Open Access Research

Mixed g-monotone property and quadruple fixed point theorems in partially ordered metric spaces

Zead Mustafa1*, Hassen Aydi2 and Erdal Karapinar3

Author Affiliations

1 Department of Mathematics, The Hashemite University, P.O. 330127, Zarqa 13115, Jordan

2 Institut Supérieur d'Informatique et des Technologies de Communication de Hammam Sousse, Université de Sousse, Route GP1-4011, Hammam Sousse, Tunisie

3 Department of Mathematics, Atilim University, 06836, İncek, Ankara, Turkey

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Fixed Point Theory and Applications 2012, 2012:71  doi:10.1186/1687-1812-2012-71


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/71


Received:19 December 2011
Accepted:1 May 2012
Published:1 May 2012

© 2012 Mustafa et al; 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.

Keywords:
quadruple coincidence point; quadruple common fixed point; ordered set; metric space; generalized contraction

Abstract

In this manuscript, we prove some quadruple coincidence and common fixed point theorems for F : X4 X and g : X X satisfying generalized contractions in partially ordered metric spaces. Our results unify, generalize and complement various known results from the current literature. Also, an application to matrix equations is given.

2000 Mathematics subject Classifications: 46T99; 54H25; 47H10; 54E50.

1 Introduction and preliminaries

Existence of fixed points in partially ordered metric spaces was first investigated by Turinici [1], where he extended the Banach contraction principle in partially ordered sets. In 2004, Ran and Reurings [2] presented some applications of Turinici's theorem to matrix equations. Following these initial articles, some remarkable results were reported see, e.g., [3-13].

Gnana Bhashkar and Lakshmikantham in [14] introduced the concept of a coupled fixed point of a mapping F : X × X X and investigated some coupled fixed point theorems in partially ordered complete metric spaces. Later, Lakshmikantham and Ćirić [15] proved coupled coincidence and coupled common fixed point theorems for nonlinear mappings F : X × X X and g : X X in partially ordered complete metric spaces. Various results on coupled fixed point have been obtained, since then see, e.g., [6,9,16-33]. Recently, Berinde and Borcut [34] introduced the concept of tripled fixed point in ordered sets.

For simplicity, we denote<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M1">View MathML</a> by Xk where k ∈ ℕ. Let us recall some basic definitions.

Definition 1.1 (See [34]) Let (X, ≤) be a partially ordered set and F: X3 → X. The mapping F is said to has the mixed monotone property if for any x, y, z X

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M2">View MathML</a>

Definition 1.2 Let F : X3 → X. An element (x, y, z) is called a tripled fixed point of F if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M3">View MathML</a>

Also, Berinde and Borcut [34] proved the following theorem:

Theorem 1.1 Let (X,, d) be a partially ordered set and suppose there is a metric d on X such that (X, d) is a complete metric space. Let F : X3 → X having the mixed monotone property. Suppose there exist j, r, l ≥ 0 with j + r + l < 1 such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M4">View MathML</a>

(1)

for any x, y, z X for which × ≤ u, v ≤ y and z ≤ w. Suppose either F is continuous or X has the following properties:

1. if a non-decreasing sequence xn → x, then xn ≤ x for all n,

2. if a non-increasing sequence yn → y, then y ≤ yn for all n.

If there exist x0, y0, z0 X such that x0 ≤ F (x0, y0, z0), y0 ≥ F (y0, x0, z0) and z0 F (z0, y0, x0), then there exist x, y, z X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M5">View MathML</a>

that is, F has a tripled fixed point.

Recently, Aydi et al. [35] introduced the following concepts.

Definition 1.3 Let (X, ) be a partially ordered set. Let F : X3 → X and g : X → X. The mapping F is said to has the mixed g-monotone property if for any x, y, z X

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M6">View MathML</a>

Definition 1.4 Let F : X3 → X and g : X → X. An element (x, y, z) is called a tripled coincidence point of F and g if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M7">View MathML</a>

(gx, gy, gz) is said a tripled point of coincidence of F and g.

