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n-tuplet fixed point theorems for contractive type mappings in partially ordered metric spaces

Müzeyyen Ertürk1* and Vatan Karakaya2

Author Affiliations

1 Department of Mathematics, The Faculty of Arts and Sciences, Yildiz Technical University, Davutpasa Campus, Esenler, Istanbul, 34210, Turkey

2 Current address: Department of Mathematical Engineering, Yildiz Technical University, Davutpasa Campus, Esenler, Istanbul, 34220, Turkey

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Journal of Inequalities and Applications 2013, 2013:196  doi:10.1186/1029-242X-2013-196


A correction to this article has been published: http://www.journalofinequalitiesandapplications.com/content/2013/1/368.

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/196


Received:4 January 2013
Accepted:5 April 2013
Published:22 April 2013

© 2013 Ertürk and Karakaya; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we study existence and uniquennes of fixed points of operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M1">View MathML</a> where n is an arbitrary positive integer and X is partially ordered complete metric space.

MSC: 47H10, 54H25, 54E50.

Keywords:
fixed point theorems; nonlinear contraction; partially ordered metric space; n-tuplet fixed point; mixed g-monotone

1 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 Rodriguez-Lopez [3] 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 [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 [5-10]. Lakshmikantham and Ciric [11] extended the results of [4] for monotone non-linear contractive mapping and generalized mixed monotone concept. Berinde and Borcut [12] 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.

Motivated by these studies, the quadruple fixed point theorem was given for different contractive type mappings [13-16].

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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M2">View MathML</a> be partially ordered set and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M1">View MathML</a>. We say that F has the mixed monotone property if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M4">View MathML</a> is monotone non-decreasing in its odd argument and it is monotone non-increasing in its even argument. That is, for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M5">View MathML</a>

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M6">View MathML</a>

(2.1)

Definition 2 Let X be a nonempty set and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M1">View MathML</a> a given mapping. An element <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M8">View MathML</a> is called a n-tuplet fixed point of F if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M9">View MathML</a>

(2.2)

Definition 3 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M2">View MathML</a> be partially ordered set and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M1">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M12">View MathML</a>. We say that F has the mixed g-monotone property if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M13">View MathML</a> is monotone g-non-decreasing in its odd argument and it is monotone g-non-increasing in its even argument. That is, for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M14">View MathML</a>

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M15">View MathML</a>

(2.3)

Note that if g is the identity mapping, this definition reduces to Definition 1.

Definition 4 Let X be a nonempty set and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M1">View MathML</a> a given mapping. An element <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M8">View MathML</a> is called a n-tuplet coincidence point of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M1">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M12">View MathML</a> if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M20">View MathML</a>

(2.4)

Note that if g is the identity mapping, this definition reduces to Definition 2.

Definition 5 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M2">View MathML</a> be partially ordered set and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M1">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M12">View MathML</a>. F and g called commutative if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M24">View MathML</a>

(2.5)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M5">View MathML</a>.

Let Φ denote the all functions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M26">View MathML</a>, which are continuous and satisfy that

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M27">View MathML</a>,

(ii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M28">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M29">View MathML</a>.

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 non-decreasing sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M30">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M31">View MathML</a> for all k,

(b) if non-increasing sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M32">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M33">View MathML</a> for all k.

Theorem 1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M2">View MathML</a>be partially ordered set and suppose that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M35">View MathML</a>is complete metric space. Assume<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M1">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M12">View MathML</a>are such thatFhas the mixedg-monotone property and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M38">View MathML</a>

(2.6)

for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M5">View MathML</a>for which<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M40">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M41">View MathML</a>, … , <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M42">View MathML</a> (ifnis odd), <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M43">View MathML</a> (ifnis even). Assume that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M44">View MathML</a>andgcommutes withF. Also, suppose that Condition 1 is satisfied. If there exist<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M45">View MathML</a>such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M46">View MathML</a>

(2.7)

then there exist<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M47">View MathML</a>such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M48">View MathML</a>

that is, Fandghave an-tuplet coincidence point.

Proof Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M45">View MathML</a> be such that (2.7). Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M50">View MathML</a>, we construct the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M51">View MathML</a> as follows:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M52">View MathML</a>

(2.8)

for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M53">View MathML</a> . We claim that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M54">View MathML</a>

(2.9)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M55">View MathML</a>. For this, we will use the mathematical induction. The inequalities in (2.9) hold <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M56">View MathML</a> because of (2.7), that is, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M57">View MathML</a>

Thus, our claim is true for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M56">View MathML</a>. Now, suppose that the inequalities in (2.9) hold <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M59">View MathML</a>. In this case,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M60">View MathML</a>

(2.10)

Now, we must show that the inequalities in (2.9) hold <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M61">View MathML</a>. If we consider (2.8) and mixed g-monotone property of F together with (2.10), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M62">View MathML</a>

Thus, (2.9) is satisfied for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M55">View MathML</a>. So, we have,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M64">View MathML</a>

(2.11)

For the simplicity, we define

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M65">View MathML</a>

We will show that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M66">View MathML</a>

(2.12)

By (2.6), (2.8) and (2.11), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M67">View MathML</a>

(2.13)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M68">View MathML</a>

(2.14)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M69">View MathML</a>

(2.15)

Due to (2.13)-(2.15), we conclude that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M70">View MathML</a>

(2.16)

Hence, we get (2.12).

