥_2𝑛+1,𝐹,𝑝,𝑡𝐴𝑥_2𝑛,𝑆𝑥_2𝑛,𝐹,𝑝,𝑡𝐵𝑥_2𝑛+1,𝑇𝑥_2𝑛+1,𝐹,𝑝,𝑡𝐴𝑥_2𝑛,𝑇𝑥_2𝑛+1𝐹𝑦,𝑝,𝑡=min_2𝑛,𝑦_2𝑛+1,𝐹𝑦,𝑝,𝑞𝑡_2𝑛+1,𝑦_2𝑛+2,𝑝,𝑞𝑡 (7) implies 𝐹(𝑦_2𝑛+1,𝑦_2𝑛+2,𝑝,𝑞𝑡)≥𝐹(𝑦_2𝑛,𝑦_2𝑛+1,𝑝,𝑡), because 𝐹 is nondecreasing.
Also, letting 𝑥=𝑥_2𝑛+2 and 𝑦=𝑥_2𝑛+1 in (2.3), we have that 𝐹𝑦_2𝑛+2,𝑦_2𝑛+3,𝑝,𝑞𝑡≥𝐹𝑦_2𝑛+1,𝑦_2𝑛+2,𝑝,𝑡∀𝑝∈𝑋,𝑡>0. (8) In general, we have 𝐹𝑦_𝑛,𝑦_𝑛+1,𝑝,𝑞𝑡≥𝐹𝑦_𝑛−1,𝑦_𝑛,𝑝,𝑡. (9) Thus for all 𝑝∈𝑋,𝑡>0and𝑛=1,2,3…𝐹𝑦_𝑛,𝑦_𝑛+1,𝑝,𝑡≥𝐹𝑦₀,𝑦₁𝑡,𝑝,𝑞^𝑛. (10)
We now show that {𝑦_𝑛} is a Cauchy sequence in 𝑋.
Let 𝜖∈(0,1) be given. Since the 𝑡-norm Δ is of 𝐻 type, there exists 𝜆∈(0,1) such that for all 𝑚, 𝑛∈𝑁 with 𝑚>𝑛Δ^2𝑚−𝑛(1−𝜆)>1−𝜖. (11) Since lim_𝑛→∞𝐹(𝑦₀,𝑦₁,𝑝,𝑡/𝑞^𝑛)=1, there exists 𝑛₀∈𝑁 such that for all 𝑝∈𝑋 and 𝑡>0,𝐹(𝑦₀,𝑦₁,𝑝,𝑡/𝑞^𝑛)>1−𝜆 for all 𝑛≥𝑛₀.
From (10) we have for all 𝑝∈𝑋 and 𝑡>0, 𝐹(𝑦_𝑛,𝑦_𝑛+1,𝑝,𝑡)>1−𝜆 for all 𝑛≥𝑛₀.
Let 𝑚>𝑛≥𝑛₀. Then for all 𝑝∈𝑋 and 𝑡>0, we have 𝐹𝑦_𝑚,𝑦_𝑛𝑦,𝑝,𝑡=𝐹_𝑚,𝑦_𝑛𝑡,𝑝,3+𝑡3+𝑡3≥Δ𝐹𝑦_𝑚,𝑦_𝑛,𝑦_𝑛+1,𝑡3,𝐹𝑦_𝑚,𝑦_𝑛+1𝑡,𝑝,3,𝐹𝑦_𝑛+1,𝑦_𝑛𝑡,𝑝,3≥Δ𝐹𝑦_𝑛+1,𝑦_𝑛𝑡,𝑝,3,Δ𝐹𝑦_𝑛+1,𝑦_𝑛,𝑦_𝑚,𝑡3,𝐹𝑦_𝑛+1,𝑦_𝑚𝑡,𝑝,3≥ΔΔ²(1−𝜆),𝐹𝑦_𝑛+1,𝑦_𝑚𝑡,𝑝,3. (12)
Since 𝐹𝑦_𝑛+1,𝑦_𝑚𝑡,𝑝,3≥Δ𝐹𝑦_𝑛+2,𝑦_𝑛+1𝑡,𝑝,3²,Δ𝐹𝑦_𝑛+2,𝑦_𝑛+1,𝑦_𝑚,𝑡3²,𝐹𝑦_𝑛+2,𝑦_𝑚𝑡,𝑝,3². (13)From (13), we get 𝐹𝑦_𝑚,𝑦_𝑛,𝑝,𝑡≥ΔΔ²²(1−𝜆),𝐹𝑦_𝑛+2,𝑦_𝑚𝑡,𝑝,3². (14) Inductively, we obtain 𝐹𝑦_𝑚,𝑦_𝑛,𝑝,𝑡≥ΔΔ^2𝑚−𝑛(1−𝜆),𝐹𝑦_𝑚,𝑦_𝑚𝑡,𝑝,3^𝑚−𝑛=Δ^2𝑚−𝑛(1−𝜆). (15)
From (11) and (13) we get, for all 𝑝∈𝑋 and 𝑡>0,𝐹(𝑦_𝑚,𝑦_𝑛,𝑝,𝑡)>1−𝜖 for all 𝑚>𝑛≥𝑛₀. Thus {𝑦_𝑛} is a Cauchy sequence in 𝑋. Since 𝑋 is complete, there exists a point 𝑧 in 𝑋 such that lim_𝑛→∞𝑦_𝑛=𝑧 and this gives lim_𝑛→∞𝑆𝑥_2𝑛=lim_𝑛→∞𝑇𝑥_2𝑛−1=lim_𝑛→∞𝐴𝑥_2𝑛−2=lim_𝑛→∞𝐵𝑥_2𝑛−1=𝑧∀𝑛∈𝑁. (16) Without loss of generality, we assume that 𝑆(𝑋) is a complete subspace of 𝑋; therefore, 𝑧=𝑆𝑢 for some 𝑢∈𝑋.