Definition 1.5 Let F : X3 → X and g : X → X. An element (x, y, z) is called a tripled common fixed point of F and g if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M8">View MathML</a>

Definition 1.6 Let X be a non-empty set. Then we say that the mappings F : X3 → X and

g : X → X are commutative if for all x, y, z X

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M9">View MathML</a>

The notion of fixed point of order N ≥ 3 was first introduced by Samet and Vetro [36]. Very recently, Karapinar used the concept of quadruple fixed point and proved some fixed point theorems on the topic [37]. Following this study, quadruple fixed point is developed and some related fixed point theorems are obtained in [38-41].

Definition 1.7 [38]Let X be a nonempty set and F : X4 → X be a given mapping. An element (x, y, z, w) ∈ X × X × X × X is called a quadruple fixed point of F if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M10">View MathML</a>

Let (X, d) be a metric space. The mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M11">View MathML</a> given by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M12">View MathML</a>

defines a metric on X4, which will be denoted for convenience by d.

Definition 1.8 [38]Let (X, ) be a partially ordered set and F : X4 → X be a mapping. We say that F has the mixed monotone property if F (x, y, z, w) is monotone non-decreasing in x and z and is monotone non-increasing in y and w; that is, for any x, y, z, w X,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M13">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M14">View MathML</a>

In this article, we establish some quadruple coincidence and common fixed point theorems for F : X4 → X and g : X → X satisfying nonlinear contractions in partially ordered metric spaces. Also, some interesting corollaries are derived and an application to matrix equations is given.

2 Main results

We start this section with the following definitions.

Definition 2.1 Let (X, ≤) be a partially ordered set. Let F : X4 → X and g : X → X. The mapping F is said to has the mixed g-monotone property if for any x, y, z, w X

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M15">View MathML</a>

Definition 2.2 Let F : X4 X and g : X → X. An element (x, y, z, w) is called a quadruple coincidence point of F and g if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M16">View MathML</a>

(gx, gy, gz, gw) is said a quadruple point of coincidence of F and g.

Definition 2.3 Let F : X4 X and g : X → X. An element (x, y, z, w) is called a quadruple common fixed point of F and g if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M17">View MathML</a>

Definition 2.4 Let X be a non-empty set. Then we say that the mappings F : X4 X and g : X → × are commutative if for all x, y, z, w X

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M18">View MathML</a>

Let Φ be the set of all functions ϕ : [0, ∞) [0, ∞) such that:

1. ϕ(t) < t for all t ∈ (0,+∞).

2. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M19">View MathML</a> for all t ∈ (0,+∞).

For simplicity, we define the following.

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M20">View MathML</a>

(2)

Now, we state the first main result of this article.

Theorem 2.1 Let (X, ) be a partially ordered set and suppose there is a metric d on X such that (X, d) is a complete metric space. Suppose F : X4 → X and g : X → X are such that F is continuous and has the mixed g-monotone property. Assume also that there exist ϕ ∈ Φ and L ≥ 0 such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M21">View MathML</a>

(3)

for any x, y, z, w, u, v, h, l X for which gx gu, gv gy, gz gh and gl gw. Suppose F (X4) ⊂ g(X), g is continuous and commutes with F. If there exist x0, y0, z0, w0 X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M22">View MathML</a>

then there exist x, y, z, w X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M23">View MathML</a>

that is, F and g have a quadruple coincidence point.

Proof. Let x0, y0, z0, w0 X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M24">View MathML</a>

Since F (X4) ⊂ g(X), then we can choose x1, y1, z1, w1 X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M25">View MathML</a>

(4)

Taking into account F (X4) ⊂ g(X), by continuing this process, we can construct sequences {xn}, {yn}, {zn}, and {wn} in X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M26">View MathML</a>

(5)

We shall show that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M27">View MathML</a>

(6)

For this purpose, we use the mathematical induction. Since, gx0 ≤ F (x0, y0, z0, w0), gy0 ≥ F (y0, z0, w0, x0), gz0 ≤ F (z0, w0, x0, y0), and gw0 ≥ F (w0, x0, y0, z0), then by (4), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M28">View MathML</a>

that is, (6) holds for n = 0.

