Arti Mishra et al. International Journal of Recent Research Aspects ISSN: 2349-7688, Vol. 2,
Issue 4, December 2015, pp.88-91
© 2014 IJRRA All Rights Reserved page - 88-
Extended Common fixed point
theorem for multi-valued mappings in
complex valued Metric space
Arti Mishra
1
and Nisha Sharma
2
1
Department of Mathematics, Manav Rachna International University, Faridabad, Haryana
2
Department of Mathematics, Pt. J.L.N Gov. College Faridabad, Haryana
Abstract: In this paper we are going to prove common fixed point theorem for weak compatible map. We
extend the result of (sitthikul and Saejung fixed point Theory and Application 2012:189).the main results
announced by Sintunavarat and Kumam (j.inequal. Appl :84,2012).some of the concepts of sequence of
function are already given in 2008, Dutta et. al. [7], Rouzkard and Imdad (Comput.
Math.appl.,2012,doi:10.1016/j.camwa.2012.02.063). The results announced by sitthikul and Saejung fixed
point Theory and Application 2012:189 is mainly improved in this paper.
Keywords: complex valued metric space; multi valued mapping; weak compatible mapping, common fixed
point
I. INTRODUCTION
Throughout the article denoted by ℂ is the set of all
complex numbers ℕ for set of all natural numbers and
ℝ for set of all real numbers.(X,d)(x for short),is a
metric space with a metric d.
It is well known that in the literature, there are so many
extensions of Banach contraction principle[1],which
states that every self-mapping t defined on a complete
metric space (x,d) satisfying ,For all,x,y∈X
d(Tx,Ty)≤kd(x,y),where k∈[0,1) has unique fixed point
for every x
0
∈X a sequence {T
n
x
0
}
nє
ℕ is convergent to
the fixed point. But the complex valued metric space is
a generalization of the classical metric space,
introduced by Azam,Fisher and Khan (see [2])
II. PRELIMINARIES
Let us recall a natural relation on ℂ,for z
1
,z
2
∈ℂ, define
a partial order ≾ on ℂ as follows;
z
1
≾z
2
iff Re(z1) ≤Re(z2), Im(z
1
) ≤Im(z
2
)
it follows that
z
1
≾z
2
if one of the following conditions is satisfied:
i. Re(z1)=Re(z2), Im(z
1
)<Im(z
2
)
ii. Re(z1)<Re(z2), Im(z
1
)=Im(z
2
)
iii. Re(z1)<Re(z2), Im(z
1
)<Im(z
2
)
iv. Re(z1)=Re(z2), Im(z
1
)=Im(z
2
)
In particular, we will write z
1
⪱z
2
if z
1
≠z
2
and one the
above conditions is not satisfied and we will write
z
1
≺z
2
if only iii is satisfied. Note that
0≾ z
1
⪱z
2
⇨|z
1
|<|z
2
|, z
1
≾z
2 ,
z
1
≺z
2
⇨ z
1
≺z
3
Definition 1let X be a nonempty set. A mapping
dμXxX→ ℂ is called a complex valued metric on X if
the following conditions are satisfied:
(CM1) 0 ≾ d(x,y) for all x,y∈ and d(x,y)=0⇔x=y.
(CM2) d(x,y)=d(y,x) for all x,y∈
(CM3) d(x,y) ≾d(x,z)+d(z,y) for all x,y,z∈.
In this case, we say that (X,d) is a complex valued
metric space.
It is obvious that this concept is generalization of the
classic metric. In fact, if d:XxX→ ℝ satisfies( (CM1)-
(CM3)), then this d is a metric in the classical sense,
that is, the following conditions are satisfies:
(M1) 0 ≤ d(x,y) for all x,y∈ and d(x,y)=0⇔x=y.
(M2) d(x,y)=d(y,x) for all x,y∈
(M3) d(x,y) ≤d(x,z)+d(z,y) for all x,y,z∈.
There are so many more different and interesting type
of metric spaces and classical theories of metric space
for example see[3,4].
Definition 2 Let ℂ be a complex valued metric space,
We say that a sequence {x
n
} is said to be a
Cauchy sequence be a sequence in x ∈XIf for
every c∈ℂ, with 0≺c there is n
0
∈ℕsuch that for
all n>n
0
such thatd(x
n
,x
m
)≺c.
We say that a sequence {x
n
} converges to an
element x∈. If for every c∈ℂ, with 0≺c there
exist an integer n
0
∈ℕsuch that for all n>n
0
such
that d(x
n
,x)≺c and we write x
n
x.
We say that (x,d) is complete if every Cauchy
sequence in X converges to a point in X.
The following fact is summarized from Azam,fisher
and Khan”s paper[2]. In fact,(b and c of preposition 1.3
are their lemmas 2 and 3.
Preposition 3
Let (X,d) be a complex value metric space. Suppose
that d=d
1
+id
2
where d
1
,d
2
: XxX→ ℝ,