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Coupled fixed point of generalized contractive mappings on partially ordered G-metric spaces

Mujahid Abbas1, Wutiphol Sintunavarat2 and Poom Kumam2*

Author Affiliations

1 Department of Mathematics, Lahore University of Management Sciences, Lahore 54792, Pakistan

2 Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand

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Fixed Point Theory and Applications 2012, 2012:31  doi:10.1186/1687-1812-2012-31

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/31


Received:2 December 2011
Accepted:29 February 2012
Published:29 February 2012

© 2012 Abbas 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

Coupled fixed point results for nonlinear contraction mappings having a mixed monotone property in a partially ordered G-metric space due to Choudhury and Maity are extended and unified. We also provide example to validate the main results in this article.

Mathematics Subject classification (2000): 47H10; 54H25.

Keywords:
coupled fixed points; mixed monotone property; G-metric spaces; partially order set

1. Introduction

One of the simplest and the most useful result in the fixed point theory is the Banach-Caccioppoli contraction [1] mapping principle, a power tool in analysis. This principle has been generalized in different directions in different spaces by mathematicians over the years (see [2-10] and references mentioned therein). On the other hand, fixed point theory has received much attention in metric spaces endowed with a partial ordering. The first result in this direction was given by Ran and Reurings [11] and they presented applications of their results to matrix equations. Subsequently, Nieto and Rodríguez-López [12] extended the results in [11] for nondecreasing mappings and obtained a unique solution for a first order ordinary differential equation with periodic boundary conditions (see also, [13-19]).

Bhaskar and Lakshmikantham [20] introduced the concept of a coupled fixed point and the mixed monotone property. Furthermore, they proved some coupled fixed point theorems for mappings which satisfy the mixed monotone property and gave some applications in the existence and uniqueness of a solution for a periodic boundary value problem. A number of articles in this topic have been dedicated to the improvement and generalization see in [21-24] and reference therein.

Mustafa and Sims [25,26] introduced a new concept of generalized metric spaces, called G-metric spaces. In such spaces every triplet of elements is assigned to a non-negative real number. Based on the notion of G-metric spaces, Mustafa et al. [27] established fixed point theorems in G-metric spaces. Afterward, many fixed point results were proved in this space (see [28-34]).

Recently, Choudhury and Maity [35] studied necessary conditions for existence of coupled fixed point in partially ordered G-metric spaces. They obtained the following interesting result.

Theorem 1.1 ([35]). Let (X, ≼) be a partially ordered set such that X is a complete G-metric space and F: X × X X be a mapping having the mixed monotone property on X. Suppose there exists k ∈ [0,1) such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M1">View MathML</a>

for all x, y, z, u, v, w X for which x u w and y v z, where either u w or v z. If there exists x0, y0 X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M2">View MathML</a>

and either

(a) F is continuous or

(b) X has the following property:

(i) if a non-decreasing sequence {xn} → x, then xn x for all n ∈ ℕ,

(ii) if a non-increasing sequence {yn} → y, then yn y for all n ∈ ℕ,

then F has a coupled fixed point.

The aim of this article is to extend and unify coupled fixed point results in [35] and to study necessary conditions to guarantee the uniqueness of coupled fixed point. We also provide illustrative example in support of our results.

2. Preliminaries

Throughout this article, (X, ≼) denotes a partially ordered set with the partial order ≼.

By x y, we mean x y but x y. If (X, ≼) is a partially ordered set. A mapping f: X X is said to be non-decreasing (non-increasing) if for all x, y X, x y implies f(x) ≼ f(y) (f(y) ≼ f(x), respectively).

Definition 2.1 ([20]). Let (X, ≼) be a partial ordered set. A mapping F: X × X X is said to has the a mixed monotone property if F is monotone non-decreasing in its first argument and is monotone non-increasing in its second argument, that is, for any x, y X

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M3">View MathML</a>

(2.1)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M4">View MathML</a>

(2.2)

Definition 2.2 ([20]). An element (x, y) ∈ X × X is called a coupled fixed point of mapping F: X × X X if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M5">View MathML</a>

Consistent with Mustafa and Sims [25,26], the following definitions and results will be needed in the sequel.

