Abstract
In this paper, we show that, unexpectedly, most of the coupled fixed point theorems (on ordered metric spaces) are in fact immediate consequences of wellknown fixed point theorems in the literature.
MSC: 47H10, 54H25.
Keywords:
coupled fixed point; fixed point; ordered set; metric space1 Introduction
In recent years, there has been recent interest in establishing fixed point theorems on ordered metric spaces with a contractivity condition which holds for all points that are related by partial ordering.
In [1], Ran and Reurings established the following fixed point theorem that extends the Banach contraction principle to the setting of ordered metric spaces.
Theorem 1.1 (Ran and Reurings [1])
Let
(i)
(ii) Tis continuous nondecreasing (with respect to ⪯);
(iii) there exists
(iv) there exists a constant
ThenThas a fixed point. Moreover, if for all
Nieto and López [2] extended the above result for a mapping T not necessarily continuous by assuming an additional hypothesis on
Theorem 1.2 (Nieto and López [2])
Let
(i)
(ii) if a nondecreasing sequence
(iii) Tis nondecreasing;
(iv) there exists
(v) there exists a constant
ThenThas a fixed point. Moreover, if for all
Theorems 1.1 and 1.2 are extended and generalized by many authors. Before presenting some of theses results, we need to introduce some functional sets.
Denote by Φ the set of functions
(
(
Denote by
Denote by Θ the set of functions
Denote by Ψ the set of functions
(
(
In [3], Harjani and Sadarangani established the following results.
Theorem 1.3 (Harjani and Sadarangani [3])
Let
(i)
(ii) Tis continuous nondecreasing;
(iii) there exists
(iv) there exist
ThenThas a fixed point. Moreover, if for all
Theorem 1.4 (Harjani and Sadarangani [3])
Let
(i)
(ii) if a nondecreasing sequence
(iii) Tis nondecreasing;
(iv) there exists
(v) there exist
ThenThas a fixed point. Moreover, if for all
In [4], AminiHarandi and Emami established the following results.
Theorem 1.5 (AminiHarandi and Emami [4])
Let
(i)
(ii) Tis continuous nondecreasing;
(iii) there exists
(iv) there exists
ThenThas a fixed point. Moreover, if for all
Theorem 1.6 (AminiHarandi and Emami [4])
Let
(i)
(ii) if a nondecreasing sequence
(iii) Tis nondecreasing;
(iv) there exists
(v) there exists
ThenThas a fixed point. Moreover, if for all
Remark 1.1 Jachymski [5] established that Theorem 1.5 (resp. Theorem 1.6) follows from Theorem 1.3 (resp. Theorem 1.4).
Remark 1.2 Theorems 1.3 and 1.4 hold if
The following results are special cases of Theorem 2.2 in [7].
Theorem 1.7 (Ćirić et al.[7])
Let
(i)
(ii) Tis continuous nondecreasing;
(iii) there exists
(iv) there exists a continuous function
ThenThas a fixed point.
Theorem 1.8 (Ćirić et al.[7])
Let
(i)
(ii) if a nondecreasing sequence
(iii) Tis nondecreasing;
(iv) there exists
(v) there exists a continuous function
ThenThas a fixed point.
Remark 1.3 Theorems 1.7 and 1.8 hold if we suppose that
Let X be a nonempty set and
In [9], Bhaskar and Lakshmikantham established some coupled fixed point theorems on ordered metric spaces and applied the obtained results to the study of existence and uniqueness of solutions to a class of periodic boundary value problems. The obtained results in [9] have been extended and generalized by many authors (see, for example, [8,1023]).
In this paper, we will prove that most of the coupled fixed point theorems are in fact immediate consequences of wellknown fixed point theorems in the literature.
2 Main results
Let
Definition 2.1F is said to have the mixed monotone property if
Let
for all
Now, define the mapping
It is easy to show the following.
