Some Common Fixed Point Results for Rational Type Contraction Mappings in Complex Valued Metric Spaces

Abstract

We prove some common fixed point theorems for two pairs of weakly compatible mappings satisfying a rational type contractive condition in the framework of complex valued metric spaces. The proved results generalize and extend some of the known results in the literature.

1. Introduction and Preliminaries

The famous Banach contraction principle states that if (𝑋,𝑑) is a complete metric space and 𝑇∶𝑋→𝑋 is a contraction mapping (i.e., 𝑑(𝑇𝑥,𝑇𝑦)≤𝑘𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋, where 𝑘 is a nonnegative number such that 𝑘<1), then 𝑇 has a unique fixed point. This principle is one of the cornerstones in the development of nonlinear analysis. Fixed point theorems have applications not only in the different branches of mathematics, but also in economics, chemistry, biology, computer science, engineering, and others. Due to its importance, generalizations of Banach’s contraction principle have been investigated heavily by several authors. Fixed point and common fixed point theorems for different types of nonlinear contractive mappings have been investigated extensively by various researchers (see [1–35] and references cited therein).

Recently, Azam et al. [1] introduced the complex valued metric space, which is more general than the well-known metric spaces. Many researchers have obtained fixed point, common fixed point, coupled fixed point, and coupled common fixed point results in partially ordered metric spaces, complex valued metric spaces, and other spaces. In this paper, we prove some common fixed point theorems for two pairs of weakly mappings satisfying a contractive condition of rational type in the framework of complex valued metric spaces. The proved results generalize and extend some of the results in the literature.

To begin with, we recall some basic definitions, notations, and results.

The following definitions of Azam et al. [1] are needed in the sequel.

Let ℂ be the set of complex numbers, and let 𝑧₁,𝑧₂∈ℂ. Define a partial order ≤ on ℂ as follows: 𝑧₁≤𝑧₂iffRe𝑧₁≤Re𝑧₂,Im𝑧₁≤Im𝑧₂. (1) It follows that 𝑧₁≤𝑧₂ if one of the following conditions is satisfied:(1)Re(𝑧₁)=Re(𝑧₂), and Im(𝑧₁)<Im(𝑧₂);(2)Re(𝑧₁)<Re(𝑧₂), and Im(𝑧₁)=Im(𝑧₂);(3)Re(𝑧₁)<Re(𝑧₂), and Im(𝑧₁)<Im(𝑧₂);(4)Re(𝑧₁)=Re(𝑧₂), and Im(𝑧₁)=Im(𝑧₂).In particular, we will write 𝑧₁≨𝑧₂ if 𝑧₁≠𝑧₂ and one of (1), (2), and (3) is satisfied, and we will write 𝑧₁<𝑧₂ if only (3) is satisfied.

Note. We obtained 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 imply 𝑧₁<𝑧₃.

Definition 1. Let 𝑋 be a nonempty set. Suppose that the mapping 𝑑∶𝑋×𝑋→ℂ satisfies the following conditions:(i)0≤𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋 and 𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦;(ii)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋;(iii)𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑧)+𝑑(𝑧,𝑦) for all 𝑥,𝑦,𝑧∈𝑋.Then, 𝑑 is called a complex valued metric on 𝑋, and (𝑋,𝑑) is called a complex valued metric space.

Example 2. Let 𝑋=ℂ. Define a mapping 𝑑∶𝑋×𝑋→ℂ by 𝑑𝑧₁,𝑧₂=𝑒^𝑖𝑘𝑧₁−𝑧₂, (2) where 𝑘∈[0,𝜋/2]. Then, (𝑋,𝑑) is a complex valued metric space.

A point 𝑥∈𝑋 is called an interior point of a set 𝐴⊆𝑋 whenever there exists 0<𝑟∈ℂ such that 𝐵(𝑥,𝑟)={𝑦∈𝑋∶𝑑(𝑥,𝑦)<𝑟}⊆𝐴. A subset 𝐴 in 𝑋 is called open, whenever each point of 𝐴 is an interior point of 𝐴. The family 𝐹={𝐵(𝑥,𝑟)∶𝑥∈𝑋,0<𝑟} is a subbasis for a Hausdorff topology on 𝑋.

