DOI: 10.1515/auom-2015-0035
An. Şt. Univ. Ovidius Constanţa
Vol. 23(2),2015, 179–185
Common fixed point theorems for generalized
contraction involving rational expressions in
complex valued metric spaces
Hemant Kumar Nashine and Brian Fisher
Abstract
The purpose of this paper is to study common fixed points in complex
valued metric spaces and obtain sufficient conditions for the existence
of common fixed points of a pair of mappings satisfying generalized
contraction involving rational expressions.
1
Introduction
The Banach contraction principle [4] is a very popular tool in solving existence
problems in many branches of mathematical analysis. This famous theorem
can be stated as follows.
Theorem 1.1. [4]. Let (X, d) be a complete metric space and T be a mapping
of X into itself satisfying:
d(T x, T y) ≤ kd(x, y), ∀ x, y ∈ X,
(1)
where k is a constant in (0, 1). Then, T has a unique fixed point x∗ ∈ X.
Key Words: Common fixed point, contractive type mapping, complex valued metric
space.
2010 Mathematics Subject Classification: Primary 54H25; Secondary 47H10.
Received: 20 June, 2013
Revised: 20 September, 2013
Accepted: 24 October, 2013
179
Common fixed point theorems for generalized contraction involving rational
expressions in complex valued metric spaces
180
There are in the literature a great number of generalizations of the Banach
contraction principle (see [1, 2] and others). Some generalizations of the notion
of a metric space have been proposed by some authors, such as, rectangular
metric spaces, semi metric spaces, pseudo metric spaces, probabilistic metric
spaces, fuzzy metric spaces, quasi metric spaces, quasi semi metric spaces,
D-metric spaces, and cone metric spaces (see [5]- [10]).
Recently, Azam et al. [3] introduced the notion of complex valued metric
spaces and established some fixed point results for mappings satisfying a rational inequality. In a continuation of Azam et al. [3], in this paper, we prove a
common fixed point theorem for a pair of mappings satisfying a more general
contraction involving rational expression in complex valued metric spaces.
2
Preliminaries
First of all, we introduce some notations and definitions that will be used later.
2.1
Notations and Definitions
The following definition was introduced by Azam et al. in [3].
Let C be the set of complex numbers and z1 , z2 ∈ C. Define a partial order
- on C as follows:
z1 - z2 if and only if Re(z1 ) ≤ Re(z2 ), Im(z1 ) ≤ Im(z2 ).
It follows that z1 - z2 if one of the following conditions is satisfied:
(i) Re(z1 ) = Re(z2 ), Im(z1 ) < Im(z2 ),
(ii) Re(z1 ) < Re(z2 ), Im(z1 ) = Im(z2 ),
(iii) Re(z1 ) < Re(z2 ), Im(z1 ) < Im(z2 ),
(iv) Re(z1 ) = Re(z2 ), Im(z1 ) = Im(z2 ).
In particular, we will write z1 z2 if z1 6= z2 and one of (i), (ii), and (iii)
is satisfied and we will write z1 ≺ z2 if only (iii) is satisfied. Note that
0 - z1 z2 ⇒ |z1 | < |z2 |,
z1 z2 , z2 ≺ z3 ⇒ z1 ≺ z3 .
Definition 2.1. Let X be a nonempty set. Suppose that the mapping d :
X × X → C, satisfies:
1. 0 - d(x, y), for all x, y ∈ X and d(x, y) = 0 if and only if x = y;
2. d(x, y) = d(y, x) for all x, y ∈ X;
3. d(x, y) - d(x, z) + d(z, y), for all x, y, z ∈ X.
Then d is called a complex valued metric on X, and (X, d) is called a
complex valued metric space.
Common fixed point theorems for generalized contraction involving rational
expressions in complex valued metric spaces
181
A point x ∈ X is called an interior point of a set A ⊆ X whenever there
exists 0 ≺ r ∈ C such that
B(x, r) = {y ∈ X : d(x, y) ≺ r} ⊆ A.
A point x ∈ X is called a limit point of A whenever for every 0 ≺ r ∈ C,
B(x, r) ∩ (A\X) 6= ∅.
A is called open whenever each element of A is an interior point of A. A
subset B ⊆ X is called closed whenever each limit point of B belongs to B.
The family
F = {B(x, r) : x ∈ X, 0 ≺ r}.
is a sub-basis for a Hausdorff topology τ on X.
