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N-fixed point theorems for nonlinear contractions in partially ordered metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 111 (2013)
Abstract
In present paper we introduce the concept of a new g-monotone mapping and define the notions of n-fixed point and n-coincidence point and prove some related theorems for nonlinear contractive mappings in partially ordered complete metric spaces. Our results are generalization of the main results of Lakshmikantham and Ćirić (Nonlinear Anal. 70:4341-4349, 2009) and include several recent developments. Moreover, we give an example to support our results.
MSC:Primary 47H10; secondary 54H25; 34B15.
1 Introduction and preliminaries
The notion of a coupled fixed point is introduced by Bhaskar and Lakshmikantham [1]. Afterward Lakshmikantham and Ćirić in [2] extended this notion by defining the g-monotone property in partially ordered spaces. For other results on coupled coincidence and coupled common fixed point theory, we refer the readers to ([3–8]). Many authors obtained important results for usual coincidence and common fixed points in partially ordered spaces (see, for instance, [9–13]). Recently, Berinde and Borcut [14, 15] introduced the concept of a tripled fixed point. Other authors obtained important results in this area (see, for instance, [8, 9]). Very recently Eshaghi and Ramezani [16] introduced and investigated the concept of an n-fixed point (see also Def. 2.7 [4]).
From now, is a partially ordered complete metric space. Further, the product space has the following partial order:
We summarize in the following the basic notions and results established in [1, 2, 14].
Definition 1.1 (See [1])
A mapping is said to have the mixed monotone property if is monotone non-decreasing in x and is monotone non-increasing in y, that is, for any ,
Definition 1.2 (See [1])
An element is said to be a coupled fixed point of the mapping if and .
Theorem 1.3 (See [1])
Let be a mapping having the mixed monotone property on X. Assume that there exists with
Also suppose either
-
(a)
F is continuous, or
-
(b)
X has the following property:
-
(i)
If a non-decreasing sequence , then for all n;
-
(ii)
If a non-increasing sequence , then for all n.
If there exist such that and , then F has a coupled fixed point.
Inspired by Definition 1.1, Lakshmikantan and Ćirić [2] introduced the following concept of mixed g-monotone mappings.
Definition 1.4 (See [2])
Let and be mappings. F is said to have the mixed g-monotone property if F is monotone g-non-decreasing in its first argument and is monotone g-non-increasing in its second argument, that is, for any ,
It is clear that Definition 1.4 reduces to Definition 1.1 when g is an identity mapping.
Definition 1.5 (See [2])
An element is called a coupled coincidence point of the mapping and if and .
Definition 1.6 (See [2])
Let , be mappings. We say that F and g are commutative if for all .
Theorem 1.7 (See [2])
Assume that there is a function with and for each , and also suppose that and are mappings such that F has the mixed g-monotone property and
for all , for which and .
Suppose that , g is continuous and commutes with F, and also suppose that either
-
(a)
F is continuous, or
-
(b)
X has the following property:
-
(i)
If a non-decreasing sequence , then for all n;
-
(ii)
If a non-increasing sequence , then for all n.
If there exist such that and , then there exist such that and , i.e., F and g have a coupled coincidence point.
Theorem 1.8 (See [2])
In addition to the hypothesis of Theorem 1.7, suppose that for every , there exists such that is comparable to and . Then F and g have a unique coupled common fixed point, i.e., there exists a unique such that
Recently, Berinde and Borcut [14] introduced the following partial order on the product space :
where (see also [15]).
Definition 1.9 (See [14])
Let be a mapping. We say that F has the mixed monotone property if is monotone non-decreasing in x and z, and it is monotone non-increasing in y, i.e., for any ,
Definition 1.10 (See [14])
An element is called a tripled fixed point of if
Theorem 1.11 (See [14])
Let have the mixed monotone property on X. Assume that there exist constants with , for which
Also suppose either
-
(a)
F is continuous, or
-
(b)
X has the following property:
-
(i)
If a non-decreasing sequence , then for all n;
-
(ii)
If a non-increasing sequence , then for all n.
If there exist such that
then there exist such that
The following concept of an n-fixed point was introduced by Eshaghi and Ramezani [16]. We suppose, as in [16], that k is a positive integer (odd or even) and that the product space is endowed with following partial order: for ,
Definition 1.12 (See [16])
An element is called a k-fixed point of if
Theorem 1.13 (See [16])
Let be a continuous mapping having the mixed monotone property on X. Assume that there exist with and for all such that
for all (), for which for all and for all .
If there exist such that
for all and
for all , then F has a k-fixed point.
In this paper, we present the new k-fixed point, and by defining the notion of a new g-monotone mapping, the existence of a k-coincidence point and the uniqueness of a common k-fixed point are obtained. Our definitions are thoroughly different from the ones in [14, 15].
2 The main results
Definition 2.1 Let X be a non-empty set, and let be a given mapping (). An element is said to be a k-fixed point of the mapping F if
Definition 2.2 Let X be a non-empty set, and let and () be two given mappings. F is said to have the new g-monotone property if F is monotone g-non-decreasing in its first argument. That is, for any ,
Definition 2.3 Let X be a non-empty set, and let and () be two given mappings. An element is called a k-coincidence point of and if
Note that if g is an identity mapping, then Definition 2.3 reduces to Definition 2.1.