We presume that (6) holds for some n > 0. As F has the mixed g-monotone property and gxn ≤ gxn+1, gyn+1 gyn, gzn ≤ gzn+1 and gwn+1 gwn, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M29">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M30">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M31">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M32">View MathML</a>

Thus, (6) holds for any n ∈ ℕ. Assume for some n ∈ ℕ,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M33">View MathML</a>

then, by (5), (xn, yn, zn, wn) is a quadruple coincidence point of F and g. From now on, assume for any n ∈ ℕ that at least

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M34">View MathML</a>

(7)

By (2) and (5), it is easy that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M35">View MathML</a>

(8)

Due to (3) and (8), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M36">View MathML</a>

(9)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M37">View MathML</a>

(10)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M38">View MathML</a>

(11)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M39">View MathML</a>

(12)

Having in mind that ϕ (t) < t for all t > 0, so from (9)-(12) we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M40">View MathML</a>

(13)

It follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M41">View MathML</a>

(14)

Thus, {max{d(gxn, gxn+1), d(gyn, gyn+1), d(gzn, gzn+1), d(gwn, gwn+1)}} is a positive decreasing sequence. Hence, there exists r ≥ 0 such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M42">View MathML</a>

Suppose that r > 0. Letting n → +∞ in (13), we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M43">View MathML</a>

(15)

It is a contradiction. We deduce that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M44">View MathML</a>

(16)

We shall show that {gxn}, {gyn}, {gzn}, and {gwn} are Cauchy sequences in the metric space (X, d). Assume the contrary, that is, one of the sequence {gxn}, {gyn}, {gzn} or {gwn} is not a Cauchy, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M45">View MathML</a>

or

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M46">View MathML</a>

This means that there exists ε > 0, for which we can find subsequences of integers (mk) and (nk) with nk > mk > k such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M47">View MathML</a>

(17)

Further, corresponding to mk we can choose nk in such a way that it is the smallest integer with nk > mk and satisfying (17). Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M48">View MathML</a>

(18)

By triangular inequality and (18), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M49">View MathML</a>

Thus, by (16) we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M50">View MathML</a>

(19)

Similarly, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M51">View MathML</a>

(20)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M52">View MathML</a>

(21)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M53">View MathML</a>

(22)

Again by (18), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M54">View MathML</a>

Letting k → + ∞ and using (16), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M55">View MathML</a>

(23)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M56">View MathML</a>

(24)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M57">View MathML</a>

(25)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M58">View MathML</a>

(26)

Using (17) and (23)-(26), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M59">View MathML</a>

(27)

By (16), it is easy to see that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M60">View MathML</a>

(28)

Now, using inequality (3), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M61">View MathML</a>

(29)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M62">View MathML</a>

(30)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M63">View MathML</a>

(31)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M64">View MathML</a>

(32)

From (29)-(32), we deduce that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M65">View MathML</a>

(33)

Letting k → +∞ in (33) and having in mind (27) and (28), we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M66">View MathML</a>

it is a contradiction. Thus, {gxn}, {gyn}, {gzn}, and {gwn} are Cauchy sequences in (X, d).

Since (X, d) is complete, there exist x, y, z, w X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M67">View MathML</a>

(34)

From (34) and the continuity of g, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M68">View MathML</a>

(35)

From (5) and the commutativity of F and g, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M69">View MathML</a>

(36)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M70">View MathML</a>

(37)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M71">View MathML</a>

(38)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M72">View MathML</a>

(39)

Now we shall show that gx = F (x, y, z, w), gy = F (y, z, w, x), gz = F (z, w, x, y), and gw = F (w, x, y, z).

