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Tripled coincidence point theorems for a φ-contractive mapping in a complete metric space without the mixed g-monotone property
Fixed Point Theory and Applications volume 2013, Article number: 252 (2013)
Abstract
In this paper, we show the existence of a tripled coincidence point theorem and a tripled common fixed point theorem for a φ-contractive mapping in a complete metric space without the mixed g-monotone property, using the concept of -invariant set. Further, we apply our results to the existence of a tripled coincidence point of the given mapping in a partially ordered metric space.
1 Introduction
The existence of a fixed point for contraction type mappings in partially ordered metric spaces was first considered recently by Ran and Reurings [1] where they established some new results for contractions in partially ordered metric spaces and presented applications to matrix equations. Later, Nieto and Lopez [2, 3] and Agarwal et al. [4] presented some new results for contractions in partially ordered metric spaces.
The concept of coupled fixed point was introduced by Guo and Lakshmikantham [5]. In 2006, Bhaskar and Lakshmikantham [6] introduced the concept of mixed monotone property for contractive operators of the form , where X is a partially ordered metric space. They also established coupled fixed point theorems for mappings which satisfy the mixed monotone property. After the publication of this work, many authors conducted research on the coupled fixed point theory in partially ordered metric spaces and different spaces. For example, see [7–19].
Later, Sintunavarat et al. [15, 16] proved the existence and uniqueness of coupled fixed point theorems for nonlinear contractions without the mixed monotone property and extended some coupled fixed point theorems of Bhaskar and Lakshmikantham [6] by using the concept of F-invariant set due to Samet and Vetro [10]. Recently, in 2013, Batra and Vashistha [17] introduced the concept of -invariant set which is a generalization of an F-invariant set introduced by Samet and Vetro [10] and proved the existence coupled fixed point theorems for nonlinear contractions under c-distance in cone metric spaces having an -invariant subset.
Tripled fixed point theorems for nonlinear mappings can be viewed as a generalization of coupled fixed point theorems and those can be applied to ODE problems. In 2011, Berinde and Borcut [20] introduced the concept of tripled fixed point for nonlinear mappings in partially ordered complete metric spaces and obtained existence and uniqueness theorems for contractive type mappings. Later, in 2012, Berinde and Borcut [21] introduced the concept of tripled coincidence point for a pair of nonlinear contractive mappings and and obtained tripled coincidence point theorems which generalized the results of [20]. After the publication of this work, the tripled fixed point and tripled coincidence point theory has been generalized in different directions in several spaces with applications by many mathematicians over the year (see [11, 12, 22–34]).
Since 2011, some authors have studied triple fixed point and tripled coincidence point theory. This theory has received much attention in different spaces. Aydi et al. [30] established some tripled coincidence point results for a mixed g-monotone mapping satisfying -contractions in ordered generalized metric spaces. Aydi et al. [27] introduced the concept of W-compatible mappings for mappings and , where is an abstract metric space, and established some tripled coincidence and common tripled fixed point theorems in such spaces. Based on the notion of W-compatible, Aydi and Abbas [28] obtained tripled coincidence and common tripled fixed point results in the partial metric spaces which generalized and extended several well-known comparable results in the existing literature. Afterward, Abbas et al. [23] proved the existence of tripled fixed point and tripled coincidence point theorems in intuitionistic fuzzy normed spaces. The results generalized and extended recent coupled fixed point theorems in intuitionistic fuzzy normed spaces.
On the other hand, a wide discussion on a tripled fixed point and a tripled coincidence point in partially ordered metric spaces has been dedicated to improvement and generalization. In 2011, Abbas et al. [22] obtained the existence of a tripled fixed point of multivalued nonlinear contraction mappings in partially ordered metric spaces. Amini-Harandi [11] introduced a new simple and unified approach to coupled and tripled fixed point theory and presented a new tripled fixed point result in partially ordered metric spaces. Later, Hussain et al. [12] introduced a new and simple approach to coupled and tripled coincidence point theory. They established coupled coincidence and tripled coincidence point results without any type of commutativity condition on F and g.
