Weakly Compatible Maps Using E.A. and (CLR) Properties in Complex Valued 𝐺-Metric Spaces

Abstract

We introduce the notion of complex valued 𝐺-metric spaces and prove common fixed point theorems for weakly compatible maps along with E.A. and (CLR) properties in complex valued 𝐺-metric spaces.

1. Introduction

The study of fixed points of mappings satisfying certain contractive conditions has been at the center of rigorous research activity. Recently, Mustafa and Sims [1, 2] have shown that most of the results concerning Dhage’s 𝐷-metric spaces are invalid; therefore, they introduced an improved version of the generalized metric space structure and called it 𝐺-metric spaces.

In 2006, Mustafa and Sims [2] introduced the concept of 𝐺-metric spaces as follows.

Definition 1. Let 𝑋 be a nonempty set, and let 𝐺∶𝑋×𝑋×𝑋→ℝ⁺ be a function satisfying the following properties:(G1)𝐺(𝑥,𝑦,𝑧)=0 if 𝑥=𝑦=𝑧;(G2)0<𝐺(𝑥,𝑦,𝑧) for all 𝑥, 𝑦∈𝑋 with 𝑥≠𝑦;(G3)𝐺(𝑥,𝑥,𝑦)≤𝐺(𝑥,𝑦,𝑧)for all 𝑥,𝑦,𝑧∈𝑋 with 𝑦≠𝑧;(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,𝑎∈𝑋 (rectangle inequality);

Then the function 𝐺 is called a generalized metric or, more specially a 𝐺-metric on 𝑋, and the pair (𝑋,𝐺) is called a 𝐺-metric space.

The idea of complex metric space was initiated by Azam et al. [3] to exploite the idea of complex valued normed spaces and complex valued Hilbert spaces.

Definition 2. Let ℂ be the set of complex numbers and 𝑧₁, 𝑧₂∈ℂ. Define a partial order ≾ on ℂ as follows: 𝑧₁≾𝑧₂iffRe𝑧₁≤Re𝑧₂,Im𝑧₁≤Im𝑧₂. (1)

That is, 𝑧₁≾𝑧₂ if one of the following holds:(C1) Re(𝑧₁) = Re(𝑧₂) and Im(𝑧₁) = Im(𝑧₂);(C2) Re(𝑧₁) < Re(𝑧₂) and Im(𝑧₁) = Im(𝑧₂);(C3) Re(𝑧₁) = Re(𝑧₂) and Im(𝑧₁) < Im(𝑧₂);(C4) Re(𝑧₁) < Re(𝑧₂) and Im(𝑧₁) < Im(𝑧₂).

In particular, we will write that 𝑧₁⪱𝑧₂ if 𝑧₁≠𝑧₂ and one of (C2), (C3), and (C4) is satisfied, and we will write 𝑧₁≺𝑧₂ if only (C4) is satisfied.

Remark 3. We noted that the following statements hold:(i)𝑎,𝑏∈𝑅 and 𝑎≤𝑏⇒𝑎𝑧≾𝑏𝑧 for all 𝑧∈ℂ;(ii)0≾𝑧₁⪱𝑧₂⇒|𝑧₁|<|𝑧₂|;(iii)𝑧₁≾𝑧₂ and 𝑧₂≺𝑧₃⇒𝑧₁≺𝑧₃.

Now we introduce the notion of complex valued 𝐺-metric space akin to the notion of complex valued metric spaces [3] as follows.

Definition 4. Let 𝑋 be a non-empty set. Let 𝐺∶𝑋×𝑋×𝑋→ℂ be a function satisfying the following properties:(CG1)𝐺(𝑥,𝑦,𝑧)=0 if 𝑥=𝑦=𝑧;(CG2)0≺𝐺(𝑥,𝑦,𝑧) for all 𝑥,𝑦∈𝑋 with 𝑥≠𝑦;(CG3)𝐺(𝑥,𝑥,𝑦)≾𝐺(𝑥,𝑦,𝑧)for all 𝑥,𝑦,𝑧∈𝑋 with 𝑦≠𝑧;(CG4)𝐺(𝑥,𝑦,𝑧)=𝐺(𝑥,𝑧,𝑦)=𝐺(𝑦,𝑧,𝑥)=⋯ (symmetry in all three variables);(CG5)𝐺(𝑥,𝑦,𝑧)≾𝐺(𝑥,𝑎,𝑎)+𝐺(𝑎,𝑦,𝑧) for all 𝑥,𝑦,𝑧,𝑎∈𝑋.

Then the function 𝐺 is called a complex valued generalized metric or more specially, a complex valued 𝐺-metric on 𝑋, and the pair (𝑋,𝐺) is called a complex valued 𝐺-metric space.