A point 𝑥∈𝑋 is called a limit point of 𝐴, whenever for every 0<𝑟∈ℂ, 𝐵(𝑥,𝑟)∩(𝐴⧵𝑋)≠∅. A subset 𝐵⊆𝑋 is called closed, whenever each limit point of 𝐵 belongs to 𝐵.

Let {𝑥_𝑛} be a sequence in 𝑋 and 𝑥∈𝑋. If for every 𝑐∈ℂ, with 0<𝑐, there is 𝑛₀∈ℕ such that for all 𝑛>𝑛₀, 𝑑(𝑥_𝑛,𝑥)<𝑐, then 𝑥 is called the limit of {𝑥_𝑛}, and we write lim_𝑛→∞𝑥_𝑛=𝑥.

If for every 𝑐∈ℂ, with 0<𝑐, there is an 𝑛₀∈ℕ such that for all 𝑛>𝑛₀, 𝑑(𝑥_𝑛,𝑥_𝑛+𝑚)<𝑐, then {𝑥_𝑛} is called a Cauchy sequence in (𝑋,𝑑). If every Cauchy sequence is convergent in (𝑋,𝑑), then (𝑋,𝑑) is called a complete complex valued metric space.

Lemma 3 (see [1]). Let (𝑋,𝑑) be a complex valued metric space and {𝑥_𝑛} a sequence in 𝑋. Then, {𝑥_𝑛} converges to 𝑥 if and only if |𝑑(𝑥_𝑛,𝑥)|→0 as 𝑛→∞.

Lemma 4 (see [1]). Let (𝑋,𝑑) be a complex valued metric space and {𝑥_𝑛} a sequence in 𝑋. Then, {𝑥_𝑛} is a Cauchy sequence if and only if |𝑑(𝑥_𝑛,𝑥_𝑛+𝑚)|→0 as 𝑛→∞.

Let 𝑀 be a nonempty subset of a metric space (𝑋,𝑑). Let 𝑆 and 𝑇 be mappings from a metric space (𝑋,𝑑) into itself. A point 𝑥∈𝑀 is a common fixed (resp., coincidence) point of 𝑆 and 𝑇 if 𝑥=𝑆𝑥=𝑇𝑥 (resp., 𝑆𝑥=𝑇𝑥). The set of fixed points (resp., coincidence points) of 𝑆 and 𝑇 is denoted by 𝐹(𝑆,𝑇) (resp., 𝐶(𝑆,𝑇)).

In 1986, Jungck [22] introduced the more generalized commuting mappings in metric spaces, called compatible mappings, which also are more general than the concept of weakly commuting mappings (i.e., the mappings 𝑆,𝑇∶𝑋→𝑋 are said to be weakly commuting if 𝑑(𝑆𝑇𝑥,𝑇𝑆𝑥)≤𝑑(𝑆𝑥,𝑇𝑥) for all 𝑥∈𝑋) introduced by Sessa [28].

Definition 5. Let 𝑆 and 𝑇 be mappings from a metric space (𝑋,𝑑) into itself. The mappings 𝑆 and 𝑇 are said to be compatible if lim_𝑛→∞𝑑𝑆𝑇𝑥_𝑛,𝑇𝑆𝑥_𝑛=0 (3) whenever {𝑥_𝑛} is a sequence in 𝑋 such that lim_𝑛→∞𝑆𝑥_𝑛=lim_𝑛→∞𝑇𝑥_𝑛=𝑧 for some 𝑧∈𝑋.

In general, commuting mappings are weakly commuting, and weakly commuting mappings are compatible, but the converses are not necessarily true, and some examples can be found in [22–24].

Definition 6. The mappings 𝑆 and 𝑇 are said to be weakly compatible if they commute at coincidence points of 𝑆 and 𝑇.

Definition 7. Let 𝑇,𝑆∶𝑋→𝑋 be two self-mappings of a complex valued metric space (𝑋,𝑑). The pair (𝑇,𝑆) is said to satisfy (E.A.)  property (see [35]) if there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝑇𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=𝑡, for some 𝑡∈𝑋.

Pathak et al. [27] showed that weakly compatibility and (E.A.) property are independent of each other.

