Abstract
Keywords:
fixed point theorems; nonlinear contraction; partially ordered metric space; ntuplet fixed point; mixed gmonotone1 Introduction
As it is known, fixed point theory is one of the oldest and most famous theory in mathematics, and it has become an important tool for other areas of science such as approximation theory, statistics, engineering and economics.
Among hundreds of fixed point theorems, the Banach contraction theorem [1] is particularly well known due to its simplicity and usefulness. It states that any contraction mapping of a complete metric space has a unique fixed point.
In 2004, the Banach contraction principle were extended to metric space endowed with partial order by Ran and Reuring [2]. They pointed out that the contractivity condition on the nonlinear and monotone map is only assumed to hold on elements which are comparable in the partial order. Afterward, Nieto and RodriguezLopez [3] extended results of Ran and Reuring for nondecreasing mapping and studied existence and uniqueness of firstorder differential equations.
In 2006, by following the above mentioned trend, Bhaskar and Lakshmikantham [4] introduced mixed monotone property and gave their coupled fixed point theorem for mappings with mixed monotone property. Also, they produced some applications related with the existence and uniqueness of solution for a periodic boundary value problem. This work of Bhaskar and Lakshmikantham has attracted the attention of many researchers. The concept of coupled fixed point for various contractive type mappings was studied by several authors [510]. Lakshmikantham and Ciric [11] extended the results of [4] for monotone nonlinear contractive mapping and generalized mixed monotone concept. Berinde and Borcut [12] introduced tripled fixed point theorem for nonlinear mapping in partially ordered complete metric space as a generalization and extension of the coupled fixed point theorem.
Motivated by these studies, the quadruple fixed point theorem was given for different contractive type mappings [1316].
In this paper, we generalize mentioned trend in the above for an arbitrary positive number n, that is, we introduce the concept of ntuplet fixed point theorem and prove some results.
2 Main results
Let us give new definitions for our aim.
Definition 1 Let be partially ordered set and . We say that F has the mixed monotone property if is monotone nondecreasing in its odd argument and it is monotone nonincreasing in its even argument. That is, for any
Definition 2 Let X be a nonempty set and a given mapping. An element is called a ntuplet fixed point of F if
Definition 3 Let be partially ordered set and and . We say that F has the mixed gmonotone property if is monotone gnondecreasing in its odd argument and it is monotone gnonincreasing in its even argument. That is, for any
Note that if g is the identity mapping, this definition reduces to Definition 1.
Definition 4 Let X be a nonempty set and a given mapping. An element is called a ntuplet coincidence point of and if
Note that if g is the identity mapping, this definition reduces to Definition 2.
Definition 5 Let be partially ordered set and and . F and g called commutative if
Let Φ denote the all functions , which are continuous and satisfy that
Since we want to shorten expressions in the following theorem, consider Condition 1 in the following for X an F.
Condition 1 Suppose either
(i) F is continuous, or
(ii) X has the following property:
(a) if nondecreasing sequence , then for all k,
(b) if nonincreasing sequence , then for all k.
Theorem 1Letbe partially ordered set and suppose thatis complete metric space. Assumeandare such thatFhas the mixedgmonotone property and
for allfor which, , … , (ifnis odd), (ifnis even). Assume thatandgcommutes withF. Also, suppose that Condition 1 is satisfied. If there existsuch that
that is, Fandghave antuplet coincidence point.
Proof Let be such that (2.7). Since , we construct the sequence as follows:
for all . For this, we will use the mathematical induction. The inequalities in (2.9) hold because of (2.7), that is, we have
Thus, our claim is true for . Now, suppose that the inequalities in (2.9) hold . In this case,
Now, we must show that the inequalities in (2.9) hold . If we consider (2.8) and mixed gmonotone property of F together with (2.10), we have
Thus, (2.9) is satisfied for all . So, we have,
For the simplicity, we define
We will show that
By (2.6), (2.8) and (2.11), we get
Due to (2.13)(2.15), we conclude that
Hence, we get (2.12).
