Common Fixed Point for Three Pairs of Self-Maps Satisfying Common (E.A) Property in Generalized Metric Spaces

Abstract

Using the concept of common (E.A) property, we prove a common fixed point theorem for three pairs of weakly compatible self-maps satisfying a new contractive condition in the framework of a generalized metric space. Our results do not rely on any commuting or continuity condition of mappings. An example is provided to support our result. The results obtained in this paper differ from the recent relative results in the literature.

1. Introduction

The study of fixed points and common fixed points of mappings satisfying certain contractive conditions has been at the center of rigorous research activity. In 2006, Mustafa and Sims [1] introduced the concept of generalized metric spaces or simply 𝐺-metric spaces as a generalization of the notion of a metric space. Based on the notion of generalized metric spaces, Mustafa et al. [2–5], Obiedat and Mustafa [6], Aydi et al. [7, 8], Gajié and Stojakovié [9], and Zhou and Gu [10] obtained some fixed point results for mappings satisfying different contractive conditions. Shatanawi [11] obtained some fixed point results for Φ-maps in 𝐺-metric spaces. Chugh et al. [12] obtained some fixed point results for maps satisfying property 𝑃 in 𝐺-metric spaces. Study of common fixed point problems in 𝐺-metric spaces was initiated by Abbas and Rhoades [13]. Subsequently, many authors obtained many common fixed point theorems for the mappings satisfying different contractive conditions (see [14–32] for more details). Recently, Abbas et al. [33] and Mustafa et al. [34] obtained some common fixed point results for a pair of mappings satisfying (E.A) property under certain generalized strict contractive conditions in 𝐺-metric spaces. Long et al. [35] obtained some common coincidence and common fixed points results of two pairs of mappings when only one pair satisfies (E.A) property in the framework of a generalized metric space.

The aim of this paper is to study common fixed point of three pairs of mappings for which only two pairs need to satisfy common (E.A) property in the framework of 𝐺-metric spaces. Our results do not rely on any commuting or continuity condition of mappings.

2. Definitions and Preliminary Results

In this section, we present the necessary definitions and results in 𝐺-metric spaces.

Definition 1 (see [1]). Let 𝑋 be a nonempty set, and let 𝐺∶𝑋×𝑋×𝑋→𝑅⁺ be a function satisfying the following axioms:(G1)𝐺(𝑥,𝑦,𝑧)=0 if 𝑥=𝑦=𝑧; (G2)0<𝐺(𝑥,𝑥,𝑦), for all 𝑥,𝑦∈𝑋 with 𝑥≠𝑦; (G3)𝐺(𝑥,𝑥,𝑦)≤𝐺(𝑥,𝑦,𝑧), for all 𝑥,𝑦, and 𝑧∈𝑋 with 𝑧≠𝑦; (G4)𝐺(𝑥,𝑦,𝑧)=𝐺(𝑥,𝑧,𝑦)=𝐺(𝑦,𝑧,𝑥)=⋯ (symmetry in all three variables); (G5)𝐺(𝑥,𝑦,𝑧)≤𝐺(𝑥,𝑎,𝑎)+𝐺(𝑎,𝑦,𝑧) for all 𝑥,𝑦,𝑧, and 𝑎∈𝑋, (rectangle inequality).Then the function 𝐺 is called a generalized metric, or more specifically a 𝐺-metric on 𝑋 and the pair (𝑋,𝐺) are called a 𝐺-metric space.
It is known that the function 𝐺(𝑥,𝑦,and𝑧) on 𝐺-metric space 𝑋 is jointly continuous in all three of its variables, and 𝐺(𝑥,𝑦,𝑧)=0 if and only if 𝑥=𝑦=𝑧 (see [1] for more details and the reference therein).

Definition 2 (see [1]). Let (𝑋,𝐺) be a 𝐺-metric space, and let {𝑥_𝑛} be a sequence of points in 𝑋; a point 𝑥 in 𝑋 is said to be the limit of the sequence {𝑥_𝑛} if lim_{𝑚,𝑛→∞}𝐺(𝑥,𝑥_𝑛,𝑥_𝑚)=0, and one says that sequence {𝑥_𝑛} is 𝐺-convergent to 𝑥.

Thus, if 𝑥_𝑛→𝑥 in a 𝐺-metric space (𝑋,𝐺), then for any 𝜖>0 there exists 𝑁∈ℕ (throughout this paper we mean by ℕ the set of all natural numbers) such that 𝐺(𝑥,𝑥_𝑛,and𝑥_𝑚)<𝜖, for all 𝑛,𝑚≥𝑁.

Proposition 3 (see [1]). Let (𝑋,𝐺) be a 𝐺-metric space; then the followings are equivalent:(1){𝑥_𝑛} is 𝐺-convergent to 𝑥.(2)𝐺(𝑥_𝑛,𝑥_𝑛,𝑥)→0𝑎𝑠𝑛→∞.(3)𝐺(𝑥_𝑛,𝑥,𝑥)→0𝑎𝑠𝑛→∞.(4)𝐺(𝑥_𝑛,𝑥_𝑚,𝑥)→0𝑎𝑠𝑛,𝑚→∞.

Definition 4 (see [1]). Let (𝑋,𝐺) be a 𝐺-metric space. A sequence {𝑥_𝑛} is called 𝐺-Cauchy sequence if, for each 𝜖>0, there exists 𝑁∈ℕ such that 𝐺(𝑥_𝑛,𝑥_𝑚,and𝑥_𝑙)<𝜖 for all 𝑛,𝑚,and𝑙≥𝑁, that is, if 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)→0 as 𝑛,𝑚,and𝑙→∞.

Definition 5 (see [1]). A 𝐺-metric space (𝑋,𝐺) is said to be 𝐺-complete (or a complete 𝐺-metric space) if every 𝐺-Cauchy sequence in (𝑋,𝐺) is 𝐺-convergent in 𝑋.

Proposition 6 (see [1]). Let (𝑋,𝐺) be a 𝐺-metric space. Then the following are equivalent.(1)The sequence {𝑥_𝑛}𝑖𝑠𝐺-Cauchy.(2)For every 𝜖>0, there exists 𝑘∈ℕ such that 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑚)<𝜖, for all 𝑛,𝑚≥𝑘.

Proposition 7 (see [1]). Let (𝑋,𝐺) be a 𝐺-metric space. Then the function 𝐺(𝑥,𝑦,𝑎𝑛𝑑𝑧) is jointly continuous in all three of its variables.

