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Metric-like spaces, partial metric spaces and fixed points
Fixed Point Theory and Applications volume 2012, Article number: 204 (2012)
By a metric-like space, as a generalization of a partial metric space, we mean a pair , where X is a nonempty set and satisfies all of the conditions of a metric except that may be positive for . In this paper, we initiate the fixed point theory in metric-like spaces. As an application, we derive some new fixed point results in partial metric spaces. Our results unify and generalize some well-known results in the literature.
1 Introduction and preliminaries
There exist many generalizations of the concept of metric spaces in the literature. In particular, Matthews  introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks, showing that the Banach contraction mapping theorem can be generalized to the partial metric context for applications in program verification. After that, fixed point results in partial metric spaces were studied by many other authors [2–11]. In this paper, we first introduce a new generalization of a partial metric space which is called a metric-like space. Then, we give some fixed point results in such spaces. Our fixed point theorems, even in the case of partial metric spaces, generalize and improve some well-known results in the literature.
In the rest of this section, we recall some definitions and facts which will be used throughout the paper.
Definition 1.1 A mapping , where X is a nonempty set, is said to be a partial metric on X if for any , the following four conditions hold true:
(P1) if and only if ;
The pair is then called a partial metric space. A sequence in a partial metric space converges to a point if . A sequence of elements of X is called p-Cauchy if the limit exists and is finite. The partial metric space is called complete if for each p-Cauchy sequence , there is some such that
2 Main results
We first introduce the concept of a metric-like space.
Definition 2.1 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:
(σ 1) ;
(σ 2) ;
(σ 3) .
The pair is then called a metric-like space. Then a metric-like on X satisfies all of the conditions of a metric except that may be positive for . Each metric-like σ on X generates a topology on X whose base is the family of open σ-balls
Then a sequence in the metric-like space converges to a point if and only if .
Let and be metric-like spaces, and let be a continuous mapping. Then
A sequence of elements of X is called σ-Cauchy if the limit exists and is finite. The metric-like space is called complete if for each σ-Cauchy sequence , there is some such that
Every partial metric space is a metric-like space. Below we give another example of a metric-like space.
Example 2.2 Let , and let
Then is a metric-like space, but since , then is not a partial metric space.
Remark 2.3 Let , let for each , and let for each . Then it is easy to see that and , and so in metric-like spaces the limit of a convergent sequence is not necessarily unique.
Theorem 2.4 Let be a complete metric-like space, and let be a map such that
for all , where
where is a nondecreasing function satisfying
Then T has a fixed point.
Proof Let be arbitrary, and let for . Denote
First we show that is a bounded set. We shall show that for each ,
where . Suppose, to the contrary, that there are positive integers such that
From our assumption, we have
Thus, from the above and the contractive condition on T, we have
a contradiction. Thus, (1) holds. Since by the triangle inequality,
then from (1)
From our assumption on T, we have
Now by (2),
where I is the identity map. Since the sequence is nondecreasing, there exists . Suppose that . Then from (3), we get
a contradiction. Therefore, , that is,
Now we show that is a σ-Cauchy sequence. Set
Since , then by (4) we conclude that is a nonincreasing finite nonnegative number and so it converges to some . We shall prove that . Let be arbitrary, and let r, s be any positive integers such that . Then and hence we conclude that . Then
Hence, we get
Hence, as for all , . Suppose that . Then we get
a contradiction. Therefore, . Thus, we have proved that
Hence, from the triangle inequality, we conclude that is a σ-Cauchy sequence. By the completeness of X, there is some such that , that is,
We show that . Suppose, by way of contradiction, that . Then we have
From the triangle inequality, we have
Thus, . Since , , for large enough n, we have
If , then from (5), we get
Letting n tend to infinity, we get
a contradiction. If , then we have
and so . Then from (5) and our assumptions on ψ, we get , a contradiction. Thus, and so . □
Example 2.5 Let for each , where , and let for each . Then and satisfy the conditions of Theorem 2.4.
Now we illustrate our previous result by the following example.
Example 2.6 Let . Define as follows:
Then is a complete metric-like space. Note that σ is not a partial metric on X because
Define the map by
for each . Then all the required hypotheses of Theorem 2.4 are satisfied. Then T has a unique fixed point.
Theorem 2.7 Let be a complete metric-like space, and let be a map such that
for all , where is a nondecreasing continuous function such that if and only if . Then T has a unique fixed point.
Proof Let and define for . Then by our assumption,
for each . Then is a nonnegative nonincreasing sequence and hence possesses a limit . Since φ is nondecreasing, then from (6), we get
for each . Then and so . Therefore,
Now, we show that is a Cauchy sequence. Fix and choose N such that
We show that if , then . To show the claim, let us assume first that . Then
Now we assume that . Then . Therefore, from the above, we have
Since , then from the above, we deduce that for each . Since is arbitrary, we get and so is a Cauchy sequence. Since X is complete, there is some such that , that is,
and φ is continuous, then from (7) and (8), we have
then by (7) and (9), we infer that and so . To prove the uniqueness, let v be another fixed point of T, that is, . Then
which gives and so . □
Example 2.8 Let and . Then is a complete metric-like space. Take for . Let for each . Take , without loss of generality, we may assume that . Then
Then T satisfies the hypothesis of Theorem 2.7 and so T has a fixed point ( is the unique fixed point of T). Now since , we cannot invoke Theorem 2.1 of  to show the existence of fixed point of T.
The following corollary improves Theorem 1 in .
Corollary 2.9 Let be a complete partial metric space, and let be a map such that
for all , where is a nondecreasing function satisfying
Then T has a unique fixed point.
Proof The existence of a fixed point follows immediately from Theorem 2.4. To prove the uniqueness, let us suppose that x and y are fixed points of T. Then from our assumption on T, we get
Thus, and . □
Corollary 2.10 Let be a complete partial metric space, and let be a map such that
for all , where . Then T has a unique fixed point.
Proof Let for each and apply Corollary 2.9. □
Now, we present the following version of Rakotch’s fixed point theorem  in metric-like spaces.
Theorem 2.11 Let be a complete metric-like space, and let be a mapping satisfying
for each with , where is nonincreasing. Then T has a unique fixed point.
Proof Fix and let for each . Following the lines of the proof of the Theorem 3.6 in , we get that
and so is a σ-Cauchy sequence. Since is complete, then there exists such that
From our assumption, we have
which yields . Also, notice that and hence . Thus,
By the triangle inequality, we have
and so , that is, . The uniqueness easily follows from our contractive condition on T. □
The following corollary is another new extension of Matthews’s fixed point result .
Corollary 2.12 Let be a complete partial metric space, and let be a mapping satisfying
for each with , where is nonincreasing. Then T has a unique fixed point.
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The author was partially supported by a Grant from IPM (91470412) and by the Center of Excellence for Mathematics, University of Shahrekord, Iran.
The authors declare that they have no competing interests.
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Cite this article
Amini-Harandi, A. Metric-like spaces, partial metric spaces and fixed points. Fixed Point Theory Appl 2012, 204 (2012). https://doi.org/10.1186/1687-1812-2012-204
- fixed point
- metric-like space
- partial metric space