A Unique Coupled Common Fixed Point Theorem for Symmetric(φ,ψ)-Contractive Mappings in OrderedG-Metric Spaces with Applications

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Volume 2013, Article ID 134712,13pages http://dx.doi.org/10.1155/2013/134712

Research Article

A Unique Coupled Common Fixed Point Theorem for

Symmetric

(𝜑, 𝜓)-Contractive Mappings in Ordered 𝐺-Metric

Spaces with Applications

Manish Jain

1

and Kenan Ta

G

2

1_{Department of Mathematics, Ahir College, Rewari 123401, India}

2_{Department of Mathematics and Computer Science, Cankaya University, Ankara, Turkey}

Correspondence should be addressed to Kenan Tas¸; [email protected] Received 21 July 2013; Accepted 19 October 2013

Academic Editor: Antonio J. M. Ferreira

Copyright © 2013 M. Jain and K. Tas¸. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We establish the existence and uniqueness of coupled common fixed point for symmetric(𝜑, 𝜓)-contractive mappings in the framework of ordered G-metric spaces. Present work extends, generalize, and enrich the recent results of Choudhury and Maity (2011), Nashine (2012), and Mohiuddine and Alotaibi (2012), thereby, weakening the involved contractive conditions. Our theoretical results are accompanied by suitable examples and an application to integral equations.

1. Introduction and Preliminaries

The structure of G-metric spaces introduced by Mustafa and Sims [1] is a generalization of metric spaces. The theory of fixed points in this generalized structure was initiated by Mustafa et al. [2], in which Banach contraction principle was established in G-metric spaces. After that different authors proved several fixed point results in this space. References [3–

9] are some examples of these works.

Definition 1 (see [1]). Let𝑋 be nonempty set, and let 𝐺 : 𝑋 ×

𝑋×𝑋 → 𝑅+be a function satisfying the following properties: (G1)𝐺(𝑥, 𝑦, 𝑧) = 0 if 𝑥 = 𝑦 = 𝑧,

(G2)0 < 𝐺(𝑥, 𝑥, 𝑦) 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 the function𝐺 is called a 𝐺-metric on 𝑋 and the pair (𝑋, 𝐺) is a called a 𝐺-metric space.

Definition 2 (see [1]). Let(𝑋, 𝐺) 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 the sequence{𝑥_𝑛} if lim_{𝑛,𝑚 → ∞}𝐺(𝑥, 𝑥_𝑛, 𝑥_𝑚) = 0, and then we say that the sequence{𝑥_𝑛} is 𝐺-convergent to 𝑥. Thus, if 𝑥_𝑛 → 𝑥 in G-metric space (𝑋, 𝐺) then, for any 𝜀 > 0, there exists a positive integer 𝑁 such that 𝐺(𝑥, 𝑥_𝑛, 𝑥_𝑚) < 𝜀 for all 𝑛, 𝑚 ≥ 𝑁.

In [1], the authors have shown that the𝐺-metric induces a Hausdorff topology, and the convergence described in the above definition is relative to this topology. This is a Hausdorff topology, so a sequence can converge at most to one point.

Definition 3 (see [1]). Let (𝑋, 𝐺) be a 𝐺-metric space. A

sequence{𝑥_𝑛} is called G-Cauchy if, for every 𝜀 > 0, there is a positive integer 𝑁 such that 𝐺(𝑥_𝑛, 𝑥_𝑚, 𝑥_𝑙) < 𝜀 for all 𝑛, 𝑚, 𝑙 ≥ 𝑁; that is, 𝐺(𝑥_𝑛, 𝑥_𝑚, 𝑥_𝑙) → 0 as 𝑛, 𝑚, 𝑙 → ∞.

Lemma 4 (see [1]). If (𝑋, 𝐺) is a G-metric space, then the

following are equivalent:

(1){𝑥_𝑛} is 𝐺-convergent to 𝑥, (2)𝐺(𝑥_𝑛, 𝑥_𝑛, 𝑥) → 0 as 𝑛 → ∞,

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(3)𝐺(𝑥_𝑛, 𝑥, 𝑥) → 0 as 𝑛 → ∞, (4)𝐺(𝑥_𝑚, 𝑥_𝑛, 𝑥) → 0 as 𝑚, 𝑛 → ∞.

Lemma 5 (see [1]). If (𝑋, 𝐺) is a 𝐺-metric space, then the

following are equivalent:

(1) the sequence{𝑥_𝑛} is G-Cauchy;

(2) for every𝜀 > 0, there exists a positive integer 𝑁 such

that𝐺(𝑥_𝑛, 𝑥_𝑚, 𝑥_𝑚) < 𝜀 for all 𝑛, 𝑚 > 𝑁.