Definition 2.3 ([26]). Let X be a nonempty set. Suppose that a mapping G: X × X × X → ℝ+ satisfies:

(G1) G(x,y,z) = 0 if x = y = z;

(G2) G(x, x, y) > 0 for all x, y X with x y;

(G3) G(x,x,y) ≤ G(x,y,z) for all x,y,z X with z y;

(G4) G(x,y,z) = G(x,z,y) = G(y,z,x) = ..., (symmetry in all three variables);

(G5) G(x,y,z) ≤ G(x,a,a) + G(a,y, z) for all x,y,z,a X (rectangle inequality).

Then G is called a G-metric on X and (X, G) is called a G-metric space.

Definition 2.4 ([26]). Let X be a G-metric space and let {xn} be a sequence of points of X, a point x X is said to be the limit of a sequence {xn} if G(x,xn,xm) → 0 as n, m → ∞ and sequence {xn} is said to be G-convergent to x.

From this definition, we obtain that if xn x in a G-metric space X, then for any ϵ > 0 there exists a positive integer N such that G(x,xn,xm) < ϵ, for all n,m N.

It has been shown in [26] that the G-metric induces a Hausdorff topology and the convergence described in the above definition is relative to this topology. So, a sequence can converge at the most to one point.

Definition 2.5 ([26]). Let X be a G-metric space, a sequence {xn} is called G-Cauchy if for every ϵ > 0 there is a positive integer N such that G(xn,xm,xl) < ϵ for all n, m, l N, that is, if G(xn, xm, xl) → 0, as n, m, l → ∞.

We next state the following lemmas.

Lemma 2.6 ([26]). If X is a G-metric space, then the following are equivalent:

(1) {xn} is G-convergent to x.

(2) G(xn,xn,x) → 0 as n → ∞.

(3) G(xn,x,x) → 0 as n → ∞.

(4) G(xm,xn,x) → 0 as n,m → ∞.

Lemma 2.7 ([26]). If X is a G-metric space, then the following are equivalent:

(a) The sequence {xn} is G-Cauchy.

(b) For every ϵ > 0, there exists a positive integer N such that G(xn,xm,xm) < ϵ, for all n,m N.

Lemma 2.8 ([26]). If X is a G-metric space then G(x,y,y) ≤ 2G(y,x,x) for all x,y X.

Definition 2.9 ([26]). Let (X, G), (X', G') be two generalized metric spaces. A mapping f: X X' is G-continuous at a point x X if and only if it is G sequentially continuous at x, that is, whenever {xn} is G-convergent to x, {f(xn)} is G'-convergent to f(x).

Definition 2.10 ([26]). A G-metric space X is called a symmetric G-metric space if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M6">View MathML</a>

for all x,y X.

Definition 2.11 ([26]). A G-metric space X is said to be G-complete (or a complete G-metric space) if every G-Cauchy sequence in X is convergent in X.

Definition 2.12 ([26]). Let X be a G-metric space. A mapping F: X × X X is said to be continuous if for any two G-convergent sequences {xn} and {yn} converging to x and y, respectively, {F(xn,yn)} is G-convergent to F(x,y).

3. Coupled fixed point in G-metric spaces

Let Θ denotes the class of all functions θ: [0, ∞) × [0, ∞) → [0,1) which satisfies following condition:

For any two sequences {tn} and {sn} of nonnegative real numbers,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M7">View MathML</a>

Following are examples of some function in Θ.

(1) θ1(s,t) = k for s,t ∈ [0,∞), where k ∈ [0,1).

(2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M8">View MathML</a>

where k, l ∈ (0,1)

(3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M9">View MathML</a>

Now, we prove our main result.