Definition 8 (see [36]). Let 𝑓 and 𝑔 be self-maps of a set 𝑋. If 𝑤=𝑓𝑥=𝑔𝑥 for some 𝑥 in 𝑋, then 𝑥 is called a coincidence point of 𝑓 and 𝑔, and 𝑤 is called point of coincidence of 𝑓 and 𝑔.

Definition 9 (see [36]). Two self-mappings 𝑓 and 𝑔 on 𝑋 are said to be weakly compatible if they commute at coincidence points.

Definition 10 (see [33]). Let 𝑋 be a 𝐺-metric space. Self-maps 𝑓 and 𝑔 on 𝑋 are said to satisfy the 𝐺-(E.A) property if there exists a sequence {𝑥_𝑛} in 𝑋 such that {𝑓𝑥_𝑛} and {𝑔𝑥_𝑛} are 𝐺-convergent to some 𝑡∈𝑋.

Definition 11. Let (𝑋,𝑑) be a 𝐺-metric space and 𝐴,𝐵,𝑆, and 𝑇 four self-maps on 𝑋. The pairs (𝐴,𝑆) and (𝐵,𝑇) are said to satisfy common (𝐸.𝐴) property 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, for some 𝑡∈𝑋.

Definition 12 (see [17]). Self-mappings 𝑓 and 𝑔 of a 𝐺-metric space (𝑋,𝐺) are said to be compatible if lim_𝑛→∞𝐺(𝑓𝑔𝑥_𝑛,𝑔𝑓𝑥_𝑛,𝑔𝑓𝑥_𝑛)=0 and lim_𝑛→∞𝐺(𝑔𝑓𝑥_𝑛,𝑓𝑔𝑥_𝑛,𝑓𝑔𝑥_𝑛)=0, whenever {𝑥_𝑛} is a sequence in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝑡, for some 𝑡∈𝑋.

Definition 13 (see [16]). A pair of self-mappings (𝑓,𝑔) of a 𝐺-metric space is said to be weakly commuting if 𝐺𝑓𝑔𝑥,𝑔𝑓𝑥,𝑔𝑓𝑥≤𝐺𝑓𝑥,𝑔𝑥,𝑔𝑥,∀𝑥∈𝑋. (1)

Definition 14 (see [16]). A pair of self-mappings (𝑓,𝑔) of a 𝐺-metric space is said to be 𝑅-weakly commuting, if there exists some positive real number 𝑅 such that 𝐺𝑓𝑔𝑥,𝑔𝑓𝑥,𝑔𝑓𝑥≤𝑅𝐺𝑓𝑥,𝑔𝑥,𝑔𝑥,∀𝑥∈𝑋. (2)

3. Main Results

In this section, we obtain some unique common fixed point results for six mappings satisfying certain generalized contractive conditions in the framework of a generalized metric space. We start with the following result.

Theorem 15. Let (𝑋,𝐺) be a 𝐺-metric space. Suppose mappings 𝑓,𝑔,ℎ,𝑅,𝑆,𝑎𝑛𝑑𝑇∶𝑋→𝑋 satisfying the following conditions: 𝐺𝐺,𝐺(),1𝑓𝑥,𝑔𝑦,ℎ𝑧≤𝜙max𝑅𝑥,𝑆𝑦,𝑇𝑧,𝐺𝑓𝑥,𝑅𝑥,𝑅𝑥𝑔𝑦,𝑆𝑦,𝑆𝑦,𝐺ℎ𝑧,𝑇𝑧,𝑇𝑧3𝐺,1𝑓𝑥,𝑆𝑦,𝑇𝑧+𝐺𝑅𝑥,𝑔𝑦,𝑇𝑧+𝐺𝑅𝑥,𝑆𝑦,ℎ𝑧3𝐺,𝑓𝑥,𝑔𝑦,𝑇𝑧+𝐺𝑓𝑥,𝑆𝑦,ℎ𝑧+𝐺𝑅𝑥,𝑔𝑦,ℎ𝑧 (3) for all 𝑥,𝑦,𝑎𝑛𝑑𝑧∈𝑋, where 𝜙∶[0,∞)→[0,∞) is the function satisfying 0<𝜙(𝑡)<𝑡, for all 𝑡>0. If one of the following conditions is satisfied, then the pairs (𝑓,𝑅), (𝑔,𝑆), and (ℎ,𝑇) have a common point of coincidence in 𝑋:(i)the subspace 𝑅𝑋 is closed in 𝑋, 𝑓𝑋⊆𝑆𝑋, 𝑔𝑋⊆𝑇𝑋, and two pairs of (𝑓,𝑅) and (𝑔,𝑆) satisfy common (𝐸.𝐴) property;(ii)the subspace 𝑆𝑋 is closed in 𝑋, 𝑔𝑋⊆𝑇𝑋, ℎ𝑋⊆𝑅𝑋, and two pairs of (𝑔,𝑆) and (ℎ,𝑇) satisfy common (𝐸.𝐴) property;(iii)the subspace 𝑇𝑋 is closed in 𝑋, 𝑓𝑋⊆𝑆𝑋, ℎ𝑋⊆𝑅𝑋, and two pairs of (𝑓,𝑅) and (ℎ,𝑇) satisfy common (𝐸.𝐴) property.
Moreover, if the pairs (𝑓,𝑅), (𝑔,𝑆), and (ℎ,𝑇) are weakly compatible, then 𝑓, 𝑔, ℎ, 𝑅, 𝑆, and 𝑇 have a unique common fixed point in 𝑋.