Let {xn } be a sequence in X and x ∈ X. If for every c ∈ C, with 0 ≺ c
there exists n0 ∈ N such that for all n > n0 , d(xn , x) ≺ c, then {xn } is said
to be convergent, {xn } converges to x and x is the limit point of {xn }. We
denote this by limn xn = x, or xn → x, as n → ∞. If for every c ∈ C with
0 ≺ c there exists n0 ∈ N such that for all n > n0 , d(xn , xn+m ) ≺ c, then {xn }
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.
Lemma 2.2. [3]. Let (X, d) be a complex valued metric space and let {xn }
be a sequence in X. Then {xn } converges to x if and only if |d(xn , x)| → 0 as
n → ∞.
Lemma 2.3. [3]. Let (X, d) be a complex valued metric space and let {xn } be a
sequence in X. Then {xn } is a Cauchy sequence if and only if |d(xn , xn+m )| →
0 as n → ∞.
3
Common fixed point results in complete complex valued metric space
Theorem 3.1. Let (X, d) be a complete complex valued metric space and let
the mappings S, T : X → X satisfy:
β[1 + d(x, Sx)]d(y, T y)
1 + d(x, y)
+γ[d(x, Sx) + d(y, T y)] + δ[d(x, T y) + d(y, Sx)]
d(Sx, T y) - αd(x, y) +
for all x, y ∈ X, where α, β, γ, δ are nonnegative reals with α + β + 2γ + 2δ < 1.
Then S and T have a unique common fixed point.
Common fixed point theorems for generalized contraction involving rational
expressions in complex valued metric spaces
182
Proof. Let x0 be an arbitrary point in X and define
x2k+1 = Sx2k , x2k+2 = T x2k+1 , k = 0, 1, 2, · · · .
Then,
d(x2k+1 , x2k+2 )
=
d(Sx2k , T x2k+1 )
-
αd(x2k , x2k+1 ) +
-
-
-
β[1 + d(x2k , Sx2k )]d(x2k+1 , T x2k+1 )
1 + d(x2k , x2k+1 )
+γ[d(x2k , Sx2k ) + d(x2k+1 , T x2k+1 )] + δ[d(x2k , T x2k+1 ) + d(x2k+1 , Sx2k )]
β[1 + d(x2k , x2k+1 )]d(x2k+1 , x2k+2 )
αd(x2k , x2k+1 ) +
1 + d(x2k , x2k+1 )
+γ[d(x2k , x2k+1 ) + d(x2k+1 , x2k+2 )] + δ[d(x2k , x2k+2 ) + d(x2k+1 , x2k+1 )]
αd(x2k , x2k+1 ) + βd(x2k+1 , x2k+2 ) + γ[d(x2k , x2k+1 ) + d(x2k+1 , x2k+2 )]
+δ[d(x2k , x2k+1 ) + d(x2k+1 , x2k+2 )]
α+γ+δ
d(x2k , x2k+1 ).
1−β−γ−δ
Similarly,
d(x2k+2 , x2k+3 )
=
d(Sx2k+1 , T x2k+2 )
-
α(x2k+2 , x2k+1 ) +
-
+δ[d(x2k+1 , T x2k+2 ) + d(x2k+2 , Sx2k+1 )]
β[1 + d(x2k , x2k+2 )]d(x2k+2 , x2k+3 )
αd(x2k+1 , x2k+2 ) +
1 + d(x2k , x2k+2 )
+γ[d(x2k+1 , x2k+2 ) + d(x2k+2 , x2k+3 )] + δ[d(x2k+1 , x2k+3 ) + d(x2k+2 , x2k+2 )]
-
-
β[1 + d(x2k+1 , Sx2k+1 )]d(x2k+2 , T x2k+2 )
1 + d(x2k+1 , x2k+2 )
+γ[d(x2k+1 , Sx2k+1 ) + d(x2k+2 , T x2k+2 )]
αd(x2k+1 , x2k+2 ) + βd(x2k+2 , x2k+3 ) + γ[d(x2k+1 , x2k+2 ) + d(x2k+2 , x2k+3 )]
+δ[d(x2k+1 , x2k+2 ) + d(x2k+2 , x2k+3 )]
α+γ+δ
d(x2k+1 , x2k+2 ).
1−β−γ−δ
Putting
h=
α+γ+δ
,
1−β−γ−δ
we have
d(xn+1 , xn+2 ) - hd(xn , xn+1 ) - · · · - hn+1 d(x0 , x1 ).
Common fixed point theorems for generalized contraction involving rational
expressions in complex valued metric spaces
183
Hence, for any m > n,
d(xn , xm ) - d(xn , xn+1 ) + d(xn+1 , xn+2 ) + · · · + d(xm−1 , xm )
- [hn + hn+1 + · · · + hm−1 ]d(x0 , x1 )
hn
d(x0 , x1 )
1−h
and so
hn
|d(x0 , x1 )| → 0, as m, n → ∞.
