Common Fixed Point Theorems on Generalized Ratio Contraction Mapping in Extended Rectangular b-Metric Spaces

Abstract

In this paper, we propose and prove the common fixed point theorems on generalized contraction mappings in extended rectangular b-metric spaces by utilizing the weakly compatible function property.

1. Introduction

The notion of b-metric space was introduced by Czerwik in 1993 [1], as a generalization of a metric space, by modifying the third condition in the metric space. Mohanta [2] and Husain et al. [3] utilized that space to prove existence and uniqueness of common fixed points. Recently, Kamran et al. [4] in 2017 generalized b-metric to become extended b-metric for utilizing in fixed point results and Alqahtani et al. [5] in 2018 utilized extended b-metric to prove the common fixed point. In 2015, George et al. [6] introduced the notion of rectangular b-metric space as a generalization of rectangular metric space. Kadelburg and Radenović [7] and Mitrovic [8] established and proved the common fixed point in such space. Very recently, Asim et al. [9] in 2019 generalized the rectangular b-metric and termed it the extended rectangular b-metric. By utilizing this space, Mustafa et al. [10] proved some fixed point results for contracting mappings.

Inspired of some fixed point results in extended rectangular b-metric, in the main results, we establish and prove existence and uniqueness of the common fixed point in the extended rectangular b-metric and furnish examples to clarify the results.

2. Preliminaries

In the following section, we need some definitions to govern our theorems.

Definition 1 (see [1]). Let be a nonempty set. A mapping is called a b-metric, if there exists such that satisfies the following conditions:(1), if and only if (2)(3)for all .
The pair is called a b-metric space.

Definition 2 (see [4]). Let be a nonempty set. A mapping is called an extended b-metric, if there exists a function such that satisfies the following conditions:(1), if and only if (2)(3)for all .
The pair is called an extended b-metric space.

Definition 3 (see [6]). Let be a nonempty set. A mapping is called rectangular b-metric, if there exists such that satisfies the following conditions:(1), if and only if (2)(3)for all and .
The pair is called a rectangular b-metric space.

Definition 4 (see [9]). Let be a nonempty set. A mapping is called an extended rectangular b-metric, if there exists a function such that satisfies the following conditions:for all and .
The pair is called an extended rectangular b-metric space.

Example 1. Let and , and with .
It is obvious for condition A1 and A2. For condition A3, we consider from the Jensen inequality.
Thus, we have for all .
So, this shows that is an extended rectangular b-metric with . However, in general is not continuous.

Example 2 (see [7]). Let , and choose a function , for . Define a function as follows:It is easy to show that is the complete extended rectangular b-metric on with .
If we choose a sequence , it is easy to show that sequence is convergent to 0 and 2, so it is not a Cauchy sequence. Moreover, is not continuous, since and . Thus, is not metrizable because there exists a sequence in that is not convergent to unique point in .
Our proofs in the main results need uniqueness of the limit point of sequence in as given in the following lemma.

Lemma 1 (see [9, 11]). Let be a complete extended rectangular b-metric space and is a Cauchy sequence. If for all , then limit point of is unique.

Definition 5 (see [12]). Let X be a nonempty set and be self-mapping. If for some , then y is called a point of coincidence of and and x is called a coincidence point of and .

Definition 6 (see [13]). Let X be a nonempty set and be self-mappings. is called weakly compatible. For every , if , then .

Definition 7 (see [9]). Let be an extended rectangular b-metric space, then sequence(a) is called convergent if and only if there exists such that , as .(b) is called Cauchy if and only if , as .In addition, X is called complete if every Cauchy sequence in X is convergent in X.

3. Main Results

In this section, we establish and prove the common fixed point theorems in complete extended rectangular b-metric space and furnish some examples to clarify the theorem.

Lemma 2. Let be an extended rectangular b-metric space and be a sequence in .
If , , and , then is a Cauchy.

Proof. Let , .

Case 1. For is odd, let .

Consider that from rectangular condition in (1) we have

Since , using (3) we have thatwhere

Let so by applying ratio criteria and as known that , then we have for each .

This implies that is convergent for each .

Hence, for in (4) we get ; thus, is the Cauchy sequence in .

Case 2. For is even, let .Let , so by using (5) we haveThis implies as thus is Cauchy in .