Since for all , then for all . So, is monotone decreasing. Since it is bounded below, there is some such that
We want to show that . Suppose that . Then taking the limit as of both sides of (2.12) and keeping in mind that we assume that for all , we have
which is a contradiction. Thus, , that is
Now we prove that are Cauchy sequences. Suppose that at least one of is not Cauchy. So, there exists an for which we can find subsequence of integer , with such that
Additionally, corresponding to , we can choose such that it is the smallest integer satisfying (2.20) and . Thus,
By using triangle inequality and having (2.20) and (2.21) in mind
Letting in (2.22) and using (2.20)
We apply triangle inequality to (2.20) as the following.
So, from (2.25), (2.8) and (2.6), we get
Combining (2.24) with (2.26)(2.29), we get
Letting , we obtain a contradiction. This show that are Cauchy sequences. Since X is complete metric space, there exist such that
Since g is continuous, (2.30) implies that
From (2.10) and by regarding commutativity of F and g
We shall show that
Suppose now, (i) holds. Then by (2.8), (2.32) and (2.30), we have
Analogously,
Thus, we have
Suppose now the assumption (b) holds. If n is odd since are nondecreasing and are nonincreasing, if n is even since are nondecreasing and are nonincreasing and by considering , , … , we have
for all k. Thus, by triangle inequality and (2.32)
Letting implies that . Hence, . Analogously, we can get that
Thus, we proved that F and g have a ntuplet coincidence point. □
Corollary 1The above theorem reduces to Theorem 2.1 of[2]forandif (i) is satisfied andwhere.
The following corollary is a generalization of Corollary 2.1 in [11] and Theorem 2.1 in [4].
Corollary 2Letbe a partially ordered set and suppose thatis complete metric space. Supposeand there existsuch thatFhas the mixedgmonotone property and there exist awith
for allfor which, , … , (ifnis odd), (ifnis even). Assume also Condition 1 holds, and assume that, gis continuous and commutes withF. If there existsuch that
that is, Fandghave antuplet coincidence point.
3 Uniqueness of ntuplet fixed point
We say that is equal to if and only if , , … , .
Theorem 2In addition to hypothesis Theorem 1, assume that for allthere existsuch that
is comparable to
and
ThenFandghave a uniquentuplet common fixed point, that is, there existsuch that
Proof From the Theorem 1, the set of ntuplet coincidences is nonempty. We will show that if and are ntuplet coincidence points, that is, if
and
then
By assumption there is such that
is comparable with
and
Define sequences such that , , … , and
Since (3.4) and (3.5) comparable with (3.3), we may assume that
By using (2.11), we get that
for all k. From (3.1), we have
By (3.7) and (2.6), we have
Adding (3.8)(3.10), we get
Hence, it follows
for each . It is known that and imply for each . Thus, from (3.12)
Analogously, we can show that
Combining (3.13) and (3.14) and by using the triangle inequality
Hence, we get , , … , . Thus, we proved claim of theorem.
By commutativity of F and g,
Denote , , … , . Since (3.16), we get
Thus, is a ntuplet coincidence point. Then from assumption in theorem with , … , it follows , , … , , that is
From (3.18) and (3.17),
Therefore, is ntuplet common fixed point of F and g. To prove the uniqueness, assume that is another ntuplet common fixed point. Then by assumption in theorem we have
□
Corollary 3Letbe partially ordered set and suppose thatis complete metric space. Supposeand there existsuch thatFhas the mixedgmonotone property and
for allfor which, , … , (ifnis odd), (ifnis even). Suppose there existsuch that
Assume also that Condition 1 holds. Then there existsuch that
That isFhas antuplet fixed point.
Proof Take , then the assumption in Theorem 1 are satisfied. Thus, we get the result. □
Corollary 4Letbe partially ordered set and suppose thatis complete metric space. Supposeand there existsuch thatFhas the mixedgmonotone property and there existwith
for allfor which, , … , (ifnis odd), (ifnis even). Suppose there existsuch that
Assume also that Condition 1 holds. Then there existsuch that
That isFandghaventuplet coincidence point.
Proof Taking with in above corollary we obtain this corollary. □
Competing interests
The authors declare that they have no competing interests.
Acknowledgements
This work is supported by Yildiz Technical University Scientific Research Projects Coordination Unit under the project number BAPK 20120703DOP03.
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