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M27">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M72">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M73">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M74">View MathML</a>. So, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M75">View MathML</a> is monotone decreasing. Since it is bounded below, there is some <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M76">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M77">View MathML</a>

(2.17)

We want to show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M78">View MathML</a>. Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M79">View MathML</a>. Then taking the limit as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M80">View MathML</a> of both sides of (2.12) and keeping in mind that we assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M28">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M72">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M83">View MathML</a>

(2.18)

which is a contradiction. Thus, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M78">View MathML</a>, that is

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M85">View MathML</a>

(2.19)

Now we prove that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M86">View MathML</a> are Cauchy sequences. Suppose that at least one of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M87">View MathML</a> is not Cauchy. So, there exists an <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M88">View MathML</a> for which we can find subsequence of integer <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M89">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M90">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M91">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M92">View MathML</a>

(2.20)

Additionally, corresponding to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M89">View MathML</a>, we can choose <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M94">View MathML</a> such that it is the smallest integer satisfying (2.20) and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M95">View MathML</a>. Thus,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M96">View MathML</a>

(2.21)

By using triangle inequality and having (2.20) and (2.21) in mind

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M97">View MathML</a>

(2.22)

Letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M98">View MathML</a> in (2.22) and using (2.20)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M99">View MathML</a>

(2.23)

We apply triangle inequality to (2.20) as the following.

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M100">View MathML</a>

(2.24)

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M101">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M102">View MathML</a>

(2.25)

So, from (2.25), (2.8) and (2.6), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M103">View MathML</a>

(2.26)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M104">View MathML</a>

(2.27)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M105">View MathML</a>

(2.28)

Combining (2.24) with (2.26)-(2.29), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M106">View MathML</a>

(2.29)

Letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M98">View MathML</a>, we obtain a contradiction. This show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M108">View MathML</a> are Cauchy sequences. Since X is complete metric space, there exist <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M109">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M110">View MathML</a>

(2.30)

Since g is continuous, (2.30) implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M111">View MathML</a>

(2.31)

From (2.10) and by regarding commutativity of F and g

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M112">View MathML</a>

(2.32)

We shall show that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M113">View MathML</a>

Suppose now, (i) holds. Then by (2.8), (2.32) and (2.30), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M114">View MathML</a>

(2.33)

Analogously,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M115">View MathML</a>

(2.34)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M116">View MathML</a>

(2.35)

Thus, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M117">View MathML</a>

Suppose now the assumption (b) holds. If n is odd since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M118">View MathML</a> are non-decreasing and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M119">View MathML</a> are non-increasing, if n is even since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M120">View MathML</a> are non-decreasing and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M121">View MathML</a> are non-increasing and by considering <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M122">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M123">View MathML</a>, … , <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M124">View MathML</a> we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M125">View MathML</a>

(2.36)

for all k. Thus, by triangle inequality and (2.32)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M126">View MathML</a>

(2.37)

Letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M98">View MathML</a> implies that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M128">View MathML</a>. Hence, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M129">View MathML</a>. Analogously, we can get that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M130">View MathML</a>

Thus, we proved that F and g have a n-tuplet coincidence point. □

Corollary 1The above theorem reduces to Theorem 2.1 of[2]for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M131">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M132">View MathML</a>if (i) is satisfied and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M133">View MathML</a>where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M134">View MathML</a>.

The following corollary is a generalization of Corollary 2.1 in [11] and Theorem 2.1 in [4].

Corollary 2Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M2">View MathML</a>be a partially ordered set and suppose that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M35">View MathML</a>is complete metric space. Suppose<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M1">View MathML</a>and there exist<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M138">View MathML</a>such thatFhas the mixedg-monotone property and there exist a<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M139">View MathML</a>with

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M140">View MathML</a>

(2.38)

for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M141">View MathML</a>for which<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M142">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M143">View MathML</a>, … , <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M144">View MathML</a> (ifnis odd), <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M145">View MathML</a> (ifnis even). Assume also Condition 1 holds, and assume that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M146">View MathML</a>, gis continuous and commutes withF. If there exist<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M147">View MathML</a>such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M148">View MathML</a>

(2.39)

then there exist<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M109">View MathML</a>such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M150">View MathML</a>

that is, Fandghave an-tuplet coincidence point.

