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Tripled coincidence point theorems for weak φ-contractions in partially ordered metric spaces

Hassen Aydi1*, Erdal Karapinar2 and Mihai Postolache3

Author Affiliations

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

2 Department of Mathematics, Atılım University, Incek, Ankara 06836, Turkey

3 University Politehnica of Bucharest, Faculty of Applies Sciences, 313 Splaiul Independenţei, Romania

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Fixed Point Theory and Applications 2012, 2012:44  doi:10.1186/1687-1812-2012-44

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/44


Received:13 November 2011
Accepted:21 March 2012
Published:21 March 2012

© 2012 Aydi 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.

Abstract

In this article, we present tripled coincidence point theorems for F: X3 X and g: X X satisfying weak φ-contractions in partially ordered metric spaces. We also provide nontrivial examples to illustrate our results and new concepts presented herein. Our results unify, generalize and complement various known comparable results from the current literature, Berinde and Borcut and Abbas et al.

1 Introduction

Fixed point theory has fascinated hundreds of researchers since 1922 with the celebrated Banach's fixed point theorem. This theorem provides a technique for solving a variety of applied problems in mathematical sciences and engineering. There exists a last literature on the topic and this is a very active field of research at present. There are great number of generalizations of the Banach contraction principle. Bhaskar and Lakshmikantham [1] introduced the notion of coupled fixed point and proved some coupled fixed point results under certain conditions, in a complete metric space endowed with a partial order. Later, Lakshmikantham and Ćirić [2] extended these results by defining the mixed g-monotone property. More accurately, they proved coupled coincidence and coupled common fixed point theorems for a mixed g-monotone mapping in a complete metric space endowed with a partial order. Karapınar [3,4] generalized these results on a complete cone metric space endowed with a partial order. For other results on coupled fixed point theory, we address the readers to [5-13].

To make our exposition self contained, in this section we recall some previous notations and known results.

For simplicity, we denote from now on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M1">View MathML</a> by Xk, where k ∈ ℕ and X be a non-empty set.

Let (X, ≤) be a partially ordered set. According to [1], the mapping F: X2 X is said to have mixed monotone property if F(x, y) is monotone non-decreasing in x and is monotone non-increasing in y, that is, for any x, y X,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M2">View MathML</a>

An element (x, y) ∈ X2 is said to be a coupled fixed point of the mapping F: X2 X if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M3">View MathML</a>

Theorem 1.1. ([1]) Let (X, ≤) be an ordered set such that there exists a metric d on X such that (X, d) is complete. Let F: X2 X be a continuous mapping having the mixed monotone property on X. Assume that there exists k ∈ [0, 1) with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M4">View MathML</a>

(1.1)

If there exist x0, y0 X such that x0 F(x0, y0) and F(y0, x0) ≤ y0, then, there exist x, y ∈ X such that x = F(x, y) and y = F(y, x).

Recently, Samet and Vetro [14] introduced the notion of fixed point of N-order as natural extension of that of coupled fixed point and established some new coupled fixed point theorems in complete metric spaces, using a new concept of F-invariant set. Later, Berinde and Borcut [15] obtained existence and uniqueness of triple fixed point results in a complete metric space, endowed with a partial order.

Again, let (X, ≤) be a partially ordered set. In accordance with [15], the mapping F: X3 X is said to have the mixed monotone property if for any x, y, z X

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M5">View MathML</a>

An element (x, y, z) ∈ X3 is called a tripled fixed point of F if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M6">View MathML</a>

Berinde and Borcut [15] proved the following theorem.

Theorem 1.2. ([15]) Let (X, ≤) be a partially ordered set and (X, d) be a complete metric space. Let F: X3 X be a mapping having the mixed monotone property on X. Assume that there exist constants a, b, c ∈ [0, 1) such that a + b + c < 1 for which

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M7">View MathML</a>

(1.2)

for all x u, y υ, z w. Assume either

(I) F is continuous, or

(II) X has the following properties:

(i) if non-decreasing sequence xn → x, then xn x for all n,

(ii) if non-increasing sequence yn y, then yn y for all n.

