Journal of Function Spaces
Volume 2019 (2019), Article ID 1367879, 10 pages
https://doi.org/10.1155/2019/1367879

Research Article

Fixed Point Theorems for Cyclic Weakly Contraction Mappings in Dislocated Quasi Extended -Metric Space

Budi Nurwahyu

Department of Mathematics, Hasanuddin University, Tamalanrea KM 10, Makassar, Indonesia

Correspondence should be addressed to Budi Nurwahyu; budinurwahyu@unhas.ac.id

Received 21 May 2019; Accepted 24 July 2019; Published 6 August 2019

Academic Editor: Vakhtang M. Kokilashvili

Copyright © 2019 Budi Nurwahyu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we establish and prove some theorems about existence and uniqueness of fixed point for cyclic weakly contraction mappings in dislocated quasi extended b-metric space.

1. Introduction

One of the famous generalizations of metric space which was introduced by Bakhtin in 1989 [1] is b-metric space. Many authors utilized the space for fixed point results on contraction mapping or weakly contraction mapping, such as Saluja et al. [2], Mostefaoui et al. [3], Chaudhury et al. [4] and Ansari et al. [5]. In 2012, Shah et al. [6] introduced quasi b-metric space which removed symmetric conditions in b-metric and for utilizing in common fixed point results on contraction mapping. Some authors such as Zhu et al. [7] and Cvetkovic et al. [8] gave some results in that space. In 2013, Hussain et al. [9] introduced dislocated b-metric which weakened first condition in b-metric for fixed point results, and Rasham et al. [10] utilized the space for multivalued fixed point results. In 2016, Rahman et al. [11] generalized the dislocated b-metric to be dislocated quasi b-metric. Several papers has published in dislocated quasi b-metric for containing fixed point results on generalized Banach contraction mappings, such as Klin-eam et al. [12], Suanom et al. [13], and Tiwari et al. [14]. Recently, in 2017, Kamran et al. [15] generalized triangular inequality condition on b-metric such that to be extended b-metric and utilized the space for fixed point results. Samreen et al. [16] yielded some theorems for fixed point results on nonlinear contraction mappings in the space and Alqahtani et al. [17, 18] utilized the space for common fixed point results on two self-mappings and on K-contraction mapping.

Inspired by the extended b-metric space of Samreen et al. [16]. In this work, we introduced a concept of dislocated quasi extended b-metric space as a generalization of dislocated quasi b-metric space [11]. We establish and prove some fixed point theorems in the dislocated quasi extended b-metric space, by utilizing weakly contraction mapping which was introduced by Rhoades [19] and cyclic contraction which was introduced by Zoto et al. [20]. In addition, we also provide some examples to clarify the theorems.

2. Preliminaries

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

Definition 1 (see [1]). Let X be a non-empty set and a real number . Let be a function. The pair is called b-metric space if the following conditions are satisfied:

(1) if and only if ,(2),(3),

for all .

Example 2 (see [15]). Let with , where Let be a function, which is defined as , where and . Then is a b-metric with parameter .

Definition 3 (see [11]). Let X be a nonempty set and a real number . Let be a function. The pair is called a dislocated quasi b-metric space (in short dqb- metric space) if the following conditions are satisfied:

(1) then ,(2),

for all .

Example 4 (see [11]). Let and define It is easy to show that is a dislocated quasi b-metric space with

Definition 5 (see [15]). Let X be a non-empty set and be a function. Let be a function. The pair is called an extended b-metric space if the following conditions are satisfied:

(1) if and only if ,(2),(3),

for all .

Example 6 (see [16]). Let Define and as follows:

It is easy to show that is a dislocated extended b-metric space.

Definition 7. Let X be a non-empty set and . Let be a function. The pair is called a quasi extended b-metric space (in short qeb- metric space) if the following conditions are satisfied:

for all .

Example 8. Let and for . Let for .
It is obvious that for first condition and is not symmetric. For second condition, consider that

Since , we get

If then we have , and . Therefore, we get

Since, for , . Thus we get

If then we have , and . Thus we get

Hence, we have

Thus is a quasi b-metric in .