Subsequently, we have lim_𝑛→∞𝑆𝑥_𝑛=lim_𝑛→∞𝑇𝑥_𝑛=lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝐵𝑥_𝑛=𝑧=𝑆𝑢. (17) Next, we claim that 𝐴𝑢=𝑆𝑢.
For this purpose, we put 𝑥=𝑢 and 𝑦=𝑥_𝑛 in (2.3); we have 𝐹𝐴𝑢,𝐵𝑥_𝑛𝐹,𝑝,𝑞𝑡≥min𝑆𝑢,𝑇𝑥_𝑛,𝐹,𝑝,𝑡,𝐹𝐴𝑢,𝑆𝑢,𝑝,𝑡𝐵𝑥_𝑛,𝑇𝑥_𝑛,𝑝,𝑡,𝐹𝐴𝑢,𝑇𝑥_𝑛,𝑝,𝑡. (18)
Taking limit as 𝑛→∞𝐹𝐹,𝐹.𝐴𝑢,𝑧,𝑝,𝑞𝑡≥min𝑧,𝑧,𝑝,𝑡,𝐹𝐴𝑢,𝑧,𝑝,𝑡𝑧,𝑧,𝑝,𝑡,𝐹𝐴𝑢,𝑧,𝑝,𝑡≥𝐹𝐴𝑢,𝑧,𝑝,𝑡 (19) By Lemma 13, we have 𝐴𝑢=𝑧.
Hence 𝐴𝑢=𝑆𝑢=𝑧.
Since 𝐴(𝑋)⊂𝑇(𝑋), there exists a point 𝑣∈𝑋 such that 𝐴𝑢=𝑧=𝑇𝑣.
Next we claim that 𝑇𝑣=𝐵𝑣.
Putting 𝑥=𝑢 and 𝑦=𝑣 in (2.3), we have 𝐹𝐹,𝐹𝐹,𝐹.𝐴𝑢,𝐵𝑣,𝑝,𝑞𝑡≥min𝑆𝑢,𝑇𝑣,𝑝,𝑡,𝐹𝐴𝑢,𝑆𝑢,𝑝,𝑡𝐵𝑣,𝑇𝑣,𝑝,𝑡,𝐹𝐴𝑢,𝑇𝑣,𝑝,𝑡≥min𝑧,𝑧,𝑝,𝑡,𝐹𝑧,𝑧,𝑝,𝑡𝐵𝑣,𝑇𝑣,𝑝,𝑡,𝐹𝑧,𝑧,𝑝,𝑡≥𝐹𝐵𝑣,𝑇𝑣,𝑝,𝑡 (20)
That is, 𝐹(𝑇𝑣,𝐵𝑣,𝑝,𝑞𝑡)≥𝐹(𝐵𝑣,𝑇𝑣,𝑝,𝑡).
By Lemma 13, we have 𝑇𝑣=𝐵𝑣.
Thus 𝐴𝑢=𝑆𝑢=𝑇𝑣=𝐵𝑣=𝑧.
Since the pairs (𝐴,𝑆) and (𝐵,𝑇) are weakly compatible and 𝑢 and 𝑣 are their points of coincidence, respectively, then ()()()()𝐴𝑧=𝐴𝑆𝑢=𝑆𝐴𝑢=𝑆𝑧,𝐵𝑧=𝐵𝑇𝑣=𝑇𝐵𝑣=𝑇𝑧. (21)
Now we prove that 𝑧 is a common fixed point of 𝐴,𝐵,𝑆, and 𝑇.
For this purpose, puting 𝑥=𝑧 and 𝑦=𝑣 in (2.3), we get 𝐹𝐹,𝐹𝐹,𝐹.𝐴𝑧,𝐵𝑣,𝑝,𝑞𝑡≥min𝑆𝑧,𝑇𝑣,𝑝,𝑡,𝐹𝐴𝑧,𝑆𝑧,𝑝,𝑡𝐵𝑣,𝑇𝑣,𝑝,𝑡,𝐹𝐴𝑧,𝑇𝑣,𝑝,𝑡≥min𝐴𝑧,𝐵𝑣,𝑝,𝑡,𝐹𝐴𝑧,𝐴𝑧,𝑝,𝑡𝐵𝑣,𝐵𝑣,𝑝,𝑡,𝐹𝐴𝑧,𝐵𝑣,𝑝,𝑡≥𝐹𝐴𝑧,𝐵𝑣,𝑝,𝑡 (22)
By Lemma 13, we have 𝐴𝑧=𝐵𝑣=𝑧.
Hence 𝑧=𝐴𝑧=𝑆𝑧, and 𝑧 is a common fixed point of 𝐴 and 𝑆. One can prove that 𝐵𝑣=𝑧 is also a common fixed point of 𝐵 and 𝑇.
Uniqueness. Suppose 𝑧≠𝑤 is another fixed point of 𝐴,𝐵,𝑆, and 𝑇.
Then, for all 𝑝∈𝑋 with 𝑝≠𝑧 and 𝑝≠𝑤 and 𝑡>0, 𝐹𝐹,𝐹𝐹,𝐹,𝑧,𝑤,𝑝,𝑞𝑡=𝐹𝐴𝑧,𝐵𝑤,𝑝,𝑞𝑡≥min𝑆𝑧,𝑇𝑤,𝑝,𝑡,𝐹𝐴𝑧,𝑆𝑧,𝑝,𝑡𝐵𝑤,𝑇𝑤,𝑝,𝑡,𝐹𝐴𝑧,𝑇𝑤,𝑝,𝑡≥min𝑧,𝑤,𝑝,𝑡,𝐹𝑧,𝑧,𝑝,𝑡𝑤,𝑤,𝑝,𝑡,𝐹𝑧,𝑤,𝑝,𝑡≥𝐹𝑧,𝑤,𝑝,𝑡 (23) which implies that 𝐹(𝑧,𝑤,𝑝,𝑞𝑡)≥𝐹(𝑧,𝑤,𝑝,𝑡).
Hence 𝑧=𝑤 is a unique common fixed point of 𝐴,𝐵,𝑆, and 𝑇.
This completes the proof of the theorem.