If there exist x0, y0, z0 X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M8">View MathML</a>

then there exist x, y, z X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M9">View MathML</a>

In this article, we establish tripled coincidence point theorems for F: X3 X and g: X X satisfying nonlinear contractive conditions, in partially ordered metric spaces. The presented theorems extend and improve some results in litterature.

2 Main results

We shall start this section by recalling the following basic notions, introduced by [Abbas, Aydi and Karapınar, Tripled common fixed point in partially ordered metric spaces, submitted]. In this respect, let us consider (X, ≤) a partially ordered set, F: X3 X and g: X → X two mappings. The mapping F is said to have the mixed g-monotone property if for any x, y, z X

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M10">View MathML</a>

An element (x, y, z) is called a tripled coincidence point of F and g if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M11">View MathML</a>

while (gx, gy, gz) is said a tripled point of coincidence of mappings F and g. Moreover, (x, y, z) is called a tripled common fixed point of F and g if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M12">View MathML</a>

At last, mappings F and g are called commutative if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M13">View MathML</a>

In the same paper, they proved the following result.

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. Assume there is a function φ: [0, +∞) → [0, +∞) such that φ(t) < t for each t > 0. Also suppose F: X3 X and g: X X are such that F has the mixed g-monotone property and suppose there exist p, q, r ∈ [0, 1) with p + 2q + r < 1 such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M14">View MathML</a>

(2.1)

for any x, y, z X for which gx > gu, gυ gy and gz ≥ gw.

Suppose F(X3) ⊂ g(X), g is continuous and commutes with F. Suppose either

(a) F is continuous, or

(b) X has the following properties:

(i) if non-decreasing sequence gxn x (respectively, gzn z), then gxn x (respectively, gzn z) for all n,

(ii) if non-increasing sequence gyn y, then gyn y for all n.

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

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M15">View MathML</a>

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

Before starting to introduce our results, let us consider the set of functions

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M16">View MathML</a>

Our first main result is the following:

Theorem 2.2. 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: X3 X and g: X X are such that F has the mixed g-monotone property and F(X3) ⊂ g(X). Assume there is a function φ ∈ Φ such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M17">View MathML</a>

(2.2)

for any x, y, z, u, υ, w X for which gx gu, gυ gy and gz gw. Assume that F is continuous, g is continuous and commutes with F. If there exist x0, y0, z0 X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M18">View MathML</a>

(2.3)

then there exist x, y, z X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M15">View MathML</a>

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

Proof. Let x0, y0, z0 X be such that gx0 F(x0, y0, z0), gy0 F(y0, x0, y0) and gz0 F(z0, y0, x0). We can choose x1, y1, z1 X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M19">View MathML</a>

(2.4)

This can be done because F(X3) ⊂ g(X). Continuing this process, we construct sequences {xn}, {yn}, and {zn} in X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M20">View MathML</a>

(2.5)

By induction, we will prove that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M21">View MathML</a>

(2.6)

Since gx0 F(x0, y0, z0), gy0 F(y0, x0, y0), and gz0 F(z0, y0, x0), therefore by (2.4) we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M22">View MathML</a>

Thus (2.6) is true for n = 0. We suppose that (2.6) is true for some n > 0. Since F has the mixed g-monotone property, by gxn gxn+1, gyn+1 gyn, and gzn gzn+1, we have that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M23">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M24">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M25">View MathML</a>

That is, (2.6) is true for any n ∈ ℕ. If for some k ∈ ℕ

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M26">View MathML</a>

then, by (2.5), (xk, yk, zk) is a tripled coincidence point of F and g. From now on, we assume that at least

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M27">View MathML</a>

(2.7)

for any n ∈ ℕ. From (2.6) and the inequality (2.2)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M28">View MathML</a>