Definition 8. The self mappings 𝑇 and 𝑆 from 𝑋 to 𝑋 are said to satisfy the common limit in the range of 𝑆 property (𝐶𝐿𝑅_𝑆  property) (see [31]) if lim_𝑛→∞𝑇𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=𝑆𝑥, for some 𝑥∈𝑋.

Some recent papers related to (CLR)  property and the complex valued metric spaces can be found in [1, 3, 27, 31–35] and references cited therein.

2. Main Results

2.1. Common Fixed Point Theorem Using (E.A.) Property

In this section, we prove some common fixed point theorems using (E.A.) property in the complex valued metric spaces.

Theorem 9. Let (𝑋,𝑑) be a complex valued metric space and 𝐴, 𝐵, 𝑆, 𝑇∶𝑋→𝑋 four self-mappings satisfying the following conditions: (i)𝐴(𝑋)⊆𝑇(𝑋), 𝐵(𝑋)⊆𝑆(𝑋); (ii) for all 𝑥,𝑦∈𝑋 and 0<𝑎<1, 𝑑𝑑()𝑑𝑑𝐴𝑥,𝐵𝑦≤𝑎𝑆𝑥,𝐴𝑥𝑆𝑥,𝐵𝑦+𝑑𝑇𝑦,𝐵𝑦𝑇𝑦,𝐴𝑥;1+𝑑𝑆𝑥,𝐵𝑦+𝑑𝑇𝑦,𝐴𝑥 (4) (iii) the pairs (𝐴,𝑆) and (𝐵,𝑇) are weakly compatible; (iv) one of the pairs (𝐴,𝑆) or (𝐵,𝑇) satisfies (E.A.)-property.If the range of one of the mappings 𝑆(𝑋) or 𝑇(𝑋) is a closed subspace of 𝑋, then the mappings 𝐴, 𝐵, 𝑆, and 𝑇 have a unique common fixed point in 𝑋.

