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Fixed point theorems for weakly contractive mappings in partially ordered metric-like spaces
Fixed Point Theory and Applications volume 2013, Article number: 51 (2013)
Abstract
In this article, we establish some fixed point theorems for weakly contractive mappings defined in ordered metric-like spaces. We provide an example and some applications in order to support the useability of our results. These results generalize some well-known results in the literature. We also derive some new fixed point results in ordered partial metric spaces.
MSC:54H25, 47H10.
1 Introduction and preliminaries
During the last decades many authors have worked on domain theory in order to equip semantics domain with a notion of distance. In 1994, Matthews [1] introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks and showed that the Banach contraction principle can be generalized to the partial metric context for applications in program verification. Later on, many researchers studied fixed point theorems in partial metric spaces as well as ordered partial metric spaces.
A partial metric on a nonempty set X is a function such that for all :
-
(p1)
;
-
(p2)
;
-
(p3)
;
-
(p4)
.
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
A basic example of a partial metric space is the pair , where for all . For some other examples of partial metric spaces, see [2–16].
Another important development is reported in fixed point theory via ordered metric spaces. The existence of a fixed point in partially ordered sets has been considered recently in [17–32]. Tarski’s theorem is used in [25] to show the existence of solutions for fuzzy equations and in [27] to prove existence theorems for fuzzy differential equations. In [26, 29] some applications to ordinary differential equations and to matrix equations are presented, respectively. In [19–21, 30] some fixed point theorems were proved for a mixed monotone mapping in a metric space endowed with partial order and the authors applied their results to problems of existence and uniqueness of solutions for some boundary value problems.
Recently, Amini-Harandi [33] introduced the notion of a metric-like space which is a new generalization of a partial metric space. The purpose of this paper is to present some fixed point theorems involving weakly contractive mappings in the context of ordered metric-like spaces. The presented theorems extend some recent results in the literature.
Weakly contractive mappings and mappings satisfying other weak contractive inequalities have been discussed in several works, some of which are noted in [34–40]. Alber and Guerre-Delabriere in [34] suggested a generalization of the Banach contraction mapping principle by introducing the concept of a weak contraction in Hilbert spaces. Rhoades [35] showed that the result which Alber et al. had proved in Hilbert spaces [34] was also valid in complete metric spaces.
Definition 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 1 Let and
Then is a metric-like space, but since , then is not a partial metric space.
Remark 1 Let , and for each , and 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.
2 Main results
Throughout the rest of this paper, we denote by a complete partially ordered metric-like space, i.e., ⪯ is a partial order on the set X and σ is a complete metric-like on X.
A mapping is said to be nondecreasing if , .
Theorem 1 Let be a complete partially ordered metric-like space. Let be a continuous and nondecreasing mapping such that for all comparable
where M is given by
and
-
(a)
is a continuous monotone nondecreasing function with if and only if ;
-
(b)
is a lower semi-continuous function with if and only if .
If there exists with , then F has a fixed point.
Proof Since F is a nondecreasing function, we obtain by induction that
Put . Then, for each integer , as the elements and are comparable, from (2.1) we get
which implies . Using the monotone property of the ψ-function, we get
Now, from the triangle inequality, for σ we have
If , then . By (2.2) it furthermore implies that
which is a contradiction. So, we have
Therefore, the sequence is monotone nonincreasing and bounded. Thus, there exists such that
We suppose that . Then, letting in the inequality (2.2), we get
which is a contradiction unless . Hence,
Next we prove that is a σ-Cauchy sequence. Suppose that is not a σ-Cauchy sequence. Then, there exists for which we can find subsequences and of with such that
Further, corresponding to , we can choose in such a way that it is the smallest integer with satisfying (2.7). Then
Using (2.7), (2.8) and the triangle inequality, we have
Letting and using (2.6), we obtain
Again, the triangle inequality gives us

Then we have
Letting in the above inequality and using (2.6) and (2.9), we get
Similarly, we can show that
As
using (2.6) and (2.9)-(2.11), we have
As and and are comparable, setting and in (2.1), we obtain
Letting in the above inequality and using (2.9) and (2.12), we get
which is a contradiction as . Hence is a σ-Cauchy sequence. By the completeness of X, there exists such that , that is,
Moreover, the continuity of F implies that
and this proves that z is a fixed point. □
Notice that the continuity of F in Theorem 1 is not necessary and can be dropped.
Theorem 2 Under the same hypotheses of Theorem 1 and without assuming the continuity of F, assume that whenever is a nondecreasing sequence in X such that implies for all , then F has a fixed point in X.
Proof Following similar arguments to those given in Theorem 1, we construct a nondecreasing sequence in X such that for some . Using the assumption of X, we have for every . Now, we show that . By (2.1), we have
where
Taking limit as , by (2.13), we obtain
Therefore, letting in (2.14), we get
which is a contradiction unless . Thus . □
Next theorem gives a sufficient condition for the uniqueness of the fixed point.