Theorem 3.1. Let (X, ≼) be a partially ordered set such that there exists a complete G-metric on X and F: X × X X be a continuous mapping having the mixed monotone property. Suppose that there exists θ ∈ Θ such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M10">View MathML</a>

(3.1)

for all x, y, z, u, v, w X for which x u w and y v z where either u w or v z. If there exists x0, y0 X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M11">View MathML</a>

then F has a coupled fixed point.

Proof. As F(X × X) ⊆ X, we can construct sequences {xn} and {yn} in X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M12">View MathML</a>

(3.2)

Next, we show that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M13">View MathML</a>

(3.3)

Since x0 F(x0,y0) = x1 and y0 F(y0,x0) = y1, therefore (3.3) holds for n = 0.

Suppose that (3.3) holds for some fixed n ≥ 0, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M14">View MathML</a>

(3.4)

Since F has a mixed monotone property, from (3.4) and (2.1), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M15">View MathML</a>

(3.5)

for all x, y X and from (3.4) and (2.2), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M16">View MathML</a>

(3.6)

for all x,y X. If we take y = yn and x = xn in (3.5), then we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M17">View MathML</a>

(3.7)

If we take y = yn+1 and x = xn+1 in (3.6), then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M18">View MathML</a>

(3.8)

Now, from (3.7) and (3.8), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M19">View MathML</a>

(3.9)

Therefore, by the mathematical induction, we conclude that (3.3) holds for all n ≥ 0, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M20">View MathML</a>

(3.10)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M21">View MathML</a>

(3.11)

If there exists some integer k ≥ 0 such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M22">View MathML</a>

then G(xk+1,xk+1,xk) = G(yk+1,yk+1,yk) = 0 implies that xk = xk+1 and yk = yk+1. Therefore, xk = F(xk,yk) and yk = F(yk,xk) gives that (xk,yk) is a coupled fixed point of F.

Now, we assume that G(xn+1,xn+1,xn) + G(yn+1,yn+1,yn) ≠ 0 for all n ≥ 0. Since xn xn+1 and yn yn+1 for all n ≥ 0 so from (3.1) and (3.2), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M23">View MathML</a>

(3.12)

which implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M24">View MathML</a>

(3.13)

Thus the sequence {Gn+1 := G(xn+1, xn+1, xn) + G(yn+1, yn+1, yn)} is monotone decreasing. It follows that Gn g as n → ∞ for some g ≥ 0. Next, we claim that g = 0. Assume on contrary that g > 0, then from (3.12), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M25">View MathML</a>

On taking limit as n → ∞, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M26">View MathML</a>

By property of function θ, we have G(xn, xn, xn-1) → 0, G(yn, yn, yn-1) → 0 as n → ∞ and we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M27">View MathML</a>

(3.14)

a contradiction. Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M28">View MathML</a>

Similarly, we can prove that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M29">View MathML</a>

(3.15)

Next, we show that {xn} and {yn} are Cauchy sequences. On contrary, assume that at least one of {xn} or {yn} is not a Cauchy sequence. By Lemma 2.7, there is an ϵ > 0 for which we can find subsequences {xn(k)}, {xm(k)} of {xn} and {yn(k)}, {ym(k)} of {yn} with m(k) > n(k) ≥ k such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M30">View MathML</a>

(3.16)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M31">View MathML</a>

(3.17)

Using (3.16), (3.17) and the rectangle inequality, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M32">View MathML</a>

On taking limit as k → ∞, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M33">View MathML</a>

(3.18)

By the rectangle inequality, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M34">View MathML</a>

Therefore, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M35">View MathML</a>

This further implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M36">View MathML</a>

On taking limit as k → ∞ and using (3.14), (3.15) and (3.18), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M37">View MathML</a>

Since θ ∈ Θ, we have G(xn(k), xm(k), xm(k)) → 0 and G(yn(k), ym(k), ym(k)) → 0, that is

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M38">View MathML</a>

a contradiction. Therefore, {xn} and {yn} are G-Cauchy sequence. By G-completeness of X, there exists x,y X such that {xn} and {yn} G-converges to x and y, respectively. Now, we show that F has a coupled fixed point. Since F is a continuous, taking n → ∞ in (3.2), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M39">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M40">View MathML</a>

Therefore, x = F(x, y) and y = F(y, x), that is, F has a coupled fixed point.