By letting n → +∞ in (36) - (39), by (34), (35) and the continuity of F , we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M73">View MathML</a>

(40)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M74">View MathML</a>

(41)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M75">View MathML</a>

(42)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M76">View MathML</a>

(43)

We have proved that F and g have a quadruple coincidence point. This completes the proof of Theorem 2.1.

In the following theorem, we omit the continuity hypothesis of F. We need the following definition.

Definition 2.5 Let (X, ≤) be a partially ordered metric set and d be a metric on X. We say that (X, d, ≤) is regular if the following conditions hold:

(i) if non-decreasing sequence an a, then an a for all n,

(ii) if non-increasing sequence bn b, then b bn for all n.

Theorem 2.2 Let (X, ≤) be a partially ordered set and d be a metric on X such that (X, d, ≤) is regular. Suppose F : X4 X and g : X X are such that F has the mixed g-monotone property. Assume that there exist ϕ ∈ Φ and L ≥ 0 such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M77">View MathML</a>

for any x, y, z, w, u, v, h, l X for which gx gu, gv gy, gz gh, and gl gw. Also, suppose F (X4) ⊂ g(X) and (g(X), d) is a complete metric space. If there exist x0, y0, z0, w0 X such that gx0 F (x0, y0, z0, w0), gy0 F (y0, z0, w0, x0), gz0 F (z0, w0, x0, y0) and gw0 F (w0, x0, y0, z0), then there exist x, y, z, w X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M78">View MathML</a>

that is, F and g have a quadruple coincidence point.

Proof. Proceeding exactly as in Theorem 2.1, we have that {gxn}, {gyn}, {gzn}, and {gwn} are Cauchy sequences in the complete metric space (g(X), d). Then, there exist x, y, z, w X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M79">View MathML</a>

(44)

Since {gxn}, {gzn} are non-decreasing and {gyn}, {gwn} are non-increasing, then since (X, d, ≤) is regular we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M80">View MathML</a>

for all n. If gxn = gx, gyn = gy, gzn = gz, and gwn = gw for some n ≥ 0, then gx = gxn gxn+1 gx = gxn, gy gyn+1 gyn = gy, gz = gzn gzn+1 gz = gzn, and gw gwn+1 gwn = gw, which implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M81">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M82">View MathML</a>

that is, (xn, yn, zn, wn) is a quadruple coincidence point of F and g. Then, we suppose that (gxn, gyn, gzn, gwn) ≠ (gx, gy, gz, gw) for all n ≥ 0. By (3), consider now

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M83">View MathML</a>

(45)

Taking n → ∞ and using (44), the quantity M(xn, yn, zn, wn, x, y, z, w) tends to 0 and so the right-hand side of (45) tends to 0, hence we get that d(gx, F (x, y, z, w)) = 0. Thus, gx = F (x, y, z, w). Analogously, one finds

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M84">View MathML</a>

Thus, we proved that F and g have a quartet coincidence point. This completes the proof of Theorem 2.2.

Corollary 2.1 Let (X, ≤) be a partially ordered set and suppose there is a metric d on X such that (X, d) is a complete metric space. Suppose F : X4 X and g : X X are such that F is continuous and has the mixed g-monotone property. Assume also that there exist ϕ ∈ Φ a non-decreasing function and L ≥ 0 such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M85">View MathML</a>

for any x, y, z, w, u, v, h, l,∈ X for which gx gu, gv gy, gz gw, and gl gw. Suppose F (X4) ⊂ g(X), g is continuous and commutes with F .

If there exist x0, y0, z0, w0 X such that gx0 F (x0, y0, z0, w0), gy0 F (y0, z0, w0, x0), gz0 F (z0, w0, x0, y0), and gw0 F (w0, x0, y0, z0), then there exist x, y, z, w X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M86">View MathML</a>

Proof. It suffices to remark that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M87">View MathML</a>

Then, we apply Theorem 2.1, since ϕ is assumed to be non-decreasing.