2. The Complex Valued 𝐺-Metric Topology

A point 𝑥∈𝑋 is called interior point of a set 𝐴⊆𝑋, whenever there exists 0≺𝑟∈ℂ such that 𝐵_𝐺(𝑥,𝑟)=𝑦∈𝑋∶𝐺𝑥,𝑦,𝑦≺𝑟⊆𝐴. (2)

A point 𝑥∈𝑋 is called limit point of a set 𝐴, whenever there exists 0≺𝑟∈ℂ: 𝐵_𝐺()∩𝐴𝑥,𝑟𝑋≠∅. (3)

The set 𝐴 is called open, whenever each element of 𝐴 is an interior point of 𝐴. A subset 𝐵⊆𝑋 is called closed, whenever each limit point of 𝐵 belongs to 𝐵.

Proposition 5. Let (𝑋,𝐺) be complex valued 𝐺-metric space, then for any 𝑥₀∈𝑋 and 𝑟> 0, one has the following:(1)if 𝐺(𝑥₀,𝑥,𝑦)≺𝑟, then 𝑥,𝑦∈𝐵_𝐺(𝑥₀,𝑟); (2)if 𝑦∈𝐵_𝐺(𝑥₀,𝑟),  then there exists 𝑎𝛿>0 such that 𝐵_𝐺(𝑦,𝛿)⊆𝐵_𝐺(𝑥₀,𝑟).

Proposition 6. Let (𝑋,𝐺) be complex valued 𝐺-metric space; then for all 𝑥₀∈𝑋 and 𝑟>0, one has 𝐵_𝐺𝑥₀,13𝑟⊆𝐵_𝑑𝐺𝑥₀,𝑟⊆𝐵_𝐺𝑥₀,𝑟, (4) where 𝑑_𝐺(𝑥,𝑦)=𝐺(𝑥,𝑦,𝑦)+𝐺(𝑥,𝑥,𝑦).

Proposition 7. Let (𝑋,𝐺) be a complex valued 𝐺-metric space. Then for any 𝑥,𝑦,𝑧, and 𝑎 in 𝑋 it follows that: (i)if 𝐺(𝑥,𝑦,𝑧)=0 and 𝑥=𝑦=𝑧; (ii)𝐺(𝑥,𝑦,𝑧)≾𝐺(𝑥,𝑥,𝑦)+𝐺(𝑥,𝑥,𝑧); (iii)𝐺(𝑥,𝑦,𝑦)≾2𝐺(𝑦,𝑥,𝑥);(iv)𝐺(𝑥,𝑦,𝑧)≾𝐺(𝑥,𝑎,𝑧)+𝐺(𝑎,𝑦,𝑧);(v)𝐺(𝑥,𝑦,𝑧)≾2/3(𝐺(𝑥,𝑦,𝑎)+𝐺(𝑥,𝑎,𝑧)+𝐺(𝑎,𝑦,𝑧));(vi)𝐺(𝑥,𝑦,𝑧)≾(𝐺(𝑥,𝑎,𝑎)+𝐺(𝑦,𝑎,𝑎)+𝐺(𝑧,𝑎,𝑎)).

Proposition 8. Let (𝑋,𝐺) be a complex valued 𝐺-metric space. Then the following are equivalent:(i)(𝑋,𝐺) is symmetric;(ii)𝐺(𝑥,𝑦,𝑦)≾𝐺(𝑥,𝑦,𝑎), for all 𝑥,𝑦,𝑎∈𝑋;(iii)𝐺(𝑥,𝑦,𝑧)≾𝐺(𝑥,𝑦,𝑎)+𝐺(𝑧,𝑦,𝑏) for all 𝑥,𝑦,𝑧,𝑎,𝑏∈𝑋.

3. Convergence, Continuity, and Completeness in Complex Valued 𝐺-Metric Spaces

Now we discuss some definition regarding convergence, continuity, and completeness in complex valued 𝐺-metric spaces.

Definition 9. Let (𝑋,𝐺) be a complex valued 𝐺-metric space. Let {𝑥_𝑛} be a sequence of points of 𝑋; we say that {𝑥_𝑛} is complex valued 𝐺-convergent to 𝑥 if, for any 𝜖>0, there exists 𝑘∈𝑁 such that 𝐺(𝑥,𝑥_𝑛,𝑥_𝑚)≺𝜖 for all 𝑛,𝑚≥𝑘. We refer to 𝑥 as the limit of the sequence {𝑥_𝑛}, and we write 𝑥−−−→_𝑛^(𝐺)𝑥.

