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Some results in quasi-b-metric-like spaces
Journal of Inequalities and Applications volume 2014, Article number: 437 (2014)
In this paper, we introduce a new concept of quasi-b-metric-like spaces as a generalization of b-metric-like spaces and quasi-metric-like spaces. Some fixed point theorems are investigated in quasi-b-metric-like spaces. Moreover, an example is given to support one of our results.
1 Introduction and preliminaries
It is well known that the theoretical framework of metric fixed point theory has been an active research field and the contraction mapping principle is one of the most important theorems in functional analysis. Many authors have devoted their attention to generalizing metric spaces and the contraction mapping principle. In [1, 2], Matthews introduced the notion of partial metric space as a part of the study of denotational semantics of dataflow networks. The partial metric space is a generalization of the metric space. Many other generalized metric spaces, such as b-metric spaces , partial b-metric spaces , quasi-partial metric spaces , dislocated metric spaces  and b-dislocated metric spaces , were introduced. Fixed point theorems were studied in the above generalized metric spaces (see, e.g., [8–18] and the references therein).
The notion of metric-like spaces was introduced by Amini-Harandi in .
Definition 1.1 
A mapping , where X is a nonempty set, is said to be metric-like on X if for any , the following three conditions hold true:
The pair is then called a metric-like space.
Recently, the concept of b-metric-like spaces, which is a new generalization of metric-like spaces and partial metric spaces, was introduced by Alghamdi et al. .
Definition 1.2 
A b-metric-like on a nonempty set X is a function such that for all and a constant , the following three conditions hold true:
() if then ;
The pair is then called a b-metric-like space.
In , some concepts in b-metric-like spaces were introduced as follows.
Each b-metric-like D on X generalizes a topology on X whose base is the family of open D-balls for all and .
A sequence in the b-metric-like space converges to a point if and only if .
A sequence in the b-metric-like space is called a Cauchy sequence if there exists (and is finite) .
A b-metric-like space is called to be complete if every Cauchy sequence in X converges with respect to to a point such that .
In , Zhu et al. introduced the concept of quasi-metric-like spaces and investigated some fixed point theorems in quasi-metric-like spaces.
Definition 1.3 
Let X be a nonempty set. A mapping is said to be a quasi-metric-like on X if for any the following conditions hold:
The pair is then called a quasi-metric-like space.
In this paper, inspired by Definitions 1.2 and 1.3, we define a quasi-b-metric-like which generalizes the quasi-metric-like and b-metric-like. Furthermore, we investigate some fixed point theorems in quasi-b-metric-like spaces. Also, we give an example to illustrate the usability of one of the obtained results.
2 Main results
In this section, we begin with introducing the concept of a quasi-b-metric-like space.
Definition 2.1 A quasi-b-metric-like on a nonempty set X is a function such that for all and a constant , the following conditions hold true:
The pair is then called a quasi-b-metric-like space. The number s is called to be the coefficient of .
Example 2.1 Let , and let
Then is a quasi-b-metric-like space with the coefficient , but since , then is not a b-metric-like space. It is obvious that is not a quasi-metric-like space.
Definition 2.2 Let be a quasi-b-metric-like space. Then
A sequence in converges to a point if and only if
A sequence in is called a Cauchy sequence if and exist and are finite.
A quasi-b-metric-like space is called to be complete if for every Cauchy sequence in , there exists some such that
A sequence in is called a 0-Cauchy sequence if
A quasi-b-metric-like space is called to be 0-complete if for every 0-Cauchy sequence in X, there exists some such that
It is obvious that every 0-Cauchy sequence is a Cauchy sequence in the quasi-b-metric-like space , and every complete quasi-b-metric-like space is a 0-complete quasi-b-metric-like space, but the converse assertions of these facts may not be true.
Remark 2.1 In Example 2.1, let for , then it is clear that and . Therefore, in quasi-b-metric-like spaces, the limit of a convergent sequence is not necessarily unique.
Now we prove our main results.