Proof. First, we suppose that the subspace 𝑅𝑋 is closed in 𝑋, 𝑓𝑋⊆𝑆𝑋, 𝑔𝑋⊆𝑇𝑋, and two pairs of (𝑓,𝑅) and (𝑔,𝑆) satisfy common (𝐸.𝐴) property. Then by Definition 11 we know that, there exist two sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑅𝑥_𝑛=lim_𝑛→∞𝑔𝑦_𝑛=lim_𝑛→∞𝑆𝑦_𝑛=𝑡 (4) for some 𝑡∈𝑋.
Since 𝑔𝑋⊆𝑇𝑋, there exists a sequence {𝑧_𝑛} in 𝑋 such that 𝑔𝑦_𝑛=𝑇𝑧_𝑛. Hence lim_𝑛→∞𝑇𝑧_𝑛=𝑡. Next, we will show lim_𝑛→∞ℎ𝑧_𝑛=𝑡. In fact, if lim_𝑛→∞ℎ𝑧_𝑛≠𝑡, then from condition (3), we can get 𝐺𝑓𝑥_𝑛,𝑔𝑦_𝑛,ℎ𝑧_𝑛≤𝜙max𝐺𝑅𝑥_𝑛,𝑆𝑦_𝑛,𝑇𝑧_𝑛,𝐺𝑓𝑥_𝑛,𝑅𝑥_𝑛,𝑅𝑥_𝑛,𝐺𝑔𝑦_𝑛,𝑆𝑦_𝑛,𝑆𝑦_𝑛,𝐺ℎ𝑧_𝑛,𝑇𝑧_𝑛,𝑇𝑧_𝑛,13𝐺𝑓𝑥_𝑛,𝑆𝑦_𝑛,𝑇𝑧_𝑛+𝐺𝑅𝑥_𝑛,𝑔𝑦_𝑛,𝑇𝑧_𝑛+𝐺𝑅𝑥_𝑛,𝑆𝑦_𝑛,ℎ𝑧_𝑛,13𝐺𝑓𝑥_𝑛,𝑔𝑦_𝑛,𝑇𝑧_𝑛+𝐺𝑓𝑥_𝑛,𝑆𝑦_𝑛,ℎ𝑧_𝑛+𝐺𝑅𝑥_𝑛,𝑔𝑦_𝑛,ℎ𝑧_𝑛. (5) On letting 𝑛→∞ and based on the property of 𝜙, we can obtain 𝐺𝑡,𝑡,lim_𝑛→∞ℎ𝑧_𝑛≤𝜙𝐺lim_𝑛→∞ℎ𝑧_𝑛,𝑡,𝑡<𝐺lim_𝑛→∞ℎ𝑧_𝑛,𝑡,𝑡. (6) It is contradiction; so lim_𝑛→∞ℎ𝑧_𝑛=𝑡.
Since 𝑅𝑋 is a closed subspace of 𝑋 and lim_𝑛→∞𝑅𝑥_𝑛=𝑡, there exists a 𝑝 in 𝑋 such that 𝑡=𝑅𝑝. We claim that 𝑓𝑝=𝑡. Suppose not; then by using (3) we obtain 𝐺𝑓𝑝,𝑔𝑦_𝑛,ℎ𝑧_𝑛≤𝜙max𝐺𝑅𝑝,𝑆𝑦_𝑛,𝑇𝑧_𝑛,𝐺,𝐺𝑓𝑝,𝑅𝑝,𝑅𝑝𝑔𝑦_𝑛,𝑆𝑦_𝑛,𝑆𝑦_𝑛,𝐺ℎ𝑧_𝑛,𝑇𝑧_𝑛,𝑇𝑧_𝑛,13𝐺𝑓𝑝,𝑆𝑦_𝑛,𝑇𝑧_𝑛+𝐺𝑅𝑝,𝑔𝑦_𝑛,𝑇𝑧_𝑛+𝐺𝑅𝑝,𝑆𝑦_𝑛,ℎ𝑧_𝑛,13𝐺𝑓𝑝,𝑔𝑦_𝑛,𝑇𝑧_𝑛+𝐺𝑓𝑝,𝑆𝑦_𝑛,ℎ𝑧_𝑛+𝐺𝑅𝑝,𝑔𝑦_𝑛,ℎ𝑧_𝑛. (7) Taking 𝑛→∞ on the two sides of the above inequality and using the property of 𝜙, we can get 𝐺𝑓𝑝,𝑡,𝑡≤𝜙max𝐺𝑓𝑝,𝑡,𝑡<𝐺𝑓𝑝,𝑡,𝑡. (8) Which is contradiction, and so 𝑓𝑝=𝑡=𝑅𝑝. Hence, 𝑝 is the coincidence point of pair (𝑓,𝑅).
By the condition 𝑓𝑋⊆𝑆𝑋 and 𝑓𝑝=𝑡, there exists a point 𝑢 in 𝑋 such that 𝑡=𝑆𝑢. Now, we claim that 𝑔𝑢=𝑡. In fact, if 𝑔𝑢≠𝑡, then from (3) we have 𝐺𝑓𝑝,𝑔𝑢,ℎ𝑧_𝑛≤𝜙max𝐺𝑅𝑝,𝑆𝑢,𝑇𝑧_𝑛,𝐺,𝐺𝑓𝑝,𝑅𝑝,𝑅𝑝𝑔𝑢,𝑆𝑢,𝑆𝑢,𝐺ℎ𝑧_𝑛,𝑇𝑧_𝑛,𝑇𝑧_𝑛,13𝐺𝑓𝑝,𝑆𝑢,𝑇𝑧_𝑛+𝐺𝑅𝑝,𝑔𝑢,𝑇𝑧_𝑛+𝐺𝑅𝑝,𝑆𝑢,ℎ𝑧_𝑛,13𝐺𝑓𝑝,𝑔𝑢,𝑇𝑧_𝑛+𝐺𝑓𝑝,𝑆𝑢,ℎ𝑧_𝑛+𝐺𝑅𝑝,𝑔𝑢,ℎ𝑧_𝑛. (9) Letting 𝑛→∞ on the two sides of the above inequality and using the property of 𝜙, we can obtain 𝐺𝑡,𝑔𝑢,𝑡≤𝜙𝐺𝑔𝑢,𝑡,𝑡<𝐺𝑔𝑢,𝑡,𝑡. (10) It is contradiction. Hence 𝑔𝑢=𝑡=𝑆𝑡, and so 𝑢 is the coincidence point of pair (𝑔,𝑆).
Since 𝑔𝑋⊆𝑇𝑋 and 𝑔𝑢=𝑡, there exists a point 𝑣 in 𝑋 such that 𝑡=𝑇𝑣. We claim that ℎ𝑣=𝑡. If not, from (3) and the property of 𝜙, we have 𝐺()𝐺,𝐺(),1𝑡,𝑡,ℎ𝑣=𝐺𝑓𝑝,𝑔𝑢,ℎ𝑣≤𝜙max𝑅𝑝,𝑆𝑢,𝑇𝑣,𝐺𝑓𝑝,𝑅𝑝,𝑅𝑝𝑔𝑢,𝑆𝑢,𝑆𝑢,𝐺ℎ𝑣,𝑇𝑣,𝑇𝑣3𝐺,1𝑓𝑝,𝑆𝑢,𝑇𝑣+𝐺𝑅𝑝,𝑔𝑢,𝑇𝑣+𝐺𝑅𝑝,𝑆𝑢,ℎ𝑣3𝐺(𝐺(().𝑓𝑝,g𝑢,𝑇𝑣+𝐺𝑓𝑝,𝑆𝑢,ℎ𝑣+𝐺𝑅𝑝,𝑔𝑢,ℎ𝑣=𝜙ℎ𝑣,𝑡,𝑡))<𝐺ℎ𝑣,𝑡,𝑡 (11) It is contradiction. Hence ℎ𝑣=𝑡=𝑇𝑣, and so 𝑣 is the coincidence point of pair (ℎ,𝑇).
Therefore, in all the above cases, we obtain 𝑓𝑝=𝑅𝑝=𝑔𝑢=𝑆𝑢=ℎ𝑣=𝑇𝑣=𝑡. Now, weakly compatibility of the pairs (𝑓,𝑅), (𝑔,𝑆), and (ℎ,𝑇) gives that 𝑓𝑡=𝑅𝑡, 𝑔𝑡=𝑆𝑡, and ℎ𝑡=𝑇𝑡.
Next, we show that 𝑓𝑡=𝑡. In fact, if 𝑓𝑡≠𝑡, then from (3) we have 𝐺𝐺(),𝐺(),1𝑓𝑡,𝑡,𝑡=𝐺𝑓𝑡,𝑔𝑢,ℎ𝑣≤𝜙max𝑅𝑡,𝑆𝑢,𝑇𝑣,𝐺𝑓𝑡,𝑅𝑡,𝑅𝑡𝑔𝑢,𝑆𝑢,𝑆𝑢,𝐺ℎ𝑣,𝑇𝑣,𝑇𝑣3𝐺(,1𝑓𝑡,𝑆𝑢,𝑇𝑣+𝐺𝑅𝑡,𝑔𝑢,𝑇𝑣+𝐺𝑅𝑡,𝑆𝑢,ℎ𝑣)]3𝐺𝐺𝑓𝑡,𝑔𝑢,𝑇𝑣+𝐺𝑓𝑡,𝑆𝑢,ℎ𝑣+𝐺𝑅𝑡,𝑔𝑢,ℎ𝑣=𝜙𝑓𝑡,𝑡,𝑡<𝐺𝑓𝑡,𝑡,𝑡 (12) which is contradiction; hence 𝑓𝑡=𝑡, and so 𝑓𝑡=𝑅𝑡=𝑡. Similarly, it can be shown that 𝑔𝑡=𝑆𝑡=𝑡 and ℎ𝑡=𝑇𝑡=𝑡; so we get 𝑓𝑡=𝑔𝑡=ℎ𝑡=𝑅𝑡=𝑆𝑡=𝑇𝑡=𝑡, which means that 𝑡 is a common fixed point of 𝑓, 𝑔, ℎ, 𝑅, 𝑆, and 𝑇.
Next, we will show the common fixed point of 𝑓, 𝑔, ℎ, 𝑅, 𝑆, and 𝑇 is unique. Actually, suppose that 𝑤∈𝑋; 𝑤≠𝑡 is another common fixed point of 𝑓, 𝑔, ℎ, 𝑅, 𝑆, and 𝑇; then by condition (3) we have 𝐺()𝐺(),𝐺(),1𝑤,𝑡,𝑡=𝐺𝑓𝑤,𝑔𝑡,ℎ𝑡≤𝜙max𝑅𝑤,𝑆𝑡,𝑇𝑡,𝐺𝑓𝑤,𝑅𝑤,𝑅𝑤𝑔𝑡,𝑆𝑡,𝑆𝑡,𝐺ℎ𝑡,𝑇𝑡,𝑇𝑡3𝐺(,1𝑓𝑤,𝑆𝑡,𝑇𝑡+𝐺𝑅𝑤,𝑔𝑡,𝑇𝑡+𝐺𝑅𝑤,𝑆𝑡,ℎ𝑡)]3𝐺(𝐺(().𝑓𝑤,𝑔𝑡,𝑇𝑡+𝐺𝑓𝑤,𝑆𝑡,ℎ𝑡+𝐺𝑅𝑤,𝑔𝑡,ℎ𝑡=𝜙𝑤,𝑡,𝑡))<𝐺𝑤,𝑡,𝑡 (13) It is a contradiction, unless 𝑤=𝑡; that is, mappings 𝑓,𝑔,ℎ,𝑅,𝑆, and 𝑇 have a unique common fixed point.
Finally, if condition (ii) or (iii) holds, then the argument is similar to that above; so we delete it.
This completes the proof of Theorem 15.

