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 Open Access
A note on fixed point results in complexvalued metric spaces
 Saleh A AlMezel^{1},
 Hamed H Alsulami^{1},
 Erdal Karapınar^{1, 2}Email author and
 Farshid Khojasteh^{3}
https://doi.org/10.1186/s1366001505506
© AlMezel et al.; licensee Springer. 2015
 Received: 1 October 2014
 Accepted: 6 January 2015
 Published: 28 January 2015
Abstract
In this paper, we prove that the fixed point results in the context of complexvalued metric spaces can be obtained as a consequence of corresponding existing results in the literature in the setting of associative metric spaces. In particular, we deduce that any complex metric space is a special case of cone metric spaces with a normal cone.
Keywords
 contraction mapping
 cone metric spaces
 quaternionvalued metric space
 complexvalued metric space
 fixed point
MSC
 46T99
 47H10
 54H25
 54C30
1 Introduction and preliminaries
The notion of complexvalued metric spaces was introduced by Azam et al. [1], as a generalization of metric spaces to investigate the existence and uniqueness of fixed point results for mappings satisfying a rational inequalities. Following this paper, a number of authors have reported several fixed point results for various mapping satisfying a rational inequalities in the context of complexvalued metric spaces; see e.g. [1–3] and the related references therein.
The aim of this short note is to emphasize that the complexvalued metric space is an example of the cone metric space that was introduced in [4–6] under the name Kmetric and Knormed spaces and reintroduced by Huang and Zhang [7]. It is well known that if the cone is normal then the corresponding cone metric associates a metric. There are some other approaches to induce a metric from cone metric; see e.g. [8–16]. As a consequence of these observations, we notice that fixed point results in the context of complete complexvalued metric spaces can be deduced the corresponding fixed point results on (associative) complete metric space. Based on the discussion above, for our purpose, we first prove the existence of common fixed point theorems for multivalued mapping in the context of complete metric space. Then we derive the main results of the recent paper of Ahmad et al. [2] as corollaries of our results.
For the sake of completeness we recollect some basic definitions and fundamentals results on the topic in the literature. We mainly follow the notions and notations of Azam et al. in [1].
 (h_{1}):

\(\operatorname{Re}(z_{1})=\operatorname{Re}(z_{2})\); \(\operatorname{Im}(z_{1})<\operatorname{Im}(z_{2})\),
 (h_{2}):

\(\operatorname{Re}(z_{1})<\operatorname{Re}(z_{2})\); \(\operatorname{Im}(z_{1})=\operatorname{Im}(z_{2})\),
 (h_{3}):

\(\operatorname{Re}(z_{1})<\operatorname{Re}(z_{2})\); \(\operatorname{Im}(z_{1})<\operatorname{Im}(z_{2})\),
 (h_{4}):

\(\operatorname{Re}(z_{1})=\operatorname{Re}(z_{2})\); \(\operatorname{Im}(z_{1})=\operatorname{Im}(z_{2})\).
Definition 1
[1]
 (b_{1}):

\(0 \precsim d(x,y)\) for all \(x,y\in X\) and \(d(x,y) = 0\), if and only if \(x = y\),
 (b_{2}):

\(d(x,y) = d(y,x)\), for all \(x,y\in X\),
 (b_{3}):

