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])
Letbe an ordered set endowed with a metricdandbe a given mapping. Suppose that the following conditions hold:
(ii) Tis continuous nondecreasing (with respect to ⪯);
(iv) there exists a constantsuch that for allwith,
ThenThas a fixed point. Moreover, if for allthere exists asuch thatand, we obtain uniqueness of the fixed point.
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])
Letbe an ordered set endowed with a metricdandbe a given mapping. Suppose that the following conditions hold:
(ii) if a nondecreasing sequenceinXconverges to some point, thenfor alln;
(iii) Tis nondecreasing;
(v) there exists a constantsuch that for allwith,
ThenThas a fixed point. Moreover, if for allthere exists asuch thatand, we obtain uniqueness of the fixed point.
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 satisfying the following conditions:
() φ is continuous nondecreasing;
Denote by the set of functions satisfying the following condition:
Denote by Θ the set of functions which satisfy the condition:
Denote by Ψ the set of functions satisfying the following conditions:
In [3], Harjani and Sadarangani established the following results.
Theorem 1.3 (Harjani and Sadarangani [3])
Letbe an ordered set endowed with a metricdandbe a given mapping. Suppose that the following conditions hold:
(ii) Tis continuous nondecreasing;
(iv) there existsuch that for allwith,
ThenThas a fixed point. Moreover, if for allthere exists asuch thatand, we obtain uniqueness of the fixed point.
Theorem 1.4 (Harjani and Sadarangani [3])
Letbe an ordered set endowed with a metricdandbe a given mapping. Suppose that the following conditions hold:
(ii) if a nondecreasing sequenceinXconverges to some point, thenfor alln;
(iii) Tis nondecreasing;
(v) there existsuch that for allwith,
ThenThas a fixed point. Moreover, if for allthere exists asuch thatand, we obtain uniqueness of the fixed point.
In [4], AminiHarandi and Emami established the following results.
Theorem 1.5 (AminiHarandi and Emami [4])
Letbe an ordered set endowed with a metricdandbe a given mapping. Suppose that the following conditions hold:
(ii) Tis continuous nondecreasing;
(iv) there existssuch that for allwith,
ThenThas a fixed point. Moreover, if for allthere exists asuch thatand, we obtain uniqueness of the fixed point.
Theorem 1.6 (AminiHarandi and Emami [4])
Letbe an ordered set endowed with a metricdandbe a given mapping. Suppose that the following conditions hold:
(ii) if a nondecreasing sequenceinXconverges to some point, thenfor alln;
(iii) Tis nondecreasing;
(v) there existssuch that for allwith,
ThenThas a fixed point. Moreover, if for allthere exists asuch thatand, we obtain uniqueness of the fixed point.
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 satisfies only the following conditions: φ is lower semicontinuous and (see, for example, [6]).
The following results are special cases of Theorem 2.2 in [7].
Theorem 1.7 (Ćirić et al.[7])
Letbe an ordered set endowed with a metricdandbe a given mapping. Suppose that the following conditions hold:
(ii) Tis continuous nondecreasing;
(iv) there exists a continuous functionwithfor allsuch that for allwith,
ThenThas a fixed point.
Theorem 1.8 (Ćirić et al.[7])
Letbe an ordered set endowed with a metricdandbe a given mapping. Suppose that the following conditions hold:
(ii) if a nondecreasing sequenceinXconverges to some point, thenfor alln;
(iii) Tis nondecreasing;
(v) there exists a continuous functionwithfor allsuch that for allwith,
ThenThas a fixed point.
Remark 1.3 Theorems 1.7 and 1.8 hold if we suppose that (see, for example, [8]).
Let X be a nonempty set and be a given mapping. We say that is a coupled fixed point of F if
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 be a partially ordered set endowed with a metric d and be a given mapping. We endow the product set with the partial order:
Definition 2.1F is said to have the mixed monotone property if is monotone nondecreasing in x and is monotone nonincreasing in y, that is, for any ,
Let . It is easy to show that the mappings defined by
It is easy to show the following.
Lemma 2.1The following properties hold:
(a) is complete if and only ifandare complete;
(b) Fhas the mixed monotone property if and only ifTis monotone nondecreasing with respect to ⪯_{2};
(c) is a coupled fixed point ofFif and only ifis a fixed point ofT.
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])
Letbe a partially ordered set endowed with a metricd. Letbe a given mapping. Suppose that the following conditions hold:
(ii) Fhas the mixed monotone property;
(iii) Fis continuous orXhas the following properties:
() if a nondecreasing sequenceinXconverges to some point, thenfor alln,
() if a decreasing sequenceinXconverges to some point, thenfor alln;
(v) there exists a constantsuch that for allwithand,
ThenFhas a coupled fixed point. Moreover, if for allthere existssuch thatand, we have uniqueness of the coupled fixed point and.
We will prove the following result.
Theorem 2.2Theorem 2.1 follows from Theorems 1.1 and 1.2.