Similarly, as an easy consequence of Theorem 2.2 we have the following corollary.

Corollary 2.2 Let (X, ≤) be a partially ordered set and suppose there is a metric d on X such that (X, d, ≤) is regular. Suppose F : X4 X and g : X X are such that F has the mixed g-monotone property. Assume also that there exist ϕ ∈ Φ a non-decreasing function and L ≥ 0 such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M88">View MathML</a>

for any x, y, z, w, u, v, h, l X for which gx gu, gv gy, gz gw, and gl gw. Also, suppose F (X4) ⊂ g(X) and (g(X), d) is a complete metric space.

If there exist x0, y0, z0, w0 X such that gx0 F (x0, y0, z0, w0), gy0 F (y0, z0, w0, x0), gz0 F (z0, w0, x0, y0), and gw0 F (w0, x0, y0, z0), then there exist x, y, z, w X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M89">View MathML</a>

Corollary 2.3 Let (X, ≤) be a partially ordered set and suppose there is a metric d on X such that (X, d) is a complete metric space. Suppose F : X4 X and g : X X are such that F is continuous and has the mixed g-monotone property. Assume that there exist k ∈ [0, 1) and L ≥ 0 such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M90">View MathML</a>

for any x, y, z, w, u, v, h, l X for which:gx gu, gv gy, gz gw, and gl gw. Suppose F (X4) ⊂ g(X), g is continuous and commutes with F.

If there exist x0, y0, z0, w0 X such that gx0 F (x0, y0, z0, w0), gy0 F (y0, z0, w0, x0), gz0 F (z0, w0, x0, y0), and gw0 ≥ F (w0, x0, y0, z0), then there exist x, y, z, w X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M91">View MathML</a>

Proof. It suffices to take ϕ (t) = kt in Theorem 2.1.

Corollary 2.4 Let (X, ≤) be a partially ordered set and suppose there is a metric d on X such that (X, d, ≤) is regular. Suppose F : X4 X and g : X X are such that F has the mixed g-monotone property. Assume that there exist k ∈ [0, 1) and L ≥ 0 such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M92">View MathML</a>

for any x, y, z, w, u, v, h, l X for which gx gu, gv gy, gz gw, and gl gw. Suppose F (X4) ⊂ g(X) and (g(X), d) is a complete metric space.

If there exist x0, y0, z0, w0 X such that gx0 F (x0, y0, z0, w0), gy0 F (y0, z0, w0, x0), gz0 F (z0, w0, x0, y0), and gw0 F (w0, x0, y0, z0), then there exist x, y, z, w X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M93">View MathML</a>

Proof. It suffices to take ϕ (t) = kt in Theorem 2.2.

Corollary 2.5 Let (X, ≤) be a partially ordered set and suppose there is a metric d on X such that (X, d) is a complete metric space. Suppose F : X4 X and g : X X are such that F is continuous and has the mixed g-monotone property. Assume that there exist k ∈ [0, 1) and L ≥ 0 such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M94">View MathML</a>

for any x, y, z, w, ∈ X for which :gx gu, gv gy, gz gw, and gl gw. Also, suppose F (X4) ⊂ g(X) and (g(X), g is continuous and commutes with F.

If there exist x0, y0, z0, w0 X such that gx0 F (x0, y0, z0, w0), gy0 F (y0, z0, w0, x0), gz0 F (z0, w0, x0, y0), and gw0 F (w0, x0, y0, z0), then there exist x, y, z, w X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M95">View MathML</a>

Proof. It suffices to take ϕ (t) = kt in Corollary 2.1.

Corollary 2.6 Let (X, ≤) be a partially ordered set and suppose there is a metric d on X such that (X, d, ≤) is regular. Suppose F : X4 X and g : X X are such that F has the mixed g-monotone property. Assume that there exist k ∈ [0, 1) and L ≥ 0 such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M96">View MathML</a>

for any x, y, z, w, ∈ X for which gx gu, gv gy, gz gw, and gl gw. Suppose F (X4) ⊂ g(X) and (g(X), d) is a complete metric space.