Corollary 15. Let (𝑋,𝐹,Δ) be a 2-Menger space with continuous t-norm Δ of H type. Let 𝑆 and 𝑇 be self-mappings of 𝑋. Then 𝑆 and 𝑇 have a unique common fixed point in 𝑋 if and only if there exist two self-mappings 𝐴,𝐵 of 𝑋 satisfying the following:(2.5)𝐴(𝑋)⊂𝑆(𝑋);(2.6) pair (𝐴,𝑆) is weakly compatible;(2.7) there exists 𝑞∈(0,1) such that for every 𝑥,𝑦,𝑝∈𝑋 and 𝑡>0, 𝐹𝐹,𝐹;𝐴𝑥,𝐴𝑦,𝑝,𝑞𝑡≥min𝑆𝑥,𝑆𝑦,𝑝,𝑡,𝐹𝐴𝑥,𝑆𝑥,𝑝,𝑡𝐴𝑦,𝑆𝑦,𝑝,𝑡,𝐹𝐴𝑥,𝑆𝑦,𝑝,𝑡 (24)(2.8) one of the subspaces 𝐴(𝑋) or 𝑆(𝑋) is a closed subspace of 𝑋.Indeed, 𝐴 and 𝑆 have a unique common fixed point in 𝑋.

Proof . We can easily prove the theorem by setting 𝐵=𝐴 and 𝑇=𝑆, in the proof of the Theorem 14.