Definition 2.4 Let X be a non-empty set, and let and () be two given mappings. We say F and g are commutative if
Theorem 2.5 Let be a partially ordered complete metric space, and let and be two given mappings such that F has a new g-monotone property, g is continuous, and g commutes with F. Assume that there exists a continuous function , satisfying
-
(i)
for and ;
-
(ii)
for each ,
such that
for all , () so that for all and for all , and suppose there exist such that
Also suppose that either
-
(a)
F is continuous, or
-
(b)
X has the following property:
-
(i)
If a non-decreasing sequence , then for all n;
-
(ii)
If a non-increasing sequence , then for all n.
Then there exist such that
That is, F and g have a k-coincidence point.
Proof Since , we can find an element such that
We claim that
We prove (2.5) by induction. Note that by (2.2), (2.4) we have
and
Suppose that (2.5) is true for some n.
Due to the new g-monotone property of F, for , we have
and for , we have
Thus (2.5) is true. We denote
We will show that
By (2.1), (2.3) and (2.5), we have
Summing, we get
If for some n we have , then ; otherwise, for all , then
Hence is a non-increasing sequence which is bounded below (), then there exits some such that
We will show that . If for some n, , it is obvious; otherwise, suppose that . Keeping in mind that (for all ) and taking the limit as of both sides of (2.6), we have
which is contradiction. Thus , that is,
Now, we will show that for all is a Cauchy sequence. Suppose, on the contrary, that at least one of () is not Cauchy. So, there exists for which we can find sub-sequences , of with such that
We can choose , corresponding to , such that it is the smallest integer satisfying (2.8) and . Hence
Due to (2.8), (2.9) and by using the triangle inequality, we have
Taking in (2.10) and using (2.7), we have
That is a contradiction. Therefore for all are Cauchy sequences.
Since X is a complete metric space, there exist such that
for all . Due to the continuity of g, (2.11) implies that
By (2.4) and the commutativity of F and g, we have
for all . We will show that
We consider the following two cases.
Case I: The assumption (a) holds. Then by (2.4), (2.13) and (2.11), we have
for all . Thus (2.14) is proved.
Case II: The assumption (b) holds. Since is non-decreasing for all and , and also is non-increasing for all and , then by assumption (b) we have
for all n. Thus by (2.13), (2.1) and the triangle inequality,
for all . Taking the limit as , by (2.12) and the fact that , we get . Thus
Hence we proved that F and g have a k-coincidence point. □
Corrollary 2.6 Let and be a continuous mapping such that F has a new g-monotone property, and g commutes with F. Assume that there exists with
for all , () which for all and for all , and suppose that there exist such that
and suppose either
-
(a)
F is continuous, or
-
(b)
X has the following property:
-
(i)
If a non-decreasing sequence , then for all n;
-
(ii)
If a non-increasing sequence , then for all n.
Then there exist such that
That is, F and g have a k-coincidence point.
Proof It follows from Theorem 2.5 by putting for . □
Example 2.7 Let , , , , and let be defined by
for all . It is easy to check that F satisfies Corollary 2.6 by taking and . If k is an odd positive integer, then
is the k-fixed point of F, and if k is an even positive integer, then
is the k-fixed point of F.
Theorem 2.8 In addition to the hypothesis of Theorem 2.5, suppose that for every
there exists such that
is comparable to
and
Then F and g have a unique k-coincidence point, which is a fixed point of and a k-fixed point of . That is, there exists a unique such that
Proof By Theorem 2.5, the set of k-coincidence fixed points is nonempty. Now, suppose and are two coincidence fixed points of F and g, that is,
We will show that
By assumption, there exists such that
is comparable with
and
Let for all .
Since , we can choose such that for all . By a similar reason as in the proof of Theorem 2.5, we can inductively define sequences for all such that for all ,
In addition, let and for all and, in the same way, define the sequences and for all . Since
and
are comparable, then
Now, for all , we have
Then and are comparable for all .
It follows from (2.1) that
for all and
for all .
Summing, we get
It follows that
for all . Note that , , for imply that for all . Hence from (2.16) we have
Similarly, one can prove that
It follows from (2.17), (2.18) and the triangle inequality that
as for all . Hence , therefore (2.15) is proved.
Since for all , by the commutativity of F and g, we have
Denote for all . From (2.19), we have
Hence is a k-coincidence point of F and g.
It follows from (2.15) and that
This means that
Now, from (2.20) we have
Hence, is a k-fixed point of F and a fixed point of g.
To prove the uniqueness of the fixed point, assume that is another k-fixed point. Then by (2.15) we have
Thus . This completes the proof. □
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Acknowledgements
The authors are very grateful to the Spanish Government for its support of this research through Grant DPI2012-30651 and to the Basque Government for its support of this research through Grant IT378-10. They also grateful to the University of Basque Country for Grant UFI 2011/07.
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Paknazar, M., Eshaghi Gordji, M., De La Sen, M. et al. N-fixed point theorems for nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl 2013, 111 (2013). https://doi.org/10.1186/1687-1812-2013-111
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DOI: https://doi.org/10.1186/1687-1812-2013-111