One of the simplest ways to develop tripled fixed point and tripled coincidence point theory in partially ordered metric spaces is to consider contraction mappings. Using the concept of mixed monotone, Aydi et al. [25] discussed existence and uniqueness of some tripled fixed points for mappings having the mixed monotone property in partially ordered metric spaces by using Meir-Keeler type contractions. Furthermore, Aydi and Karapinar [26] proved some new Meir-Keeler type tripled fixed point theorems on partially ordered complete partial metric spaces. Aydi and Karapinar [29] established some triple fixed point theorems for mappings having the mixed monotone property in partially ordered metric spaces dependent on another contractive condition which is a generalization of the main results of Berinde and Borcut [20]. Charoensawan [33] proved the existence and uniqueness of a tripled fixed point involving a -contractive condition for mappings having the mixed monotone property in partially ordered metric spaces.
In the case of tripled coincidence point theory, using the mixed g-monotone property, Borcut [31] established tripled coincidence point theorems for a pair of mappings and satisfying a nonlinear contractive condition and having the mixed g-monotone property in partially ordered metric spaces. The presented theorems extended existing results in the literature. Recently, Choudhury et al. [32] established some tripled coincidence point results in partially ordered metric spaces dependent on other contractions. Very recently, Aydi et al. [24] established tripled coincidence point theorems for a pair of mappings and satisfying weak ϕ-contractions in partially ordered metric spaces. The results unified, generalized and complemented various known comparable results from Berinde and Borcut [21].
The purpose of this paper is to establish some tripled coincidence point theorems without the mixed g-monotone property by using the concept of -invariant set in a complete metric space which are generalizations of the results of Aydi et al. [24].
2 Preliminaries
Let be a partially ordered set, and suppose that there is a metric d on X such that is a complete metric space. Consider on the product space the following partial order: for ,
Berinde and Borcut [21] introduced the concept of tripled coincidence point and studied existence and uniqueness theorems in partially ordered complete metric spaces.
Definition 2.1 ([21])
Let be a partially ordered set and two mappings , . We say that F has the mixed g-monotone property if for any ,
and
Definition 2.2 ([21])
An element is called a tripled coincidence point of mappings F and g if , and .
Definition 2.3 ([21])
Let X be a nonempty set and , be two mappings. We say F and g are commutative if for all .
Later, Aydi et al. [24] extended the tripled coincidence point theorems for a mixed g-monotone operator obtained by Berinde and Borcut [21]. For the sake of completeness, we recollect the main results of Aydi et al. [24] here.
Let the set of functions .
Theorem 2.4 ([24])
Let be a partially ordered set, and suppose that there is a metric d on X such that is a complete metric space. Let and be such that F has the mixed g-monotone property and . Assume that there is a function such that
for all with , and . Assume that F is continuous, g is continuous and commutes with F.
If there exist such that
then there exist such that
Definition 2.5 ([24])
Let be a partially ordered set and d be a metric on X. We say that is regular if the following conditions hold:
-
(i)
if a non-decreasing sequence in X, then for all n,
-
(ii)
if a non-increasing sequence in X, then for all n.
Theorem 2.6 ([24])
Let be a partially ordered set, and suppose that there is a metric d on X such that is regular. Suppose that there exist and mapping and are such that (5) hold for any with , and . Suppose also that is complete, F has the mixed g-monotone property and .
If there exist such that
then there exist such that
Batra and Vashistha [17] introduced an -invariant set which is a generalization of an F-invariant set introduced by Samet and Vetro [10].
Definition 2.7 ([13])
Let be a metric space and , be given mappings. Let M be a nonempty subset of . We say that M is an -invariant subset of if and only if, for all ,
-
(i)
,
-
(ii)
.
In this article, we establish tripled coincidence point theorems for and satisfying nonlinear contractive conditions without the mixed g-monotone property by using the concept of -invariant set in complete metric spaces.
