Keywords:fixed point theorems; nonlinear contraction; partially ordered metric space; n-tuplet fixed point; mixed g-monotone
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  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 . 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 Rodriguez-Lopez  extended results of Ran and Reuring for non-decreasing mapping and studied existence and uniqueness of first-order differential equations.
In 2006, by following the above mentioned trend, Bhaskar and Lakshmikantham  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 [5-10]. Lakshmikantham and Ciric  extended the results of  for monotone non-linear contractive mapping and generalized mixed monotone concept. Berinde and Borcut  introduced tripled fixed point theorem for non-linear mapping in partially ordered complete metric space as a generalization and extension of the coupled fixed point theorem.
In this paper, we generalize mentioned trend in the above for an arbitrary positive number n, that is, we introduce the concept of n-tuplet 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 non-decreasing in its odd argument and it is monotone non-increasing in its even argument. That is, for any
Definition 3 Let be partially ordered set and and . We say that F has the mixed g-monotone property if is monotone g-non-decreasing in its odd argument and it is monotone g-non-increasing in its even argument. That is, for any
Note that if g is the identity mapping, this definition reduces to Definition 1.
Note that if g is the identity mapping, this definition reduces to Definition 2.
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:
that is, Fandghave an-tuplet coincidence point.
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).
By using triangle inequality and having (2.20) and (2.21) in mind
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
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
Thus, we have
for all k. Thus, by triangle inequality and (2.32)
Thus, we proved that F and g have a n-tuplet coincidence point. □
Corollary 1The above theorem reduces to Theorem 2.1 offorandif (i) is satisfied andwhere.
that is, Fandghave an-tuplet coincidence point.
3 Uniqueness of n-tuplet fixed point
is comparable to
is comparable with
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
Analogously, we can show that
Combining (3.13) and (3.14) and by using the triangle inequality
By commutativity of F and g,
From (3.18) and (3.17),
That isFhas an-tuplet fixed point.
That isFandghaven-tuplet coincidence point.
The authors declare that they have no competing interests.
This work is supported by Yildiz Technical University Scientific Research Projects Coordination Unit under the project number BAPK 2012-07-03-DOP03.
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