Abstract

The aim of this paper is to prove common fixed point theorems for variants of weak compatible maps in a complex valued-metric space. In this paper, we generalize various known results in the literature using (CLRg) property. The concept of (CLRg) does not require a more natural condition of closeness of range.

Keywords

Weakly Compatible Maps; (CLRg) Property; Common Fixed Point

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1. Introduction

Recently, Azam et al. [1] introduced complex-valued metric space which is more general than classical metric space. Sastry et al. [2] proved that every complex-valued metric space is metrizable and hence is not real generalizations of metric spaces. But indeed it is a metric space and it is well known that complex numbers have many applications in Control theory, Fluid dynamics, Dynamic equations, Electromagnetism, Signal analysis, Quantum mechanics, Relativity, Geometry, Fractals, Analytic number theory, Algebraic number theory etc. For more detail, one can refer to [3] -[5] . The aim of this paper is to prove a common fixed point theorem for variants of weak compatible maps in a complex valued-metric space. As a consequence, we extend and generalize various known results in the literature using (CLRg) property in complex valued metric space.

2. Preliminaries

Let ℂ be the set of complex numbers and z₁, z₂ ∊ ℂ, recall a natural partial order relation ≾ on ℂ as follows:

if and only if and

if and only if and

Definition 2.1. [1] . Let X be a nonempty set such that the map d: X × X → ℂ satisfies the following conditions:

(C1) for all and if and only if x = y;

(C2) for all

(C3) for all

Then d is called a complex-valued metric on X, and (X, d) is called a complex-valued metric space.

Example 2.1. [1] Define complex-valued metric d: X × X → ℂ by. Then (X, d) is a complex-valued metric space.

Definition 2.2. [1] . Let (X, d) complex -valued metric space and x ∈ X. Then sequence {x_n} sequence is i) convergent if for every there is a natural number N such that, for all We write it as

ii) a Cauchy sequence, if for every there is a natural number N such that, for all

Definition 2.3. [5] [6] . A pair of self-maps f and g of a complex-valued metric space are weakly compatible if fgz = gfz for all z ∊ X at which fz = gz.

Example 2.2. [6] . Define complex-valued metric defined by

where a is any real constant. Then (X, d) is a complex-valued metric space. Suppose self maps f and g be defined as:

and

Clearly, f and g are weakly compatible self maps.

In 2011, Sintunavarat and Kumam [7] introduced a new property called as “common limit in the range of g property” i.e., (CLR_g) property, defined as:

Definition 2.4. A pair (f, g) of self-mappings is said to be satisfy the common limit in the range of g property if there exists a sequence in X such that for some

3. Main Results

Definition 3.1. Let (X, d) be a complex valued metric space and (f, g) be a pair of self mappings on X and Let us consider the following sets:

and define the following conditions:

A) For arbitrary there exists

such that

;

B) For arbitrary there exists

such that

;

C) For arbitrary there exists

such that

.

Conditions A), B) and C) are called strict contractive conditions.

Theorem 3.1. Let f and g be two weakly compatible self mappings of a complex valued metric space (X, d) such that

(3.1) f, g satisfy (CLRg) property;

(3.2) for all there exists

such that

.

Then f and g have a unique common fixed point in X.

Proof. Since f and g satisfy the (CLRg) property, there exists a sequence {x_n} in X such that for some x X.

We first show that fx = gx. Suppose not, i.e., fx ≠ gx.

From (3.2),

(3.3)

where

Three cases arises:

i) If

then (3.3) implies

Taking limit as n→∞,

which gives, contradiction.

ii) If

then (3.3) implies,

.

Taking limit as n→∞,

i.e., which gives, , a contradiction.

iii) If

then (3.3) gives,

Making limit as n→∞,

i.e., which gives, , a contradiction.

Hence, from all three cases, gx = fx.

Now let z = fx = gx. Since f and g are weakly compatible mappings fgx = gfx which implies that fz = fgx = gfx = gz.

We claim that fz = z. Let, if possible, fz ≠ z.

Now

(3.4)

where

Two cases arises:

i) If, then (3.4) gives,

which gives, , a contradiction.

ii) If, then (3.4) gives,

which gives, a contradiction.

Hence, from two cases, it is clear that fz = z = gz.

Hence z is a common fixed point of f and g.

For uniqueness, suppose that w is another common fixed point of f and g.

We shall prove that z = w. Let, if possible, z ≠ w.

Then

(3.5)

where

Again, two possible cases i) If, then by (3.5), we have

which gives, , a contradiction.

ii) If, then by (3.5), we have

which gives, , a contradiction.

Hence, z = w.

So, we can say that f and g have a unique common fixed point.

Remark 3.1. Theorem 3.1 also holds true if is replaced by.

Definition 3.2. Let (X, d) be a complex valued metric space, and let f, g: X→X. Then f is called a g-quasicontraction, if for some constant and for every x, yX, there exists

such that

Theorem 3.2. Let f and g be two weakly compatible self mappings of a complex valued metric space (X, d) such that

(3.6) f is a g-quasi-contraction;

(3.7) f and g satisfy (CLRg) property.

Then f and g have a unique common fixed point.

Proof. Since f and g satisfy the (CLRg) property, there exists a sequence {x_n} in X such that

We first claim that fx = gx. Suppose not. Since, f is a g-quasi-contraction, therefore

(3.8)

for some

Following five cases arises:

i) If then by (3.8), we have

taking limit as n→∞,we have

which gives, , a contradiction.

ii) If then by (3.8), we have

taking limit as n→∞,we have

which gives, , a contradiction.

iii) If then by (3.8), we have

taking limit as n→∞, we have

which gives, , a contradiction.

iv) If then by (3.8), we have

taking limit as n→∞,we have

which gives, , a contradiction.

v) If then by (3.8), we have

taking limit as n→∞,we have

which gives, , a contradiction.

Thus from all fives possible cases, gx = fx.

Now, let z = fx = gx. Since f and g are weakly compatible mappings fgx = gfx which implies that fz = fgx = gfx = gz.

We claim that fz = z. Suppose not, then by (3.6), we have

(3.9)

where

Two cases arises:

i) If then by (3.9), we have

which gives, a contradiction.

ii) If, then by (3.9), we have

which gives, a contradiction.

Thus, fz = z = gz.

Hence, z is a common fixed point of f and g.

For uniqueness, suppose that w is another common fixed point of f and g in X.

By (3.6), we have

(3.10)

where

Two possible cases arises:

i) If, then by (3.9), we have

which gives, a contradiction.

ii) If then by (3.9), we have

which gives, a contradiction.

Hence, z = w i.e., f and g have a unique common fixed point.

NOTES

^*Corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

References