Proof It is sufficient to take <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M151">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M152">View MathML</a> in previous theorem. □

3 Uniqueness of n-tuplet fixed point

For all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M153">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M154">View MathML</a>

(3.1)

We say that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M155">View MathML</a> is equal to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M156">View MathML</a> if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M157">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M158">View MathML</a>, … , <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M159">View MathML</a>.

Theorem 2In addition to hypothesis Theorem 1, assume that for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M160">View MathML</a>there exist<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M161">View MathML</a>such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M162">View MathML</a>

is comparable to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M163">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M164">View MathML</a>

ThenFandghave a uniquen-tuplet common fixed point, that is, there exist<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M165">View MathML</a>such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M166">View MathML</a>

Proof From the Theorem 1, the set of n-tuplet coincidences is non-empty. We will show that if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M155">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M156">View MathML</a> are n-tuplet coincidence points, that is, if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M169">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M170">View MathML</a>

then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M171">View MathML</a>

(3.2)

By assumption there is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M172">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M173">View MathML</a>

(3.3)

is comparable with

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M174">View MathML</a>

(3.4)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M175">View MathML</a>

(3.5)

Define sequences <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M176">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M177">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M178">View MathML</a>, … , <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M179">View MathML</a> and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M180">View MathML</a>

(3.6)

Since (3.4) and (3.5) comparable with (3.3), we may assume that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M181">View MathML</a>

By using (2.11), we get that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M182">View MathML</a>

for all k. From (3.1), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M183">View MathML</a>

(3.7)

By (3.7) and (2.6), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M184">View MathML</a>

(3.8)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M185">View MathML</a>

(3.9)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M186">View MathML</a>

(3.10)

Adding (3.8)-(3.10), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M187">View MathML</a>

(3.11)

Hence, it follows

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M188">View MathML</a>

(3.12)

for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M55">View MathML</a>. It is known that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M27">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M191">View MathML</a> imply <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M192">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M72">View MathML</a>. Thus, from (3.12)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M194">View MathML</a>

(3.13)

Analogously, we can show that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M195">View MathML</a>

(3.14)

Combining (3.13) and (3.14) and by using the triangle inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M196">View MathML</a>

(3.15)

Hence, we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M197">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M198">View MathML</a>, … , <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M199">View MathML</a>. Thus, we proved claim of theorem.

By commutativity of F and g,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M200">View MathML</a>

(3.16)

Denote <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M201">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M202">View MathML</a>, … , <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M203">View MathML</a>. Since (3.16), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M204">View MathML</a>

(3.17)

Thus, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M205">View MathML</a> is a n-tuplet coincidence point. Then from assumption in theorem with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M206">View MathML</a>, … , <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M207">View MathML</a> it follows <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M208">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M209">View MathML</a>, … , <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M210">View MathML</a>, that is

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M211">View MathML</a>

(3.18)

From (3.18) and (3.17),

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M212">View MathML</a>

Therefore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M205">View MathML</a> is n-tuplet common fixed point of F and g. To prove the uniqueness, assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M214">View MathML</a> is another n-tuplet common fixed point. Then by assumption in theorem we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M215">View MathML</a>

 □

Corollary 3Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M2">View MathML</a>be partially ordered set and suppose that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M35">View MathML</a>is complete metric space. Suppose<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M1">View MathML</a>and there exist<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M138">View MathML</a>such thatFhas the mixedg-monotone property and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M220">View MathML</a>

(3.19)

for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M221">View MathML</a>for which<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M222">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M223">View MathML</a>, … , <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M224">View MathML</a> (ifnis odd), <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M225">View MathML</a> (ifnis even). Suppose there exist<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M45">View MathML</a>such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M227">View MathML</a>

(3.20)

Assume also that Condition 1 holds. Then there exist<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M228">View MathML</a>such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M229">View MathML</a>

(3.21)

That isFhas an-tuplet fixed point.

Proof Take <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M230">View MathML</a>, then the assumption in Theorem 1 are satisfied. Thus, we get the result. □

Corollary 4Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M2">View MathML</a>be partially ordered set and suppose that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M35">View MathML</a>is complete metric space. Suppose<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M1">View MathML</a>and there exist<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M138">View MathML</a>such thatFhas the mixedg-monotone property and there exist<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M152">View MathML</a>with

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M236">View MathML</a>

(3.22)

for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M141">View MathML</a>for which<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M238">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M223">View MathML</a>, … , <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M224">View MathML</a> (ifnis odd), <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M225">View MathML</a> (ifnis even). Suppose there exist<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M45">View MathML</a>such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M243">View MathML</a>

Assume also that Condition 1 holds. Then there exist<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M228">View MathML</a>such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M245">View MathML</a>

(3.23)

That isFandghaven-tuplet coincidence point.

Proof Taking <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M246">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/196/mathml/M152">View MathML</a> 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 2012-07-03-DOP03.

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