If there exist x0, y0, z0, w0 X such that gx0 F (x0, y0, z0, w0), gy0 F (y0, z0, w0, x0), gz0 F (z0, w0, x0, y0), and gw0 F (w0, x0, y0, z0), then there exist x, y, z, w X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M97">View MathML</a>

Proof. It suffices to take ϕ (t) = kt in Corollary 2.2.

Remark 1 Corollary 2.4 of Karapinar [39]is a particular case of Corollary 2.5 by taking L = 0 and g = IX the identity on X.

Corollary 2.4 of Karapinar [39]is a particular case of Corollary 2.6 by taking L = 0 and g = IX .

Theorem 2.6 of Berinde and Karapinar [40]is a particular case of Corollary 2.1 by taking L = 0.

Theorem 2.6 of Berinde and Karapinar [40]is a particular case of Corollary 2.1 by taking L = 0.

Now, we shall prove the existence and uniqueness of quadruple common fixed point. For a product X4 of a partial ordered set (X, ≤), we define a partial ordering in the following way: For all (x, y, z, w), (u, v, r, h) ∈ X4

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M98">View MathML</a>

(46)

We say that (x, y, z, w) and (u, v, r, l) are comparable if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M99">View MathML</a>

Also, we say that (x, y, z, w) is equal to (u, v, r, l) if and only if x = u, y = v, z = r and w = l.

Theorem 2.3 In addition to hypotheses of Theorem 2.1, suppose that for all (x, y, z, w), (u, v, r, l) ∈ X4, there exists(a, b, c, d) ∈ X4 such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M100">View MathML</a>

is comparable to

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M101">View MathML</a>

Then, F and g have a unique quadruple common fixed point (x, y, z, w) such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M102">View MathML</a>

Proof. The set of quadruple coincidence points of F and g is not empty due to Theorem 2.1. Assume, now, (x, y, z, w) and (u, v, r, l) are two quadruple coincidence points of F and g, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M103">View MathML</a>

(47)

We shall show that (gx, gy, gz, gw) and (gu, gv, gr, gl) are equal. By assumption, there exists (a, b, c, d) ∈ X4 such that (F (a, b, c, d), F (b, c, d, a), F (c, d, a, b), F (d, a, b, c)) is comparable to (F (x, y, z, w), F (y, z, w, x), F (z, w, x, y), F (w, x, y, z)) and (F (u, v, r, l), F (v, r, l, u), F (r, l, u, v), F (l, u, v, r)).

Define sequences {gan}, {gbn}, {gcn}, and {gdn} such that

a0 = a, b0 = b, c0 = c, d0 = d and for any n ≥ 1

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M104">View MathML</a>

(48)

for all n. Further, set x0 = x, y0 = y, z0 = z, w0 = w and u0 = u, v0 = v, r0 = r, l0 = l and on the same way define the sequences {gxn}, {gyn}, {gzn}, {gwn} and {gun}, {gvn}, {grn}, {gln}. Then, it is easy that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M105">View MathML</a>

(49)

for all n ≥ 1. Since

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M106">View MathML</a>

is comparable to

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M107">View MathML</a>

then it is easy to show (gx, gy, gz, gw) ≥ (ga1, gb1, gc1, gd1). Recursively, we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M108">View MathML</a>

(50)

From (2) and (47), it is obvious that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M109">View MathML</a>

(51)

By (50), (51), and (3), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M110">View MathML</a>

(52)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M111">View MathML</a>

(53)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M112">View MathML</a>

(54)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M113">View MathML</a>

(55)

From (52)-(55), it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M114">View MathML</a>

(56)

Therefore, for each n ≥ 1,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M115">View MathML</a>

(57)

It is known that ϕ(t) < t and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M19">View MathML</a> imply <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M116">View MathML</a> for each t > 0. Thus, from (57)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M117">View MathML</a>

This yields that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M118">View MathML</a>

(58)

Analogously, we may show that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M119">View MathML</a>

(59)

Combining (58) and (59) yields that (gx, gy, gz, gw) and (gu, gv, gr, gl) are equal.

