Abstract
Owning the concept of complex valued metric spaces introduced by Azam et al., we prove several fixed point theorems for mappings satisfying certain point-dependent contractive conditions. The main results announced by Sintunavarat and Kumam (J. Inequal. Appl. 2012:84, 2012), Rouzkard and Imdad (Comput. Math. Appl., 2012, doi:10.1016/j.camwa.2012.02.063), and Dass and Gupta (Indian J. Pure Appl. Math. 6(12):1455-1458, 1975) are deduced from our results under weaker assumptions.
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1 Introduction
The concept of a complex valued metric space which is a generalization of the classical metric space was recently introduced by Azam, Fisher and Khan (see [1]). To mention this, let us recall a natural relation ≾ on ℂ, the set of complex numbers as follows: for
Definition 1.1 Let X be a nonempty set. A mapping is called a complex valued metric on X if the following conditions are satisfied:
(CM1) for all and ;
(CM2) for all ;
(CM3) for all .
In this case, we say that is a complex valued metric space.
It is obvious that this concept is a generalization of the classical metric. In fact, if satisfies (CM1)-(CM3), then this d is a metric in the classical sense; that is, the following conditions are satisfied:
(M1) for all and ;
(M2) for all ;
(M3) for all .
The following definition is an analogue of several concepts in the classical theory of metric spaces and they are discussed in [1]. There are also other interesting types of generalization of metric spaces; for example, see [2, 3].
Definition 1.2 Suppose that is a complex valued metric space.
-
We say that a sequence is a Cauchy sequence if for every there exists an integer N such that for all .
-
We say that converges to an element if for every there exists an integer N such that for all . In this case, we write .
-
We say that 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 [1]. In fact, (b) and (c) of Proposition 1.3 are their Lemmas 2 and 3.
Proposition 1.3 Let be a complex value metric space. Suppose that where , that is, and . Then the following assertions hold.
-
(a)
is a (classical) metric on X.
-
(b)
If is a sequence in X and , then if and only if .
-
(c)
is complete if and only if 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 ([1])
Let be a complete complex valued metric space, and let λ, μ be nonnegative real numbers such that . Suppose that are mappings satisfying:
Then S and T have a unique common 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 Sintunavarat and Kumam [4], Rouzkard and Imdad [5]. 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].
2 Main result
Throughout the paper, let be a complete complex valued metric space and .
Proposition 2.1 Let and define the sequence by
Assume that there exists a mapping satisfying
Then and for all and .
Proof Let and . Then we have
Similarly, we have
□
Lemma 2.2 Let and . If S and T satisfy
then
respectively.
Proof We can write
Similarly, we get
□
Lemma 2.3 Let be a sequence in X and . If satisfies
then is a Cauchy sequence.
Proof Let . Then
For such that , we have
Thus, we have as , and hence is a Cauchy sequence. □
Theorem 2.4 Let be a complete complex valued metric space and . If there exist mappings such that for all :
-
(a)
and ,
and ,
and ;
-
(b)
;(c)
(6)
Then S and T have a unique common fixed point.
Proof Let . From (6), we consider
From Lemma 2.2, we have
Similarly, we get
From Lemma 2.2, we have
Let and the sequence be defined by (2). We show that is a Cauchy sequence. From Proposition 2.1, (7), (8) and for all , we have
which implies that
Similarly, we get
which implies that
Let . Then we have
From Lemma 2.3, we have is a Cauchy sequence in . By the completeness of X, there exists such that as .
Next, we show that z is a fixed point of S. By (6) and Proposition 2.1, we have
Thus, and hence .
We also show that z is a fixed point of T. By (6), we have
Thus, and hence . Therefore, z is a common fixed point of S and T.
Finally, we show the uniqueness. Suppose that there is such that . Then
Therefore, we have
Since , we have . Thus . □
By setting in Theorem 2.4, we deduce the following corollary.
Corollary 2.5 Let be a complete complex valued metric space and . If there exist mappings such that for all :
-
(a)
and ,
and ,
and ;
-
(b)
;(c)
(12)
Then T has a unique common fixed point.