3 Main results
We give the notions of F-invariant set and -invariant set which are useful for main results.
Definition 3.1 Let be a metric space and be a given mapping. Let M be a nonempty subset of . We say that M is an F-invariant subset of if and only if, for all ,
Definition 3.2 Let be a metric space and M be a subset of . We say that M satisfies the transitive property if and only if, for all ,
Definition 3.3 Let be a metric space and , be given mappings. Let M be a nonempty subset of . We say that M is an -invariant subset of if and only if, for all ,
Remark 3.4
-
1.
The set is trivially -invariant, which satisfies the transitive property.
-
2.
Every F-invariant set is -invariant, when denotes an identity map on X.
Example 3.5 Let be a partially ordered set, and suppose that there is a metric d on X such that is a complete metric space. Let and be two mappings such that F satisfies mixed g-monotone property. Define a subset by . Then M is an F-invariant and -invariant subset of , which satisfies the transitive property.
The following example shows that we may have an -invariant set which is not F-invariant set.
Example 3.6 Let and be defined by . Let be given by . Then it is easy to show that is an -invariant subset of but not an F-invariant subset of as , but .
Our first main result is the following.
Theorem 3.7 Let be a complete metric space and M be a nonempty subset of . Assume that there is a function with and for each , and also suppose that and are two continuous functions such that
for all with or . Suppose that , g commutes with F.
If there exists such that
and M is an -invariant set which satisfies the transitive property, then there exist such that
Proof Let satisfy . Since , we can choose such that , and . Again from we can choose such that , and .
Continuing this process, we can construct sequences , and in X such that
If there exists such that , and , then , and . Thus, is a tripled coincidence point of F. This finishes the proof.
Therefore, we may assume that or , or for all . Since = ∈ M and M is an -invariant set, we have = ∈ M.
By repeating this argument, we get = ∈ M for all .
Consider now the sequence of nonnegative real numbers given by
Since , we have
The right-hand side of (6) will be equal to
Therefore, the sequence satisfies
From (10) it follows that the sequence is decreasing. Therefore, there exists some such that
We shall prove that . Assume, to the contrary, that . Then, by letting in (10), from and , for each , we have
a contradiction. Thus and hence
We now prove that , and are Cauchy sequences in . Suppose, to the contrary, that at least one of , or is not a Cauchy sequence. Then there exists for which we can find subsequences and of , and of and and of with such that
Further, corresponding to , we can choose in such a way that it is the smallest integer with and satisfies (14). Then
Using (14) and (15) and the triangle inequality, we have
Letting in the above inequality and using (13), we get
that is,
Since and M satisfies the transitive property, we get
Hence we can use (6) to obtain
By (18), we have
Letting in (19) and using (13) and (17) and for each , we have
which is a contradiction. This shows that , and are Cauchy sequences. Since X is a complete metric space, there exist such that
From (20) and continuity of g,
From (7) and commutativity of F and g,
We now show that , and . Taking the limit as in (22), (23) and (24), by (20), (21), continuity of F and commutativity of F and g, we get
and
Thus we prove that , and . □
Theorem 3.8 Let be a complete metric space and M be a nonempty subset of . Assume that there is a function with and for each , and also suppose that and are two continuous functions such that
for all with or . Suppose that , g commutes with F and is a complete metric space and for any three sequences , , with , , and for all imply for all . If there exists such that and M is an -invariant set which satisfies the transitive property, then there exist such that
Proof Consider the Cauchy sequences , and as in the proof of Theorem 3.7. Since is a complete metric space, there exist such that , and . By assumption, we have for all ; by the triangle inequality and (6), we get
Taking the limit as in the inequality above and using the fact that , we obtain
which implies that , and . □
Example 3.9 Let , and be defined by
The mapping F does not satisfy the mixed g-monotone property.