Since gx = F(x, y, z, w), gy = F(y, z, w, x), gz = F(z, w, x. y), and gz = F(z, w, x, y), by commutativity of F and g we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M120">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M121">View MathML</a>

where gx = x', gy = y', gz = z', and gw = w'. Thus, (x', y', z', w') is a quadruple coincidence point of F and g. Consequently, (gx', gy', gz', gz') and (gx, gy, gz, gw) are equal. We deduce

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M122">View MathML</a>

Therefore, (x', y', z', w') is a quadruple common fixed of F and g. Its uniqueness follows easily from (3).

Example 2.1 Let X = ℝ be endowed with the usual ordering and the usual metric, which is complete.

Let g: X X and F: X4X be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M123">View MathML</a>

Take ϕ : [0, ∞) [0, ∞) be given by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M124">View MathML</a> for all t ∈ [0, ∞).

We will check that the contraction (3) is satisfied for all x, y, z, w, u, v, h, l X satisfying gx ≤ gu, gv ≤ gy, gz ≤ gh, and gl ≤ gw. In this case, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M125">View MathML</a>

for arbitrary L ≥ 0.

It is obvious that the other hypotheses of Theorem 2.3 are satisfied. We deduce that (0, 0, 0, 0) is the unique quadruple common fixed point of F and g.

3 Application to matrix equations

In this section, we study the existence and uniqueness of solutions (X, Y, Z, T) to the system of matrix equations

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M126">View MathML</a>

(60)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M127">View MathML</a>: the set of all n × n matrices, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M128">View MathML</a> the set of all n × n positive definite matrices, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M129">View MathML</a> is the set of all n × n Hermitian matrices.

We endow <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M129">View MathML</a> with the partial order ≼ given by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M130">View MathML</a>

For a fixed <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M131">View MathML</a>, we consider

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M132">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M133">View MathML</a>, where tr is the trace operator. The space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M129">View MathML</a> equipped with the metric induced by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M134">View MathML</a> is a complete metric space for any positive definite matrix P (see [42]).

The following lemma will be useful for our application.

Lemma 3.1 Let A ≽ 0 and B ≽ 0 be n × n matrices. Then, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M135">View MathML</a>

where ||.|| is the spectral norm.

Theorem 3.1 Suppose that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M131">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M136">View MathML</a>

(61)

Suppose also that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M137">View MathML</a>

(62)

Then, the system (60) has one and only one solution <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M138">View MathML</a>.

Proof. Consider the mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M139">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M140">View MathML</a> defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M141">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M142">View MathML</a>i =1, . . . , 4.

For all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M143">View MathML</a>i = 1. . . , 4 with gX1 gY1, gY2 gX2, gX3 gY3 and gY4 gX4, by using Lemma 3.1, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M144">View MathML</a>

Thus, we proved that the contractive condition given in Corollary 2.5 is satisfied for all L ≥ 0. Moreover, from (62), we have letting gQ F (Q, 0, Q, 0) and g0 ≽ F (0, Q, 0, Q). Applying Corollary 2.5, F and g have a coupled coincidence point (and so a quadrupled fixed point since g is the identity on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M129">View MathML</a>). Then, there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M145">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M146">View MathML</a>

On the other hand, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/71/mathml/M147">View MathML</a> there is a greatest lower bound and a least upper bound, hence it is obvious that the hypotheses of Theorem 2.3 hold, so the uniqueness of that quadrupled fixed point of F, which is also the unique solution of the system (60).

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors have contributed in obtaining the new results presented in this article. All authors read and approve the final manuscript.

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