Abstract
The purpose of this paper is to present some multidimensional fixed point theorems for isotone mappings in a complete metric space endowed with a partial order. Our results include the corresponding unidimensional, coupled, tripled, and k-dimensional fixed point theorems as particular cases.
MSC:47H10, 54H25.
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1 Introduction
A very recent trend in metrical fixed point theory, initiated by Ran and Reurings [1] and continued by Nieto and Rodriguez-López [2, 3], is to consider a partial order on the ambient metric space and to transfer a part of the contractive property of the nonlinear operators into its monotonicity properties. Their results were extended in other contexts: L-spaces [4], probabilistic metric spaces [5], and metric spaces with a graph [6]. Some fixed point theorems are proved for a mixed monotone mapping in a metric space endowed with partial order, and the obtained results found important applications to the existence of solutions for matrix equations or ordinary differential equations and integral equations; see [7–20] and the references therein.
In 2006, Bhaskar and Lakshmikantham [7] introduced the notion of coupled fixed point and proved some fixed point theorems under certain conditions. Later, Berinde and Borcut [8] introduced the concept of tripled fixed point and proved some related theorems. These results were then extended and generalized by several authors in the last years; see [9–20] and the references therein. Amongst these generalizations, we refer to the one obtained by Berinde [9], who considered the more general contractive condition
where are functions satisfying some appropriate conditions, and , .
Currently, the authors of [21–23] obtained some k-dimensional fixed point theorems for mixed monotone mappings, which extended the corresponding coupled, tripled and quadrupled fixed point theorems. In order to guarantee the existence of k-dimensional fixed points, the authors proved that more than one sequences are simultaneously Cauchy. Besides, some authors focused on obtaining some fixed point theorems for a monotone mapping in the context of ordered metric space; see [24, 25] and the references therein. In 2010, Harjani and Sadarangani [24] proved some fixed point theorems for a one-variable monotone mapping in a partially ordered metric space.
It is well known that the fixed point problems for isotone mappings are easier than that of mixed monotone mappings. So, the above results bring us to the following natural question: can we obtain some multidimensional fixed point theorems for a k-variable isotone mapping, which include the unidimensional, coupled, tripled, and k-dimensional fixed point theorems as particular cases?
Motivated and inspired by the above results, we establish some fixed point theorems for a k-variable isotone mapping, which include the unidimensional, coupled, tripled, and k-dimensional fixed point theorems as particular cases. Moreover, we show that some control conditions in the recent corresponding literature are not necessary.
2 Preliminaries
In order to fix the framework needed to state our main results, we recall the following notions. For simplicity, we denote from now on by , where and X is a non-empty set. If elements x, y of a partially ordered set are comparable (i.e. or holds), we will write . Let be a partition of the set , that is, and , and . Henceforth, let be k mappings from into itself, and let ϒ be the k-tuple .
Definition 2.1 ([12])
Let be a partially ordered set and d be a metric on R. We say that is regular if the following conditions hold:
-
(i)
if a non-decreasing sequence is such that , then for all n,
-
(ii)
if a non-increasing sequence is such that , then for all n.
Definition 2.2 (Altering distance function [26])
A function is called an altering distance function if the following properties are satisfied:
-
(i)
ψ is continuous and non-decreasing,
-
(ii)
if and only if .
Definition 2.3 ([7])
Let be a partially ordered set and . We say F has the mixed monotone property if F is monotone non-decreasing in its first argument and is monotone non-increasing in its second argument, that is, for any ,
and
Definition 2.4 ([8])
Let be a partially ordered set and . We say that F has the mixed monotone property if is monotone non-decreasing in x, it is monotone non-increasing in y and is monotone non-decreasing in z, that is, for any ,
and
Definition 2.5 ([22])
Let be a partially ordered space. We say that F has the mixed g-monotone property if F is monotone non-decreasing in arguments of A and monotone non-increasing in arguments of B, i.e., for all and all i
Definition 2.6 Let be a mapping. A point is:
-
(i)
a coupled fixed point [7] if , and ;
-
(ii)
a tripled fixed point [8] if , , and ;
-
(iii)
a ϒ-fixed point [23] of F if for .
Let be a partially ordered set and d be a metric on X such that is a metric space. We use the next notation from [22]:
The product space is endowed with the following natural partial order: for ,
which will be denoted in the sequel, for convenience, by ≤, also. Obviously, is a partially ordered set. In particular, we denote by A the odd numbers in and by B its even numbers if . The mapping , given by
where , defines a metric on . It is easy to see that
where .
Let Φ denote the set of all continuous and strictly increasing functions , and Ψ denote the set of all functions such that for all .
Inspired by Definitions 2.3-2.6, we give Definitions 2.7 and 2.9.
Definition 2.7 Let be a partially ordered set and T be a self-mapping on . It is said that T is an isotone property if, for any ,
Remark 2.8 Note that if in Definition 2.7, then T is a non-decreasing mapping (see [2]).
Definition 2.9 An element is called a fixed point of the mapping if .
In order to prove our main results, we need the following lemmas.