Now, we give an example to support Theorem 15.

Example 16. Let 𝑋=[0,1] be a 𝐺-metric space with 𝐺𝑥,𝑦,𝑧=𝑥−𝑦+𝑦−𝑧+|𝑧−𝑥|. (14) We define mappings 𝑓,𝑔,ℎ,𝑅,𝑆, and 𝑇 on 𝑋 by 1𝑓𝑥=1,𝑥∈0,2561,𝑥∈2,7,1𝑔𝑥=81,𝑥∈0,2561,𝑥∈2,6,1ℎ𝑥=71,𝑥∈0,2561,𝑥∈2,4,1𝑅𝑥=51,𝑥∈0,2561,𝑥∈26,171,𝑥=1,𝑆𝑥=1,𝑥∈0,2561,𝑥∈21,10,𝑥=1,𝑇𝑥=1,𝑥∈0,2561,𝑥∈27,18,𝑥=1. (15) Note that 𝑓, 𝑔, ℎ, 𝑅, 𝑆, and 𝑇 are discontinuous mappings. Clearly, the subspace 𝑅𝑋 is closed in 𝑋, 𝑓𝑋⊆𝑆𝑋, 𝑔𝑋⊆𝑇𝑋, ℎ𝑋⊆𝑅𝑋, and the pairs (𝑓,𝑅), (𝑔,𝑆), and (ℎ,𝑇) are weakly compatible. Also, the pairs (𝑓,𝑅) and (𝑔,𝑆) satisfy common (𝐸.𝐴) property; indeed, 𝑥_𝑛=4/5 and 𝑦_𝑛=3/4 for each 𝑛∈𝑁 are the required sequences. The control function 𝜙∶[0,∞)→[0,∞) is defined by 𝜙(𝑡)=11𝑡/12.
First, we let 𝑀𝐺,𝐺(),1𝑥,𝑦,𝑧=max𝑅𝑥,𝑆𝑦,𝑇𝑧,𝐺𝑓𝑥,𝑅𝑥,𝑅𝑥𝑔𝑦,𝑆𝑦,𝑆𝑦,𝐺ℎ𝑧,𝑇𝑧,𝑇𝑧3𝐺,1𝑓𝑥,𝑆𝑦,𝑇𝑧+𝐺𝑅𝑥,𝑔𝑦,𝑇𝑧+𝐺𝑅𝑥,𝑆𝑦,ℎ𝑧3𝐺.𝑓𝑥,𝑔𝑦,𝑇𝑧+𝐺𝑓𝑥,𝑆𝑦,ℎ𝑧+𝐺𝑅𝑥,𝑔𝑦,ℎ𝑧 (16) To prove (3), let us discuss the following cases.