Theorem 3.2. Let (X, ≼) be a partially ordered set such that there exists a complete G-metric on X and F: X × X X be a mapping having the mixed monotone property. Suppose that there exists θ ∈ Θ such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M41">View MathML</a>

(3.19)

for all x, y, z, u, v, w X for which x u w and y v z where either u w or v z. If there exists x0,y0 X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M2">View MathML</a>

and X has the following property:

(i) if a non-decreasing sequence {xn} → x, then xn x for all n ∈ ℕ,

(ii) if a non-increasing sequence {yn} → y, then yn y for all n ∈ ℕ,

then F has a coupled fixed point.

Proof. Following arguments similar to those given in Theorem 3.1, we obtain a non-decreasing sequence {xn} converges to x and a non-increasing sequence {yn} converges to y for some x,y X. By using (i) and (ii), we have xn x and yn y for all n.

If xn = x and yn = y for some n ≥ 0, then, by construction, xn+1 = x and yn+1 = y. Thus (x, y) is a coupled fixed point of F. So we may assume either xn x or yn y, for all n ≥ 0. Then by the rectangle inequality, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M42">View MathML</a>

On taking limit as n → ∞, we have G(F(x,y),x,x) + G(F(y,x),y,y) = 0. Thus x = F(x,y) and y = F(x, y) and so (x, y) is a coupled fixed point of F.

Corollary 3.3. Let (X, ≼) be a partially ordered set such that there exists a complete G-metric on X and F: X × X X be a mapping having the mixed monotone property. Suppose that there exists η ∈ Θ such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M43">View MathML</a>

(3.20)

for all x, y, z, u, v, w X for which x u w and y v z, where either u w or v z. If there exists x0,y0 X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M2">View MathML</a>

and either

(a) F is continuous or

(b) X has the following property:

(i) if a non-decreasing sequence {xn} → x, then xn x for all n ∈ ℕ,

(ii) if a non-increasing sequence {yn} → y, then yn y for all n ∈ ℕ,

then F has a coupled fixed point.

Proof. For x,y,z,u,v,w X with x u w and y v z, where either u w or v z, from (3.20), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M43">View MathML</a>

(3.21)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M44">View MathML</a>

(3.22)

From (3.21) and (3.22), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M45">View MathML</a>

(3.23)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M46">View MathML</a>

(3.24)

for x,y,z,u,v,w X with x u w and y v z where either u w or v z, where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M47">View MathML</a>

for all t1,t2 ∈ [0, ∞). It is easy to verify that θ ∈ Θ and we can apply Theorems 3.1 and 3.2. Hence F has a coupled fixed point.

Corollary 3.4. [[35], Theorems 3.1 and 3.2] Let (X,≼) be a partially ordered set such that there exists a complete G-metric on X and F: X × X X be a mapping having the mixed monotone property. Suppose that there exists a k ∈ [0,1) such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M1">View MathML</a>

for all x, y, z, u, v, w X for which x u w and y v z, where either u w or v z. If there exists x0, y0 X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M2">View MathML</a>

and either

(a) F is continuous or

(b) X has the following property:

(i) if a non-decreasing sequence {xn} → x, then xn x for all n ∈ ℕ,

(ii) if a non-increasing sequence {yn} → y, then yn y for all n ∈ ℕ,

then F has a coupled fixed point.

Proof. Taking η(t1, t2) = k with k ∈ [0,1) for all t1, t2 ∈ [0, ∞) in Theorems 3.1 and 3.2, result follows immediately.

Let Ω denotes the class of those functions ω: [0, ∞) → [0,1) which satisfies the condition: For any sequences {tn} of nonnegative real numbers, ω(tn) → 1 implies tn → 0.

