Complex Valued -Metric Spaces and Common Fixed Point Theorems under Rational Contractions

Journal of Complex Analysis, Jun 2016

The aim of this paper is to prove the existence and uniqueness of a common fixed point for a pair of mappings satisfying certain rational contraction conditions in complex valued -metric space. The obtained results generalize and extend some of the well-known results in the literature.

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Complex Valued -Metric Spaces and Common Fixed Point Theorems under Rational Contractions

Complex Valued -Metric Spaces and Common Fixed Point Theorems under Rational Contractions Anil Kumar Dubey Department of Mathematics, Bhilai Institute of Technology, Bhilai House, Durg, Chhattisgarh 491001, India Received 16 November 2015; Revised 2 March 2016; Accepted 10 May 2016 Academic Editor: Arcadii Z. Grinshpan Copyright © 2016 Anil Kumar Dubey. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The aim of this paper is to prove the existence and uniqueness of a common fixed point for a pair of mappings satisfying certain rational contraction conditions in complex valued -metric space. The obtained results generalize and extend some of the well-known results in the literature. 1. Introduction Banach contraction principle in [1] gives appropriate and simple conditions to establish the existence and uniqueness of a solution of an operator equation . Later, a number of papers were devoted to the improvement and generalization of that result. Most of these results deal with the generalizations of the different contractive conditions in metric spaces. There have been a number of generalizations of metric spaces such as vector valued metric spaces, -metric spaces, pseudometric spaces, fuzzy metric spaces, -metric spaces, cone metric spaces, and modular metric spaces. Bakhtin [2] introduced the notion of -metric space which is a generalized form of metric spaces. In [3], Czerwik proved the contraction mapping principle in -metric spaces. Subsequently, many authors obtained fixed point results for single valued and multivalued operators in -metric spaces. A new space called the complex valued metric space which is more general than the well-known metric space has been introduced by Azam et al. [4]. They proved some fixed point results for a pair of mappings for contraction condition satisfying a rational expression. Azam et al. [4] improved the Banach contraction principle in the context of complex valued metric space involving rational inequality which could not be meaningful in cone metric spaces. Several authors studied many common fixed point theorems on complex valued metric spaces (see [5–9]). The concept of complex valued -metric spaces was introduced in 2013 by Rao et al. [10]. In sequel, Mukheimer [11] proved some common fixed point theorems in complex valued -metric spaces. In this paper, we continue the study of fixed point theorems in complex valued -metric spaces. The obtained results are generalizations of recent results proved by Dubey et al. [12, 13], Nashine et al. [5, 6], and Rao et al. [10]. 2. Preliminaries Let be the set of complex numbers and . Define a partial order on as follows: if and only if , . Thus if one of the following holds:(1) and ;(2) and ;(3) and ;(4) and .We will write if and one of (2), (3), and (4) is satisfied; also we will write if only (4) is satisfied. It follows that(i) implies ;(ii) and imply ;(iii) implies ;(iv)if , and , then for all .The following definition is recently introduced by Rao et al. [10]. Definition 1. Let be a nonempty set and let be a given real number. A function is called a complex valued -metric on if for all the following conditions are satisfied:(i) and if and only if ;(ii);(iii).The pair is called a complex valued -metric space. Example 2 (see [10]). If , define the mapping by , for all . Then is complex valued -metric space with . Definition 3 (see [10]). Let be a complex valued -metric space.(i)A point is called interior point of a set whenever there exists such that .(ii)A point is called a limit point of a set whenever for every , .(iii)A subset is called open whenever each element of is an interior point of .(iv)A subset is called closed whenever each element of belongs to .(v)A subbasis for a Hausdorff topology on is a family . Definition 4 (see [10]). Let be a complex valued -metric space and let be a sequence in and .(i)If for every , with , there is such that for all , , then is said to be convergent, converges to , and is the limit point of . We denote this by or .(ii)If for every , with , there is such that for all , , where , then is said to be Cauchy sequence.(iii)If every Cauchy sequence in is convergent, then is said to be a complete complex valued -metric space. Lemma 5 (see [10]). Let be a complex valued -metric space and let be a sequence in . Then converges to if and only if as . Lemma 6 (see [10]). Let be a complex valued -metric space and let be a sequence in . Then is a Cauchy sequence if and only if as , where . 3. Main Results Theorem 7. Let be a complete complex valued -metric space with the coefficient and let be mappings satisfyingfor all , such that , , where , are nonnegative reals with or if . Then and have a unique common fixed point. Proof. For any arbitrary (...truncated)


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Anil Kumar Dubey. Complex Valued -Metric Spaces and Common Fixed Point Theorems under Rational Contractions, Journal of Complex Analysis, 2016, 2016, DOI: 10.1155/2016/9786063