Case 1. For 𝑥,𝑦,and𝑧∈((1/2),1], then we have 𝐺(𝑓𝑥,𝑔𝑦,ℎ𝑧)=0, and hence (3) is obviously satisfied.

Case 2. For 𝑥,𝑦,and𝑧∈[0,(1/2)], then we have 𝐺7𝑓𝑥,𝑔𝑦,ℎ𝑧=𝐺1,8,67=27,𝜙𝑀𝐺4𝑥,𝑦,𝑧=𝜙max54,1,1,𝐺1,5,457,𝐺8,𝐺6,1,17,1,1,13𝐺()41,1,1+𝐺5,784,1+𝐺56,1,7,13𝐺71,86,1+𝐺1,1,74+𝐺5,78,672=𝜙max5,25,14,4,815235=𝜙5=11.30 (17) Therefore, 𝐺=2𝑓𝑥,𝑔𝑦,ℎ𝑧7<11𝑀.30=𝜙𝑥,𝑦,𝑧 (18)

Case 3. For 𝑥,𝑦∈[0,(1/2)], 𝑧∈((1/2),1], then we have 𝐺7𝑓𝑥,𝑔𝑦,ℎ𝑧=𝐺1,8,56=13. (19) Next we divide the study in two subcases.(a)If 𝑧∈((1/2),1), then we have 𝜙𝑀𝐺4𝑥,𝑦,𝑧=𝜙max55,1,64,𝐺1,5,457,𝐺8,𝐺5,1,16,56,56,13𝐺51,1,64+𝐺5,78,564+𝐺55,1,6,13𝐺71,8,565+𝐺1,1,64+𝐺5,78,562=𝜙max5,25,14,0,53,180492180=𝜙5=11.30 (20)(b)If 𝑧=1, then we get 𝜙𝑀𝐺4𝑥,𝑦,𝑧=𝜙max57,1,84,𝐺1,5,457,𝐺8,𝐺5,1,16,78,78,13𝐺71,1,84+𝐺5,78,784+𝐺55,1,6,13𝐺71,8,785+𝐺1,1,64+𝐺5,78,562=𝜙max5,25,14,1,412,1511245=𝜙5=11.30 (21)Hence, we can get 𝐺=1𝑓𝑥,𝑔𝑦,ℎ𝑧3<11𝑀.30=𝜙𝑥,𝑦,𝑧 (22)

Case 4. For 𝑥,𝑧∈[0,(1/2)], 𝑦∈((1/2),1], then we have 𝐺5𝑓𝑥,𝑔𝑦,ℎ𝑧=𝐺1,6,67=13. (23) Next we divide the study in two subcases.(a) If 𝑦∈((1/2),1), then we obtain 𝜙𝑀𝐺4𝑥,𝑦,𝑧=𝜙max5,564,1,𝐺1,5,455,𝐺6,56,56,𝐺67,1,1,13𝐺51,64,1+𝐺5,564,1+𝐺5,56,67,13𝐺51,65,1+𝐺1,6,674+𝐺5,56,672=𝜙max5,252,0,7,89,315822315=𝜙5=11.30 (24)Thus we have 𝐺=1𝑓𝑥,𝑔𝑦,ℎ𝑧3<11𝑀.30=𝜙𝑥,𝑦,𝑧 (25)(b) If 𝑦=1, then we can get 𝜙𝑀𝐺4𝑥,𝑦,𝑧=𝜙max54,0,1,𝐺1,5,455,𝐺6,𝐺6,0,07,1,1,13𝐺()41,0,1+𝐺5,564,1+𝐺56,0,7,13𝐺51,66,1+𝐺1,0,74+𝐺5,56,672=𝜙max2,5,53,27,142,10577(2)=315=𝜙116. (26)So, we have 𝐺=1𝑓𝑥,𝑔𝑦,ℎ𝑧3<116=𝜙𝑀𝑥,𝑦,𝑧. (27)

Case 5. For 𝑥∈((1/2),1], 𝑦,𝑧∈[0,(1/2)], then we have 𝐺5𝑓𝑥,𝑔𝑦,ℎ𝑧=𝐺6,78,67=1.12 (28) Next we divide the study in two subcases.(a) If 𝑥∈((1/2),1), then we can get 𝜙𝑀𝐺5𝑥,𝑦,𝑧=𝜙max65,1,1,𝐺6,56,567,𝐺8,𝐺6,1,17,1,1,13𝐺565,1,1+𝐺6,785,1+𝐺66,1,7,13𝐺56,785,1+𝐺66,1,75+𝐺6,78,671=𝜙max31,0,4,27,13,141=𝜙3=11.36 (29)Thus we have 𝐺=1𝑓𝑥,𝑔𝑦,ℎ𝑧<1211𝑀.36=𝜙𝑥,𝑦,𝑧 (30)(b) If 𝑥=1, then we obtain 𝜙𝑀𝐺6𝑥,𝑦,𝑧=𝜙max75,1,1,𝐺6,67,677,𝐺8,𝐺6,1,17,1,1,13𝐺566,1,1+𝐺7,786,1+𝐺76,1,7,13𝐺56,785,1+𝐺66,1,76+𝐺7,78,672=𝜙max7,1,1214,27,19,6359252=𝜙19=63209.756 (31)Hence we get 𝐺=1𝑓𝑥,𝑔𝑦,ℎ𝑧<12209𝑀.756=𝜙𝑥,𝑦,𝑧 (32)