Proof. First, we suppose that the pair (𝐵,𝑇) satisfies (E.A.) property. Then, by Definition 7 there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝐵𝑥_𝑛=lim_𝑛→∞𝑇𝑥_𝑛=𝑡 for some 𝑡∈𝑋.
Further, since 𝐵(𝑋)⊆𝑆(𝑋), there exists a sequence {𝑦_𝑛} in 𝑋 such that 𝐵𝑥_𝑛=𝑆𝑦_𝑛. Hence, lim_𝑛→∞𝑆𝑦_𝑛=𝑡. We claim that lim_𝑛→∞𝐴𝑦_𝑛=𝑡. Let lim_𝑛→∞𝐴𝑦_𝑛=𝑡₁≠𝑡, then putting 𝑥=𝑦_𝑛, 𝑦=𝑥_𝑛 in condition (ii), we have 𝑑𝐴𝑦_𝑛,𝐵𝑥_𝑛𝑑≤𝑎𝑆𝑦_𝑛,𝐴𝑦_𝑛𝑑𝑆𝑦_𝑛,𝐵𝑥_𝑛+𝑑𝑇𝑥_𝑛,𝐵𝑥_𝑛𝑑𝑇𝑥_𝑛,𝐴𝑦_𝑛1+𝑑𝑆𝑦_𝑛,𝐵𝑥_𝑛+𝑑𝑇𝑥_𝑛,𝐴𝑦_𝑛. (5) Letting 𝑛→∞, we have 𝑑𝑡₁𝑑,𝑡≤𝑎𝑡,𝑡₁𝑑(𝑡,𝑡)+𝑑(𝑡,𝑡)𝑑𝑡,𝑡₁1+𝑑(𝑡,𝑡)+𝑑𝑡,𝑡₁. (6) Then, |𝑑(𝑡₁,𝑡)|≤0; hence, 𝑡₁=𝑡 and that is, lim_𝑛→∞𝐴𝑦_𝑛=lim_𝑛→∞𝐵𝑥_𝑛=𝑡.
Now suppose that 𝑆(𝑋) is a closed subspace of 𝑋, then 𝑡=𝑆𝑢 for some 𝑢∈𝑋. Subsequently, we have lim_𝑛→∞𝐴𝑦_𝑛=lim_𝑛→∞𝐵𝑥_𝑛=lim_𝑛→∞𝑇𝑥_𝑛=lim_𝑛→∞𝑆𝑦_𝑛=𝑡=𝑆𝑢. (7) We claim that 𝐴𝑢=𝑆𝑢. Put 𝑥=𝑢 and 𝑦=𝑥_𝑛 in contractive condition (ii), and we have 𝑑𝐴𝑢,𝐵𝑥_𝑛𝑑()𝑑≤𝑎𝑆𝑢,𝐴𝑢𝑆𝑢,𝐵𝑥_𝑛+𝑑𝑇𝑥_𝑛,𝐵𝑥_𝑛𝑑𝑇𝑥_𝑛,𝐴𝑢1+𝑑𝑆𝑢,𝐵𝑥_𝑛+𝑑𝑇𝑥_𝑛.,𝐴𝑢 (8) Letting 𝑛→∞ and using (7), we have 𝑑()𝑑()𝑑()()𝑑()𝐴𝑢,𝑡≤𝑎𝑡,𝐴𝑢𝑡,𝑡+𝑑𝑡,𝑡𝑡,𝐴𝑢()().1+𝑑𝑡,𝑡+𝑑𝑡,𝐴𝑢 (9) Then, |𝑑(𝐴𝑢,𝑡)|≤0, which is contradiction. Hence, 𝑢 is a coincidence point of (𝐴,𝑆).
Now the weak compatibility of pair (𝐴,𝑆) implies that 𝐴𝑆𝑢=𝑆𝐴𝑢 or 𝐴𝑡=𝑆𝑡.
On the other hand, since 𝐴(𝑋)⊆𝑇(𝑋), there exists 𝑣 in 𝑋 such that 𝐴𝑢=𝑇𝑣. Thus, 𝐴𝑢=𝑆𝑢=𝑇𝑣=𝑡. Now, we show that 𝑣 is a coincidence point of (𝐵,𝑇); that is, 𝐵𝑣=𝑇𝑣=𝑡. Put 𝑥=𝑢, 𝑦=𝑣 in contractive condition (ii), and we have 𝑑()𝑑()𝑑()()𝑑()𝐴𝑢,𝐵𝑣≤𝑎𝑆𝑢,𝐴𝑢𝑆𝑢,𝐵𝑣+𝑑𝑇𝑣,𝐵𝑣𝑇𝑣,𝐴𝑢()(),1+𝑑𝑆𝑢,𝐵𝑣+𝑑𝑇𝑣,𝐴𝑢 (10) or 𝑑()𝑑()𝑑()()𝑑()𝑡,𝐵𝑣≤𝑎𝑡,𝑡𝑡,𝐵𝑣+𝑑𝑡,𝐵𝑣𝑡,𝑡()();1+𝑑𝑡,𝐵𝑣+𝑑𝑡,𝑡 (11) whence |𝑑(𝑡,𝐵𝑣)|≤0, which is a contradiction. Thus, 𝐵𝑣=𝑡. Hence, 𝐵𝑣=𝑇𝑣=𝑡, and 𝑣 is the coincidence point of 𝐵 and 𝑇.
Further, the weak compatibility of pair (𝐵,𝑇) implies that 𝐵𝑇𝑣=𝑇𝐵𝑣, or 𝐵𝑡=𝑇𝑡. Therefore, 𝑡 is a common coincidence point of 𝐴, 𝐵, 𝑆, and 𝑇.
Now, we show that 𝑡 is a common fixed point. Put 𝑥=𝑢 and 𝑦=𝑡 in contractive condition (ii), and we have 𝑑()()𝑑()𝑑()()𝑑()𝑡,𝐵𝑡=𝑑𝐴𝑢,𝐵𝑡≤𝑎𝑆𝑢,𝐴𝑢𝑆𝑢,𝐵𝑡+𝑑𝑇𝑡,𝐵𝑡𝑇𝑡,𝐴𝑢()(),1+𝑑𝑆𝑢,𝐵𝑡+𝑑𝑇𝑡,𝐴𝑢 (12) or |𝑑(𝑡,𝐵𝑡)|≤0, which is a contradiction. Thus, 𝐵𝑡=𝑡. Hence, 𝐴𝑡=𝐵𝑡=𝑆𝑡=𝑇𝑡=𝑡.
Similar argument arises if we assume that 𝑇(𝑋) is closed subspace of 𝑋. Similarly, the (E.A.)-property of the pair (𝐴,𝑆) will give a similar result.
For uniqueness of the common fixed point, let us assume that 𝑤 is another common fixed point of 𝐴, 𝐵, 𝑆, and 𝑇. Then, put 𝑥=𝑤, 𝑦=𝑡 in contractive condition (ii), and we have 𝑑()()𝑑()𝑑()()𝑑()𝑤,𝑡=𝑑𝐴𝑤,𝐵𝑡≤𝑎𝑆𝑤,𝐴𝑤𝑆𝑤,𝐵𝑡+𝑑𝑇𝑡,𝐵𝑡𝑇𝑡,𝐴𝑤()(),1+𝑑𝑆𝑤,𝐵𝑡+𝑑𝑇𝑡,𝐴𝑤 (13) or |𝑑(𝑤,𝑡)|≤0, which is a contradiction. Thus, 𝑤=𝑡. Hence, 𝐴𝑡=𝐵𝑡=𝑆𝑡=𝑇𝑡=𝑡, and 𝑡 is the unique common fixed point of 𝐴, 𝐵, 𝑆, and 𝑇.