Proof From (v), for all with and , we have
and
This implies that for all with and ,
that is,
for all with . From Lemma 2.1, since is complete, is also complete. Since F has the mixed monotone property, T is a nondecreasing mapping with respect to ⪯_{2}. From (iv), we have . Now, if F is continuous, then T is continuous. In this case, applying Theorem 1.1, we get that T has a fixed point, which implies from Lemma 2.1 that F has a coupled fixed point. If conditions () and () are satisfied, then Y satisfies the following property: if a nondecreasing (with respect to ⪯_{2}) sequence in Y converges to some point , then for all n. Applying Theorem 1.2, we get that T has a fixed point, which implies that F has a coupled fixed point. If, in addition, we suppose that for all there exists such that and , from the last part of Theorems 1.1 and 1.2, we obtain the uniqueness of the fixed point of T, which implies the uniqueness of the coupled fixed point of F. Now, let be a unique coupled fixed point of F. Since is also a coupled fixed point of F, we get . □
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])
Letbe a partially ordered set endowed with a metricd. Letbe a given mapping. Suppose that the following conditions hold:
(ii) Fhas the mixed monotone property;
(iii) Fis continuous orXhas the following properties:
() if a nondecreasing sequenceinXconverges to some point, thenfor alln,
() if a decreasing sequenceinXconverges to some point, thenfor alln;
(v) there existsuch that for allwithand,
ThenFhas a coupled fixed point. Moreover, if for allthere existssuch thatand, we have uniqueness of the coupled fixed point and.
We will prove the following result.
Theorem 2.4Theorem 2.3 follows from Theorems 1.3 and 1.4.
Proof From (v), for all with and , we have
and
This implies (since ψ is nondecreasing) that for all with and ,
that is,
for all with . Thus we proved that the mapping T satisfies the condition (iv) (resp. (v)) of Theorem 1.3 (resp. Theorem 1.4). The rest of the proof is similar to the above proof. □
2.3 Lakshmikantham and Ćirić’s coupled fixed point results
In [8], putting (the identity mapping), we obtain the following result.
Theorem 2.5 (see Lakshmikantham and Ćirić’s [8])
Letbe a partially ordered set endowed with a metricd. Letbe a given mapping. Suppose that the following conditions hold:
(ii) Fhas the mixed monotone property;
(iii) Fis continuous orXhas the following properties:
() if a nondecreasing sequenceinXconverges to some point, thenfor alln,
() if a decreasing sequenceinXconverges to some point, thenfor alln;
(v) there existssuch that for allwithand,
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 with and , we have
and
This implies that for all with and ,
that is,
for all with . Here, is the metric on Y given by
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 be a metric space. A mapping is said to be ICS if S is injective, continuous and has the property: for every sequence in X, if is convergent, then is also convergent.
We have the following result.
Lemma 2.2Letbe a metric space andbe an ICS mapping. Then the mappingdefined by
is a metric onX. Moreover, ifis complete, thenis also complete.
Proof Let us prove that is a metric on X. Let such that . This implies that . Since S is injective, we obtain that . Other properties of the metric can be easily checked. Now, suppose that is complete and let be a Cauchy sequence in the metric space . This implies that is Cauchy in . Since is complete, is convergent in to some point . Since S is an ICS mapping, is also convergent in to some point . On the other hand, the continuity of S implies the convergence of in to Sx. By the uniqueness of the limit in , we get that , which implies that as . Thus is a convergent sequence in . This proves that is complete. □
The obtained result in [18] is the following.
Theorem 2.7 (see Luong and Thuan [18])
Letbe a partially ordered set endowed with a metricd. Letbe an ICS mapping. Letbe a given mapping. Suppose that the following conditions hold:
(ii) Fhas the mixed monotone property;
(iii) Fis continuous orXhas the following properties:
() if a nondecreasing sequenceinXconverges to some point, thenfor alln,
() if a decreasing sequenceinXconverges to some point, thenfor alln;
(v) there existssuch that for allwithand,
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 with and ,
that is,
for all with , where is the metric (see Lemma 2.2) on Y defined by
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])
Letbe a partially ordered set endowed with a metric d. Letbe a given mapping. Suppose that the following conditions hold:
(ii) Fhas the mixed monotone property;
(iii) Fis continuous orXhas the following properties:
() if a nondecreasing sequenceinXconverges to some point, thenfor alln,
() if a decreasing sequenceinXconverges to some point, thenfor alln;
(v) there exists a constantsuch that for allwithand,
ThenFhas a coupled fixed point. Moreover, if for allthere existssuch thatand, we have uniqueness of the coupled fixed point and.
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 with . Thus we proved that the mapping T satisfies the condition (iv) (resp. (v)) of Theorem 1.1 (resp. Theorem 1.2). Then T has a fixed point, which implies that F has a coupled fixed point. The rest of the proof is similar to the above proofs. □
2.6 Rasouli and Bahrampour’s coupled fixed point results
Theorem 2.11 (see Rasouli and Bahrampour [20])
Letbe a partially ordered set endowed with a metricd. Letbe a given mapping. Suppose that the following conditions hold:
(ii) Fhas the mixed monotone property;
(iii) Fis continuous orXhas the following properties:
() if a nondecreasing sequenceinXconverges to some point, thenfor alln,
() if a decreasing sequenceinXconverges to some point, thenfor alln;
(v) there existssuch that for allwithand,
ThenFhas a coupled fixed point. Moreover, if for allthere existssuch thatand, we have uniqueness of the coupled fixed point and.
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 with . Thus we proved that the mapping T satisfies the condition (iv) (resp. (v)) of Theorem 1.5 (resp. Theorem 1.6). Then T has a fixed point, which implies that F has a coupled fixed point. The rest of the proof is similar to the above proofs. □
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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