Proposition 10. Let (𝑋,𝐺) be complex valued 𝐺-metric spaces. For a sequence {𝑥_𝑛}⊆𝑋 and point 𝑥∈𝑋, the following are equivalent:(1){𝑥_𝑛} is complex valued 𝐺-convergent to𝑥;(2)|𝐺(𝑥_𝑛,𝑥_𝑛,𝑥)|→0 as 𝑛→∞;(3)|𝐺(𝑥_𝑛,𝑥,𝑥)|→0 as 𝑛→∞;(4)|𝐺(𝑥_𝑚,𝑥_𝑛,𝑥)|→0 as 𝑛,𝑚→∞.

Definition 11. Let (𝑋,𝐺) and (𝑋^′,𝐺^′) be two complex valued 𝐺-metric spaces. Then a function 𝑓∶𝑋→𝑋^′ is complex valued 𝐺-continuous at a point 𝑥₀∈𝑋 if 𝑓⁻¹(𝐵_𝐺′(𝑓(𝑥₀),𝑟))∈𝜏(𝐺) for all 𝑟>0. We say 𝑓 is complex valued 𝐺-continuous if it complex valued 𝐺-continuous at all points of 𝑋; that is, continuous as a function from 𝑋 with the 𝜏(𝐺)-topology to 𝑋^′with 𝜏(𝐺^′)-topology.

Since complex valued 𝐺-metric topologies are metric topologies, therefore we have some proposition in this regard.

Proposition 12. Let (𝑋,𝐺) and (𝑋^′,𝐺^′) be two complex valued 𝐺-metric spaces. Then a function 𝑓∶𝑋→𝑋^′ is complex valued 𝐺-continuous at a point 𝑥∈𝑋 if and only if it is complex valued 𝐺-sequentially continuous at 𝑥; that is, whenever {𝑥_𝑛} is complex valued 𝐺-convergent to 𝑥, one has (𝑓{𝑥_𝑛}) is complex valued 𝐺-convergent to 𝑓(𝑥).

Proposition 13. Let (𝑋,𝐺) be a complex valued 𝐺-metric spaces. The function 𝐺(𝑥,𝑦,𝑧) is jointly continuous in all three of its variables.

Proof. Suppose that {𝑥_𝑘}, {𝑦_𝑚}, and {𝑧_𝑛} are complex valued 𝐺-convergent to 𝑥,𝑦, and 𝑧, respectively. Then by (CG5), we have 𝐺𝑥,𝑦,𝑧≾𝐺𝑦,𝑦_𝑚,𝑦_𝑚+𝐺𝑦_𝑚,𝐺,𝑥,𝑧𝑧,𝑥,𝑦_𝑚≾𝐺𝑥,𝑥_𝑘,𝑥_𝑘+𝐺𝑥_𝑘,𝑦_𝑚,𝐺,𝑧𝑧,𝑥_𝑘,𝑦_𝑚≾𝐺𝑧,𝑧_𝑛,𝑧_𝑛+𝐺𝑧_𝑛,𝑦_𝑚,𝑥_𝑘, (5) so 𝐺𝑥,𝑦,𝑧−𝐺𝑥_𝑘,𝑦_𝑚,𝑧_𝑛≾𝐺𝑦,𝑦_𝑚,𝑦_𝑚+𝐺𝑥,𝑥_𝑘,𝑥_𝑘+𝐺𝑧,𝑧_𝑛,𝑧_𝑛. (6) Similarly, 𝐺𝑥_𝑘,𝑦_𝑚,𝑧_𝑛𝑥−𝐺𝑥,𝑦,𝑧≾𝐺_𝑘,𝑥,𝑥+𝐺𝑦_𝑚,𝑦,𝑦+𝐺𝑧_𝑛,𝑧,𝑧. (7) Combining the above inequality and using (iii) of Proposition 7, we have 𝐺𝑥_𝑘,𝑦_𝑚,𝑧_𝑛𝐺−𝐺𝑥,𝑦,𝑧≤2𝑥,𝑥_𝑘,𝑥_𝑘+𝐺𝑦,𝑦_𝑚,𝑦_𝑚+𝐺𝑧,𝑧_𝑛,𝑧_𝑛, (8)|𝐺(𝑥_𝑘,𝑦_𝑚,𝑧_𝑛)−𝐺(𝑥,𝑦,𝑧)|→ 0, as 𝑘,𝑚,𝑛,→∞, and the result follows by Proposition 12.

Definition 14. Let (𝑋,𝐺) be a complex valued 𝐺-metric space. A sequence {𝑥_𝑛} is complex valued 𝐺-Cauchy if, given 𝜖>0, there exists 𝑘∈𝑁 such that 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)≺𝜖 for all 𝑛,𝑚,𝑙≥𝑘.