By choosing in Theorem 2.4, we deduce the following corollary.
Corollary 2.6 Let be a complete complex valued metric space and . If there exist mappings such that for all :
-
(a)
and ,
and ;
-
(b)
;(c)
(13)
Then S and T have a unique common fixed point.
By choosing in Theorem 2.4, we deduce the following corollary.
Corollary 2.7 Let be a complete complex valued metric space and . If there exist mappings such that for all :
-
(a)
and ,
and ;
-
(b)
;(c)
(14)
Then S and T have a unique common fixed point.
The following result is closely related to Corollary 2.5 with . The real valued metric space version of this result is an extension of Dass and Gupta’s result [7].
Theorem 2.8 Let be a complete complex valued metric space and . If there exist mappings such that for all :
-
(a)
and ,
and ;
-
(b)
;(c)
(15)
Then T has a unique fixed point.
Proof Let and the sequence be defined by
We show that is a Cauchy sequence. From (15), we have
It follows from (a) that
Therefore,
and hence
Let . Then
From Lemma 2.3, we have is a Cauchy sequence in . By the completeness of X, there exists such that as . Next, we show that z is a fixed point of T. Then
Notice that . Therefore, we get , that is, .
Finally, we show the uniqueness. Suppose that there is such that . Then
Since , we have , that is, . This completes the proof. □
3 Deduced results
3.1 Sintunavarat and Kumam’s results
We deduce the main result of [4] as follows.
Theorem 3.1 ([[4], Theorem 3.1])
Let be a complete complex valued metric space and . If there exist mappings such that for all :
-
(i)
and ;
-
(ii)
and ;
-
(iii)
;(iv)
Then S and T have a unique common fixed point.
Proof Define by
Then for all ,
-
(a)
and ;
and ;
-
(b)
;(c)
By Corollary 2.6, S and T have a unique common fixed point. □
Remark 1 It is worth mentioning that (i) and (ii) of Theorem 3.1 above can be weakened by the condition
3.2 Rouzkard and Imdad’s results
The following corollary is easily obtained from our Theorem 2.4.
Corollary 3.2 Let be a complete complex valued metric space and . If there exist mappings such that for all :
-
(a)
, and ;
-
(b)
;(c)
(19)
Then S and T have a unique common fixed point.
Proof Define by
Then for all ,
-
(a)
and ;
and ;
and ;
-
(b)
;(c)
By Theorem 2.4, S and T have a unique common fixed point. □
Letting , and in Corollary 3.2 gives the following result proved by Rouzkard and Imdad in [5].
Corollary 3.3 ([5])
If S and T are self-mappings defined on a complete complex valued metric space satisfying the condition
for all , where λ, μ, γ are nonnegative reals with , then S and T have a unique common fixed point.
3.3 Dass and Gupta’s results
Applying the proof of our Theorem 2.8, we can deduce the following result of Dass and Gupta [7] in the context of real valued metric spaces.
Theorem 3.4 ([7])
Let be a real valued metric space. Let be such that (i)
for all , , , , and
-
(ii)
for some , the sequence of iterates has a subsequence with .
Then z is a unique fixed point of T.
Proof Define by
Then the conditions (a), (b) and (c) of Theorem 2.8 are satisfied. Hence, we have is a Cauchy sequence in . By (ii), the whole sequence as . It follows again from the proof of Theorem 2.8 that z is a unique fixed point of T as desired. □
References
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Acknowledgements
The authors thank the referee for their comments and suggestions on the manuscript. The first author is supported by Khon Kaen University-Integrated Multidisciplinary Research Cluster (Sciences and Technologies). The research work of the second author was also supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission, through the Cluster of Research to Enhance the Quality of Basic Education.
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Sitthikul, K., Saejung, S. Some fixed point theorems in complex valued metric spaces. Fixed Point Theory Appl 2012, 189 (2012). https://doi.org/10.1186/1687-1812-2012-189
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DOI: https://doi.org/10.1186/1687-1812-2012-189