Common Fixed Point Theorems in Complex Valued
Metric Spaces
T.Senthil Kumar*, R.Jahir Hussain
*P.G. Department of Mathematics, Arignar Anna Government Arts College, Musiri,
P.G. and Research Department of Mathematics, Jamal Mohamed College (Autonomous),
Tiruchirappalli
TamilNadu, India.
* Corresponding Author: [email protected]
Abstract:- Recently, Azam et al. introduced new spaces called the complex valued metric spaces and established the existence of fixed point theorems under the contraction condition. In this paper, we extend and improve the condition of contraction of results of Sitthikul, Saejung [6] for two single-valued mappings in such spaces.
Key words: Complex valued metric, families of self mappings, common fixed point.
I. INTRODUCTION
The fixed point theorem, generally known as the Banach’s contraction mapping principle, appeared in explicit form in Banach’s thesis in 1922. Recently, Azam et al. first introduced the complex valued metric space which is more general than well-known metric spaces and also gave common fixed point theorems for mappings satisfying generalized contraction condition.
Preliminaries
Let C the set of complex numbers and z1, z2 . We define a partial order on as follows: z1 z2 if and
only if Re (z1) ≤ Re (z2) and Im (z1) ≤ Im (z2) that is z1 z2 if one of the following holds
C1: Re (z1) = Re (z2) and Im (z1) = Im (z2)
C2: Re (z1) < Re (z2) and Im (z1) = Im (z2)
C3: Re (z1) = Re (z2) and Im (z1) < Im (z2)
C4: Re (z1) < Re (z2) and Im (z1) < Im (z2)
In particular, we will write z1 z2 if z1 z2 and one of (C2), (C3), and (C4) is satisfied and we will write z1
z2 if only (C4) is satisfied.
Definition 1.1 Let X be a non empty set .A mapping is called a complex valued matrix on X if the following conditions are satisfied:
(CM1) 0 d(x, y) for all x, y X and d(x, y) = 0 if and only if x = y;
(CM2) d(x, y) = d(y, x) for all x, y X;
(CM3) d(x, y) d(x, z) + d (z, y), for all x, y, z X. Then d is called a complex valued metric space.
Definition 1.2 Let (X, d) be a complex valued metric space.
(i) A point x X is called interior point of set A X whenever there exist 0 r such that B(x, r) {γ │d(x, y) r}
(ii) A point x X is called a limit of A whenever for every 0 r , B(x, r) (A–X) .
(iii) A subset A X is called open whenever each element A is an interior point of A.
(iv) A sub set A X is called closed whenever each limit point of A belongs to A.
(v) A sub-basis for a Hausdorff topology τ on X is a family F = {B(x, r) │x Х and 0 < r}.
Definition1.3 Let (x, d) be a complex valued metric space,{xn} be a sequence in X and x X.
(i) If for every c C, with 0 c there is N such that for all n ,
d(xn, x) , then {xn} is said to be convergent, {xn} converges to x and x is the limit point of {xn}, we denote
this by (or) {xn} x as n .
(ii) If for every c C, with 0 c there is N such that for all n ,d(xn, xn+m) , where m , then
{xn} is said to be Cauchy sequence.
ISSN (Online) 2347-3207
Proposition1.4 Let (X, d) be a complex valued metric space. Suppose that where , that is, ( ) ( ). Then the following assertions hold.
(a)│d│= ( ) : ( )
(b) If {xn}is a sequence in X and x X, then
│ │ →
(c) (X, d) is complete if and only if (X, │d│) is complete.
The following common fixed point theorem was also proved by Azam, Fisher and Khan. This can be viewed as a generalization of the well known Banach fixed point theorem.
Theorem 1.4 (Azam et al). Let (X, d) be a complete complex valued metric Space and S, T: X → X. If S and T satisfy
( ) ( ) ( ) ( )
( ) ( )
Then S and T have a unique fixed point.
In this paper, we continue the study of fixed point theorems in complex valued metric spaces. The obtained results are generalizations of recent results proved by Sitthikul, Saejung [6]. Moreover, we improve several assumptions on the involved mappings. It should be noted that there are also some different fixed point theorems recently proved in [6].
II. RESULT
Throughout the paper, let (X, d) be a complete complex valued metric space and S, T: X→X.