Remark 10. (a) Continuity of mappings 𝐴, 𝐵, 𝑆, and   𝑇 is relaxed in Theorem 9.
(b) Completeness of space 𝑋 is relaxed in Theorem 9.
If 𝐴=𝐵 and 𝑆=𝑇 in Theorem 9, we have the following result.

Corollary 11. Let (𝑋,𝑑) be a complex valued metric space and 𝐴,𝑇∶𝑋→𝑋 self-mappings satisfying the following conditions: (i)𝐴(𝑋)⊆𝑇(𝑋); (ii) for all 𝑥,𝑦∈𝑋 and 0<𝑎<1, 𝑑𝑑()𝑑𝑑𝐴𝑥,𝐴𝑦≤𝑎𝑇𝑥,𝐴𝑥𝑇𝑥,𝐴𝑦+𝑑𝑇𝑦,𝐴𝑦𝑇𝑦,𝐴𝑥;1+𝑑𝑇𝑥,𝐴𝑦+𝑑𝑇𝑦,𝐴𝑥 (14) (iii) the pair (𝐴,𝑇) is weakly compatible; (iv) the pair (𝐴,𝑇) satisfies (E.A.)-property.If the range of the mapping 𝑇(𝑋) is a closed subspace of 𝑋, then 𝐴 and 𝑇 have a unique common fixed point in 𝑋.

Theorem 12. Let (𝑋,𝑑) be a complex valued metric space and 𝐴, 𝐵, 𝑆, 𝑇∶𝑋→𝑋 four self-mappings satisfying the following conditions: (i)𝐴(𝑋)⊆𝑇(𝑋), 𝐵(𝑋)⊆𝑆(𝑋); (ii) for all 𝑥,𝑦∈𝑋 and 0<𝑎<1, 𝑑≤𝑎𝑑()𝑑𝑑𝐴𝑥,𝐵𝑦𝑆𝑥,𝐴𝑥𝑆𝑥,𝐵𝑦+𝑑𝑇𝑦,𝐵𝑦𝑇𝑦,𝐴𝑥𝑑,𝑆𝑥,𝐵𝑦+𝑑𝑇𝑦,𝐴𝑥𝑖𝑓𝐷≠00,𝑖𝑓𝐷=0, (15)where 𝐷=𝑑(𝑆𝑥,𝐵𝑦)+𝑑(𝑇𝑦,𝐴𝑥); (iii) the pairs (𝐴,𝑆) and (𝐵,𝑇) are weakly compatible; (iv) one of the pairs (𝐴,𝑆) or (𝐵,𝑇) satisfies (E.A.)-property.If the range of one of the mappings 𝑆(𝑋) or 𝑇(𝑋) is a closed subspace of 𝑋, then the mappings 𝐴, 𝐵, 𝑆, and 𝑇 have a unique common fixed point in 𝑋.

Proof. Using the same arguments as in Theorem 9, we have the following result.