Lemma 2.10 ([27])
Let be a partially ordered set and d be a metric on X. If is regular, then is regular.
3 Main results
Now, we state and prove our main results.
Theorem 3.1 Let be a partially ordered set and suppose that there is a metric d on X such that is a complete metric space. Let be an isotone mapping for which there exist and such that, for all with ,
where is defined by (2.2). Suppose either
-
(a)
T is continuous or
-
(b)
is regular.
If there exists such that , then T has a fixed point.
Proof Consider the Picard iteration associated to T and the initial approximation , that is, the sequence defined by for . Obviously, if for some , then is a fixed point of T. So, we may assume that for every .
Since , without loss of generality, we assume that (the case is treated similarly), that is, . Since T is an isotone mapping, we see that the sequence is non-decreasing. Taking and in (3.1), we obtain
In view of the fact that ,
Since φ is strictly increasing, we have
Hence, the sequence given by is monotone decreasing and bounded below. Therefore, there exists some such that . We shall prove that . Assume that . Then by letting in (3.2) and using the properties of φ and ψ, we have
which is a contradiction. Thus,
We claim that is a Cauchy sequence. Indeed, if not, then there would exist an and subsequences and of such that is the minimal in the sense that and . Therefore, .
Using the triangle inequality, we obtain
Letting in the above inequality and using (3.3), we get
Since , we have . Hence, using (3.1) with and , we obtain
Letting in the above inequality and using (3.4), we have
where , which is a contradiction. Hence, the sequence is a Cauchy sequence in the metric space . Since is a complete metric space, by (2.2), we find that is complete. Therefore, there exists such that .
Now suppose that (a) holds. It follows from that is a fixed point of T, that is, .
Suppose that (b) holds. Using Lemma 2.10, we find that is regular. Thus, by is non-decreasing sequence that converges to , we have for all . From (3.1) and the fact that , we obtain
for all . From (3.5) and the strict monotonicity of φ, we have
Letting in (3.6) and using , we get and so , which implies . □
Taking in Theorem 3.1, we obtain the following result immediately.
Corollary 3.2 Let be a partially ordered set and suppose that there is a metric d on X such that is a complete metric space. Let be a non-decreasing mapping for which there exist and such that, for all with ,
Suppose either
-
(a)
T is continuous or
-
(b)
is regular.
If there exists such that , then T has a fixed point.
Remark 3.3 Corollary 3.2 includes Theorems 2.1 and 2.2 in [24] as particular cases. Note that in [24] the authors use only condition , although the alternative assumption is also acceptable. In addition, the condition in [24], if and only if , is not necessary in Corollary 3.2. Moreover, the control condition is weaker than the condition in [24] (i.e., ψ is an altering distance function).
Taking and for in Theorem 3.1, we obtain the following result.
Corollary 3.4 Let be a partially ordered set and suppose that there is a metric d on X such that is a complete metric space. Let be a mixed monotone mapping for which there exist and such that, for all with , ,
Suppose either
-
(a)
F is continuous or
-
(b)
is regular.
If there exist such that
or
then there exist such that and .
Proof Now we use the result of Theorem 3.1 () to prove Corollary 3.4. For simplicity, we denote , , , and for all . The product space is endowed with the following partial order:
Consider the function defined by
Obviously, and are two particular cases of and defined by (2.1) and (2.2), respectively.
Let the operator be defined by
Now we show that T is an isotone mapping. Suppose that for . From (3.10), we have , . Thus, by the mixed monotonicity of F, we have
for any . From (3.10), (3.12), and (3.13), we have
for any . Therefore, T is an isotone mapping.
From (3.11) and (3.12), we have
and
for . Thus, by (3.7), we have , . By (3.8)-(3.10), there exists such that .
Now suppose that (a) holds. It is easy to see that T is continuous. Indeed, let be a sequence in such that (as ), where . From (3.11) and (2.3), we have and as . Since F is continuous, this implies and as . Thus, by (3.11) and (3.12), we have
which implies that T is continuous.
Hence, there is no doubt that all conditions of Theorem 3.1 are satisfied. Hence, T has a fixed point . From (3.12), we have and . This completes the proof. □
Remark 3.5 It is worth pointing out that we omit the control conditions: for all , for all and , which are necessary in the proof of Theorem 2 in Berinde [9].
Taking and for in Theorem 3.1, by a similar argument to the proof of Corollary 3.4, we obtain the following result.
Corollary 3.6 Let be a partially ordered set and suppose that there is a metric d on X such that is a complete metric space. Let be a mixed monotone mapping for which there exist and such that, for all with , , ,
Suppose either
-
(a)
F is continuous or
-
(b)
is regular.
If there exist such that
or
then there exist such that
that is, F has a tripled fixed point.
Theorem 3.7 Let be a partially ordered set and suppose that there is a metric d on X such that is a complete metric space. Let be an isotone mapping for which there exists a constant such that, for all with ,
where is defined by (2.2). Suppose either T is continuous or is regular. If there exists such that , then T has a fixed point.
Proof Taking and for and in Theorem 3.1, we obtain the above result immediately. □
Remark 3.8 The metric in Theorem 3.1 can be replaced by some other metrics on , for example, by the next one:
where and , and the result will also be true.
