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Fixed point theorems for weakly C-contractive mappings in partial metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 107 (2013)
Abstract
In this work, we establish some fixed point theorems for weakly C-contractive mappings in partial metric spaces. Presented theorems extend and generalize some existence results in the literature. Also, an example is given to support our results.
MSC:47H10, 54H25.
1 Introduction and preliminaries
Fixed point theory has fascinated many mathematicians since 1922 with the celebrated Banach’s fixed point theorem. Fixed point theory plays a major role within as well as outside mathematics, so the attraction of fixed point theory to large numbers of researchers is understandable, and the problem of fixed point has been studied in several directions; see for example, [1–4]. The study of metric fixed point theory has been researched extensively in the past decades. Recently, some generalizations of the notion of a metric space have been proposed by some authors. In 1992, Matthews introduced a new notion of generalized metric space called partial metric space (for short PMS) [5, 6], in which the distance of a point from itself may not be zero. After the appearance of partial metric spaces, some authors started to generalize Banach contraction mapping theorem to partial metric spaces and focus on fixed point theory on partial metric spaces (see, e.g., [7–24]). A new category of fixed point problems was addressed by Khan et al. [25]. In this study, they introduced the concept of altering distance function. In [26], Choudhury introduced the concept of weakly C-contractive mapping as follows.
Definition 1.1 [26]
Let be a metric space and be a mapping. Then T is said to be weakly C-contractive (or a weakly C-contraction) if for all , the following inequality holds:
where is a continuous function such that if and only if .
Shatanawi [27] investigated some fixed point theorems and coupled fixed point theorems for weakly C-contractive mapping by using an altering distance function in metric and partially ordered metric spaces.
Recently, Haghi et al. [28] pointed that many fixed point generalizations to partial metric spaces can be obtained from the corresponding results in metric spaces and considered some cases to demonstrate this fact. The aim of this paper is to research fixed point and common fixed point theorems for weakly C-contractive type mappings in partial metric spaces. Our results extend and generalize some results of [27] to partial metric spaces; all of our results cannot be obtained from the corresponding results in metric spaces. Moreover, even in metric spaces, our results are the generalizations of some results of [27]. Also, we give an example to illustrate our results.
Throughout this paper, the letters N and denote the set of all nonnegative integer numbers and the set of all positive integer numbers, respectively. Let us recall some definitions and properties of partial metric spaces.
Definition 1.2 [6]
Let X be a nonempty set. The mapping is said to be a partial metric on X if the following conditions hold:
(P1) ,
(P2) ,
(P3) ,
(P4) ,
for any . The pair is then called a partial metric space.
It is clear that, if , then from (P1) and (P2), . But if , may not be 0.
For a partial metric p on X, the function given by
is a (usual) metric on X. Each partial metric p on X generates a -topology on X with a base of the family of open p-balls , where for all and .
Let be a partial metric space. Then:
A sequence in a partial metric space converges to a point if and only if .
A sequence in a partial metric space is called a Cauchy sequence if there exists (and is finite) .
A partial metric space is said to be complete if every Cauchy sequence in X converges, with respect to , to a point such that .
The following lemmas play a major role in proving our main results.
Lemma 1.1 [29]
Let be a partial metric space.
-
(A)
A sequence is a Cauchy sequence in if and only if is a Cauchy sequence in .
-
(B)
is complete if and only if is complete. Moreover,
(1.1)
Assume that as in a PMS such that . Then for every .
Lemma 1.3 [31]
Let be a partial metric space and let be a sequence in X such that
If is not a Cauchy sequence in , then there exist and two sequences and of positive integers such that and the following four sequences tend to ε when :
2 Main results
We start this section with the following definition, which can be seen in [9, 16, 17, 30].
Definition 2.1 Let be a partial metric space. A mapping is said to be continuous at if for every , there exists such that .
Definition 2.2 [25]
The function is called an altering distance function, if the following properties are satisfied:
-
(1)
φ is continuous and nondecreasing;
-
(2)
if and only if .
Lemma 2.1 [31]
Let be a partial metric space, be a given mapping. Suppose that T is continuous at . Then, for each sequence in X, in in holds.
Theorem 2.1 Let be a partially ordered set and suppose that there exists a partial metric p on X such that is complete. Let be a continuous nondecreasing mapping. Suppose that for comparable , we have
where ψ and φ are altering distance functions with
for all , and is a continuous function with if and only if . If there exists such that , then f has a fixed point.
Proof If , then is a fixed point of f. Suppose that , we can choose such that . Since f is a nondecreasing function, we have
Continuing this process, we can construct a sequence in X such that with
It is clear that if for some , then f has a fixed point. Taking for all , now let us prove the following inequality:
Suppose this is not true, then for some , that is,
From (2.1) and (2.4), we obtain that