Corollary 16. Let (𝑋,𝐹,Δ) be a 2-Menger space with continuous t-norm Δ of 𝐻 type. Let 𝑆 and 𝑇 be self-mappings of 𝑋. Then 𝑆 and 𝑇 have a unique common fixed point in 𝑋 if and only if there exist two self-mappings 𝐴,𝐵 of 𝑋 satisfying (2.1), (2.2), (2.4), and the following: (2.9) there exists q∈(0,1) such that for every 𝑥,𝑦,𝑝∈𝑋 and t>0, 𝐹(𝐴𝑥,𝐵𝑦,𝑝,𝑞𝑡)≥𝐹(𝑆𝑥,𝑇𝑦,𝑝,𝑡).Indeed, 𝐴,𝐵,𝑆, and T have a unique common fixed point in 𝑋.

Proof. Proof easily follows by setting min{𝐹(𝑆𝑥,𝑇𝑦,𝑝,𝑡), 𝐹(𝐴𝑥,𝑆𝑥,𝑝,𝑡), 𝐹(𝐵𝑦,𝑇𝑦,𝑝,𝑡), 𝐹(𝐴𝑥,𝑇𝑦,𝑝,𝑡)}=𝐹(𝑆𝑥,𝑇𝑦,𝑝,𝑡) in the proof of Theorem 14.

3. E.A. Property and Weakly Compatible Maps

Now we prove our main result for weakly compatible maps along with the E.A. property as follows.

Theorem 17. Let (𝑋,𝐹,Δ) be a 2-Menger space with continuous t-norm Δ of H type. Let 𝐴, 𝐵, 𝑆, and 𝑇 be self-mappings of 𝑋 satisfying (2.1), (2.2), (2.3), and (2.4) and the following:(3.1) pair (𝐴,𝑆) or pair (𝐵,𝑇) satisfies E.A. property.Then 𝐴,𝐵,𝑆, and 𝑇 have a unique common fixed point in 𝑋.

