1 Introduction and preliminaries
The notion of complex-valued 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 complex-valued metric spaces; see
e.g. [
1‐
3] and the related references therein.
The aim of this short note is to emphasize that the complex-valued metric space is an example of the cone metric space that was introduced in [
4‐
6] under the name
K-metric and
K-normed spaces and re-introduced 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 complex-valued 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 multi-valued 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].
Let ℂ be the set of complex numbers and
\(z_{1},z_{2}\in \mathbb{C}\). Define a partial order ≾ on ℂ as follows:
$$z_{1} \precsim z_{2} \quad\mbox{if and only if} \quad \operatorname{Re}(z_{1}) \leq \operatorname{Re}(z_{2}), \operatorname{Im}(z_{1}) \leq \operatorname{Im}(z_{2}). $$
It follows that
$$z_{1} \precsim z_{2} $$
if one of the following conditions is satisfied:
(h1)
\(\operatorname{Re}(z_{1})=\operatorname{Re}(z_{2})\); \(\operatorname{Im}(z_{1})<\operatorname{Im}(z_{2})\),
(h2)
\(\operatorname{Re}(z_{1})<\operatorname{Re}(z_{2})\); \(\operatorname{Im}(z_{1})=\operatorname{Im}(z_{2})\),
(h3)
\(\operatorname{Re}(z_{1})<\operatorname{Re}(z_{2})\); \(\operatorname{Im}(z_{1})<\operatorname{Im}(z_{2})\),
(h4)
\(\operatorname{Re}(z_{1})=\operatorname{Re}(z_{2})\); \(\operatorname{Im}(z_{1})=\operatorname{Im}(z_{2})\).
In particular, we shall write
\(z_{1} \precnsim z_{2}\) if
\(z_{1} \neq z_{2}\) and one of (h
1), (h
2), and (h
3) is satisfied. Further we write
\(z_{1} \prec z_{2}\) if only (h
3) is satisfied. Note that
$$0\precsim z_{1} \precnsim z_{2} \quad \Longrightarrow\quad |z_{1}|<|z_{2}|, $$
where
\(|\cdot|\) represents the modulus or magnitude of
z, and
$$z_{1}\precsim z_{2},\qquad z_{2} \prec z_{3} \quad \Longrightarrow\quad z_{1} \prec z_{3}. $$
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 complex-valued metric space.
Let
E be a real Banach space. A subset
P of
E is called a cone, if the followings hold:
(a1)
P is closed, nonempty, and \(P \neq\{0\}\),
(a2)
\(a, b \in \mathbb{R}\), \(a,b\geq0\), and \(x, y \in P\) imply that \(ax+by\in P\),
(a3)
\(x \in P\) and \(-x \in P\) imply that \(x = 0\).
Given a cone
\(P\subset E\), we define a partial ordering ≤ with respect to
P by
\(x \leq y\), if
\(y - x \in P\). We write
\(x < y\) to indicate that
\(x \leq y\) but
\(x\neq y\), while
\(x \ll y\) stands for
\(y - x \in \operatorname{int} P\), where int
P denotes the interior of
P.
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].
The following definitions and lemmas have been taken from [
7,
18].
2 Main result
In this section, we represent our main results. First of all, we represent some simple observations. Let
\((X,d_{\mathbb{C}})\) be a complex-valued metric space. Now, we define the following set:
$$\mathcal{P}_{\mathbb{C}}=\{x+iy:x\geq0,y\geq0\}. $$
It is apparent that
\(\mathcal{P}_{\mathbb{C}}\subset \mathbb{C}\). Note that
\((\mathbb{C},|\cdot|)\) is a real Banach space.
Finally, we recall some fundamental definition for multi-valued mappings and related metric spaces. Let
\((X,d)\) be a metric space. Let
\(\mathcal{P}(X)= \{ Y\mid Y\subset X \}\) and
\(P(X):= \{ Y\in\mathcal{P} ( X ) \mid Y\neq\emptyset \}\). Let us define
the gap functional
\(D:P(X)\times P(X)\rightarrow \mathbb{R}_{+}\cup\{+\infty\}\), as
$$ D(A,B)=\operatorname{inf} \bigl\{ d(a,b) \mid a\in A, b\in B \bigr\} . $$
In particular, if \(x_{0}\in X\), then \(D ( x_{0},B ) :=D ( \{ x_{0} \} ,B ) \).
We denote by
\(\mathcal{C}(X)\) the family of all nonempty closed subsets of
X and
\(\mathcal{CB}(X)\) the family of all nonempty closed and bounded subsets of
X. A function
\(\mathcal{H}:\mathcal{CB}(X)\times \mathcal{CB}(X)\rightarrow{}[0,\infty)\) defined by
$$\mathcal{H}(A,B)= \max \Bigl\{ \sup_{x\in B}D(x,A)\textit{,}\sup _{x\in A}D(x,B) \Bigr\} $$
is said to be the Hausdorff metric on
\(\mathcal{CB}(X)\) induced by the metric
d on
X where
\(D(x,A)=\inf\{d(x,y):y\in A\}\) for each
\(A\in \mathcal{CB}(X)\). A point
v in
X is a fixed point of a map
T if
\(v=Tv\) (when
\(T:X\rightarrow X\) is a single-valued map) or
\(v\in Tv\) (when
\(T:X\rightarrow\mathcal{P}(X)\) is a multi-valued map). We say that
T has an endpoint if there exists
\(v\in X\) such that
\(Tv=\{v\}\). The set of fixed points of
T is denoted by
\(\mathcal{F}(T)\) and the set of common fixed points of two multi-valued mappings
T,
S is denoted by
\(\mathcal{F}(T,S)\).
Throughout the paper, we assume that \(\{a,b,c,d,e\} \subset[0,1)\).
The following is the fundamental theorem of this paper.
The following results, the main results of Ahmad
et al. [
2], can be considered as a consequence of Theorem
20.
In what follows we state a theorem that is just a variation of Theorem
20.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.