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Common Fixed Point Theorems for a Rational Inequality in Complex Valued Metric Spaces

DOI: 10.1155/2013/942058

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Abstract:

We prove a common fixed point theorem for a pair of mappings. Also, we prove a common fixed point theorem for pairs of self-mappings along with weakly commuting property. 1. Introduction Azam et al. [1] introduced the notion of complex valued metric spaces and established some fixed point theorems for the mappings satisfying a rational inequality. The definition of a cone metric space banks on the underlying Banach space which is not a division Ring. The idea of rational expressions is not meaningful in cone metric spaces, and therefore many results of analysis cannot be generalized to cone metric spaces. The complex valued metric spaces form a special class of cone metric space, and we can study improvements of host results of analysis involving divisions. A complex number is an ordered pair of real numbers, whose first coordinate is called and second coordinate is called . Let be the set of complex numbers and , . Define a partial order on as follows: if and only if and ; that is, , if one of the following holds:(C1) and ;(C2) and ;(C3) and ;(C4) and .In particular, we will write if and one of (C2), (C3), and (C4) is satisfied, and we will write if only (C4) is satisfied. Remark 1. We note that the following statements hold:(i) , and ?? ,(ii) ,(iii) and .Azam et al. [1] defined the complex valued metric space ( , ) as follows. Definition 2. 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 3. Let . Define the mapping by Then, is a complex valued metric space. Definition 4. Let be a complex valued metric space. A sequence in is said to be (i)convergent to , if for every with there is such that, for all , . We denote this by as or ;(ii)Cauchy, if for every with there is such that for all , , where ;(iii)complete, if every Cauchy sequence in converges in . Lemma 5. Let be a complex valued metric space, and let be a sequence in . Then, converges to if and only if as . Lemma 6. Let be a complex valued metric space, and let be a sequence in . Then, is a Cauchy sequence if and only if as , where . In 1982, Sessa [2] introduced the notion of weak commutativity as follows. Definition 7. Two self-maps and of a metric space are said to be weakly commuting if , for all in . In a similar mode, we introduce the notion of weak commutativity in complex valued metric spaces as follows. Definition 8. Two self maps and of a complex valued metric space are said

References

[1]  A. Azam, B. Fisher, and M. Khan, “Common fixed point theorems in complex valued metric spaces,” Numerical Functional Analysis and Optimization, vol. 32, no. 3, pp. 243–253, 2011.
[2]  S. Sessa, “On a weak commutativity condition of mappings in fixed point considerations,” Publications de l'Institut Mathématique, vol. 32, no. 46, pp. 149–153, 1982.

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