Proof. If the pair (𝐵,𝑇) satisfies the property E.A., then there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝐵𝑥_𝑛=lim_𝑛→∞𝑇𝑥_𝑛=𝑧 for some 𝑧∈𝑋.
Since 𝐵(𝑋)⊂𝑆(𝑋), there exists a sequence {𝑦_𝑛} in 𝑋 such that 𝐵𝑥_𝑛=𝑆𝑦_𝑛. Hence lim_𝑛→∞𝑆𝑦_𝑛=𝑧. Also 𝐴(𝑋)⊂𝑇(𝑋); there exists a sequence {𝑦^′_𝑛} in 𝑋 such that 𝐴𝑦^′_𝑛=𝑇𝑥_𝑛. Hence lim_𝑛→∞𝐴𝑦^′_𝑛=𝑧.
Suppose that 𝑆(𝑋) is a complete subspace of 𝑋. Then 𝑧=𝑆𝑢 for some 𝑢∈𝑋; subsequently; we have lim_𝑛→∞𝐵𝑥_𝑛=lim_𝑛→∞𝑇𝑥_𝑛=𝑧=lim_𝑛→∞𝐴𝑦^′_𝑛=lim_𝑛→∞𝑆𝑦_𝑛=𝑧=𝑆𝑢forsome𝑢∈𝑋. (25)
Next, we claim that 𝐴𝑢=𝑆𝑢.
For this purpose, we put 𝑥=𝑢 and 𝑦=𝑥_𝑛 in (2.3); this gives 𝐹𝐴𝑢,𝐵𝑥_𝑛𝐹,𝑝,𝑞𝑡≥min𝑆𝑢,𝑇𝑥_𝑛,𝐹,𝑝,𝑡,𝐹𝐴𝑢,𝑆𝑢,𝑝,𝑡𝐵𝑥_𝑛,𝑇𝑥_𝑛,𝑝,𝑡,𝐹𝐴𝑢,𝑇𝑥_𝑛,𝑝,𝑡. (26)
Taking limit as 𝑛→∞𝐹𝐹,𝐹.𝐴𝑢,𝑧,𝑝,𝑞𝑡≥min𝑧,𝑧,𝑝,𝑡,𝐹𝐴𝑢,𝑧,𝑝,𝑡𝑧,𝑧,𝑝,𝑡,𝐹𝐴𝑢,𝑧,𝑝,𝑡≥𝐹𝐴𝑢,𝑧,𝑝,𝑡 (27)
By Lemma 13, we have 𝐴𝑢=𝑧.
Hence 𝐴𝑢=𝑆𝑢=𝑧.
Since 𝐴(𝑋)⊂𝑇(𝑋), there exists a point 𝑣∈𝑋 such that 𝐴𝑢=𝑧=𝑇𝑣.
Next we claim that 𝑇𝑣=𝐵𝑣.
Putting 𝑥=𝑢 and 𝑦=𝑣 in (2.3), we have 𝐹𝐹,𝐹𝐹,𝐹.𝐴𝑢,𝐵𝑣,𝑝,𝑞𝑡≥min𝑆𝑢,𝑇𝑣,𝑝,𝑡,𝐹𝐴𝑢,𝑆𝑢,𝑝,𝑡𝐵𝑣,𝑇𝑣,𝑝,𝑡,𝐹𝐴𝑢,𝑇𝑣,𝑝,𝑡≥min𝑧,𝑧,𝑝,𝑡,𝐹𝑧,𝑧,𝑝,𝑡𝐵𝑣,𝑇𝑣,𝑝,𝑡,𝐹𝑧,𝑧,𝑝,𝑡≥𝐹𝐵𝑣,𝑇𝑣,𝑝,𝑡 (28) That is, 𝐹(𝑇𝑣,𝐵𝑣,𝑝,𝑞𝑡)≥𝐹(𝐵𝑣,𝑇𝑣,𝑝,𝑡).
By Lemma 13, we have 𝑇𝑣=𝐵𝑣.
Thus 𝐴𝑢=𝑆𝑢=𝑇𝑣=𝐵𝑣=𝑧.
Since the pairs (𝐴,𝑆) and (𝐵,𝑇) are weakly compatible and 𝑢 and 𝑣 are their points of coincidence, respectively, then 𝐴𝑧=𝐴(𝑆𝑢)=𝑆(𝐴𝑢)=𝑆𝑧 and 𝐵𝑧=𝐵(𝑇𝑣)=𝑇(𝐵𝑣)=𝑇𝑧.
Now we prove that 𝑧 is a common fixed point of 𝐴,𝐵,𝑆, and 𝑇.
For this purpose, we put 𝑥=𝑧 and 𝑦=𝑣 in (2.3); we get 𝐹𝐹,𝐹𝐹,𝐹.𝐴𝑧,𝐵𝑣,𝑝,𝑞𝑡≥min𝑆𝑧,𝑇𝑣,𝑝,𝑡,𝐹𝐴𝑧,𝑆𝑧,𝑝,𝑡𝐵𝑣,𝑇𝑣,𝑝,𝑡,𝐹𝐴𝑧,𝑇𝑣,𝑝,𝑡≥min𝐴𝑧,𝐵𝑣,𝑝,𝑡,𝐹𝐴𝑧,𝐴𝑧,𝑝,𝑡𝐵𝑣,𝐵𝑣,𝑝,𝑡,𝐹𝐴𝑧,𝐵𝑣,𝑝,𝑡≥𝐹𝐴𝑧,𝐵𝑣,𝑝,𝑡 (29)
By Lemma 13, we have 𝐴𝑧=𝐵𝑣=𝑧.
Hence 𝑧=𝐴𝑧=𝑆𝑧 and 𝑧 is a common fixed point of 𝐴 and 𝑆. One can prove that 𝐵𝑣=𝑧 is also a common fixed point of 𝐵 and 𝑇.
Uniqueness. Suppose 𝑧≠𝑤 is another fixed point of 𝐴,𝐵,𝑆, and 𝑇.
Then, for all 𝑝∈𝑋 with 𝑝≠𝑧 and 𝑝≠𝑤 and 𝑡>0, 𝐹𝐹,𝐹𝐹,𝐹,𝑧,𝑤,𝑝,𝑞𝑡=𝐹𝐴𝑧,𝐵𝑤,𝑝,𝑞𝑡≥min𝑆𝑧,𝑇𝑤,𝑝,𝑡,𝐹𝐴𝑧,𝑆𝑧,𝑝,𝑡𝐵𝑤,𝑇𝑤,𝑝,𝑡,𝐹𝐴𝑧,𝑇𝑤,𝑝,𝑡≥min𝑧,𝑤,𝑝,𝑡,𝐹𝑧,𝑧,𝑝,𝑡𝑤,𝑤,𝑝,𝑡,𝐹𝑧,𝑤,𝑝,𝑡≥𝐹𝑧,𝑤,𝑝,𝑡 (30) which implies that 𝐹(𝑧,𝑤,𝑝,𝑞𝑡)≥𝐹(𝑧,𝑤,𝑝,𝑡).
Hence 𝑧=𝑤 is a unique common fixed point of 𝐴,𝐵,𝑆, and 𝑇.