Theorem 3.5. Let (X, ≼) be a partially ordered set such that there exists a complete G-metric on X and F: X × X X be a mapping having the mixed monotone property. Suppose that there exists ω ∈ Ω such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M48">View MathML</a>

(3.25)

for all x, y, z, u, v, w X for which x u w and y v z where either u w or v z. If there exists x0,y0 X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M2">View MathML</a>

and either

(a) F is continuous or

(b) X has the following property:

(i) if a non-decreasing sequence {xn} → x, then xn x for all n ∈ ℕ,

(ii) if a non-increasing sequence {yn} → y, then yn y for all n ∈ ℕ,

then F has a coupled fixed point.

Proof. Taking θ(t1,t2) = ω(t1 + t2) for all t1,t2 ∈ [0,∞) in Theorems 3.1 and 3.2, result follows.

Taking ω(t) = k with k ∈ [0,1) for all t ∈ [0, ∞) in Theorem 3.5, we obtain the following corollary.

Corollary 3.6. Let (X, ≼) be a partially ordered set such that there exists a complete G-metric on X and F: X × X X be a mapping having the mixed monotone property. Suppose that there exists ∈ [0,1) such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M49">View MathML</a>

for all x, y, z, u, v, w X for which x u w and y v z where either u w or v z. If there exists x0,y0 X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M2">View MathML</a>

and either

(a) F is continuous or

(b) X has the following property:

(i) if a non-decreasing sequence {xn} → x, then xn x for all n ∈ ℕ,

(ii) if a non-increasing sequence {yn} → y, then yn y for all n ∈ ℕ,

then F has a coupled fixed point.

Theorem 3.7. Let (X, ≼) be a partially ordered set such that there exists a complete G-metric on X and F: X × X X be a mapping having the mixed monotone property and F(x,y) ≼ F(y,x), whenever x y. Suppose that there exists θ ∈ Θ such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M50">View MathML</a>

(3.26)

for all x, y, z, u, v, w X for which w u xy v z, where either u w or v z. If there exists x0,y0 X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M51">View MathML</a>

and either

(a) F is continuous or

(b) X has the following property:

(i) if a non-decreasing sequence {xn} → x, then xn x for all n ∈ ℕ,

(ii) if a non-increasing sequence {yn} → y, then yn y for all n ∈ ℕ,

then F has a coupled fixed point.

Proof. By given hypothesis, there exist x0,y0 X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M52">View MathML</a>

We define x1,y1 X by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M53">View MathML</a>

Since x0 y0, by given assumptions, we have F(x0,y0) ≼ F(y0,x0). Hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M54">View MathML</a>

Continuing the above process, we have two sequences {xn} and {yn} such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M55">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M56">View MathML</a>

for all n ≥ 0. If there is k ∈ ℕ such that xk = yk = α (say), then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M57">View MathML</a>

that is, α = F(α, α). Therefore, (α, α) is a coupled fixed point of F. Next, assume that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M58">View MathML</a>

(3.27)

for all n ∈ ℕ. Further, using similar arguments as stated in Theorem 3.1, we may assume that (xn,yn) ≠ (xn+1,yn+1). Then, in view of (3.27), for all n ≥ 0, the inequality (3.26) holds with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M59">View MathML</a>

The rest of the proof follows by following the same steps as given in Theorem 3.1 for case (a). For case (b), we follow the same steps as given in Theorem 3.2.

Example 3.8. Let X = ℕ ∪ {0} and G: X × X × X X be define by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M60">View MathML</a>

Then X is a complete G-metric space. Let partial order ≼ on X be defined as follows: For x,y X,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M61">View MathML</a>

Let F: X × X X be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M62">View MathML</a>

If w u x y v z holds, then we have w u x > y v z. Therefore F(x,y) = F(u,v) = F(w,z) = 1 and F(y,x) = F(v,u) = F(z,w) = 0. So the left side of (3.26) becomes

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M63">View MathML</a>

and (3.26) is satisfied for all θ ∈ Θ. Thus Theorem 3.7 is applicable to this example with x0 = 0 and y0 = 81. Moreover, F has coupled fixed points (0,0) and (1,0).