For each n ≥ 1, take

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M29">View MathML</a>

(2.8)

One can write

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M30">View MathML</a>

(2.9)

By (2.7), we have δn > 0. Having in mind φ(t) < t for each t > 0, so we have φ(δn) < δn. From (2.9), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M31">View MathML</a>

that is, the sequence {δn} is non-negative and decreasing. Therefore, there exists some δ ≥ 0 such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M32">View MathML</a>

(2.10)

We shall prove that δ = 0. Assume, on the contrary, that δ > 0. Then by letting n → +∞ in (2.9) we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M33">View MathML</a>

which is a contradiction. Thus, δ = 0, and by (2.10), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M34">View MathML</a>

(2.11)

We now prove that {gxn}, {gyn}, and {gzn} are Cauchy sequences in (X,d).

Suppose, on the contrary, that at least one of {gxn}, {gyn}, and {gzn} is not a Cauchy sequence. So, there exists ε > 0 for which we can find subsequences {gxn(k)}, {gxm(k)} of {gxn}, {gyn(k)}, {gym(k)} of {gyn}, and {gzn(k)}, {gzm(k)} of {gzn} with n(k) > m(k) ≥ k such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M35">View MathML</a>

(2.12)

Additionally, corresponding to m(k), we may choose n(k) such that it is the smallest integer satisfying (2.12) and n(k) > m(k) ≥ k. Thus,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M36">View MathML</a>

(2.13)

By using triangle inequality and having in mind (2.12) and (2.13)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M37">View MathML</a>

(2.14)

Letting k → ∞ in (2.14) and using (2.11)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M38">View MathML</a>

(2.15)

Again by triangle inequality,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M39">View MathML</a>

(2.16)

Since n(k) > m(k), then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M40">View MathML</a>

(2.17)

Take (2.17) in (2.2) to get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M41">View MathML</a>

Combining this in (2.16), we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M42">View MathML</a>

Letting k → ∞ and having in mind (2.11) and (2.15), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M43">View MathML</a>

which is a contradiction. This shows that {gxn}, {gyn}, and {gzn} are Cauchy sequences in (X, d).

Since X is complete, there exist x, y, z X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M44">View MathML</a>

(2.18)

From (2.18) and the continuity of g.

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M45">View MathML</a>

(2.19)

From the commutativity of F and g, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M46">View MathML</a>

(2.20)

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

Suppose that F is continuous. Letting n → +∞ in (2.20), therefore by (2.18) and (2.19), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M47">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M48">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M49">View MathML</a>

We have proved that F and g have a tripled coincidence point.

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: X3 X and g: X X are such that F has the mixed g-monotone property and F(X3) ⊂ g(X). Assume there exists α ∈ [0,1) such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M50">View MathML</a>

for any x, y, z, u, υ, w X for which gx gu, gυ gy, and gz gw. Assume that F is continuous, g is continuous and commutes with F. If there exist x0, y0, z0 X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M51">View MathML</a>

then there exist x, y, z X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M52">View MathML</a>

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

Proof. It follows by taking φ(t) = αt in Theorem 2.2.

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

Definition 2.1. Let (X, ≤) be a partially ordered set and d be a metric on X. We say that (X, d,≤) is regular if the following conditions hold:

(i) if a non-decreasing sequence (xn) is such that xn x, then xn x for all n,

(ii) if a non-increasing sequence (yn) is such that yn y, then y yn for all n.