Case 6. For 𝑥∈[0,(1/2)], 𝑦∈((1/2),1], and 𝑧∈((1/2),1], then we have 𝐺5𝑓𝑥,𝑔𝑦,ℎ𝑧=𝐺1,6,56=13. (33) Next we divide the study in four subcases.(a) If 𝑦∈((1/2),1) and 𝑧∈((1/2),1), then we have 𝜙𝑀𝐺4𝑥,𝑦,𝑧=𝜙max5,56,564,𝐺1,5,455,𝐺6,56,56,𝐺56,56,56,13𝐺51,6,564+𝐺5,56,564+𝐺5,56,56,13𝐺51,6,565+𝐺1,6,564+𝐺5,56,561=𝜙max,21557,0,0,,3011230=𝜙5=11.30 (34)Thus we have 𝐺=1𝑓𝑥,𝑔𝑦,ℎ𝑧3<11𝑀.30=𝜙𝑥,𝑦,𝑧 (35)(b) If 𝑦=1 and 𝑧∈((1/2),1), then we have 𝜙𝑀𝐺4𝑥,𝑦,𝑧=𝜙max55,0,64,𝐺1,5,455,𝐺6,𝐺5,0,06,56,56,13𝐺51,0,64+𝐺5,56,564+𝐺55,0,6,13𝐺51,6,565+𝐺1,0,64+𝐺5,56,565=𝜙max3,25,53,0,56,44555=𝜙3=55.36 (36)Therefore, we can get 𝐺=1𝑓𝑥,𝑔𝑦,ℎ𝑧3<55𝑀.36=𝜙𝑥,𝑦,𝑧 (37)(c) If 𝑦∈((1/2),1) and 𝑧=1, then we get 𝜙𝑀𝐺4𝑥,𝑦,𝑧=𝜙max5,56,784,𝐺1,5,455,𝐺6,56,56,𝐺56,78,78,13𝐺51,6,784+𝐺5,56,784+𝐺5,56,56,13𝐺51,6,785+𝐺1,6,564+𝐺5,56,563=𝜙max,22051,0,,1211,6011245=𝜙5=11.30 (38)Therefore, we can get 𝐺=1𝑓𝑥,𝑔𝑦,ℎ𝑧3<11𝑀.30=𝜙𝑥,𝑦,𝑧 (39)(d) If 𝑦=1 and 𝑧=1, then we have 𝜙𝑀𝐺4𝑥,𝑦,𝑧=𝜙max57,0,84,𝐺1,5,455,𝐺6,𝐺5,0,06,78,78,13𝐺71,0,84+𝐺5,56,784+𝐺55,0,6,13𝐺51,6,785+𝐺1,0,64+𝐺5,56,567=𝜙max4,25,53,1,12229,418057=𝜙4=77.48 (40)Thus, we can get 𝐺=1𝑓𝑥,𝑔𝑦,ℎ𝑧3<77𝑀.48=𝜙𝑥,𝑦,𝑧 (41)

Case 7. For 𝑥∈((1/2),1], 𝑦∈[0,(1/2)], and 𝑧∈((1/2),1], then we have 𝐺5𝑓𝑥,𝑔𝑦,ℎ𝑧=𝐺6,78,56=1.12 (42) Next we divide the study in four subcases.(a) If 𝑥∈((1/2),1) and 𝑧∈((1/2),1), then we have 𝜙𝑀𝐺5𝑥,𝑦,𝑧=𝜙max65,1,65,𝐺6,56,567,𝐺8,𝐺5,1,16,56,56,13𝐺565,1,65+𝐺6,78,565+𝐺65,1,6,13𝐺56,78,565+𝐺65,1,65+𝐺6,78,561=𝜙max31,0,41,0,4,161=𝜙3=11.36 (43)Thus, we can get 𝐺=1𝑓𝑥,𝑔𝑦,ℎ𝑧<1211𝑀.36=𝜙𝑥,𝑦,𝑧 (44)(b) If 𝑥=1 and 𝑧∈((1/2),1), then we have 𝜙𝑀𝐺6𝑥,𝑦,𝑧=𝜙max75,1,65,𝐺6,67,677,𝐺8,𝐺5,1,16,56,56,13𝐺565,1,66+𝐺7,78,566+𝐺75,1,6,13𝐺56,78,565+𝐺65,1,66+𝐺7,78,561=𝜙max3,1,12141,0,4,161=𝜙3=11.36 (45)Hence, we obtain 𝐺=1𝑓𝑥,𝑔𝑦,ℎ𝑧<1211𝑀.36=𝜙𝑥,𝑦,𝑧 (46)(c)If 𝑥∈((1/2),1) and 𝑧=1, then we get 𝜙𝑀𝐺5𝑥,𝑦,𝑧=𝜙max67,1,85,𝐺6,56,567,𝐺8,𝐺5,1,16,78,78,13𝐺567,1,85+𝐺6,78,785+𝐺65,1,6,13𝐺56,78,785+𝐺65,1,65+𝐺6,78,561=𝜙max31,0,4,1,1124,161=𝜙3=11.36 (47)Therefore, we have 𝐺=1𝑓𝑥,𝑔𝑦,ℎ𝑧<1211𝑀.36=𝜙𝑥,𝑦,𝑧 (48)(d) If 𝑥=1 and 𝑧=1, then we have 𝜙𝑀𝐺6𝑥,𝑦,𝑧=𝜙max77,1,85,𝐺6,67,677,𝐺8,𝐺5,1,16,78,78,13𝐺567,1,86+𝐺7,78,786+𝐺75,1,6,13𝐺56,78,785+𝐺65,1,66+𝐺7,78,562=𝜙max7,1,1124,1,1259,125262=𝜙7=11.42 (49)Thus, we obtain 𝐺=1𝑓𝑥,𝑔𝑦,ℎ𝑧<1211𝑀.42=𝜙𝑥,𝑦,𝑧 (50)

