Some Fixed Point Theorems in Complex Valued b-Metric Spaces

Abstract

Recently, Azam et al. introduced the notion of complex valued metric spaces and proved fixed point theorems under the contraction condition. Rao et al. introduced the notion of complex valued b-metric spaces. In this paper, we obtain some fixed point results for the mapping satisfying rational expressions in complex valued b-metric spaces. Also, an example is given to illustrate our obtained result.

1. Introduction

Banach contraction principle [1] is one of the most important results in fixed point theory. Later, a large number of articles have been devoted to the improvement and generalization of the Banach contraction principle by using different form of contraction condition in various spaces. Bakhtin [2] introduced the notion of b-metric space which is a generalized form of metric spaces. Azam et al. [3] introduced the notion of complex valued metric spaces which is more general than well-known metric spaces and also proved common fixed point theorems for mappings satisfying rational expression. Afterwards, the concept of complex valued b-metric spaces was introduced in 2013 by Rao et al. [4]. In the sequel, Mukheimer [5] proved some common fixed point theorems in complex valued b-metric spaces.

The aim of this paper is to prove some fixed point theorems for map with different type of rational expressions in complex valued b-metric spaces. Our results unify, generalize, and complement the comparable results from the current literature.

2. Preliminaries

Let be the set of complex numbers and . Define a partial order on as follows.

if and only if , . Thus if one of the following holds:(1) and ,(2) and ,(3) and ,(4) and .We will write if and one of (2), (3), and (4) is satisfied; also we will write if only (4) is satisfied.

It follows that(i) implies ,(ii) and imply ,(iii) implies ,(iv) and imply for all .The following definitions and results [4] will be needed in the sequel.

Definition 1 (see [4]). Let be a nonempty set and let be a given real number. A function is called a complex valued b-metric on if, for all , the following conditions are satisfied:(i) and if and only if ;(ii); (iii).
The pair is called a complex valued b-metric space.

Example 2 (see [4]). Let . Define the mapping by for all . Then is a complex valued b-metric space with .

Definition 3 (see [4]). Let be a complex valued b-metric space.(i)A point is called interior point of a set whenever there exists such that .(ii)A point is called limit point of a set whenever, for every , .(iii)A subset is called open whenever each element of is an interior point of .(iv)A subset is called closed whenever each element of belongs to .(v)A subbasis for Hausdorff topology on is a family .

Definition 4 (see [4]). Let be a complex valued b-metric space; let be a sequence in and .(i)If, for every , with , there is such that, for all , , then is said to be convergent, converges to , and is the limit point of . One denotes this by or as .(ii)If, for every , with , there is such that, for all , , where , then is said to be Cauchy sequence.(iii)If every Cauchy sequence in is convergent, then is said to be a complete complex valued b-metric space.

Lemma 5 (see [4]). Let be a complex valued b-metric space and let be a sequence in . Then converges to if and only if as .

Lemma 6 (see [4]). Let be a complex valued b-metric space and let be a sequence in . Then is a Cauchy sequence if and only if as , where .

3. Main Results

Theorem 7. Let be a complete complex valued b-metric space with the coefficient and let be a mapping satisfying for all , where are nonnegative reals with . Then has a unique fixed point in .

Proof. For any arbitrary point, . Define sequence in such that Now, we show that the sequence is Cauchy: which implies that Since , we get and hence Similarly, we obtain Since and , we get .
Therefore, with and for all and consequently, we have Thus for any , and since , we get By using (8), we get Therefore, and hence Thus, is a Cauchy sequence in . Since is complete, there exists some such that as . Suppose this is not possible; then there exists such that Now, which implies that Taking the limit of (15) as , we obtain that , a contradiction with (13).
So . Hence .
Now, we show that has a unique fixed point in . To show this, assume that is another fixed point of . Then, So since Therefore, So , which proves the uniqueness of fixed point in . This completes the proof.

Corollary 8. Let be a complete complex valued b-metric space with the coefficient and let be a mapping satisfying (for some fixed ) for all , where are nonnegative reals with . Then has a unique fixed point in .

Proof. From Theorem 7, we obtain such that The uniqueness follows from By taking modulus (22), we have since Therefore, So . Hence .
Therefore, the fixed point of is unique. This completes the proof.

Theorem 9. Let be a complete complex valued b-metric space with the coefficient and let be a mapping satisfying for all such that , , where are nonnegative reals with or if . Then has a unique fixed point in .

Proof. For any arbitrary point, . Define sequence in such that Now, we show that the sequence is Cauchy: which implies that since Therefore, Similarly, we obtain Since and , we get .
Therefore, with and for all and consequently, we have Thus, for any , , we have By using (33), we get Therefore, and hence Thus, is a Cauchy sequence in . Since is complete, there exists some such that as . Suppose this is not possible; then there exists such that Now, which implies that Taking the limit of (40) as , we obtain that , a contradiction with (38).
So . Hence .
Now, we show that has a unique fixed point in . To show this, assume that is another fixed point of . Then, so that So , which proves the uniqueness of fixed point in .
Now, we consider the following case: (for any ) implies , so that . Thus we have , so there exist and such that . Using foregoing arguments, one can also show that there exist and such that . As (due to definition) implies , , which in turn yields that . Similarly, one can also have . As implies , therefore is fixed point of .
We now prove that has unique fixed point. For this, assume that in is another fixed point of . Then we have . As , therefore .
This implies that . This completes the proof of the theorem.

Corollary 10. Let be a complete complex valued b-metric space with the coefficient and let be a mapping satisfying (for some fixed ): for all such that , , where are nonnegative reals with or if . Then has a unique fixed point in .

Proof. From Theorem 9, we obtain such that The uniqueness follows from By taking modulus (45), we get So . Hence .
Therefore, the fixed point of is unique. This completes the proof.

Example 11. Let be the set of complex numbers.
Define as where and .
Obviously, is complete complex valued b-metric space with . Define as Now, for and , we get Thus which is a contradiction as .
However, notice that for , so for all and with . So all conditions of Corollary 10 are satisfied to get a unique fixed point 0 of .

Example 12. Let . Consider a b-metric defined as To verify that is a complete complex valued b-metric space with , it is enough to verify the triangular inequality condition: That is, Therefore, . Define as , for all . Then Here  It is easily and clearly verified that the map satisfies contractive condition (26) of Theorem 9 with the coefficients , , and . Observe that the point remains fixed under and is indeed unique.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.