Definition 9. Let X be a non-empty set and and let be a function. The pair is called a dislocated quasi extended b-metric space (in short dqeb- metric space) if the following conditions are satisfied:

for all .

Remark 10. If , then dqeb is dqb.

Example 11. Let and , for .
Let for .
In fact, it is clear that if , then , which is satisfied for first condition. For second condition, we consider,

Thus is a dislocated quasi extended b-metric space, with

Definition 12 (see [12, 13]). Let be a dislocated quasi extended b-metric space and let be a sequence in .

(i) convergent sequence to , if (ii) is called Cauchy in , if (iii) is called complete if every Cauchy sequence in is convergent in

Definition 13 (see [6]). Let X be nonempty set, and are subsets of . A function is called a cyclic map if and .

Definition 14 (see [17]). Let be a dislocated quasi extended b-metric space, and be subsets of . A function is called dqeb-cyclic weakly contraction if there exists continuous and non-decreasing function such that for every , ,

where if and only if

Definition 15 (see [20]). Let be a dislocated quasi extended b-metric space, and be subsets of . A function is called a cyclic contraction if is a cyclic and there exists a continuous and non-decreasing function such that for every ,

3. Main Results

In this section, we show some theorems and examples of the existence and uniqueness of fixed point for generalized dqeb-cyclic weakly contraction mapping in complete dislocated quasi extended b-metric space.

Theorem 16. Let be a complete dislocated quasi extended b-metric space, and be closed subsets of . If is a cyclic map that satisfies the condition of dqeb-cyclic weakly contraction and , then has a unique fixed point in .

Proof. Since is a cyclic map, if taking , then and . Define a sequence , where . So we have and for
Since for all , then for all we have

Thus we have is a nonincreasing sequence of non-negative real numbers.
Claim that Suppose .
Since is nondecreasing and , we have

Since is continuous then for , we have . Since , thus we get . Hence we have Similarly we have .
Now, we have to prove that is a Cauchy sequence in .
Suppose is not a Cauchy sequence. Then there exists such that every , there exists , such that and .
From (11) we have

This implies

We also have that

It implies . Therefore we have , so we have , thus we obtain . From (15) and (16) we have

For and using continuity of , we get

Since and , we have . Thus we have , this implies Since then we obtain , which is a contradiction.
Hence is a Cauchy sequence in . Since complete, there exists such that for Similarly we can have
Since the sequence , and be closed, then we have .
Now we prove that is a fixed point of Using (2) and (11) we have

Using continuity of and for , we have .
Thus , hence .
Now we have to show that has unique fixed point in . Suppose that is an another fixed point of ,

Thus we get . Since , we have . Which implies that , so we have

Example 17. Let and be a dislocated quasi extended b-metric space which in Example 8. Let be a function defined by , where . Let be a function which is defined by
In fact, It is clear that is a cyclic map, indeed and .
Now, we have to show that

Hence, has a deqb-weak contraction property of Theorem 16 and is the unique fixed point of .

Theorem 18. Let be a complete dislocated quasi extended b-metric space, and be closed subsets of . If is a cyclic map, continuous mapping and , such that

where is a nondecreasing, continuous mapping and iff
Then has a unique fixed point in .

Proof. Since is a cyclic map, if taking , then and . Define a sequence , where . So we have and for
By using (23) and for all , we have

Thus we have is a nonincreasing sequence of non-negative real numbers. Claim that Suppose .
Since is non-decreasing and , then we have

Since is a continuous mapping then for , we have . Since , thus we get . Hence we have Similarly we have .

Now, we have to prove that is a Cauchy sequence in .

Suppose is not a Cauchy, then there exists such that every , there exists , such that

From (23) we have

By using (26) and (27), then we have

By using (2) and (26) we also have that

It implies . By using (28) and (29), we have , so we get , thus we obtain .

From (23) and (28) we have

Since is a non-decreasing and , we have

For and using continuity of , we get

Since and , thus we have However, from (32) we have , this implies Since then we obtain which is a contradiction.