Definition 15. A complex valued 𝐺-metric space (𝑋,𝐺) is said to be complex valued 𝐺-complete if every complex valued 𝐺-Cauchy sequence is complex valued 𝐺-convergent in (𝑋,𝐺).

Proposition 16. Let (𝑋,𝐺) be a complex valued 𝐺-metric space. Then the following are equivalent:(1)the sequence {𝑥_𝑛} is a complex valued 𝐺-Cauchy in X;(2)for every 𝜖>0, there exists 𝑘∈𝑁 such that 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑚)≺𝜖 for all 𝑛,𝑚≥𝑘;(3){𝑥_𝑛} is a Cauchy sequence in the complex valued metric space (𝑋, 𝑑_𝐺).

Proposition 17. Let (𝑋,𝐺) be a complex valued 𝐺-metric space, and let {𝑥_𝑛} be a sequence in 𝑋. Then {𝑥_𝑛} is complex valued 𝐺-convergent to 𝑥 if and only if |𝐺(𝑥,𝑥_𝑛,𝑥_𝑚)|→0 as 𝑛,𝑚→∞.

Proof. Suppose that {𝑥_𝑛} is complex valued 𝐺-convergent to 𝑥. For a given real number 𝜖>0, let 𝜖𝑐=√2𝜖+𝑖√2. (9) Then 0≺𝑟∈ℂ, and there is a natural number 𝑘, such that 𝐺(𝑥,𝑥_𝑛,𝑥_𝑚)≺𝜖 for all 𝑛,𝑚≥𝑘.
Therefore, 𝐺𝑥,𝑥_𝑛,𝑥_𝑚<|𝑐|=𝜖,∀𝑛,𝑚≥𝑘. (10) It follows that |𝐺(𝑥,𝑥_𝑛,𝑥_𝑚)|→0 as 𝑛,𝑚→∞.
Conversely, suppose that |𝐺(𝑥,𝑥_𝑛,𝑥_𝑚)|→0 as 𝑛,𝑚→∞. Then given 𝑟∈ℂ with 0≺𝑐, there exists a real number 𝛿>0, such that, for 𝑧∈ℂ, |𝑧|<𝛿⟹𝑧≺𝑐. (11) For this 𝛿, there is a natural number 𝑘 such that 𝐺𝑥,𝑥_𝑛,𝑥_𝑚<𝛿,∀𝑛,𝑚≥𝑘. (12) This means that 𝐺(𝑥,𝑥_𝑛,𝑥_𝑚)≺𝜖 for all 𝑛,𝑚≥𝑘. Hence {𝑥_𝑛} is complex valued 𝐺-convergent to 𝑥.

Proposition 18. Let (𝑋,𝐺) be a complex valued 𝐺-metric space, and let {𝑥_𝑛} be a sequence in 𝑋. Then {𝑥_𝑛} is complex valued 𝐺-Cauchy sequence if and only if |𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)|→0 as 𝑛,𝑚→∞.

Proof. Suppose that {𝑥_𝑛} is complex valued 𝐺-Cauchy sequence. For a given real number 𝜖>0, let 𝜖𝑐=√2𝜖+𝑖√2. (13) Then 0≺𝑟∈ℂ, and there is a natural number 𝑘, such that 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)≺𝜖 for all 𝑛,𝑚≥𝑘.
Therefore, 𝐺𝑥_𝑛,𝑥_𝑚,𝑥_𝑙<|𝑐|=𝜖,∀𝑛,𝑚≥𝑘. (14) It follows that |𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)|→0 as 𝑛,𝑚→∞.
Conversely, suppose that |𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)|→0 as 𝑛,𝑚→∞. Then given 𝑐∈ℂ with 0≺𝑐, there exists a real number 𝛿>0, such that, for 𝑧∈ℂ, |𝑧|<𝛿⟹𝑧≺𝑐. (15) For this 𝛿, there is a natural number 𝑘 such that 𝐺𝑥_𝑛,𝑥_𝑚,𝑥_𝑙<𝛿,∀𝑛,𝑚≥𝑘. (16) This means that 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)≺𝜖 for all 𝑛,𝑚 ≥ 𝑘. Hence {𝑥_𝑛} is complex valued 𝐺-Cauchy sequence.

4. Weakly Compatible Maps

In 1996, Jungck [4] introduced the concept of weakly compatible maps as follows.

Definition 19. Two self-maps 𝑓 and 𝑔 are said to be weakly compatible if they commute at coincidence points.
Now we prove our main result for a pair of self-mappings.