\(d(x,y) \precsim d(x,z) + d(y,z)\), for all \(x,y,z\in X\).
Let \(\{x_{n}\}\) be a sequence in X and \(x\in X\). If for every \(c\in \mathbb{C}\), with \(0\prec c\) there is \(n_{0}\in \mathbb{N}\) such that for all \(n > n_{0}\), \(d(x_{n},x) \prec c\), then \(\{x_{n}\}\) is said to be convergent, \(\{x_{n}\}\) converges to x and x is the limit point of \(\{ x_{n}\}\). We denote this by \(\lim_{n}x_{n} = x\), or \(x_{n}\to x\), as \(n\to\infty\). If for every \(c\in \mathbb{C}\) with \(0\prec c\) there is \(n_{0}\in \mathbb{N}\) such that for all \(n > n_{0}\), \(d(x_{n}, x_{n+m})\prec c\), then \(\{x_{n}\}\) is called a Cauchy sequence in \((X, d)\). If every Cauchy sequence is convergent in \((X, d)\), then \((X, d)\) is called a complete complexvalued metric space.
Lemma 2
[1, Lemma 2, Azam et al.]
Let \((X, d)\) be a complexvalued metric space and let \(\{x_{n}\}\) be a sequence in X. Then \(\{x_{n}\}\) converges to x if and only if \(d(x_{n},x)\to0\) as \(n\to\infty\).
Lemma 3
[1, Lemma 3, Azam et al.]
Let \((X, d)\) be a complexvalued metric space and let \(\{x_{n}\}\) be a sequence in X. Then \(\{x_{n}\}\) is a Cauchy sequence if and only if \(d(x_{n},x_{n+m})\to0\) as \(n\to\infty\).
 (a_{1}):

P is closed, nonempty, and \(P \neq\{0\}\),
 (a_{2}):

\(a, b \in \mathbb{R}\), \(a,b\geq0\), and \(x, y \in P\) imply that \(ax+by\in P\),
 (a_{3}):

\(x \in P\) and \(x \in P\) imply that \(x = 0\).
The cone P is called normal, if there exist a number \(K\geq1\) such that \(0 \leq x \leq y\) implies \(\x\ \leq K\y\\), for all \(x, y \in E\). The least positive number satisfying this, called the normal constant [7, 17].
In this paper, E denotes a real Banach space, P denotes a cone in E with \(\operatorname{int} P\neq\emptyset\), and ≤ denotes a partial ordering with respect to P. For more details on the cone metric, we refer e.g. to [7, 17, 18].
Definition 4
[7]
 (b_{1}):

\(d(x,y) \geq0\) for all \(x,y\in X\) and \(d(x,y) = 0\), if and only if \(x = y\),
 (b_{2}):

\(d(x,y) = d(y,x)\), for all \(x,y\in X\),
 (b_{3}):