Hence is a Cauchy sequence in .

Since complete, there exists such that and for

Since the sequence , and closed, we have .

Now we have to prove that is a fixed point of By using (2) and (23), we have

Thus for , we have , hence

Now we have to show that has unique fixed point in Suppose that is an another fixed point ,

Thus we get . Since , we have . Which implies that , so we have

Example 19. Let and be a dislocated quasi extended b-metric space which in Example 8. Let be a function defined by , where Let be a function and defined as, .
In fact, it is clear that is cyclic map, indeed and .
Now, for all we have to show that

Hence, has a deqb-weak contraction property of Theorem 18 and is the unique fixed point of

Theorem 20. Let be a complete dislocated quasi extended b-metric space, and be closed subsets of . If is a cyclic, continuous mapping and , such that

where be a nondecreasing, continuous function and iff
Then has a unique fixed point in .

Proof. Since is a cyclic map, taking , then and . Define a sequence , where . So we have and for
From (37), then for all we have

Thus we have be a nonincreasing sequence of non-negative real numbers. Claim that Suppose .
Since is a nondecreasing and , then we have

Since is continuous then for , we have . Since , thus we get . Hence we have Similarly we have .
Now, we have to prove that is a Cauchy sequence in .
Suppose is not a Cauchy, then there exists such that every , there exists , such that and .
By using (37) we have

Since , we get

By using (2), we also have that

It implies that .
Therefore, from (40) and (41), we have

Thus we have , and we obtain

From (40) and (44) and , we have

By using (45) and continuity of , then for we get

Since and , thus we have However, from (46) we have , this implies and since then we obtain which is a contradiction.
Hence is a Cauchy sequence in .
Since complete, there exists such that for Since the sequence , and closed, it implies that .
Now, we have to prove that is a fixed point of

Thus for , we have , hence
Now we have to show that has unique fixed point in . Suppose is an another fixed point of ,

Thus we get . Since , we have . Which implies that , so we have

Example 21. Let and be a dislocated quasi extended b-metric space which in Example 8. Let be a function defined by , where , , and let be a function and defined as, .
In fact it clear is cyclic, since and .
Now, we have to show that

Hence, has a deqb- weak contraction property of Theorem 20 and is the unique fixed point of

Theorem 22. Let be a complete dislocated quasi extended b-metric space, and be closed subsets of and let If is a cyclic, continuous function which satisfy the conditions

where be a nondecreasing and continuous function, if only if and , , , for , and
Then has unique fixed point in .

Proof. Since is a cyclic map, for , then and . Define a sequence , where . So we have and for
Since and then for all , we have

We have

where
By using (2) and (52), we have

We have

Let
Since we have

By using the ratio test criteria, we get convergence.
Let , then from (54), we get

Thus for we get . Hence is a Cauchy sequence in
Since complete, there exists such that for
Similarly, we can have
Since the sequence , and closed, thus we have .
Now we prove that is a fixed point of Using (2) and (11) we have

Using continuity of , and , then for , we have .
Thus , hence
Now we have to show that has unique fixed point in Suppose is an another fixed point of ,

We have

Since thus we get . Since , then . Which implies that , so we have

Example 23. Let and be a dislocated quasi extended b-metric space which in Example 8. Let be a function defined by , where , Let be a function and defined as, and
In fact, it clear that , and is cyclic, since and
Since and , it is easy to show that
Now, we have to show that

Hence, has a deqb-weak contraction property of Theorem 22 and is the unique fixed point of

4. Conclusion

In this article, we considered and proved the fixed point theorems for cyclic weakly contraction mapping in complete dislocated quasi extended b-metric space. These results generalize the recent results of Samreen [14] and Rahman [9], which was in our results more general in the sense by utilizing dislocated quasi extended b-metric and cyclic weakly contraction. Furthermore, In Theorems 16, 18, 20, and 22 one can derive several consequences in dislocated quasi b-metric by letting and in dislocated quasi metric by letting .

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 of this work by BMIS Research Project 2017 no. 3556/UN.4.3.2/LK.23/2017.

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