Theorem 20. Let (𝑋,𝐺) be a complete complex valued 𝐺-metric space. Let 𝑆,𝑇∶𝑋→𝑋 be self-mappings satisfying the following conditions:(2.1)𝑆(𝑋)⊆𝑇(𝑋); (2.2) any one of the subspace 𝑆(𝑋) or 𝑇(𝑋) is complete;(2.3)𝐺(𝑆𝑥,𝑆𝑦,𝑆𝑧)≾𝑘  𝐺(𝑇𝑥,𝑇𝑦,𝑇𝑧) for all 𝑥,𝑦,𝑧∈𝑋, where 0≤𝑘<1;(2.4)𝑆 and 𝑇 are weakly compatible self-maps.
Then 𝑆 and 𝑇 have a unique common fixed point in X.

Proof. Let 𝑥₀∈𝑋 be an arbitrary point in 𝑋. By (2.1), one can choose a point 𝑥₁ in 𝑋 such that 𝑆𝑥₀=𝑇𝑥₁. In general, choose 𝑥_𝑛+1 such that 𝑦_𝑛=𝑆𝑥_𝑛=𝑇𝑥_𝑛+1. (17) Now, we prove that {𝑦_𝑛} is a complex valued 𝐺-Cauchy sequence in X.
Putting 𝑥=𝑥_𝑛,  𝑦=𝑥_𝑛+1,  and  𝑧=𝑥_𝑛+1 in (2.3), we have 𝐺𝑆𝑥_𝑛,𝑆𝑥_𝑛+1,𝑆𝑥_𝑛+1≾𝑘𝐺𝑇𝑥_𝑛,𝑇𝑥_𝑛+1,𝑇𝑥_𝑛+1=𝑘𝐺𝑆𝑥_𝑛−1,𝑆𝑥_𝑛,𝑆𝑥_𝑛. (18) Continuing in the same way, we have 𝐺𝑆𝑥_𝑛,𝑆𝑥_𝑛+1,𝑆𝑥_𝑛+1≾𝑘^𝑛𝐺𝑆𝑥₀,𝑆𝑥₁,𝑆𝑥₁. (19) This implies that 𝐺(𝑦_𝑛,𝑦_𝑛+1,𝑦_𝑛+1)≾𝑘^𝑛𝐺(𝑦₀,𝑦₁,𝑦₁).
Then, for all 𝑛,𝑚∈𝑁,  𝑛<𝑚, we have by (CG5) 𝐺𝑦_𝑛,𝑦_𝑚,𝑦_𝑚≾𝐺𝑦_𝑛,𝑦_𝑛+1,𝑦_𝑛+1+𝐺𝑦_𝑛+1,𝑦_𝑛+2,𝑦_𝑛+2+𝐺𝑦_𝑛+2,𝑦_𝑛+3,𝑦_𝑛+3+⋯+𝐺𝑦_𝑚−1,𝑦_𝑚,𝑦_𝑚≾𝑘^𝑛+𝑘^𝑛+1+𝑘^𝑛+2+⋯+𝑘^𝑚−1×𝐺𝑦₀,𝑦₁,𝑦₁≾𝑘^𝑛𝐺𝑦1−𝑘⁰,𝑦₁,𝑦₁. (20) Therefore, 𝐺𝑦_𝑛,𝑦_𝑚,𝑦_𝑚≤𝑘^𝑛𝐺𝑦1−𝑘⁰,𝑦₁,𝑦₁. (21) Proceeding limit as 𝑛,𝑚→∞ and since 0 ≤𝑘<1, then (𝑘^𝑛/(1−𝑘))|𝐺(𝑦₀,𝑦₁,𝑦₁)|→0; that is, |𝐺(𝑦_𝑛,𝑦_𝑚,𝑦_𝑚)|→0.
For 𝑛,𝑚,𝑙∈𝑁, (CG5) implies that 𝐺𝑦_𝑛,𝑦_𝑚,𝑦_𝑙≾𝐺𝑦_𝑛,𝑦_𝑚,𝑦_𝑚+𝐺𝑦_𝑙,𝑦_𝑚,𝑦_𝑚. (22) Therefore, 𝐺𝑦_𝑛,𝑦_𝑚,𝑦_𝑙≤𝐺𝑦_𝑛,𝑦_𝑚,𝑦_𝑚+𝐺𝑦_𝑙,𝑦_𝑚,𝑦_𝑚. (23) Taking limit as 𝑛,𝑚,𝑙→∞, we get |𝐺(𝑦_𝑛,𝑦_𝑚,𝑦_𝑙)|→ 0; that is, 𝐺(𝑦_𝑛,𝑦_𝑚,𝑦_𝑙→0). So {𝑦_𝑛} is complex valued 𝐺-Cauchy sequence. Since either 𝑆(𝑋) or 𝑇(𝑋) is complete. Without loss of generality, we assume that 𝑇(𝑋) is complete subspace of 𝑋, and then the subsequence of {𝑦_𝑛} must get a limit in 𝑇(𝑋), say 𝑧. Then 𝑇𝑢=𝑧 for some 𝑢∈𝑋, as {𝑦_𝑛} is a complex valued 𝐺-Cauchy sequence containing a convergent subsequence.
Next we show that 𝑆𝑢=𝑧. On setting 𝑥=𝑢,𝑦=𝑥_𝑛, and 𝑧=𝑥_𝑛 in (2.3), we have 𝐺𝑆𝑢,𝑆𝑥_𝑛,𝑆𝑥_𝑛≾𝑘𝐺𝑇𝑢,𝑇𝑥_𝑛,𝑇𝑥_𝑛. (24) Taking limit as 𝑛→∞, we have 𝐺(𝑆𝑢,𝑧,𝑧)≾𝑘𝐺(𝑇𝑢,𝑧,𝑧).
Therefore, |𝐺(𝑆𝑢,𝑧,𝑧)|≤𝑘|𝐺(𝑇𝑢,𝑧,𝑧)| implies that 𝑆𝑢=𝑧.
Therefore, 𝑆𝑢=𝑇𝑢=𝑧. That is, 𝑢 is coincidence point of 𝑆 and 𝑇. Since 𝑆 and 𝑇 are weakly compatible, it follows that 𝑆𝑇𝑢=𝑇𝑆𝑢; that is, 𝑆𝑧=𝑇𝑧.
We now show that 𝑆𝑧=𝑧. Suppose that 𝑆(𝑧)≠𝑧; therefore 0≺𝐺(𝑆𝑧,𝑧,𝑧) implies that |𝐺(𝑆𝑧,𝑧,𝑧)|>0.
Putting 𝑥=𝑧, 𝑦=𝑢, and 𝑧=𝑢 in (2.3), we have 𝐺(𝑆𝑧,𝑆𝑢,𝑆𝑢)≾𝑘𝐺(𝑇𝑧,𝑇𝑢,𝑇𝑢)=𝑘𝐺(𝑆𝑧,𝑧,𝑧); (25) That is, |𝐺(𝑆𝑧,𝑧,𝑧)|≤𝑘|𝐺(𝑆𝑧,𝑧,𝑧)|<|𝐺(𝑆𝑧,𝑧,𝑧)|, which is a contradiction; therefore 𝑆𝑧=𝑧. Thus 𝑆𝑧=𝑇𝑧=𝑧; that is, 𝑧 is a common fixed point of 𝑆 and 𝑇.
Uniqueness. To prove uniqueness, suppose that 𝑤≠𝑧 be another common fixed point of 𝑆 and 𝑇. Then 0≺G(𝑧,𝑤,𝑤) implies that |𝐺(𝑧,𝑤,𝑤)|>0.
Putting 𝑥=𝑧,𝑦=𝑢,  and  𝑧=𝑢 in (2.3), we have 𝐺()()()();𝑧,𝑤,𝑤=𝐺𝑆𝑧,𝑆𝑤,𝑆𝑤≾𝑘𝐺𝑇𝑧,𝑇𝑤,𝑇𝑤=𝑘𝐺𝑧,𝑤,𝑤 (26) that is, |𝐺(𝑧,𝑤,𝑤)|≤𝑘|𝐺(𝑧,𝑤,𝑤)|<|𝐺(𝑧,𝑤,𝑤)|, which is a contradiction; therefore 𝑧=𝑤. Thus 𝑆𝑧=𝑇𝑧=𝑧; that is, 𝑧 is a unique common fixed point of 𝑆 and 𝑇.