Let and , we have
and
We get
When and , we have
It is easy to check that F satisfies (6) and is the tripled coincidence point of F.
If we take the mapping in Theorem 3.7 and Theorem 3.8, then we get the following.
Corollary 3.10 Let be a complete metric space and M be a nonempty subset of . Assume that there is a function with and for each , and also suppose that is a mapping such that
for all with or . Suppose that either
-
(a)
F is continuous, or
-
(b)
if for any three sequences , , with , , and for all , then for all .
If there exists such that and M is an F-invariant set which satisfies the transitive property, then there exist such that
Now we shall prove the uniqueness of a tripled fixed point.
Theorem 3.11 In addition to the hypotheses of Theorem 3.7, suppose that for every , there exists such that and . Then F and g have a unique tripled common fixed point, that is, there exists a unique such that
Proof From Theorem 3.7, the set of tripled coincidence points is nonempty. Suppose and are tripled coincidence points of F, that is, , , , , and .
We shall show that
By assumption, there is such that and .
Put , , and choose .
So that , and . Then, similarly as in Theorem 3.7, we can inductively define sequences , and such that
It easy to show that
Thus, from (6) and (27), we have
for all . Thus, by (28), we deduce that the sequence is defined by
From (28) and (29), we have that is non-increasing. Hence, there exists such that
Suppose, to the contrary, that . Letting in (28), we get
which is a contradiction. Thus , that is,
which implies
Similarly, we obtain that
By (30) and (31) and the triangle inequality, we have
Let in (32). Hence , and . Since , and , by commutativity of F and g, we have
Denote , and . Then from (33),
Thus is a tripled coincidence point. Then from (26) with , and it follows that , and , that is,
From (34) and (35), we have
Therefore, is a tripled common fixed point of F and g. To prove the uniqueness, assume that is another coupled common fixed point. Then by (26) we have
□
Next, we give a simple application of our results to tripled coincidence point theorems in partially ordered metric spaces.
Corollary 3.12 Let be a partially ordered set, and suppose that there is a metric d on X such that is a complete metric space. Assume that there is a function with and for each , and also suppose that and are such that F has the mixed g-monotone property and
for all with , and . Suppose that , g is continuous and commutes with F and is a complete metric space and X has the following properties:
-
(i)
if a non-decreasing sequence in X, then for all n,
-
(ii)
if a non-increasing sequence in X, then for all n.
If there exist such that
then there exist such that
Proof Define a subset by , then M is an -invariant set which satisfies the transitive property. By (36), we have
for all with or . Since such that
we get . For any three sequences , , such that is a non-decreasing sequence in X with , is a non-increasing sequence in X with and is a non-decreasing sequence in X with , we have
and
for all . Therefore, we have for all , and so the assumption of Theorem 3.8 holds, thus F has a triple coincidence point. □
Corollary 3.13 In addition to the hypotheses of Corollary 3.12, suppose that for all , there exists such that , , and , , . Then F and g have a unique tripled common fixed point.
Proof Define a subset by , then M is an -invariant set which satisfies the transitive property. Thus, the proof of the existence of a tripled common fixed point is straightforward by following the same lines as in the proof of Corollary 3.12.
Next, we show the uniqueness of a tripled common fixed point. Since for all , there exists such that , , and , , , we can conclude that and . Therefore, since all the hypotheses of Theorem 3.11 hold, F and g have a unique tripled common fixed point. This completes the proof. □
Remark This research can be extended and improved to the more general case; see [9, 35–37].
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Acknowledgements
This research was supported by Chiang Mai University and the author would like to express sincere appreciation to Prof. Suthep Suantai and the referees for very helpful suggestions and many kind comments.
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Charoensawan, P. Tripled coincidence point theorems for a φ-contractive mapping in a complete metric space without the mixed g-monotone property. Fixed Point Theory Appl 2013, 252 (2013). https://doi.org/10.1186/1687-1812-2013-252
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DOI: https://doi.org/10.1186/1687-1812-2013-252