Remark 3.9. A G-metric naturally induces a metric dG given by dG(x,y) = G(x,y,y) + G(x,x,y) [25]. From the condition that either u w or v z, the inequality (3.1), (3.19), (3.25) and (3.26) do not reduce to any metric inequality with the metric dG. Therefore, the corresponding metric space (X, dG) results are not applicable to Example 3.8.

Remark 3.10. Example 3.8 is not supported by Theorems 3.1, 3.2 and 3.5. This is evident by the fact that the inequality (3.1), (3.19) and (3.25) are not satisfied when w = u = x = y = 3, v = 0 and z = 1. Moreover, the coupled fixed point is not unique.

4. Uniqueness of coupled fixed point in G-metric spaces

In this section, we study necessary conditions to obtain the uniqueness of a coupled fixed point in the setting of partially ordered G-metric spaces. If (X, ≼) is a partially ordered set, then we endow the product of X × X with the following partial order:

For (x, y), (u, v) ∈ X × X, (x, y) ⊴ (u, v) if and only if x u and y v.

Theorem 4.1. In addition to the hypotheses in Theorem 3.1, suppose that for every (x, y), (z, t) ∈ X × X, there exists a point (u,v) ∈ X × X that is comparable to (x,y) and (z,t). Then F has a unique coupled fixed point.

Proof. From Theorem 3.1, F has a coupled fixed points. Suppose (x, y) and (z, t) are coupled fixed points of F, that is, x = F(x,y),y = F(y,x),z = F(z,t) and t = F(t,z). Next, we claim that x = z and y = t. By given hypothesis, there exists (u,v) ∈ X × X that is comparable to (x,y) and (z,t). We put u0 = u and v0 = v and construct sequences {un} and {vn} by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M64">View MathML</a>

Since (u,v) is comparable with (x,y), we assume that (u0,v0) = (u,v) ⊴ (x,y). Using the mathematical induction, it is straight forward to prove that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M65">View MathML</a>

From (3.1), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M66">View MathML</a>

(4.1)

Consequently, sequence {G(x,x,un) + G(y,y,vn)} is non-negative and decreasing, so

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M67">View MathML</a>

for some g ≥ 0. We claim that g = 0. Indeed, if g > 0 then following similar arguments to those given in the proof of Theorem 3.1, we conclude that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M68">View MathML</a>

Since θ ∈ Θ, we obtain G(x,x,un-1) → 0 and G(vn-1,y,y) → 0. Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M69">View MathML</a>

which is a contradiction. Hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M70">View MathML</a>

(4.2)

Similarly, one can prove that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M71">View MathML</a>

(4.3)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M72">View MathML</a>

(4.4)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M73">View MathML</a>

(4.5)

From rectangular inequality, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M74">View MathML</a>

(4.6)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M75">View MathML</a>

(4.7)

Combine (4.6) and (4.7), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/31/mathml/M76">View MathML</a>

Taking n → ∞, by (4.2), (4.3), (4.4) and (4.5), we have G(z,x,x) + G(y,t,t) ≤ 0. So G(z,x,x) = 0 and G(y,t,t) = 0, that is, z = x and y = t. Therefore, F has a unique coupled fixed point. This completes the proof.

Theorem 4.2. In addition to the hypotheses in Theorem 3.2, suppose that for every (x, y), (z, t) ∈ X × X, there exists a point (u,v) ∈ X × X that is comparable to (x,y) and (z,t). Then F has a unique coupled fixed point.

Proof. Proof is similar to the one given in Theorems 4.1 and 3.2.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors read and approved the final manuscript.

5. Acknowledgements

This study was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission under the Computational Science and Engineering Research Cluster (CSEC Grant No. 54000267).

W. Sintunavarat would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST). The authors thank the referee for comments and suggestions on this manuscript.

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