this together with (2.2) shows that
Using the property of Ï•, we have
Since

applying (2.5), we get
From the property of ψ, we have , which contradicts with for all ; hence (2.3) holds. Therefore, is a nonincreasing sequence, and thus there exists such that
Using (2.1), we obtain

it means that
Letting in the above inequality, we get
the continuity of Ï• guarantees that
and the property of Ï• gives that
Since

on taking inferior limit in the above inequalities and using (2.8), we obtain that and so , therefore,
moreover, we have
Now, we claim that is a Cauchy sequence in the metric space (and so also in the space by Lemma 1.1). For this, it is sufficient to show that is a Cauchy sequence in . Suppose that this is not the case, then using Lemma 1.1 we have that is not a Cauchy sequence in . By Lemma 1.3, we obtain that there exist and two sequences and of positive integers such that and sequences in (1.2) tend to ε when . For two comparable elements and , we can obtain, from (2.1), that

Taking in (2.9), we get
which implies that , hence , a contradiction. Thus, is a Cauchy sequence in and so is a Cauchy sequence both in and in . Since is complete then the sequence converges to some , that is
Moreover, since is a Cauchy sequence in , we have . By and , we have . Then (2.10) yields that
Applying the triangular inequality, we have
taking in the above inequalities, then the continuity of f and Lemma 2.1 give that
hence
By combining (2.1) and (2.12), we have
which yields that , and thus , that is . Therefore, z is a fixed point of f. □
Theorem 2.2 Suppose that X, f, ψ, φ, and ϕ are the same as in Theorem 2.1 except the continuity of f. Suppose that for a nondecreasing sequence in X with , we have for all . If there exists such that , then f has a fixed point.
Proof As in the proof of Theorem 2.1, we have a Cauchy sequence in X. Since is complete, there exists such that , that is,
due to the hypothesis, we get . Similar to the proof of Theorem 2.1, we have that
From (2.1), we obtain that
Letting in the above inequalities, and by Lemma 1.2, we have
which implies, from (2.2), that , hence , and thus . Therefore, f has a fixed point. □
Theorem 2.3 Let be a complete partial metric space, f and g be self-mappings on X. Suppose that for all
where ψ and φ are altering distance functions with
for all , and is a continuous function with if and only if .
Then f and g have a unique common fixed point.
Proof Let be an arbitrary point in X. One can choose such that . Also, one can choose such that . Continuing this process, one can construct a sequence in X such that
Now, we discuss the following two cases.
Case 1. If for some , then f and g have at least one common fixed point. In fact, if for some , that is , which implies that . If (), then . Using (2.13), we have
With the help of (2.14) and (2.16), we conclude that , hence, using the property of Ï•, we get , that is . By similar arguments, we obtain , and so on. Thus, becomes a constant from , that is,
Equations (2.15) and (2.17) yield that
which implies that is the common fixed point of f and g. Similarly, one can show that if (), then f and g have at least one common fixed point. Therefore, we have proved that if for some , then f and g have at least one common fixed point.
Case 2. If for some , then f and g have at least one common fixed point. Indeed, if (), then . Hence, , due to (2.13), we have
Applying (2.14) and (2.19), we obtain . Using the property of Ï•, we have
From (2.20) and using , we get that
which implies that , and thus . Hence we obtain that f and g have at least one common fixed point from case 1. Similarly, it is easy to show that if for some (), then f and g have at least one common fixed point, this completes the proof of case 2.
Taking and for all . Now we prove that for every , we have
Suppose this is not true, then for some , that is,
Using (2.13) and (2.15), we obtain that
Equations (2.14) and (2.23) give that . Using the property of Ï•, we get , which contradicts with for , hence (2.22) holds.
Similarly, one can show that for every , the following inequality holds.
Equations (2.22) and (2.24) imply that the sequence is nonincreasing, and consequently there exists some such that
