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

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 is a complete metric space and is a contraction mapping (i.e., for all , where is a nonnegative number such that ), then 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 be the set of complex numbers, and let . Define a partial order on as follows: It follows that if one of the following conditions is satisfied:(1) , and ;(2) , and ;(3) , and ;(4) , and .In particular, we will write if and one of (1), (2), and (3) is satisfied, and we will write if only (3) is satisfied. Note. We obtained that the following statements hold:(i) and ,？？for all ;(ii) ;(iii) and imply . Definition 1. Let be a nonempty set. Suppose that the mapping satisfies the following conditions:(i) for all and if and only if ;(ii) for all ;(iii) for all .Then, is called a complex valued metric on , and is called a complex valued metric space. Example 2. Let . Define a mapping by where . Then, is a complex valued metric space. A point is called