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Quadruple fixed point theorems in partially ordered metric spaces with mixed g-monotone property
Fixed Point Theory and Applications volume 2013, Article number: 147 (2013)
Abstract
We prove quadruple fixed point theorems in partially ordered metric spaces with mixed g-monotone property. Also, we state some examples to show that our results are real generalization of known ones in quadruple fixed point theory.
MSC:46T99, 54H25, 47H10, 54E50.
1 Introduction
In 1987, the notion of coupled fixed point was introduced by Guo and Lakshmikantham [1]. Later, Bhaskar and Lakshmikantham [2] introduced the concept of mixed monotone property for contractive operators of the form , where X is a partially ordered metric space, and then established some coupled fixed point theorems. They also illustrated these results by proving the existence and uniqueness of the solution for a periodic boundary value problem. Recently, Lakshmikantham and Ćirić in [3] defined a g-monotone property and proved coupled coincidence and coupled common fixed point results for nonlinear mappings satisfying certain contractive conditions in partially ordered metric spaces. They also proved related fixed point theorems. Many authors focused on coupled fixed point theory and proved remarkable results (see [4–17]).
Very recently, Berinde and Borcut [18] introduced the concept of triple fixed point and proved some tripled point theorems by virtue of mixed monotone mappings. Their contributions generalize and extend Bhaskar and Lakshmikantham’s research for nonlinear mappings. The notion of fixed point of order was first introduced by Samet and Vetro [19]. Karapinar used the concept of quadruple fixed point and proved some fixed point theorems on the topic [20]. Following this study, a quadruple fixed point is developed and some related fixed point theorems are obtained in [21–24]. Recently, Karapinar et al. [25] have proved a number of quadruple fixed point theorems under ϕ-contractive conditions for a mapping in ordered metric spaces.
Let us recall some basic definitions from [21].
Definition 1.1 (See [21])
Let X be a nonempty set and let be a given mapping. An element is called a quadruple fixed point of F if
Let be a metric space. The mapping , given by
defines a metric on , which will be denoted for convenience by d.
Definition 1.2 (See [21])
Let be a partially ordered set and let be a mapping. We say that F has the mixed monotone property if is monotone nondecreasing in x and z and is monotone non-increasing in y and w; that is, for any ,
In this article, we establish some quadruple coincidence and common fixed point theorems for and satisfying nonlinear contractions in partially ordered metric spaces. Also, some examples are given to support our results.
2 Preliminary
We start this section with the following definitions.
Definition 2.1 Let be a partially ordered set. Let and . The mapping F is said to have the mixed g-monotone property if for any ,
Definition 2.2 Let and . An element is called a quadruple coincidence point of F and g if
Definition 2.3 Let and . An element is called a quadruple common fixed point of F and g if
Definition 2.4 Let X be a nonempty set. Then we say that the mappings and are commutative if for all ,
Let Φ denote all the functions which satisfy that for all and .
Let Ψ denote all the functions which satisfy
-
(i)
if and only if ,
-
(ii)
ψ is continuous and nondecreasing,
-
(iii)
, .
Examples of typical functions ϕ and ψ are given in [4]. The aim of this paper is to prove the following theorem.
3 Main results
Now, we present the main results of this paper.
Theorem 3.1 Let be a partially ordered set and suppose there is a metric d on X such that is a complete metric space. Suppose that and are such that F is continuous and has the mixed g-monotone property. Assume also that there exist and such that
for any , for which , , , and . Suppose that , g is continuous and commutes with F. If there exist such that
then there exist such that
that is, F and g have a quadruple coincidence point.
Proof Let such that
Since , then we can choose such that
Taking into account , by continuing this process, we can construct sequences , , , and in X such that
We shall show that
For this purpose, we use the mathematical induction. Since , , , and , then by (2), we get
that is, (4) holds for . We presume that (4) holds for some . As F has the mixed g-monotone property and , , , and , we obtain
Thus, (4) holds for any . Assume, for some , that
then, by (3), is a quadruple coincidence point of F and g. From now on, assume for any that at least or , or , or .