By (2.25) and the following inequalities,
we get that is bounded, and hence it has some subsequence converging to some , that is,
Taking (P2) into account, we have
which combining with (2.25) shows that is bounded, and hence there exists subsequence of such that converges to some , that is,
By (2.13), we have
Letting in (2.28), and using (2.25)-(2.27), we obtain that
which means that , hence and .
Since
taking the limit as , we have , which implies that , that is,
Now, we claim that is a Cauchy sequence in the metric space (and so also in the space by Lemma 1.1). For this, it is sufficient to show that is a Cauchy sequence in . Suppose that this is not the case, then using Lemma 1.1, we have that is not a Cauchy sequence in . By Lemma 1.3, we obtain that there exist and two sequences and of positive integers such that and sequences in (1.2) tend to ε when .
From (2.13), we get that
Letting in the above inequalities and using the continuity of ψ, φ and ϕ, we get that
therefore, we get that . Hence, which is a contradiction. Thus, is a Cauchy sequence in , and is also a Cauchy sequence in . Since is complete, then the sequence converges to some , that is,
Moreover, the sequence and converge to , that is,
and
Using the fact that is a Cauchy sequence in , we have , which together with yields that . Hence, we have
By substituting , in (2.13), we get that
letting and applying Lemma 1.2, we conclude that
which yields that ; hence, , and thus . Similarly, one can easily show that , therefore, z is the common fixed point of f and g.
Now we prove the uniqueness of common fixed point. Let us suppose that u is also the common fixed point of f and g. Since
which means that ; hence, , and so . Thus, the uniqueness of the common fixed point is proved. □
By taking in Theorems 2.1-2.3, respectively, we have the following results.
Corollary 2.1 Let be a partially ordered set and suppose that there exists a partial metric p on X such that is complete. Let be a continuous nondecreasing mapping. Suppose that for comparable , we have
where ψ is an altering distance function and is a continuous function with if and only if . If there exists such that , then f has a fixed point.
Corollary 2.2 Suppose that X, f, ψ, and ϕ are the same as in Corollary 2.1 except the continuity of f. Suppose that for a nondecreasing sequence in X with , we have for all . If there exists such that , then f has a fixed point.
Corollary 2.3 Let be a complete partial metric space, f and g be self-mappings on X. Suppose that there exist functions ψ and ϕ such that for all
where ψ is an altering distance function and is a continuous function with if and only if .
Then f and g have a unique common fixed point.
Remark 2.1 If we replace the partial metric p by (usual) metric d in Corollaries 2.1-2.3, then we get Theorems 2.1-2.3 of [27].
Now, we introduce an example to support the usability of our results.
Example 2.1 Let be endowed with the usual partial metric defined by . It is easy to show that the partial metric space is complete. Also, define the mappings by and , respectively. Let us take such that and , respectively, and take such that . If , then
and
So, we have
If , then
and
So, we have
From the above arguments, we conclude that (2.13) holds; hence, all the required hypotheses of Theorem 2.3 are satisfied. Thus, we deduce the existence and uniqueness of a common fixed point of f and g. Here, 0 is the unique common fixed point.
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Acknowledgements
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 (11071108) and supported by the Provincial Natural Science Foundation of Jiangxi, China (20114BAB201007, 2010GZS0147).
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Chen, C., Zhu, C. Fixed point theorems for weakly C-contractive mappings in partial metric spaces. Fixed Point Theory Appl 2013, 107 (2013). https://doi.org/10.1186/1687-1812-2013-107
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DOI: https://doi.org/10.1186/1687-1812-2013-107
Keywords
- fixed point
- common fixed point
- partial metric space
- weakly C-contraction