Corollary 18. Let (𝑋,𝐹,Δ) be a 2-Menger space with continuous t-norm Δ of 𝐻 type. Let 𝐴,𝐵,𝑆, and 𝑇 be self-mappings of 𝑋 satisfying (2.1), (2.2), (2.4), and (2.9).
Then 𝐴,𝐵,𝑆, and 𝑇 have a unique common fixed point in 𝑋.

4. CLR Property and Weakly Compatible Maps

Now we prove our main result for weakly compatible maps along with the CLR_S property as follows.

Theorem 19. Let (𝑋,𝐹,Δ) be a 2-Menger space with continuous t-norm Δ of 𝐻 type. Let 𝐴,𝐵,𝑆, and 𝑇 be self mapping of 𝑋 satisfying (2.1), (2.2), (2.3), and the following:(4.1) pair (𝐴,𝑆) or pair (𝐵,𝑇) satisfies CLR_S property;(4.2) one of the subspaces 𝐴(𝑋),𝐵(𝑋),𝑆(𝑋), or 𝑇(𝑋) is a closed subspace of 𝑋.Then 𝐴,𝐵,𝑆, and 𝑇 have a unique common fixed point in 𝑋.

Proof. If the pair (𝐴,𝑆) satisfies the CLR_S property then there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=𝑧, where 𝑧∈𝑆(𝑋). Therefore, there exists a point 𝑢∈𝑋 such that 𝑆𝑢=𝑧. Since 𝑇(𝑋) is a closed subspace of 𝑋 and 𝐴(𝑋)⊂𝑇(𝑋), for each {𝑥_𝑛}⊂𝑋 there corresponds a sequence {𝑦_𝑛}⊂𝑋 such that 𝐴𝑥_𝑛=𝑇𝑦_𝑛.
Therefore, lim_𝑛→∞𝑇𝑥_𝑛=lim_𝑛→∞𝐴𝑥_𝑛=𝑧, where 𝑧∈𝑆(𝑋).
Thus in all, we have lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=lim_𝑛→∞𝑇𝑦_𝑛=𝑧. (31)
Now we are required to show that lim_𝑛→∞𝐵𝑦_𝑛=𝑧.
Putting 𝑥=𝑥_𝑛 and 𝑦=𝑦_𝑛 in (2.3), we get 𝐹𝐴𝑥_𝑛,𝐵𝑦_𝑛𝐹,𝑝,𝑞𝑡≥min𝑆𝑥_𝑛,𝑇𝑦_𝑛,𝑝,𝑡,𝐹𝐴𝑥_𝑛,𝑆𝑥_𝑛,𝐹,𝑝,𝑡𝐵𝑦_𝑛,𝑇𝑦_𝑛,𝑝,𝑡,𝐹𝐴𝑥_𝑛,𝑇𝑦_𝑛,𝑝,𝑡. (32)
Let lim_𝑛→∞𝐵𝑦_𝑛=𝑙≠𝑧 for 𝑡>0.Then taking limit as 𝑛→∞, we have 𝐹𝐹,𝐹.𝑧,𝑙,𝑝,𝑞𝑡≥min𝑧,𝑧,𝑝,𝑡,𝐹𝑧,𝑧,𝑝,𝑡𝑙,𝑧,𝑝,𝑡,𝐹𝑧,𝑧,𝑝,𝑡≥𝐹𝑙,𝑧,𝑝,𝑡 (33)
By Lemma 13, we have 𝑧=𝑙, then lim_𝑛→∞𝐵𝑦_𝑛=𝑧.
Hence lim_𝑛→∞𝐴𝑥_𝑛 = lim_𝑛→∞𝑆𝑥_𝑛=lim_𝑛→∞𝑇𝑦_𝑛=lim_𝑛→∞𝐵𝑦_𝑛=𝑧=𝑆𝑢, for some 𝑢∈𝑋.
Uniqueness follows from Theorem 14.