2.2. Fixed Point Theorem Using (CLR)-Property

In this section, we prove some common fixed point theorems using (CLR) property in the complex valued metric spaces.

Theorem 13. Let (𝑋,𝑑) be a complex valued metric space and 𝐴, 𝐵, 𝑆, and 𝑇∶𝑋→𝑋 four self-mappings satisfying the following conditions: (i)𝐴(𝑋)⊆𝑇(𝑋), 𝐵(𝑋)⊆𝑆(𝑋); (ii) for all 𝑥,𝑦∈𝑋 and 0<𝑎<1, 𝑑𝑑()𝑑𝑑𝐴𝑥,𝐵𝑦≤𝑎𝑆𝑥,𝐴𝑥𝑆𝑥,𝐵𝑦+𝑑𝑇𝑦,𝐵𝑦𝑇𝑦,𝐴𝑥;1+𝑑𝑆𝑥,𝐵𝑦+𝑑𝑇𝑦,𝐴𝑥 (16) (iii) the pairs (𝐴,𝑆) and (𝐵,𝑇) are weakly compatible.If the pair (𝐴,𝑆) satisfies (𝐶𝐿𝑅_𝐴) property or (𝐵,𝑇) satisfies (𝐶𝐿𝑅_𝐵) property, then 𝐴, 𝐵, 𝑆 and 𝑇 have a unique common fixed point in 𝑋.

Proof. First, we suppose that the pair (𝐵,𝑇) satisfies (CLR_𝐵) property. Then, by Definition 8, there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝐵𝑥_𝑛=lim_𝑛→∞𝑇𝑥_𝑛=𝐵𝑥, (17) for some 𝑥∈𝑋.
Further, since 𝐵(𝑋)⊆𝑆(𝑋), we have 𝐵𝑥=𝑆𝑢, for some 𝑢∈𝑋. We claim that 𝐴𝑢=𝑆𝑢=𝑡(say). Put 𝑥=𝑢 and 𝑦=𝑥_𝑛 in contractive condition (ii), and we have 𝑑𝐴𝑢,𝐵𝑥_𝑛𝑑()𝑑≤𝑎𝑆𝑢,𝐴𝑢𝑆𝑢,𝐵𝑥_𝑛+𝑑𝑇𝑥_𝑛,𝐵𝑥_𝑛𝑑𝑇𝑥_𝑛,𝐴𝑢1+𝑑𝑆𝑢,𝐵𝑥_𝑛+𝑑𝑇𝑥_𝑛;,𝐴𝑢 (18) letting 𝑛→∞ and using (17), we have 𝑑()𝑑()𝑑()()𝑑()𝐴𝑢,𝐵𝑥≤𝑎𝑆𝑢,𝐴𝑢𝑆𝑢,𝐵𝑥+𝑑𝐵𝑥,𝐵𝑥𝐵𝑥,𝐴𝑢()().1+𝑑𝑆𝑢,𝐵𝑥+𝑑𝐵𝑥,𝐴𝑢 (19) Then, |𝑑(𝐴𝑢,𝐵𝑥)|≤0, which is contradiction. Thus, 𝐴𝑢=𝑆𝑢. Hence, 𝐴𝑢=𝑆𝑢=𝐵𝑥=𝑡.
Now, the weak compatibility of pair (𝐴,𝑆) implies that, 𝐴𝑆𝑢=𝑆𝐴𝑢 or 𝐴𝑡=𝑆𝑡.
Further, since 𝐴(𝑋)⊆𝑇(𝑋), there exists 𝑣 in 𝑋 such that 𝐴𝑢=𝑇𝑣. Thus, 𝐴𝑢=𝑆𝑢=𝑇𝑣=𝑡.
Now, we show that 𝑣 is a coincidence point of (𝐵,𝑇) that is, 𝐵𝑣=𝑇𝑣=𝑡. Put 𝑥=𝑢, 𝑦=𝑣 in contractive condition (ii), and we have 𝑑()𝑑()𝑑()()𝑑()𝐴𝑢,𝐵𝑣≤𝑎𝑆𝑢,𝐴𝑢𝑆𝑢,𝐵𝑣+𝑑𝑇𝑣,𝐵𝑣𝑇𝑣,𝐴𝑢()(),1+𝑑𝑆𝑢,𝐵𝑣+𝑑𝑇𝑣,𝐴𝑢 (20) or 𝑑()𝑑()𝑑()()𝑑()𝑡,𝐵𝑣≤𝑎𝑡,𝑡𝑡,𝐵𝑣+𝑑𝑡,𝐵𝑣𝑡,𝑡()();1+𝑑𝑡,𝐵𝑣+𝑑𝑡,𝑡 (21) whence |𝑑(𝑡,𝐵𝑣)|≤0, which is a contradiction. Thus, 𝐵𝑣=𝑡. Hence, 𝐵𝑣=𝑇𝑣=𝑡, and 𝑣 is coincidence point of 𝐵 and 𝑇.
Further, the weak compatibility of pair (𝐵,𝑇) implies that 𝐵𝑇𝑣=𝑇𝐵𝑣, or 𝐵𝑡=𝑇𝑡. Therefore, 𝑡 is a common coincidence point of 𝐴, 𝐵, 𝑆, and 𝑇.
Now, we show that 𝑡 is a common fixed point. Put 𝑥=𝑢 and 𝑦=𝑡 in contractive condition (ii), and we have 𝑑()()𝑑()𝑑()()𝑑()𝑡,𝐵𝑡=𝑑𝐴𝑢,𝐵𝑡≤𝑎𝑆𝑢,𝐴𝑢𝑆𝑢,𝐵𝑡+𝑑𝑇𝑡,𝐵𝑡𝑇𝑡,𝐴𝑢()(),1+𝑑𝑆𝑢,𝐵𝑡+𝑑𝑇𝑡,𝐴𝑢 (22) or |𝑑(𝑡,𝐵𝑡)|≤0, which is a contradiction. Thus, 𝐵𝑡=𝑡. Hence, 𝐴𝑡=𝐵𝑡=𝑆𝑡=𝑇𝑡=𝑡. The uniqueness of the common fixed point follows easily.
In a similar way, the argument that the pair (𝐴,𝑆) satisfies property (CLR_𝐴) will also give the unique common fixed point of 𝐴, 𝐵, 𝑆 and 𝑇. Hence the result follows.