1−h
This implies that {xn } is a Cauchy sequence. Since X is complete, there exists
u ∈ X such that xn → u. It follows that u = Su, otherwise d(u, Su) = z > 0
and we would then have
|d(xm , xn )| ≤
z
- d(u, x2k+2 ) + d(x2k+2 , Su)
- d(u, x2k+2 ) + d(T x2k+1 , Su)
β[1 + d(u, Su)]d(x2k+1 , T x2k+1 )
1 + d(u, x2k+1 )
+γ[d(u, Su) + d(x2k+1 , T x2k+1 )]
- d(u, x2k+2 ) + αd(x2k+1 , u) +
+δ[d(u, T x2k+1 ) + d(x2k+1 , Su)]
β[1 + d(u, Su)]d(x2k+1 , x2k+2 )
- d(u, x2k+2 ) + αd(x2k+1 , u) +
1 + d(u, x2k+1 )
+γ[d(u, Su) + d(x2k+1 , x2k+2 )] + δ[d(u, x2k+2 ) + d(x2k+1 , Su)].
This implies that
|z|
≤
β|1 + z||d(x2k+1 , x2k+2 )|
|1 + d(u, x2k+1 )|
+γ[|z| + |d(x2k+1 , x2k+2 )|] + δ|d(u, x2k+2 ) + d(x2k+1 , Su)|.
|d(u, x2k+2 )| + α|d(x2k+1 , u)| +
Letting n → ∞, it follows that
|z| ≤ (γ + δ)|z| ≤ (α + β + 2γ + 2δ)|z| < |z|,
a contradiction and so |z| = 0, that is, u = Su.
It follows similarly that u = T u.
We now show that S and T have unique common fixed point. For this,
assume that u∗ in X is a second common fixed point of S and T . Then
d(u, u∗ )
=
-
-
d(Su, T u∗ )
β[1 + d(u, Su)]d(u∗ , T u∗ )
+ γ[d(u, Su) + d(u∗ , T u∗ )]
1 + d(u, u∗ )
+δ[d(u, T u∗ ) + d(u∗ , Su)]
αd(u, u∗ ) +
(α + 2δ)d(u, u∗ )
Common fixed point theorems for generalized contraction involving rational
expressions in complex valued metric spaces
184
and so d(u, u∗ ) = 0, since (α + 2δ) < 1. This implies that u∗ = u, completing
the proof of the theorem.
Putting S = T , we have
Corollary 3.2. Let (X, d) be a complete complex valued metric space and let
the mappings T : X → X satisfy:
d(T x, T y)
-
β[1 + d(x, T x)]d(y, T y)
+ γ[d(x, T x) + d(y, T y)]
1 + d(x, y)
+δ[d(x, T y) + d(y, Sx)]
αd(x, y) +
for all x, y ∈ X, where α, β, γ, δ are nonnegative reals with α+β +2γ +2δ < 1.
Then T has a unique fixed point.
Corollary 3.3. Let (X, d) be a complete complex valued metric space and let
the mappings T : X → X satisfy:
d(T n x, T n y)
-
β[1 + d(x, T n x)]d(y, T n y)
+ γ[d(x, T n x) + d(y, T n y)]
1 + d(x, y)
+δ[d(x, T n y) + d(y, T n x)]
αd(x, y) +
for all x, y ∈ X, where α, β, γ, δ are nonnegative reals with α+β +2γ +2δ < 1.
Then T has a unique fixed point.
Proof. By Corollary 3.2 there exists v ∈ X such that T n v = v. Then
d(T v, v)
= d(T T n v, T n v) = d(T n T v, T n v)
β[1 + d(T v, T n T v)]d(v, T n v)
- αd(T v, v) +
1 + d(T v, v)
n
+γ[d(T v, T T v) + d(v, T n v)] + δ[d(T v, T n v) + d(v, T n T v)]
β[1 + d(T v, T T n v)]d(v, v)
- αd(T v, v) +
1 + d(T v, v)
+γd(T v, T T n v) + δ[d(T v, v) + d(v, T T n v)]
=
(α + 2δ)d(T v, v)
and so d(T v, v) = 0.
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Hemant Kumar NASHINE,
Department of Mathematics,
Disha Institute of Management and Technology,
Satya Vihar, Vidhansabha-Chandrakhuri Marg,
Mandir Hasaud, Raipur-492101(Chhattisgarh), India.
Email: drhknashine@gmail.com
Brian FISHER,
Department of Mathematics,
University of Leicester,
Leicester, England.
Email: fbr@leicester.ac.uk
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expressions in complex valued metric spaces
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