Example 21. Let 𝑋=[−1,1], and let 𝐺∶𝑋×𝑋×𝑋→ℂ be complex valued 𝐺-metric space defined as follows: 𝐺𝑥,𝑦,𝑧=𝑥−𝑦+𝑦−𝑧+|𝑧−𝑥|,∀𝑥,𝑦,𝑧∈𝑋. (27) Then (𝑋,𝐺) is complex valued 𝐺-metric space. Define 𝑆,𝑇∶𝑋→𝑋 as 𝑆𝑥=𝑥/6 and 𝑇𝑥=𝑥/2. Here we note that (2.1) 𝑆(𝑋)⊆𝑇(𝑋), (2.3) 𝐺(𝑆𝑥,𝑆𝑦,𝑆𝑧)≾𝑘𝐺(𝑇𝑥,𝑇𝑦,𝑇𝑧) holds for all 𝑥,𝑦,𝑧∈𝑋,  1/3≤𝑘<1, and (2.4) 𝑆 and 𝑇 are weakly compatible because 𝑆 and 𝑇 commute at their coincidence point, that is, at 𝑥=0, and 𝑥=0 is the unique common fixed point of 𝑆 and 𝑇 and 𝑆 and 𝑇 also satisfy the condition (2.2).

5. E.A. Property and Weakly Compatible Maps

In 2002, Aamri and Moutawakil [5] introduced the notion of E.A. property as follows.

Definition 22. Two self-mappings 𝑓 and 𝑔 of a metric space (𝑋,𝑑) are said to satisfy E.A. property if there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝑡 for some 𝑡 in 𝑋.

In a similar mode, we use these notions in complex valued 𝐺-metric spaces.

Example 23. Let 𝑋=ℂ. Let 𝐺∶𝑋×𝑋×𝑋→ℂ be complex valued 𝐺-metric space defined as follows: 𝐺𝑥,𝑦,𝑧=𝑥−𝑦+𝑦−𝑧+|𝑧−𝑥|,∀𝑥,𝑦,𝑧∈𝑋. (28) Then (𝑋,𝐺) is complex valued 𝐺-metric space. Define 𝑆, 𝑇: 𝑋→𝑋 as 𝑆𝑥=𝑥+1 and 𝑇𝑥=2𝑥; for all 𝑥∈𝑋.
Consider a sequence {𝑥_𝑛}={1−1/𝑛},  𝑛∈ℕ, in 𝑋; then lim_𝑛→∞𝑆𝑥_𝑛=lim_𝑛→∞𝑥_𝑛+1=lim_𝑛→∞11−𝑛+1=2,lim_𝑛→∞𝑇𝑥_𝑛=lim_𝑛→∞2𝑥_𝑛=lim_𝑛→∞211−𝑛=2. (29) Thus 𝑆 and 𝑇 satisfy E.A. property.

Now we prove a common fixed point theorem for weakly compatible maps along with E.A. property.

Theorem 24. Let 𝑆 and 𝑇 be self-mappings of a complex valued 𝐺-metric space (𝑋,𝐺) satisfying (2.3), (2.4), and the following:(3.1)𝑆 and 𝑇 satisfy E.A. property;(3.2)𝑇(𝑋) is a closed subspace of 𝑋.Then 𝑆 and 𝑇 have a unique common fixed point.

Proof. Since 𝑆 and 𝑇 satisfy the E.A. property, therefore, there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝑆𝑥_𝑛=lim_𝑛→∞𝑇𝑥_𝑛=𝑢∈𝑋. Since 𝑇(𝑋) is a closed subspace of 𝑋, therefore every convergent sequence of points of 𝑇(𝑋) has a limit point in 𝑇(𝑋).
Then lim_𝑛→∞𝑆𝑥_𝑛=𝑢=𝑇𝑣=lim_𝑛→∞𝑇𝑥_𝑛 for some 𝑣 in 𝑋. This implies that 𝑢=𝑇𝑣∈𝑇(𝑋).
On setting 𝑥=𝑣,𝑦=𝑥_𝑛, and 𝑧=𝑥_𝑛, in (2.3), we have 𝐺𝑆𝑣,𝑆𝑥_𝑛,𝑆𝑥_𝑛≾𝑘𝐺𝑇𝑣,𝑇𝑥_𝑛,𝑇𝑥_𝑛. (30) Taking limit as 𝑛→∞, we have 𝐺(𝑆𝑣,𝑢,𝑢)≾𝑘𝐺(𝑇𝑣,𝑢,𝑢).
Therefore, |𝐺(𝑆𝑣,𝑢,𝑢)|≤𝑘|𝐺(𝑇𝑣,𝑢,𝑢)| implies that 𝑆𝑣=𝑢.
Therefore, 𝑆𝑣=𝑇𝑣=𝑢. That is, 𝑣 is coincidence point of 𝑆 and 𝑇. Since 𝑆 and 𝑇 are weakly compatible, it follows that 𝑆𝑇𝑣=𝑇𝑆𝑣;thatis,𝑆𝑢=𝑇𝑢.
Next we show that 𝑆𝑢=𝑢. On setting 𝑥=𝑢,  𝑦=𝑣, and 𝑧=𝑣 in (2.3), we have 𝐺(𝑆𝑢,𝑆𝑣,𝑆𝑣)≾𝑘𝐺(𝑇𝑢,𝑇𝑣,𝑇𝑣). (31) Therefore, |𝐺(𝑆𝑢,𝑢,𝑢)|≤𝑘|𝐺(𝑇𝑢,𝑢,𝑢)| implies that 𝑆𝑢=𝑢. Hence 𝑢 is a common fixed point of 𝑆 and 𝑇.

Uniqueness easily follows from Theorem 20. Hence 𝑢 is a unique common fixed point of 𝑆 and 𝑇.