Due to (1)-(4), we have
Due to (5)-(8), we conclude that
From the property (iii) of ψ, we have
Combining with (9) and (10), we get that
Set . Then we have
which yields that for all n.
Since ψ is nondecreasing, we get that for all n. Hence is a non-increasing sequence. Since it is bounded below from 0, there is some such that
We shall show that . Suppose, on the contrary, that .
Letting in (12) and having in mind that we suppose that for all and , we have
which is a contraction. Thus, , that is,
Now, we shall show that , , , and are Cauchy sequences in the metric space . Assume the contrary, that is, one of the sequences , , or is not a Cauchy sequence, that is,
This means that there exists , for which we can find subsequences , of and , of and , of and , of with such that
In addition, by virtue of , we can choose in such a way that it is the smallest integer with and satisfying (16). It follows that
By use of the triangle inequality, we have
Similarly, we get that
Adding both sides to (18), (19), (20), (21) and using (16) and (17), we have that
Letting and by use of (15), we get
Again, by the triangle inequality, we have
Since , then
Hence from (1), (3), and (22), we get that
Combining (22) and (24)-(27), we have that
Letting , we get a contradiction. This shows that , , , and are Cauchy sequences in the metric space . Since is complete, there exist such that
From (28) and the continuity of g, we have
It follows from (3) and the commutativity of F and g that
Now we shall show that , , , .
By letting in (30)-(33), by (28), (29), and the continuity of F, we obtain
We have shown that F and g have a quadruple coincidence point. □
In the following theorem, the continuity of F is removed. We state the following definition.
Definition 3.1 Let be a partially ordered metric space and d be a metric on X. We say that is regular if the following conditions hold:
-
(i)
if a nondecreasing sequence , then for all n,
-
(ii)
if a non-increasing sequence , then for all n.
Theorem 3.2 Let be a partially ordered set and suppose there is a metric d on X such that is regular. Suppose that and are such that F has the mixed g-monotone property. Assume also that there exist and such that
for any , for which , , , and . Suppose that , is a complete metric space. If there exist such that
then there exist such that
that is, F and g have a quadruple coincidence point.
Proof Proceeding exactly as in Theorem 3.1, we have that , , , and are Cauchy sequences in the complete metric space . Then there exist such that
Since , are nondecreasing and , are non-increasing, then since is regular, we get that
for all n. If , , , and for some , then , , , and , which implies that
that is, is a quadruple coincidence point of F and g. Then, we suppose that for all . By use of (1), consider now
Letting and by (38), then the right-hand side of (39) tends to 0, thus . By the property (i) of ψ, we have . It follows that . Similarly, we can show that
Therefore, we have proved that F and g have a quadruple coincidence point. □
Corollary 3.1 Let be a partially ordered set and suppose there is a metric d on X such that is a complete metric space. Suppose that and are such that F is continuous and has the mixed g-monotone property. Assume also that there exist and such that
for any , for which , , , and . Suppose that , g is continuous and commutes with F. If there exist such that
then there exist such that
that is, F and g have a quadruple coincidence point.
Proof Since
then we apply Theorem 3.1, since ψ is assumed to be nondecreasing. □
Similarly, as an easy consequence of Theorem 3.2, we have the following corollary.
Corollary 3.2 Let be a partially ordered set and suppose there is a metric d on X such that is regular. Suppose that and are such that F has the mixed g-monotone property. Assume also that there exist and such that
for any , for which , , , and . Suppose that , is a complete metric space. If there exist such that
then there exist such that
that is, F and g have a quadruple coincidence point.
Corollary 3.3 Let be a partially ordered set and suppose there is a metric d on X such that is a complete metric space. Suppose that and are such that F is continuous and has the mixed g-monotone property. Assume also that there exists such that
for any , for which , , , and . Suppose that , g is continuous and commutes with F. If there exist such that
then there exist such that
that is, F and g have a quadruple coincidence point.