Theorem 2.4. Let (X, ≤) be a partially ordered set and d be a metric on X such that (X, d, ≤) is regular. Suppose that there exist φ ∈ Φ and mappings F: X3 X and g: X X such that (2.2) holds for any x, y, z, u, υ, w X for which gx gu, gυ gy and gz gw. Suppose also that (g(X), d) is complete, F has the mixed g-monotone property and F(X3) ⊂ g(X). If there exist x0, y0, z0 X such that gx0 F(x0, y0, z0), gy0 F(y0, x0, y0), and gz0 F(z0, y0, x0), then there exist x, y, z X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M52">View MathML</a>

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

Proof. Proceeding exactly as in Theorem 2.2, we have that (gxn), (gyn), and (gzn) are Cauchy sequences in the complete metric space (g(X), d). Then, there exist x, y, z X such that gxn gx, gyn gy, and gzn gz. Since (gxn) and (gzn) are non-decreasing and (gyn) is non-increasing, using the regularity of (X, d, ≤), we have gxn gx, gzn gz, and gy gyn for all n ≥ 0. If gxn = gx, gyn = gy, and gzn = gz for some n ≥ 0, then gx = gxn gxn+1 gx = gxn, gz = gzn gzn+1 gz = gzn, and gy gyn+1 gyn = gy, which implies that gxn = gxn+1 = F(xn, yn, zn), gyn = gyn+1 = F(yn, xn, yn), and gzn = gzn+1 = F(zn, yn, xn), that is, (xn, yn, zn) is a tripled coincidence point of F and g. Then, we suppose that (gxn, gyn, gzn) ≠ (gx, gy, gz) for all n ≥ 0. Using the triangle inequality, (2.2) and the property φ(t) < t for all t > 0,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M53">View MathML</a>

(2.21)

Taking n → ∞ in the above inequality we obtain that d(gx,F(x, y, z)) = 0, so gx = F(x, y, z).

Analogously, we find that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M54">View MathML</a>

thus, we have proved that F and g have a tripled coincidence point.

Corollary 2.5. Let (X, ≤) be a partially ordered set and suppose there is a metric d on X such that (X, ≤,d) is regular. Suppose F: X3 X and g: X X are such that F has the mixed g-monotone property and F(X3) ⊂ g(X). Assume there exists α ∈ [0,1) such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M55">View MathML</a>

for any x, y, z, u, υ, w X for which gx gu, gυ gy, and gz gw. Suppose also that (g(X), d) is complete. If there exist x0, y0, z0 X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M56">View MathML</a>

then there exist x, y, z X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M82">View MathML</a>

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

Proof. It follows by taking φ(t) = αt in Theorem 2.4.

Now, we shall prove the existence and the uniqueness of a tripled common fixed point theorem. For a product X3 = X × X × X of a partial ordered set (X, ≤), we define a partial ordering in the following way: For all (x, y, z), (u, υ, r) ∈ X3

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M57">View MathML</a>

(2.22)

We say that (x, y, z) and (u, υ, w) are comparable if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M58">View MathML</a>

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

Theorem 2.6. In addition to hypothesis of Theorem 2.2, suppose that for all (x, y, z) and (u, υ, r) in X3, there exists (a, b, c) in X3 such that (F(a, b, c), F(b, a, b), F(c, b, a)) is comparable to (F(x, y, z), F(y, x, y), F(z, y, x)) and (F(u, υ, r), F(υ, u, υ), F(r, υ, u)). Also, assume that φ is non-decreasing. Then, F and g have a unique tripled common fixed point (x, y, z), that is

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M59">View MathML</a>

Proof. Due to Theorem 2.2, the set of tripled coincidence points of F and g is not empty. Assume now, that (x, y, z) and (u,υ,r) are two tripled coincidence points of F and g, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M60">View MathML</a>

We shall show that (gx, gy, gz) and (gu, gυ, gr) are equal.

By assumption, there is (a, b, c) in X3 such that (F(a, b, c), F(b, a, b), F(c, b, a)) is comparable to (F(x, y, z), F(y, x, y), F(z, y, x)) and (F(u, υ, r), F(υ, u, v), F(r, υ, u)).

