Hindawi Publishing Corporation
Journal of Complex Systems
Volume 2013, Article ID 942058, 7 pages
http://dx.doi.org/10.1155/2013/942058
Research Article
Common Fixed Point Theorems for a Rational Inequality in
Complex Valued Metric Spaces
Pankaj Kumar,
1
Manoj Kumar,
1
and Sanjay Kumar
2
1
Department of Mathematics, Guru Jambheshwar University of Science and Technology, Hisar 125001, India
2
Department of Mathematics, Deenbandhu Chhotu Ram University of Science and Technology, Murthal 131039, India
Correspondence should be addressed to Sanjay Kumar; sanjuciet@redifmail.com
Received 30 April 2013; Accepted 10 September 2013
Academic Editor: Fuwen Yang
Copyright © 2013 Pankaj Kumar et al. Tis 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.
We prove a common fxed point theorem for a pair of mappings. Also, we prove a common fxed 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 fxed point theorems for
the mappings satisfying a rational inequality. Te defnition
of a cone metric space banks on the underlying Banach space
which is not a division Ring. Te idea of rational expressions
is not meaningful in cone metric spaces, and therefore many
results of analysis cannot be generalized to cone metric
spaces. Te 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 ∈ C is an ordered pair of real
numbers, whose frst coordinate is called Re() and second
coordinate is called Im().
Let C be the set of complex numbers and
1
,
2
∈ C.
Defne a partial order ≾ on C as follows:
1
≾
2
if and only if
Re(
1
)≤ Re(
2
) and Im(
1
)≤ Im(
2
); that is,
1
≾
2
, if one
of the following holds:
(C1) Re(
1
)= Re(
2
) and Im(
1
)= Im(
2
);
(C2) Re(
1
)< Re(
2
) and Im(
1
)= Im(
2
);
(C3) Re(
1
)= Re(
2
) and Im(
1
)< Im(
2
);
(C4) Re(
1
)< Re(
2
) and Im(
1
)< Im(
2
).
In particular, we will write
1
⋨
2
if
1
̸ =
2
and one of (C2),
(C3), and (C4) is satisfed, and we will write
1
≺
2
if only
(C4) is satisfed.
Remark 1. We note that the following statements hold:
(i) , ∈ R and ≤ ⇒ ≾ ∀ ∈ C,
(ii) 0≾
1
⋨
2
⇒ |
1
| < |
2
|,
(iii)
1
≾
2
and
2
≺
3
⇒
1
≺
3
.
Azam et al. [1] defned the complex valued metric space (,
) as follows.
Defnition 2. Let be a nonempty set. Suppose that the
mapping :×→ C satisfes the following conditions:
(i) 0 ≾ (, ), for all , ∈ and (, ) = 0 if and
only if =;
(ii) (, ) = (, ) for all , ∈;
(iii) (, ) ≾ (, ) + (, ), for all , , ∈.
Ten, is called a complex valued metric on , and (, ) is
called a complex valued metric space.
Example 3. Let = C. Defne the mapping :×→ C
by
(, ) = i
−
, ∀, ∈ . (1)
Ten, (, ) is a complex valued metric space.
Defnition 4. Let (, ) be a complex valued metric space. A
sequence {
} in is said to be