6. (CLR) Property and Weakly Compatible Maps

In 2011, Kumam and Sintunavarat [6] introduced the notion of (CLR_𝑔) property as follows.

Definition 25. Two self-mappings 𝑓 and 𝑔 of a metric space (𝑋,𝑑) are said to satisfy (CLR_𝑔) property if there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝑔𝑥 for some 𝑥 in 𝑋.
In a similar mode, we use these notions in complex valued 𝐺-metric spaces.

Example 26. Let 𝑋=ℂ. Let 𝐺∶𝑋×𝑋×𝑋→ℂ be complex valued 𝐺-metric space defined as follows: 𝐺𝑥,𝑦,𝑧=𝑥−𝑦+𝑦−𝑧+|𝑧−𝑥|,∀𝑥,𝑦,𝑧∈𝑋. (32) Then (𝑋,𝐺) is complex valued 𝐺-metric space. Define 𝑆,𝑇∶𝑋→𝑋 as 𝑆𝑥=𝑥+1 and 𝑇𝑥=2𝑥 for all 𝑥∈𝑋.
Consider a sequence {𝑥_𝑛}={1−1/𝑛},  𝑛∈ℕ, in 𝑋; then lim_𝑛→∞𝑆𝑥_𝑛=lim_𝑛→∞𝑥_𝑛+1=lim_𝑛→∞11−𝑛+1=2,lim_𝑛→∞𝑇𝑥_𝑛=lim_𝑛→∞2𝑥_𝑛=lim_𝑛→∞211−𝑛=2. (33) Also, we have lim_𝑛→∞𝑆𝑥_𝑛=lim_𝑛→∞𝑇𝑥_𝑛=2=𝑆(1). (34) Thus 𝑆 and 𝑇 satisfy (CLR_𝑆) property
Now we prove a common fixed point theorem for weakly compatible maps along with (CLR_𝑇) property.

Theorem 27. Let 𝑆 and 𝑇 be self-mappings of a complex valued 𝐺-metric space (𝑋,𝐺) satisfying (2.1), (2.3), (2.4), and the following:(4.1)𝑆 and 𝑇 satisfy (CLR_𝑇) property.
Then 𝑆 and 𝑇 have a unique common fixed point.

Proof. Since 𝑆 and 𝑇 satisfy the (CLR_𝑇) property, therefore, there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝑆𝑥_𝑛=lim_𝑛→∞𝑇𝑥_𝑛=𝑢∈𝑋. Then lim_𝑛→∞𝑆𝑥_𝑛=𝑢=𝑇𝑣=lim_𝑛→∞𝑇𝑥_𝑛 for some 𝑣 in 𝑋. This implies that 𝑢=𝑇𝑣∈𝑇(𝑋).
On setting 𝑥=𝑣,  𝑦=𝑥_𝑛, and 𝑧=𝑥_𝑛 in (2.3), we have 𝐺𝑆𝑣,𝑆𝑥_𝑛,𝑆𝑥_𝑛≾𝑘𝐺𝑇𝑣,𝑇𝑥_𝑛,𝑇𝑥_𝑛. (35) Taking limit as 𝑛→∞, we have 𝐺(𝑆𝑣,𝑢,𝑢)≾𝑘𝐺(𝑇𝑣,𝑢,𝑢).
Therefore, |𝐺(𝑆𝑣,𝑢,𝑢)|≤𝑘|𝐺(𝑇𝑣,𝑢,𝑢)| implies that 𝑆𝑣=𝑢.
Therefore, 𝑆𝑣=𝑇𝑣=𝑢. That is, 𝑣 is coincidence point of 𝑆 and 𝑇. Since 𝑆 and 𝑇 are weakly compatible, it follows that 𝑆𝑇𝑣=𝑇𝑆𝑣; that is,  𝑆𝑢=𝑇𝑢.
Next we show that 𝑆𝑢=𝑢. On setting 𝑥=𝑢,𝑦=𝑣, and 𝑧=𝑣 in (2.3), we have 𝐺(𝑆𝑢,𝑆𝑣,𝑆𝑣)≾𝑘𝐺(𝑇𝑢,𝑇𝑣,𝑇𝑣). (36) Therefore, |𝐺(𝑆𝑢,𝑢,𝑢)|≤𝑘|𝐺(𝑢,𝑢,𝑢)| implies that 𝑆𝑢=𝑢. Hence 𝑢 is a common fixed point of 𝑆 and 𝑇.

Uniqueness easily follows from Theorem 20. Hence 𝑢 is a unique common fixed point of 𝑆 and 𝑇.

Acknowledgment

S. Kumar would like to acknowledge UGC for providing Major Research Project Financial Grant under Reference (39-41/2010(SR)).