Case 8. For 𝑥∈((1/2),1], 𝑦∈((1/2),1], and 𝑧∈[0,(1/2)], then we have 𝐺5𝑓𝑥,𝑔𝑦,ℎ𝑧=𝐺6,56,67=1.21 (51) Next we divide the study in four subcases.(a) If 𝑥∈((1/2),1) and 𝑦∈((1/2),1), then we get 𝜙𝑀𝐺5𝑥,𝑦,𝑧=𝜙max6,565,1,𝐺6,56,565,𝐺6,56,56,𝐺67,1,1,13𝐺56,565,1+𝐺6,565,1+𝐺6,56,67,13𝐺56,565,1+𝐺6,56,675+𝐺6,56,671=𝜙max32,0,0,7,5,12171=𝜙3=11.36 (52)Thus, we obtain 𝐺=1𝑓𝑥,𝑔𝑦,ℎ𝑧<2111𝑀.36=𝜙𝑥,𝑦,𝑧 (53)(b) If 𝑥=1 and 𝑦∈((1/2),1), then we have 𝜙𝑀𝐺6𝑥,𝑦,𝑧=𝜙max7,565,1,𝐺6,67,675,𝐺6,56,56,𝐺67,1,1,13𝐺56,566,1+𝐺7,566,1+𝐺7,56,67,13𝐺56,565,1+𝐺6,56,676+𝐺7,56,671=𝜙max3,1221,0,7,5,12171=𝜙3=11.36 (54)Thus, we obtain 𝐺=1𝑓𝑥,𝑔𝑦,ℎ𝑧<2111𝑀.36=𝜙𝑥,𝑦,𝑧 (55)(c) If 𝑥∈((1/2),1) and 𝑦=1, then we obtain 𝜙𝑀𝐺5𝑥,𝑦,𝑧=𝜙max65,0,1,𝐺6,56,565,𝐺6,𝐺6,0,07,1,1,13𝐺565,0,1+𝐺6,565,1+𝐺66,0,7,13𝐺56,565,1+𝐺66,0,75+𝐺6,56,675=𝜙max2,0,3,27,85,6344(2)=63=𝜙116. (56)Thus, we obtain 𝐺=1𝑓𝑥,𝑔𝑦,ℎ𝑧<21116=𝜙𝑀𝑥,𝑦,𝑧. (57)(d) If 𝑥=1 and 𝑦=1, then we have 𝜙𝑀𝐺6𝑥,𝑦,𝑧=𝜙max75,0,1,𝐺6,67,675,𝐺6,𝐺6,0,07,1,1,13𝐺566,0,1+𝐺7,566,1+𝐺76,0,7,13𝐺56,565,1+𝐺66,0,76+𝐺7,56,671=𝜙max2,,5213,27,85,6344(2)=63=𝜙116. (58)Thus, we obtain 𝐺=1𝑓𝑥,𝑔𝑦,ℎ𝑧<21116=𝜙𝑀𝑥,𝑦,𝑧. (59)
Then in all the above cases, the mappings 𝑓, 𝑔, ℎ, 𝑅, 𝑆, and 𝑇 are satisfying conditions (3) and (i) of Theorem 15, so that all the conditions of Theorem 15 are satisfied. Moreover, 5/6 is the unique common fixed point of 𝑓, 𝑔, ℎ, 𝑅, 𝑆, and 𝑇.

Remark 17. In this paper, we get the new common fixed point theorem using the common (𝐸.𝐴) property with three pairs of weakly compatible mappings. This is a new result has never been discussed by other authors.

Remark 18. If we take (1) 𝑅=𝑆=𝑇; (2) 𝑓=𝑔=ℎ; (3) 𝑅=𝑆=𝑇=𝐼 (𝐼 is identity mapping); (4) 𝑆=𝑇 and 𝑔=ℎ; (5) 𝑆=𝑇, 𝑔=ℎ=𝐼 in Theorem 15, then several new results can be obtained.

Corollary 19. Let (𝑋,𝐺) be a 𝐺-metric space. Suppose mappings 𝑓,𝑔,ℎ,𝑅,𝑆,𝑎𝑛𝑑𝑇∶𝑋→𝑋 satisfy the following conditions: 𝐺𝐺,𝐺(),1𝑓𝑥,𝑔𝑦,ℎ𝑧≤𝜆max𝑅𝑥,𝑆𝑦,𝑇𝑧,𝐺𝑓𝑥,𝑅𝑥,𝑅𝑥,𝐺𝑔𝑦,𝑆𝑦,𝑆𝑦ℎ𝑧,𝑇𝑧,𝑇𝑧3𝐺,1𝑓𝑥,𝑆𝑦,𝑇𝑧+𝐺𝑅𝑥,𝑔𝑦,𝑇𝑧+𝐺𝑅𝑥,𝑆𝑦,ℎ𝑧3𝐺𝑓𝑥,𝑔𝑦,𝑇𝑧+𝐺𝑓𝑥,𝑆𝑦,ℎ𝑧+𝐺𝑅𝑥,𝑔𝑦,ℎ𝑧 (60) for all 𝑥,𝑦,𝑎𝑛𝑑𝑧∈𝑋, where 𝜆∈[0,1). If one of the following conditions is satisfied, then the pairs (𝑓,𝑅), (𝑔,𝑆), and (ℎ,𝑇) have a common point of coincidence in 𝑋.(i)The subspace 𝑅𝑋 is closed in 𝑋, 𝑓𝑋⊆𝑆𝑋, 𝑔𝑋⊆𝑇𝑋, and two pairs of (𝑓,𝑅) and (𝑔,𝑆) satisfy common (𝐸.𝐴) property.(ii)The subspace 𝑆𝑋 is closed in 𝑋, 𝑔𝑋⊆𝑇𝑋, ℎ𝑋⊆𝑅𝑋, and two pairs of (𝑔,𝑆) and (ℎ,𝑇) satisfy common (𝐸.𝐴) property.(iii)The subspace 𝑇𝑋 is closed in 𝑋, 𝑓𝑋⊆𝑆𝑋, ℎ𝑋⊆𝑅𝑋, and two pairs of (𝑓,𝑅) and (ℎ,𝑇) satisfy common (𝐸.𝐴) property.
Moreover, if the pairs (𝑓,𝑅), (𝑔,𝑆), and (ℎ,𝑇) are weakly compatible, then 𝑓,𝑔,ℎ,𝑅,𝑆, and 𝑇 have a unique common fixed point in 𝑋.

Proof. Taking 𝜙(𝑡)=𝜆𝑡, 𝜆∈[0,1) in Theorem 15, the conclusion of Corollary 19 can be obtained from Theorem 15 immediately. This completes the proof of Corollary 19.