Define the sequences {gan},{gbn}, and {gcn} such that a = a0, b = b0, c = c0 and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M61">View MathML</a>

for all n. Further, set x0 = x, y0 = y, z0 = z and u0 = u, υ0 = υ, r0 = r, and similar define the sequences {gxn},{gyn}, {gzn} and {gun},{n}, {grn}. Then,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M62">View MathML</a>

(2.23)

for all n ≥ 1. Since (F(x, y, z), F(y, x, y), F(z, y, x)) = (gx1, gy1, gz1) = (gx, gy, gz) is comparable to (F(a, b, c), F(b, a, b), F(c, b, a)) = (ga1, gb1, gc1), then it is easy to show that (gx, gy, gz) ≥ (ga1, gb1, gc1). Recursively, we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M63">View MathML</a>

(2.24)

By (2.24) and (2.2), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M64">View MathML</a>

(2.25)

Set

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M65">View MathML</a>

From (2.25), we deduce that γn+1 φ(γn). Since φ is non-decreasing, it follows

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M66">View MathML</a>

From the definition of Φ, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M67">View MathML</a>. Then, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M68">View MathML</a>. Thus,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M69">View MathML</a>

(2.26)

By analogy, we show that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M70">View MathML</a>

(2.27)

Combining (2.26) and (2.27) yields that (gx, gy, gz) and (gu, gυ, gr) are equal.

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

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M71">View MathML</a>

Denote gx = x', gy = y', and gz = z'. From the precedent identities,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M72">View MathML</a>

that is, (x', y', z') is a tripled coincidence point of F and g. Consequently, (gx', gy', gz') and (gx, gy, gz) are equal, that is, gx = gx', gy = gy', and gz = gz'.

We deduce gx' = gx = x', gy' = gy = y', and gz' = gz = z'. Therefore, (x', y', z') is a tripled common fixed of F and g. Its uniqueness follows from Theorem 2.2.

3 Examples

Remark that Theorem 2.2 is more general than Theorem 2.1, since the contractive condition (2.2) is weaker than (2.1), a fact which is clearly illustrated by the following example.

Example 3.1. Let X = ℝ with d(x, y) = |x - y| and natural ordering and let g: X X, F: X3 X be given by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M73">View MathML</a>

It is clear that F is continuous and has the mixed g-monotone property. We now take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M74">View MathML</a>. We shall show that (2.2) holds for all gx gu, gy , and gz gw.

Let x, y, z, u, υ, and w such that gx gu, gy , and gz gw, and by definition of g, it means that x u, y υ and z w, so we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M75">View MathML</a>

which is the contractive condition (2.2). On the other hand, x0 = 0, y0 = 0, z0 = 0 satisfy (2.3). All the hypotheses of Theorem 2.2 are verified, and (0,0,0) is a tripled coincidence point of F and g.

On the other hand, assume that (2.1) holds. Then, there exist p,q,r ≥ 0 such that p + 2q + r < 1 and φ satisfying φ(t) < t for each t > 0. If x > u, z = w and y = υ, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M76">View MathML</a>

which implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M77">View MathML</a> for any n ≥ 1, and letting n → +∞, we get p ≥ 1, that is a contradiction. Thus, Theorem 2.1 is not applicable in this case.

Following example shows that Theorem 2.2 is more general than Theorem 1.2.

Example 3.2. Let X = ℝ be endowed with the usual ordering and the usual metric. Consider g: X X and F: X3 X be given by the formulas

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M78">View MathML</a>

Take φ: [0, ∞) → [0, ∞) be given by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M79">View MathML</a>for all t ∈ [0, ∞).

It is clear that all conditions of Theorem 2.2 are satisfied. Moreover, (0,0,0) is a tripled coincidence point (also a common fixed point) of F and g.

Now, for x = u, z = w and υ > y, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M80">View MathML</a>

for any k ∈ [0,1), that is the result of Berinde and Borcut [15]given by Theorem 1.2 is not applicable <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/44/mathml/M81">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors contributed equally and significantly in writing this article. All authors read and approve the final manuscript.

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