\(d(x,y) \leq d(x,z) + d(y,z)\), for all \(x,y,z\in X\).
The following definitions and lemmas have been taken from [7, 18].
Definition 5
Let \((X,d)\) be a cone metric space and \(\{x_{n}\}_{n\in \mathbb{N}}\) be a sequence in X and \(x\in X\). If for all \(c \in E\) with \(0\ll c\), there is \(n_{0} \in \mathbb{N}\) such that for all \(n > n_{0}\), \(d(x_{n}, x_{0})\ll c\), then \(\{x_{n}\}_{n\in \mathbb{N}}\) is said to be convergent and \(\{x_{n}\}_{n\in \mathbb{N}}\) converges to x and x is the limit of \(\{x_{n}\}_{n\in \mathbb{N}}\).
Definition 6
Let \((X,d)\) be a cone metric space and \(\{x_{n}\}_{n\in \mathbb{N}}\) be a sequence in X. If for all \(c \in E\) with \(0\ll c\), there is \(n_{0} \in \mathbb{N}\) such that for all \(m,n > n_{0}\), \(d(x_{n},x_{m})\ll c\), then \(\{x_{n}\}_{n\in \mathbb{N}}\) is called a Cauchy sequence in X.
Definition 7
Let \((X,d)\) be a cone metric space. If every Cauchy sequence is convergent in X, then X is called a complete cone metric space.
Definition 8
Let \((X,d)\) be a cone metric space. A selfmap T on X is said to be continuous, if \(\lim_{n\rightarrow\infty}x_{n}=x\) implies \(\lim_{n\rightarrow\infty}T(x_{n})=T(x)\) for all sequences \(\{x_{n}\}_{n\in \mathbb{N}}\) in X.
Lemma 9
Lemma 10
Let \((X,d)\) be a normal cone metric space and \(\{x_{n}\}_{n\in \mathbb{N}}\) be a sequence in X. If \(\{x_{n}\}_{n\in \mathbb{N}}\) is convergent, then it is a Cauchy sequence.
Lemma 11
Let \((X,d)\) be a cone metric space and P be a cone in E. Let \(\{x_{n}\}_{n\in \mathbb{N}}\) be a sequence in X. Then \(\{x_{n}\}_{n\in \mathbb{N}}\) is a Cauchy sequence, if and only if \(\lim_{m,n\rightarrow\infty }d(x_{m},x_{n}) = 0\).
2 Main result
Lemma 12
\(\mathcal{P}_{\mathbb{C}}\) is a normal cone in a real Banach space \((\mathbb{C},\cdot)\).
Proof
Precisely, \(\mathcal{P}_{\mathbb{C}}\) is nonempty, closed and \(\mathcal {P}_{\mathbb{C}}\neq(\mathbf{0}_{\mathbb{C}})\). Also for all \(\alpha,\beta\in \mathbb{R}^{+}\) and \(p,q\in\mathcal{P}_{\mathbb{C}}\) we have \(\alpha p+\beta q\in\mathcal{P}_{\mathbb{C}}\) and \(\mathcal {P}_{\mathbb{C}}\cap(\mathcal{P}_{\mathbb{C}})=(\mathbf{0}_{\mathbb{C}})\). Also the normality of \(\mathcal{P}_{\mathbb{C}}\) is apparent. □
Lemma 13
Any complexvalued metric space \((X,d_{\mathbb{C}})\) is a cone metric space.
Proof
Lemma 14
The partial ordered ⫅ defined in Lemma 13 is equivalent to ≲.
Proof
Assume \(p=p_{1}+ip_{2}\) and \(q=q_{1}+iq_{2}\). \(p \subseteqq q\), if and only if \(qp\in\mathcal{P}_{\mathbb{C}}\), if and only if \(q_{1}p_{1}\geq0\), \(q_{2}p_{2}\geq0\). In other words, \(\operatorname{Re}(p)\leq \operatorname{Re}(q)\), \(\operatorname{Im}(p)\leq \operatorname{Im}(q)\), if and only if \(p\lesssim q\). □
Lemma 15
A sequence \(\{x_{n}\}\) in \((X,d_{\mathbb{C}})\) is convergent according to the concept of complexvalued metric space if and only if \(\{x_{n}\}\) is convergent according to the concept of a cone metric space.
Proof
Let \(\{x_{n}\}\) be sequence in X. Sequence \(\{x_{n}\}\) converges to \(x\in X\) according to the concept of complexvalued metric space if and only if \(d(x_{n},x)\to0\) as \(n\to\infty\) if and only if \(\{x_{n}\}\) converges to x according to the concept of a cone metric space by considering ℂ as the Banach space endowed with the cone \(\mathcal{P}_{\mathbb{C}}\) (see Lemma 9). □
In particular, if \(x_{0}\in X\), then \(D ( x_{0},B ) :=D ( \{ x_{0} \} ,B ) \).
Definition 16
Definition 17
Throughout the paper, we assume that \(\{a,b,c,d,e\} \subset[0,1)\).
The following is the fundamental theorem of this paper.
Theorem 18
Proof
Theorem 19
Proof
Theorem 20
Proof
Corollary 21
Proof
Theorem 22
Proof
Taking \(\rho(x,y)=d(x,y)\), \((X,\rho)\) is a complete metric space and applying Theorem 20 we conclude that f and g have a common fixed point. As in the proof of Corollary 21, uniqueness of the common fixed point of f and g can be derived easily by reductio ad absurdum. □
The following results, the main results of Ahmad et al. [2], can be considered as a consequence of Theorem 20.
Theorem 23
Proof
Remark 24
By Theorem 20, one can derive the other results in [2] but we prefer not to list these here.
In what follows we state a theorem that is just a variation of Theorem 20.
Corollary 25
Proof
Declarations
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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