Using a similar argument to the proof of Corollaries 3.4 and 3.6, we deduce the following corollaries from Theorem 3.7, respectively.
Corollary 3.9 Let be a partially ordered set and suppose that there is a metric d on X such that is a complete metric space. Let be a mixed monotone mapping for which there exists a constant such that, for each , ,
Suppose either F is continuous or is regular. If there exist such that (3.8) or (3.9), then there exist such that and .
Corollary 3.10 Let be a partially ordered set and suppose that there is a metric d on X such that is a complete metric space. Let be a mixed monotone mapping for which there exists a constant such that, for each , , ,
Suppose either F is continuous or is regular. If there exist such that (3.14) or (3.15), then F has a tripled fixed point.
Now we give a k-dimensional fixed point theorem for mixed monotone mappings.
Corollary 3.11 Let be a partially ordered set and suppose that there is a metric d on X such that is a complete metric space. Let be a k-tuple of mapping from into itself such that if and if . Let be a mixed monotone mapping. Assume that there exists verifying
for which for . Suppose either F is continuous or is regular. If there exist verifying
then F has one ϒ-fixed point.
Proof For brevity, , , and will be denoted by Y, V and , respectively. Consider the mapping defined by
for .
Now, we show that T is an isotone mapping. Indeed, suppose that for . By (2.1), we have when and when . For each , we have . So, , and , . Thus, by the mixed monotonicity of F, we have, for fixed ,
when . Similarly, if , then the inequality (3.21) holds for fixed . So, for fixed , inequality (3.21) holds for . From this, we have
for . Similarly, we have
for . From (3.20), (3.22), and (3.23), we deduce that T is an isotone mapping.
Suppose that for . For fixed , we have for . From (3.18), we have
for all . Similarly, for fixed , we have for . It follows from (3.18) that
for all . By (2.1), (3.20), (3.24), and (3.25), we have , where is defined by (3.17). From (3.19), there exists such that . Using the same argument as in the proof of Corollary 3.4, we deduce that T is continuous. Using Theorem 3.7, we see that T has a fixed point. So, F has a ϒ-fixed point. □
Remark 3.12 In order to obtain the multidimensional fixed point theorems, the authors of [21–23] constructed some Cauchy sequences using the properties of multidimensional mixed monotone mappings. To prove that more than one sequence is simultaneously Cauchy does not seem so easy. It is also known that the fixed point problems for isotone mappings are easier than for mixed monotone mappings. Using the properties of isotone mappings, we obtain some multidimensional fixed point theorems by constructing only one Cauchy sequence. As an application, we give a simple proof of Corollary 3.11, which is similar to Theorem 9 in [22].
Theorem 3.13 In addition to the hypotheses of Theorem 3.1, suppose that, for all fixed points of T, there exists such that Z is comparable to and to . Then T has a unique fixed point.
Proof From Theorem 3.1, the set of fixed points of T is non-empty. Assume that and are two fixed points of T. Put and for . Since Z is comparable to , we may assume . Since T is an isotone mapping, we obtain inductively for . Therefore, by (3.1), we have
which, by the fact that , implies
By (3.27) and the strict monotonicity of φ, we see that the sequence defined by is non-increasing. Hence, there exists such that . We shall prove that . Suppose, conversely, that . Letting in (3.26), we get
which is a contradiction. Thus , that is,
Similarly, we obtain
Combining (3.28) and (3.29) yields . □
Using a similar argument to the proof of Corollary 3.4, we deduce the following corollaries from Theorem 3.13, respectively.
Corollary 3.14 In addition to the hypotheses of Corollary 3.4 (or Corollary 3.9), suppose that, for all coupled fixed points of F, there exists such that Z is comparable to and to . Then F has a unique coupled fixed point.
Corollary 3.15 In addition to the hypotheses of Corollary 3.6 (or Corollary 3.10), suppose that, for all tripled fixed points of F, there exists such that Z is comparable to and to . Then F has a unique tripled fixed point.
Corollary 3.16 In addition to the hypotheses of Corollary 3.11, suppose that, for all ϒ-fixed points of F, there exists such that Z is comparable to and to . Then F has a unique tripled fixed point.
Remark 3.17 In this paper, the results show how to extend unidimensional fixed point results to the multidimensional case and give a simple and unified approach to some coupled, tripled, and k-dimensional fixed points. It is worth pointing out that we extend the interesting technique of proof in [10] from to k arbitrary. The results in [19] and [20] are the unidimensional and coupled cases, respectively.
Author’s contributions
SW completed the paper herself. The author read and approved the final manuscript.
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The author thanks the editor and the referees for their useful comments and suggestions. This work was supported by the Natural Science Foundation of Jiangsu Province under Grant (13KJB110028).
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Wang, S. Multidimensional fixed point theorems for isotone mappings in partially ordered metric spaces. Fixed Point Theory Appl 2014, 137 (2014). https://doi.org/10.1186/1687-1812-2014-137
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DOI: https://doi.org/10.1186/1687-1812-2014-137