Remark 20. If we take (1) 𝑅=𝑆=𝑇; (2) 𝑓=𝑔=ℎ; (3) 𝑅=𝑆=𝑇=𝐼 (𝐼 is identity mapping); (4) 𝑆=𝑇 and 𝑔=ℎ; (5) 𝑆=𝑇, 𝑔=ℎ=𝐼 in Corollary 19, then several new results can be obtained.

Corollary 21. Let (𝑋,𝐺) be a 𝐺-metric space. Suppose mappings 𝑓,𝑔,ℎ,𝑅,𝑆,𝑎𝑛𝑑𝑇∶𝑋→𝑋 satisfy the following conditions: 𝐺𝑓𝑥,𝑔𝑦,ℎ𝑧≤𝑎₁𝐺𝑅𝑥,𝑆𝑦,𝑇𝑧+𝑎₂𝐺𝑓𝑥,𝑅𝑥,𝑅𝑥+𝑎₃𝐺𝑔𝑦,𝑆𝑦,𝑆𝑦+𝑎₄𝐺(ℎ𝑧,𝑇𝑧,𝑇𝑧)+𝑎₅𝐺𝑓𝑥,𝑆𝑦,𝑇𝑧+𝐺𝑅𝑥,𝑔𝑦,𝑇𝑧+𝐺𝑅𝑥,𝑆𝑦,ℎ𝑧+𝑎₆𝐺𝑓𝑥,𝑔𝑦,𝑇𝑧+𝐺𝑓𝑥,𝑆𝑦,ℎ𝑧+𝐺𝑅𝑥,𝑔𝑦,ℎ𝑧 (61) for all 𝑥,𝑦,𝑎𝑛𝑑𝑧∈𝑋, where 𝑎_𝑖≥0(𝑖=1,2,3,4,5,𝑎𝑛𝑑6) and 0≤𝑎₁+𝑎₂+𝑎₃+𝑎₄+3𝑎₅+3𝑎₆<1. If one of the following conditions is satisfied, then the pairs (𝑓,𝑅), (𝑔,𝑆), and (ℎ,𝑇) have a common point of coincidence in 𝑋:(i)the subspace 𝑅𝑋 is closed in X, 𝑓𝑋⊆𝑆𝑋, 𝑔𝑋⊆𝑇𝑋, and two pairs of (𝑓,𝑅) and (𝑔,𝑆) satisfy common (𝐸.𝐴) property;(ii)the subspace 𝑆𝑋 is closed in 𝑋, 𝑔𝑋⊆𝑇𝑋, ℎ𝑋⊆𝑅𝑋, and two pairs of (𝑔,𝑆) and (ℎ,𝑇) satisfy common (𝐸.𝐴) property;(iii)the subspace 𝑇𝑋 is closed in 𝑋, 𝑓𝑋⊆𝑆𝑋, ℎ𝑋⊆𝑅𝑋, and two pairs of (𝑓,𝑅) and (ℎ,𝑇) satisfy common (𝐸.𝐴) property.
Moreover, if the pairs (𝑓,𝑅), (𝑔,𝑆), and (ℎ,𝑇) are weakly compatible, then 𝑓,𝑔,ℎ,𝑅,𝑆, and 𝑇 have a unique common fixed point in 𝑋.

Proof. Suppose that 𝑀𝐺,𝐺(),1𝑥,𝑦,𝑧=max𝑅𝑥,𝑆𝑦,𝑇𝑧,𝐺𝑓𝑥,𝑅𝑥,𝑅𝑥,𝐺𝑔𝑦,𝑆𝑦,𝑆𝑦ℎ𝑧,𝑇𝑧,𝑇𝑧3𝐺,1𝑓𝑥,𝑆𝑦,𝑇𝑧+𝐺𝑅𝑥,𝑔𝑦,𝑇𝑧+𝐺𝑅𝑥,𝑆𝑦,ℎ𝑧3𝐺.𝑓𝑥,𝑔𝑦,𝑇𝑧+𝐺𝑓𝑥,𝑆𝑦,ℎ𝑧+𝐺𝑅𝑥,𝑔𝑦,ℎ𝑧 (62) Then 𝑎₁𝐺𝑅𝑥,𝑆𝑦,𝑇𝑧+𝑎₂𝐺𝑓𝑥,𝑅𝑥,𝑅𝑥+𝑎₃𝐺𝑔𝑦,𝑆𝑦,𝑆𝑦+𝑎₄𝐺(ℎ𝑧,𝑇𝑧,𝑇𝑧)+𝑎₅𝐺𝑓𝑥,𝑆𝑦,𝑇𝑧+𝐺𝑅𝑥,𝑔𝑦,𝑇𝑧+𝐺𝑅𝑥,𝑆𝑦,ℎ𝑧+𝑎₆𝐺≤𝑎𝑓𝑥,𝑔𝑦,𝑇𝑧+𝐺𝑓𝑥,𝑆𝑦,ℎ𝑧+𝐺𝑅𝑥,𝑔𝑦,ℎ𝑧₁+𝑎₂+𝑎₃+𝑎₄+3𝑎₅+3𝑎₆𝑀𝑥,𝑦,𝑧. (63) So, if condition (61) holds, then 𝐺(𝑓𝑥,𝑔𝑦,ℎ𝑧)≤(𝑎₁+𝑎₂+𝑎₃+𝑎₄+3𝑎₅+3𝑎₆)𝑀(𝑥,𝑦,𝑧). Taking 𝜆=𝑎₁+𝑎₂+𝑎₃+𝑎₄+3𝑎₅+3𝑎₆ in Corollary 19, the conclusion of Corollary 21 can be obtained from Corollary 19 immediately.

Remark 22. If we take (1) 𝑅=𝑆=𝑇; (2) 𝑓=𝑔=ℎ; (3) 𝑅=𝑆=𝑇=𝐼 (𝐼 is identity mapping); (4) 𝑆=𝑇 and 𝑔=ℎ; (5) 𝑆=𝑇, 𝑔=ℎ=𝐼 in Corollary 21, then several new results can be obtained.

Acknowledgments

The present studies are supported by the National Natural Science Foundation of China (11071169, 11271105), the Natural Science Foundation of Zhejiang Province (Y6110287, LY12A01030), and the Physical Experiment Center of Hangzhou Normal University.