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.

Weakly Compatible Maps; (CLRg) Property; Common Fixed Point
1. Introduction

Recently, Azam et al.  introduced complex-valued metric space which is more general than classical metric space. Sastry et al.  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  - . 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.  . 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.  Define complex-valued metric d: X × X → ℂ by. Then (X, d) is a complex-valued metric space.

Definition 2.2.  . Let (X, d) complex -valued metric space and x ∈ X. Then sequence {xn} 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.   . 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.  . 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  introduced a new property called as “common limit in the range of g property” i.e., (CLRg) 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 {xn} in X such that for some x X.

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

From (3.2),

where  Three cases arises:

i) If then (3.3) implies Taking limit as n→∞, 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

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

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 {xn} in X such that We first claim that fx = gx. Suppose not. Since, f is a g-quasi-contraction, therefore

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

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

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.

NOTESReferencesAzam, A., Fisher, B. and Khan, M. (2011) Common Fixed Point Theorems in Complex Valued Metric Spaces. Numerical Functional Analysis and Optimization, 32, 243-253. http://dx.doi.org/10.1080/01630563.2011.533046Sastry K.P.R., Naidu G.A. and Bekeshie, T. ,et al. (2012)Metrizability of Complex Valued Metric Spaces and Remarks on Fixed Point Theorems in Complex Valued Metric Spaces International Journal of Mathematical Archive 3, 2686-2690.Manro S. ,et al. (2013)Some Common Fixed Point Theorems in Complex Valued Metric Spaces Using Implicit Relation International Journal of Analysis and Applications 2, 62-70.Sintunavarat W. and Kumam, P. ,et al. (2012)Generalized Common Fixed Point Theorems in Complex Valued Metric Spaces with Applications Journal of Inequalities and Applications 1, 1-12.Verma R.K. and Pathak, H.K. ,et al. (2012)Common Fixed Point Theorems Using Property (E.A) in Complex Valued Metric Spaces Thai Journal of Mathematics 11, 347-355.Jungck G. and Rhoades, B.E. ,et al. (1998)Fixed Point for Set Valued Functions without Continuous Indian Journal of Pure and Applied Mathematics 29, 227-238.Sintunavarat, W. and Kumam, P. (2011) Common Fixed Point Theorems for a Pair of Weakly Compatible Mappings in Fuzzy Metric Spaces. Journal of Applied Mathematics, 2011, Article ID: 637958.