Proposition 2.1 Let x0 X and define the sequence {xn} by
, ……….. (2) , for all n = 0, 1, 2, …
Assume there exist a mapping , )
( ) ( ) and ( ) ( ) for all x, y X.
Then ( ) ( ) ( ) ( )
Proof Let X and n = 0, 1, 2, ….Then we have
( ) ( ) ( ) ( ) ( ). Similarly, we have
( ) ( ) ( ) ( ) ( ).
Theorem 2.2 Let (X, d) be complex valued metric space and . If there exist mappings A, B, C, D, E:
, ) such that for all x, y X:
1. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ); 2. ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
Then S and T have a unique common fixed point.
Proof: Let x, y X. From (3), we consider
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )| ( )|
( )| ( )|| ( )| | ( )|
( )| ( )|| ( )| | ( )|
| ( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
|( )| ( )| ( )| ( )| ( )|| ( )| | ( )|
( )| ( )|| ( )| | ( )|
|( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )
Let x0 X and the sequence {xn} be defined by (2). We show that {xn} is a Cauchy sequence. From proposition
2.1, (4), (5), and for all k = 0, 1 …we have
| ( )| | ( )|
( )| ( )|
( )| ( )| ( )| ( )|
| ( )| | ( )| ( )| ( )|
( )| ( )| ( )| ( )|
( )| ( )| ( )| ( )| ( )| ( )|
( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )|
( )| ( )| ( )| ( )|
| ( )|( ( ( ) ( )) ( ( ) ( ))| ( )| Which implies that
| ( )|
( ) ( )
( ( ( ) ( ))) | ( )| ( )
Similarly, we get
| ( )| ( )| ( )| ( )| ( )| ( )| ( )| ( ) | ( )| | ( )| ( )| ( )|
( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )|
= ( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )| ( )|
( ( ) ( ))| ( )| ( ( ) ( ))| ( )|
| ( )|( ( ( ) ( ))) ( ( ) ( ))| ( )|
| ( )|
( ( ) ( ))
( ( ( ) ( )))| ( )| ( )
( ( ) ( ))
ISSN (Online) 2347-3207
Then we have | ( )| | ( )|, for all ………….... (8) If bn = | ( )| , then
, for all For such that , we have
| ( )|
( )
Thus, we have | ( )| as , and hence * + is a Cauchy sequence in (X, d). By the completeness of X, there exist such that as .
Next, we show that is a fixed point of S. By (3) and proposition 2.1, we have
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) Which implies that
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) Thus, ( ) and hence .
We also show that fixed point of T, we have
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
Thus, ( ) and hence . It follows that similarly .Therefore; z is a common fixed point of S and T. Finally we show the uniqueness. Suppose that there is such that . Consider
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( )
So that
| ( )| ( )| ( )| ( )| ( )|| ( )| | ( )|
| ( )| ( )| ( )| ( )| ( )| | ( )| | ( )|
Therefore
| ( )| A( ) | ( )| ( )| ( )|
( ( ) ( ))| ( )|. This is contraction to ( ) ( ) Hence;
There is a unique common fixed point of S and T.
By putting S = T in theorem 2.2, we deduce the following corollary.
Corollary 2.3 Let (X, d) be complex valued metric space and . If there exist mappings A, B, C, D, E:
, ) such that for all x, y X: 1. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( );
2. ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
Then T has a unique common fixed point.
REFERENCES
[1] A. Azam, B. Fisher, and M. Khan, “common fixed point theorems in complex valued metric spaces” Numerical Functional Analysis and Optimization, vol.32, no.3, pp.243-253, 2011.
[2] Ding, HS, Kadelburg, Z, Nashine, HK: Common fixed point theorems of weakly contractions in cone metric spaces. Abstr. Appl. Anal. 2012, Article ID 793862 (2012). doi:10.1155/2012/793862
[3] Ding, HS, Kadelburg, Z, Nashine, HK: Common fixed point theorems of weakly increasing mappings on ordered orbit ally complete metric spaces . Fixed point theory Appll. 2012, 85 (2012) doi: 10.1186/1687-1812-2012-85 [4] W. Sintunavarat and P. Kumam, “Generalized common fixed point in complex valued metric spaces and
applications”. Journal of inequalities and Applications, vol. 2012, article 84, 2012.
[5] F. Rouzkard and Imdad, “Some Common Fixed Theorems in Complex Valued metric spaces, “Computers & Mathematics with Applications, vol. 64, no.6,pp. 1866-1874,2012.