Following the similar steps as in Theorem 13, we have the following result.

Theorem 14. Let (𝑋,𝑑) be a complex valued metric space and 𝐴, 𝐵, 𝑆, and 𝑇∶𝑋→𝑋 four self-mappings satisfying the following conditions: (i)𝐴(𝑋)⊆𝑇(𝑋), 𝐵(𝑋)⊆𝑆(𝑋); (ii) for all 𝑥,𝑦∈𝑋 and 0<𝑎<1, 𝑑≤𝑎𝑑()𝑑𝑑𝐴𝑥,𝐵𝑦𝑆𝑥,𝐴𝑥𝑆𝑥,𝐵𝑦+𝑑𝑇𝑦,𝐵𝑦𝑇𝑦,𝐴𝑥𝑑,𝑆𝑥,𝐵𝑦+𝑑𝑇𝑦,𝐴𝑥𝑖𝑓𝐷≠00,𝑖𝑓𝐷=0, (23)where 𝐷=𝑑(𝑆𝑥,𝐵𝑦)+𝑑(𝑇𝑦,𝐴𝑥); (iii) the pairs (𝐴,𝑆) and (𝐵,𝑇) are weakly compatible.If the pair (𝐴,𝑆) satisfy (𝐶𝐿𝑅_𝐴) property or (𝐵,𝑇) satisfies (𝐶𝐿𝑅_𝐵) property, then 𝐴, 𝐵, 𝑆, and 𝑇 have a unique common fixed point in 𝑋.

Remark 15. In this paper, we used the (E.A.) property and CLR property to claim the existence of common fixed point of some rational type contraction mappings. (E.A.) property requires the condition of closedness of subspace. However, CLR property never requires any condition on closedness of subspace, continuity of one or more mappings and containment of ranges of involved mappings. So, CLR property is an interesting auxiliary tool to claim the existence of a common fixed point.

Acknowledgment

The authors are thankful to the learned referees for the very careful reading and valuable suggestions.