Theorem 3 Let all the conditions of Theorem 1 (resp. Theorem 2) be fulfilled and let the following condition be satisfied: For arbitrary two points , there exists which is comparable with both x and y. Then the fixed point of F is unique.
Proof Suppose that there exist which are fixed points. We distinguish two cases.
Case 1. If x is comparable to z, then is comparable to for and
where
Using (2.15) and (2.16), we have
which is a contradiction unless . This implies that .
Case 2. If x is not comparable to z, then there exists comparable to x and z. The monotonicity of F implies that is comparable to and , for . Moreover,
where
for n sufficiently large, because and when . Similarly as in the proof of Theorem 1, it can be shown that . It follows that the sequence is nonnegative decreasing and, consequently, there exists such that
We suppose that . Then letting in (2.17), we have
which is a contradiction. Hence . Similarly, it can be proved that
Now, passing to the limit in , it follows that , so , and the uniqueness of the fixed point is proved. □
Now, we present an example to support the useability of our results.
Example 2 Let and a partial order be defined as whenever , and define as follows:

Then is a complete partially ordered metric-like space.
Let be defined by
Define by and . We next verify that the function F satisfies the inequality (2.1). For that, given with , so . Then we have the following cases.
Case 1. If , , then
and
As , the inequality (2.1) is satisfied in this case.
Case 2. If , , then
and
As , the inequality (2.1) is satisfied in this case.
Case 3. If , , then as and , the inequality (2.1) is satisfied in this case.
Case 4. If , , then as and , the inequality (2.1) is satisfied in this case.
Case 5. If , , then as and , the inequality (2.1) is satisfied in this case.
Case 6. If , , then as and , the inequality (2.1) is satisfied in this case.
So, F, ψ and ϕ satisfy all the hypotheses of Theorem 1. Therefore F has a unique fixed point. Here 2 is the unique fixed point of F.
If we take in Theorem 1, we have the following corollary.
Corollary 1 Let be a complete partially ordered metric-like space. Let be a nondecreasing mapping such that for all comparable with
where M is given by
is lower semi-continuous, and if and only if . If there exists with and one of the following two conditions is satisfied:
-
(a)
F is continuous in ;
-
(b)
is a nondecreasing sequence in X such that implies for all .
Then F has a fixed point. Moreover, if the following condition is satisfied: For arbitrary two points , there exists which is comparable with both x and y, then the fixed point of F is unique.
If we take for in Corollary 1, we have the following corollary.
Corollary 2 Let be a complete partially ordered metric-like space. Let be a nondecreasing mapping such that for all comparable
where M is given by
and . If there exists with and one of the following two conditions is satisfied:
-
(a)
F is continuous in ;
-
(b)
is a nondecreasing sequence in X such that implies for all .
Then F has a fixed point. Moreover, if the following condition is satisfied: For arbitrary two points , there exists which is comparable with both x and y, then the fixed point of F is unique.
The following corollary improves Theorem 2.7 in [33].
Corollary 3 Let be a complete partially ordered metric-like space. Let be a nondecreasing mapping such that for all comparable
where is a lower semi-continuous, and if and only if . If there exists with and one of the following two conditions is satisfied:
-
(a)
F is continuous in ;
-
(b)
is a nondecreasing sequence in X such that implies for all .
Then F has a fixed point. Moreover, if the following condition is satisfied: For arbitrary two points , there exists which is comparable with both x and y, then the fixed point of F is unique.
The following corollary improves Theorem 2.1 in [10].
Corollary 4 Let be a complete partially ordered partial metric space. Let be a nondecreasing mapping such that for all comparable
where M is given by
, ψ is continuous monotone nondecreasing, ϕ is lower semi-continuous, and if and only if . If there exists with and one of the following two conditions is satisfied:
-
(a)
F is continuous in ;
-
(b)
is a nondecreasing sequence in X such that implies for all .
Then F has a fixed point. Moreover, the set of fixed points of F is well ordered if and only if F has one and only one fixed point.
3 Applications
Denote by Λ the set of functions satisfying the following hypotheses:
-
(h1)
α is a Lebesgue-integrable mapping on each compact subset of ;
-
(h2)
For every , we have
We have the following results.
Corollary 5 Let be a complete partially ordered metric-like space. Let be a continuous and nondecreasing mapping such that for all comparable
where . If there exists with , then F has a fixed point.
Proof Follows from Theorem 1 by taking and . □
Corollary 6 Under the same hypotheses of Corollary 5 and without assuming the continuity of F, assume that whenever is a nondecreasing sequence in X such that implies for all , then F has a fixed point in X.
Proof Follows from Theorem 2 by taking and . □
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Acknowledgements
Dedicated to Professor Hari M Srivastava.
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Işık, H., Türkoğlu, D. Fixed point theorems for weakly contractive mappings in partially ordered metric-like spaces. Fixed Point Theory Appl 2013, 51 (2013). https://doi.org/10.1186/1687-1812-2013-51
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DOI: https://doi.org/10.1186/1687-1812-2013-51
Keywords
- fixed point
- weak contraction
- partially ordered set
- partial metric space
- metric-like space