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

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 getIf then we have , and . Therefore, we get Since, for , . Thus we getIf 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 haveThus 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 haveUsing 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 thatHence, 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 thatFrom (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 haveFor 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 haveThus 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