Theorem 2.1 Let be a 0-complete quasi-b-metric-like space with the coefficient , and let be a mapping such that
for all , where is a continuous mapping such that if and only if and for all . If converges for all , where is the nth iterate of φ, then f has a unique fixed point. Moreover, for any , the iterative sequence converges to the fixed point.
Proof Let be an arbitrary point in X. From (2.1), we have
If or , then , which means that is a fixed point of f. Suppose that and . Now we show that is a 0-Cauchy sequence. For any integer (the set of positive integers), the property () implies that
Equations (2.2) and (2.4) yield that
Since converges for all , then , which means that for ,
Also, applying (2.3), we proceed similarly as above and obtain , which means that for ,
From (2.6) and (2.7), we get that is a 0-Cauchy sequence. Since is 0-complete, then the sequence converges to some point , that is,
We now show that z is a fixed point of f. By the triangle inequality, we have
Using (2.8) in the above inequalities, we obtain , that is, , hence z is a fixed point of f. Next, we show that z is the unique fixed point of f. Suppose that u is also a fixed point of f, then we claim . Suppose that this is not the case, then
It is a contradiction, hence , which implies , therefore f has a unique fixed point. □
In Theorem 2.1, taking with , we can get the following corollary.
Corollary 2.1 Let be a 0-complete quasi-b-metric-like space with the coefficient , and let be a mapping such that
for all , where . Then f has a unique fixed point in X. Moreover, for any , the iterative sequence converges to the fixed point.
Theorem 2.2 Let be a 0-complete quasi-b-metric-like space with the coefficient , and let be a mapping. If there exists such that
for each , then F has a coupled fixed point, that is, there exists such that and .
Proof Let and define
for . It is straightforward to show that is a 0-complete quasi-b-metric-like space with the coefficient s. Define by . Let , . From (2.10), we have . Applying Corollary 2.1, we obtain that T has a unique fixed point in , hence there exists a unique such that , that is, . Therefore, and , which implies that F has a unique coupled fixed point. □
Lemma 2.1 
Let X be a nonempty set and be a mapping. Then there exists a subset such that and is one-to-one.
Definition 2.3 Let f and g be two self-mappings on a set X. If for some x in X, then x is called a coincidence point of f and g, where ω is called a point of coincidence of f and g.
Definition 2.4 Let f and g be two self-mappings defined on a set X. Then f and g are said to be weakly compatible if they commute at every coincidence point, i.e., if for some , then .
Theorem 2.3 Let be a quasi-b-metric-like space with the coefficient , and let f, g be self-mappings on X which satisfy the following condition:
for all , where . If and is a 0-complete subset of X, then f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point.
Proof By Lemma 2.1, there exists such that and is one-to-one. Now, define a mapping by . Since g is one-to-one on E, h is well defined. Note that for all , where . Since is 0-complete, by using Corollary 2.1, there exists a unique such that , hence , which means that f and g have a unique point of coincidence in X. Let , since f and g are weakly compatible, then , which together with the uniqueness of the point of coincidence implies that . Therefore, z is the unique common fixed point of f and g. □
Now, we give an example to illustrate the validity of one of our main results.
Example 2.2 Let . Define as follows:
Then is a complete quasi-b-metric-like space with the coefficient . Define the mapping by
It is easy to prove that f satisfies all the conditions of Corollary 2.1 with . Now, by Corollary 2.1, f has a unique fixed point. In fact, 2 is the unique fixed point of f.
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The authors are thankful to the referees for their valuable comments and suggestions to improve this paper. The research was supported by the National Natural Science Foundation of China (11361042, 11071108) and supported by the Provincial Natural Science Foundation of Jiangxi, China (20114BAB201007, 20142BAB201007, 20142BAB211004) and the Science and Technology Project of Educational Commission of Jiangxi Province, China (GJJ13081).
The authors declare that they have no competing interests.
All authors contributed equally to the work. All authors read and approved the final manuscript.
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Zhu, C., Chen, C. & Zhang, X. Some results in quasi-b-metric-like spaces. J Inequal Appl 2014, 437 (2014). https://doi.org/10.1186/1029-242X-2014-437
- quasi-b-metric-like spaces
- common fixed point
- fixed point