Corollary 20. Let (𝑋,𝐹,Δ) be a 2-Menger space with continuous t-norm Δ of H type. Let 𝐴,𝐵,𝑆, and 𝑇 be self-mappings of X satisfying (2.1), (2.2), (2.9), (4.1), and (4.2).
Then 𝐴,𝐵,𝑆, and 𝑇 have a unique common fixed point in 𝑋.

5. Application

As an application of Theorem 14, we prove a common fixed point theorem for four finite families of mappings which runs as follows.

Theorem 21. Let {𝐴₁,𝐴₂,…,𝐴_𝑚}, {𝐵₁,𝐵₂,…,𝐵_𝑛}, {𝑆₁,𝑆₂,…,𝑆_𝑝}, and {𝑇₁,𝑇₂,…,𝑇_𝑞} be four finite families of self-mappings of a 2-Menger space (𝑋,𝐹,Δ) with continuous t-norm of Hadzic type such that 𝐴=𝐴₁𝐴₂⋯𝐴_𝑚, 𝐵=𝐵₁𝐵₂⋯𝐵_𝑛, 𝑆=𝑆₁𝑆₂⋯𝑆_𝑝, and 𝑇=𝑇₁𝑇₂⋯𝑇_𝑞 satisfy condition (2.1), (2.2), (2.3), and (2.4); then(i)A and 𝑆 have a point of coincidence;(ii)B and 𝑇 have a point of coincidence.Moreover, if 𝐴_𝑖𝐴_𝑗=𝐴_𝑗𝐴_𝑖, 𝐵_𝑘𝐵_𝑙=𝐵_𝑙𝐵_𝑘, 𝑆_𝑟𝑆_𝑠=𝑆_𝑠𝑆_𝑟, 𝑇_𝑡𝑇_𝑢=𝑇_𝑢𝑇_𝑡, 𝐴_𝑖𝑆_𝑟=𝑆_𝑟𝐴_𝑖, and 𝐵_𝑘𝑇_𝑡=𝑇_𝑡𝐵_𝑘 for all 𝑖,𝑗∈𝐼₁={1,2,…,𝑚}, 𝑘, 𝑙∈𝐼₂={1,2,…,𝑛}, 𝑟, 𝑠∈𝐼₃={1,2,…,𝑝}, and 𝑡, 𝑢∈I₄={1,2,…,𝑞}, then (for all 𝑖∈𝐼₁,𝑘∈𝐼₂,𝑟∈𝐼₃, and 𝑡∈𝐼₄)𝐴_𝑖, 𝑆_𝑟, 𝐵_𝑘, and 𝑇_𝑡 have a common fixed point.