Lemma 2.1The following properties hold:
(a)
(b) Fhas the mixed monotone property if and only ifTis monotone nondecreasing with respect to ⪯_{2};
(c)
2.1 Bhaskar and Lakshmikantham’s coupled fixed point results
We present the obtained results in [9] in the following theorem.
Theorem 2.1 (see Bhaskar and Lakshmikantham [9])
Let
(i)
(ii) Fhas the mixed monotone property;
(iii) Fis continuous orXhas the following properties:
(
(
(iv) there exist
(v) there exists a constant
ThenFhas a coupled fixed point
We will prove the following result.
Theorem 2.2Theorem 2.1 follows from Theorems 1.1 and 1.2.
Proof From (v), for all
and
This implies that for all
that is,
for all
2.2 Harjani, López and Sadarangani’s coupled fixed point results
We present the results obtained in [16] in the following theorem.
Theorem 2.3 (see Harjani et al.[16])
Let
(i)
(ii) Fhas the mixed monotone property;
(iii) Fis continuous orXhas the following properties:
(
(
(iv) there exist
(v) there exist
ThenFhas a coupled fixed point
We will prove the following result.
Theorem 2.4Theorem 2.3 follows from Theorems 1.3 and 1.4.
Proof From (v), for all
and
This implies (since ψ is nondecreasing) that for all
that is,
for all
2.3 Lakshmikantham and Ćirić’s coupled fixed point results
In [8], putting
Theorem 2.5 (see Lakshmikantham and Ćirić’s [8])
Let
(i)
(ii) Fhas the mixed monotone property;
(iii) Fis continuous orXhas the following properties:
(
(
(iv) there exist
(v) there exists
ThenFhas a coupled fixed point.
We will prove the following result.
Theorem 2.6Theorem 2.5 follows from Theorems 1.7 and 1.8.
Proof From (v), for all
and
This implies that for all
that is,
for all
Thus we proved that the mapping T satisfies the condition (iv) (resp. (v)) of Theorem 1.7 (resp. Theorem 1.8). Then T has a fixed point, which implies that F has a coupled fixed point. □
2.4 Luong and Thuan’s coupled fixed point results
Luong and Thuan [18] presented a coupled fixed point result involving an ICS mapping.
Definition 2.2 Let
We have the following result.
Lemma 2.2Let
is a metric onX. Moreover, if
Proof Let us prove that
The obtained result in [18] is the following.
Theorem 2.7 (see Luong and Thuan [18])
Let
(i)
(ii) Fhas the mixed monotone property;
(iii) Fis continuous orXhas the following properties:
(
(
(iv) there exist
(v) there exists
ThenFhas a coupled fixed point.
We will prove the following result.
Theorem 2.8Theorem 2.7 follows from Theorems 1.7 and 1.8.
Proof The condition (v) implies that for all
that is,
for all
Thus we proved that the mapping T satisfies the condition (iv) (resp. (v)) of Theorem 1.7 (resp. Theorem 1.8). Then T has a fixed point, which implies that F has a coupled fixed point. □
2.5 Berind’s coupled fixed point results
The following result was established in [11].
Theorem 2.9 (see Berinde [11])
Let
(i)
(ii) Fhas the mixed monotone property;
(iii) Fis continuous orXhas the following properties:
(
(
(iv) there exist
(v) there exists a constant
ThenFhas a coupled fixed point
We have the following result.
Theorem 2.10Theorem 2.9 follows from Theorems 1.1 and 1.2.
Proof From the condition (v), the mapping T satisfies
for all
2.6 Rasouli and Bahrampour’s coupled fixed point results
Theorem 2.11 (see Rasouli and Bahrampour [20])
Let
(i)
(ii) Fhas the mixed monotone property;
(iii) Fis continuous orXhas the following properties:
(
(
(iv) there exist
(v) there exists
ThenFhas a coupled fixed point
We have the following result.
Theorem 2.12Theorem 2.11 follows from Theorems 1.5 and 1.6.
Proof From the condition (v), the mapping T satisfies
for all
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgements
This work is supported by the Research Center, College of Science, King Saud University.
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