Theorem 1. Let be a complete extended rectangular b-metric space and let be self-mappings such that and . Ifholds for all , , , , and , then and have a unique coincidence in .

Moreover, if and are weakly compatible, then and have a unique common fixed point.

Proof. Let as an initial point, define sequence and in such that , . Suppose that , , because if for some , then , so and have a coincidence point.
Considering (7), we haveSince , then we haveThus, we getwhere . Since , we have , and as known that , then by using Lemma 2, we get is a Cauchy sequence in .
Since is complete, then there exists such thatand since and is a closed set, then and for some .
Furthermore, we will show that and have a point coincidence as follows.
From (1) and (7) we haveIt follows from (11) to (12) that if , then we getThis implies that , so then we get . Hence, is a coincidence point of and .
To show uniqueness of the coincidence point of and , we suppose there exists such that .
From (1) and (7), we haveThus, we getSince , we have and thus we obtain ; hence, .
Furthermore, we have to show that is a common unique fixed point of and .
From (1) and (7), we haveSince and are weakly compatible and , so we have . Thus, from (16) we haveThus, we getSince , we have and thus we obtain ; hence, . This implies that .
Hence, is a common fixed point of and .
Uniqueness of the common fixed point of and are shown as follows.
Suppose is another common fixed point of and ,Thus, we getSince , we have and thus we obtain ; hence, .
Hence, and have a unique common fixed point in .
By taking for all in Theorem 1, we obtain the following.

Corollary 1. Let be a complete extended rectangular b-metric space, and let be self-mappings such thatholds for all , , , , and , then has a unique fixed point in .

Example 3. Let and Let , and define for all , and define as follows:It is easy that is an extended rectangular space with .
Let and , for all . It is clear that .
By using , , and , we haveGenerally, we haveSince , the sequence is convergent to 0. Indeed, .(1)For or , it is clear that (2)For and The largest value of occurs if and . So, we have(3)For and ,The largest value of occurs if and . So, we haveHence, we get that conditionis hold.
Hence, from Theorem 1, we have is a unique common fixed point of and .

Theorem 2. Let be a complete extended rectangular b-metric space, and let be self-mappings such that and . If the conditionshold for all , where , , , and , and then and have a unique coincidence in .
Moreover, if and have weakly compatible property, then and have a unique fixed point.

Proof. Let taking as initial point define sequence and in such that , . Suppose that , , because if for some , then , so and have a coincidence point.
Consider from (1) and (32), we haveSince , we getSo, we getwhere . Since as known and , then by using Lemma 2, we get is a Cauchy sequence in .
Since is complete, then there exists such thatSince and closed, then and for some .
Hence, for , in (16) we get ; thus, is the Cauchy sequence in .
Furthermore, we will show that and have a point coincidence, as follows.
From (1) and (32), we haveBy using (36) and (37), if , then we getSince , this implies that , so then we get . Hence, is a coincidence point of and .
To show uniqueness of the coincidence point of and , we suppose there exists such that :Thus, we getSince , then we get ; hence, .
Next, we have to show that is a common fixed point of . Consider thatSince and are weakly compatible and , we have . Thus, from (41) we getThus, we obtain
Since , we obtain that . This implies that .
Hence, is a common fixed point of and .
Uniqueness of the common fixed point of and are shown as follows.
Suppose is another common fixed point of and , that is, .Since , we obtain that , and it implies that .
Hence, and have a unique common fixed point in .
By taking in Theorem 2 we obtain

Corollary 2. Let be a complete extended rectangular b-metric space, and let be a self-mapping such thatholds for all , where , , and , and then has a unique fixed point in .

Example 4. We use and in Example 3, and taking , . These parameters satisfy for conditionsFrom Example 3 we haveThus, we haveSince , the sequence is convergent to 0. Indeed, .(1)For or , it is clear that (2)For and The largest value of occurs if and . So, we haveAnd we haveSo, we get that(3)For and Similarly, we also obtain thatHence, we get that the conditionfor all holds.
Hence, from Theorem 2, we can conclude that is a unique common fixed point of and .

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

The author is grateful to Hasanuddin University for financial support to this work by BMIS Research Project 2017 (no. 3556/UN.4.3.2/LK.23/2017).