Proof. The conclusions (i) and (ii) are immediate as 𝐴,𝑆,𝐵, and 𝑇 satisfy all the conditions of Theorem 14. Now appealing to component wise commutativity of various pairs, one can immediately prove that 𝐴𝑆=𝑆𝐴 and 𝐵𝑇=𝑇𝐵, and hence, obviously both pairs (𝐴,𝑆) and (𝐵,𝑇) are coincidently commuting. Note that all the conditions of Theorem 14 (for mappings 𝐴,𝑆,𝐵, and 𝑇) are satisfied ensuring the existence of a unique common fixed point, say 𝑧. Now one needs to show that 𝑧 remains the fixed point of all the component maps. For this consider 𝐴𝐴_𝑖𝑧=𝐴₁𝐴₂⋯𝐴_𝑚𝐴_𝑖𝑧=𝐴₁𝐴₂⋯𝐴_𝑚−1𝐴_𝑚𝐴_𝑖𝑧=𝐴₁𝐴₂⋯𝐴_𝑚−1𝐴_𝑖𝐴_𝑚𝑧=𝐴₁𝐴₂⋯𝐴_𝑚−2𝐴_𝑚−1𝐴_𝑖𝐴_𝑚𝑧=𝐴₁𝐴₂⋯𝐴_𝑚−2𝐴_𝑖𝐴_𝑚−1𝐴_𝑚𝑧=⋯=𝐴₁𝐴_𝑖𝐴₂𝐴₃𝐴₄⋯𝐴_𝑚𝑧=𝐴_𝑖𝐴₁𝐴₂𝐴₃𝐴₄⋯𝐴_𝑚𝑧=𝐴_𝑖(𝐴𝑧)=𝐴_𝑖𝑧. (34) Similarly, one can show that 𝐴𝑆_𝑟𝑧=𝑆_𝑟(𝐴𝑧)=𝑆_𝑟𝑧,𝑆𝑆_𝑟𝑧=𝑆_𝑟(𝑆𝑧)=𝑆_𝑟𝑆𝐴𝑧,_𝑖𝑧=𝐴_𝑖(𝑆𝑧)=𝐴_𝑖𝑧,𝐵𝐵_𝑘𝑧=𝐵_𝑘(𝐵𝑧)=𝐵_𝑘𝐵𝑇𝑧,_𝑡𝑧=𝑇_𝑡(𝐵𝑧)=𝑇_𝑡𝑧,𝑇𝑇_𝑡𝑧=𝑇_𝑡(𝑇𝑧)=𝑇_𝑡𝑇𝐵𝑧,_𝑘𝑧=𝐵_𝑘(𝑇𝑧)=𝐵_𝑘𝑧, (35) which show that (for all 𝑖,𝑟,𝑘, and 𝑡) 𝐴_𝑖𝑧and𝑆_𝑟𝑧 are other fixed points of the pair (𝐴,𝑆) whereas 𝐵_𝑘𝑧and𝑇_𝑡𝑧 are other fixed points of the pair (𝐵,𝑇). Now appealing to the uniqueness of common fixed points of both pairs separately, we get 𝑧=𝐴_𝑖𝑧=𝑆_𝑟𝑧=𝐵_𝑘𝑧=𝑇_𝑡𝑧, (36) which shows that 𝑧 is a common fixed point of 𝐴_𝑖,𝑆_𝑟,𝐵_𝑘, and 𝑇_𝑡 for all 𝑖,𝑟,𝑘, and 𝑡.

By setting 𝐴=𝐴₁=𝐴₂=⋯=𝐴_𝑚,𝐵=𝐵₁=𝐵₂=⋯=𝐵_𝑛,𝑆=𝑆₁=𝑆₂=⋯=𝑆_𝑝, and 𝑇=𝑇₁=𝑇₂=⋯=𝑇_𝑞, one deduces the following for certain iterates of maps, which run as follows.

Corollary 22. Let 𝐴,𝐵,𝑆, and 𝑇 be four self-mappings of a 2-Menger space (𝑋,𝐹,Δ) such that 𝐴^𝑚,𝐵^𝑛,𝑆^𝑝, and 𝑇^𝑞 satisfy the conditions (2.1) and (2.3). If one of 𝐴^𝑚(𝑋),𝐵^𝑛(𝑋),𝑆^𝑝(𝑋), or 𝑇^𝑞(𝑋) is a closed subset of 𝑋, then 𝐴,𝐵,𝑆, and 𝑇 have a unique common fixed point provided (𝐴,𝑆) and (𝐵,𝑇) commute.

Acknowledgments

One of the authors (S. Kumar) would like to acknowledge the UGC for providing financial grant through Major Research Project under Ref. 39-41/2010 (SR). I am also thankful to the referee for careful reading of the paper and giving useful comments to improve it.