Proof It is sufficient to set and in Theorem 3.1. □
Corollary 3.4 Let be a partially ordered set and suppose there is a metric d on X such that is regular. Suppose that and are such that F has the mixed g-monotone property. Assume also that there exists such that
for any , for which , , , and . Suppose that , is a complete metric space. If there exist such that
then there exist such that
that is, F and g have a quadruple coincidence point.
Proof It is sufficient to set and in Theorem 3.2. □
Corollary 3.5 Let be a partially ordered set and suppose there is a metric d on X such that is a complete metric space. Suppose that and are such that F is continuous and has the mixed g-monotone property. Assume also that there exists such that
for any , for which , , , and . Suppose that , g is continuous and commutes with F. If there exist such that
then there exist such that
that is, F and g have a quadruple coincidence point.
Proof It suffices to remark that
Then we apply Corollary 3.3. □
Corollary 3.6 Let be a partially ordered set and suppose there is a metric d on X such that is regular. Suppose that and are such that F has the mixed g-monotone property. Assume also that there exists such that
for any , for which , , , and . Suppose that , is a complete metric space. If there exist such that
then there exist such that
that is, F and g have a quadruple coincidence point.
Remark 3.1 (i) Theorem 11 of Karapinar and Luong [21] is a particular case of Theorem 3.1 and Theorem 3.2 by taking , respectively. Corollary 12 of Karapinar and Luong [21] is a particular case of Theorem 3.1 and Theorem 3.2 by taking , , .
(ii) Theorem 2.3 of Karapinar [22] is a particular case of Theorem 3.1 and Theorem 3.2 by taking and , respectively. Corollary 2.4 of Karapinar [22] is a particular case of Theorem 3.1 and Theorem 3.2 by taking , , .
Now, we shall prove the existence and uniqueness of a quadruple common fixed point. For a product of a partial ordered set , we define a partial ordering in the following way: For all ,
We say that and are comparable if
Also, we say that is equal to if and only if , , and .
Theorem 3.3 In addition to the hypotheses of Theorem 3.1, suppose that for all , there exists such that
is comparable to
and
Then F and g have a unique quadruple common fixed point such that
Proof The set of quadruple coincidence points of F and g is not empty due to Theorem 3.1. Assume now that and are two quadruple coincidence points of F and g, i.e.,
We shall show that and are equal. By assumption, there exists such that is comparable to and . Define sequences , , , and such that , , , , and for any ,
for all n. Further, set , , , and , , , , and in the same way define the sequences , , , and , , , . Then it is easy to see that
for all .
Since is comparable to
then it is easy to show . Recursively, we get that
and
From (44)-(47), it follows that
By the property (iii) of ψ, we obtain that
Set . Then due to (48), we have
which implies that . By the property of ψ, we obtain that . Thus, the sequence is decreasing and bounded below from 0. Therefore, there exists such that
Now, we shall show that . Suppose to the contrary that . Letting in (49), we obtain that
which is a contradiction. It yields that . That is, .
Consequently, we have
Similarly, we can prove that
Combining (50) and (51) yields that and are equal.
Since , , , and , by the commutativity of F and g, we obtain that
where , , , and . Thus, is a quadruple coincidence point of F and g. Therefore, and are equal. We obtain that
Thus, is a quadruple common fixed point of F and g. Its uniqueness follows from contraction (1). □
Example 3.1 Let with the metric for all and the usual ordering. Let and be given by
Let be given by
We will check that the condition (1) is satisfied for all satisfying , , , . In this case, we have
It is easy to check that all the conditions of Theorem 3.3 are satisfied and is the unique quadruple fixed point of F and g.
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Acknowledgements
This work is partially supported by the Scientific Research Fund of Sichuan Provincial Education Department (12ZA098), Scientific Research Fund of Sichuan University of Science and Engineering (2012KY08).
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Liu, Xl. Quadruple fixed point theorems in partially ordered metric spaces with mixed g-monotone property. Fixed Point Theory Appl 2013, 147 (2013). https://doi.org/10.1186/1687-1812-2013-147
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DOI: https://doi.org/10.1186/1687-1812-2013-147