Lemma 6 (see [1]). If (𝑋, 𝐺) is a 𝐺-metric space, then

𝐺(𝑥, 𝑦, 𝑦) ≤ 2𝐺(𝑦, 𝑥, 𝑥) for all 𝑥, 𝑦 ∈ 𝑋.

Lemma 7. If (𝑋, 𝐺) is a 𝐺-metric space, then 𝐺(𝑥, 𝑥, 𝑦) ≤

𝐺(𝑥, 𝑥, 𝑧) + 𝐺(𝑧, 𝑧, 𝑦) for all 𝑥, 𝑦, 𝑧 ∈ 𝑋.

Definition 8 (see [1]). Let(𝑋, 𝐺), (𝑋󸀠, 𝐺󸀠) be two G-metric

spaces. Then a function𝑓 : 𝑋 → 𝑋󸀠is G-continuous at a point𝑥 ∈ 𝑋 if and only if it is G-sequentially continuous at 𝑥; that is, whenever {𝑥_𝑛} is 𝐺-convergent to 𝑥, {𝑓(𝑥_𝑛)} is 𝐺󸀠 -convergent to𝑓(𝑥).

Lemma 9 (see [1]). Let (𝑋, 𝐺) be a 𝐺-metric space; then

the function𝐺(𝑥, 𝑦, 𝑧) is jointly continuous in all three of its

variables.

Definition 10 (see [1]). A 𝐺-metric space (𝑋, 𝐺) is said to

becomplete (or a complete G-metric space) if every 𝐺-Cauchy sequence in(𝑋, 𝐺) is convergent in 𝑋.

Definition 11 (see [10]). Let (𝑋, 𝐺) be a 𝐺-metric space. A

mapping𝐹 : 𝑋 × 𝑋 → 𝑋 is said to be continuous if for any two G-convergent sequences{𝑥_𝑛} and {𝑦_𝑛} converging to 𝑥 and 𝑦, respectively, {𝐹(𝑥_𝑛, 𝑦_𝑛)} is G-convergent to 𝐹(𝑥, 𝑦).

Recently, fixed point theorems under different contrac-tive conditions in metric spaces endowed with the partial ordering have been studied by various authors. Works noted in [11–25] are some examples in this direction. Bhaskar and Lakshmikantham [11] introduced the notion of coupled fixed points and proved some coupled fixed point theorems for a mapping having mixed monotone property. The work [11] was illustrated by proving the existence and uniqueness of the solution for a periodic boundary value problem.

Lakshmikantham and ́Ciri ́c [12] extended the notion of mixed monotone property by introducing the notion of mixed𝑔-monotone property in partially ordered metric spaces.

Definition 12 (see [11]). Let(𝑋, ≤) be a partially ordered set

and𝐹 : 𝑋×𝑋 → 𝑋. The mapping 𝐹 is said to have the mixed monotone property if𝐹(𝑥, 𝑦) is monotone nondecreasing in 𝑥 and monotone nonincreasing in 𝑦; that is, for any 𝑥, 𝑦 ∈ 𝑋,

𝑥₁, 𝑥₂∈ 𝑋, 𝑥₁≤ 𝑥₂ implies 𝐹 (𝑥₁, 𝑦) ≤ 𝐹 (𝑥₂, 𝑦) , 𝑦₁, 𝑦₂∈ 𝑋, 𝑦₁≤ 𝑦₂ implies 𝐹 (𝑥, 𝑦₁) ≥ 𝐹 (𝑥, 𝑦₂) . (1)

Definition 13 (see [12]). Let (𝑋, ≤) be a partially ordered

set 𝐹 : 𝑋 × 𝑋 → 𝑋, and 𝑔 : 𝑋 → 𝑋. We say

that the mapping𝐹 has the mixed 𝑔-monotone property if 𝐹(𝑥, 𝑦) is monotone 𝑔-nondecreasing in its first argument and monotone𝑔-nonincreasing in its second argument; that is, for any𝑥, 𝑦 ∈ 𝑋,

𝑥₁, 𝑥₂∈ 𝑋, 𝑔𝑥₁≤ 𝑔𝑥₂ implies𝐹 (𝑥₁, 𝑦) ≤ 𝐹 (𝑥₂, 𝑦) , 𝑦₁, 𝑦₂∈ 𝑋, 𝑔𝑦₁≤ 𝑔𝑦₂ implies𝐹 (𝑥, 𝑦₁) ≥ 𝐹 (𝑥, 𝑦₂) . (2)

Definition 14 (see [11]). An element(𝑥, 𝑦) ∈ 𝑋 × 𝑋 is called

a coupled fixed point of the mapping𝐹 : 𝑋 × 𝑋 → 𝑋 if 𝐹(𝑥, 𝑦) = 𝑥 and 𝐹(𝑦, 𝑥) = 𝑦.

Definition 15 (see [12]). An element(𝑥, 𝑦) ∈ 𝑋 × 𝑋 is called a

coupled coincidence point of the mappings𝐹 : 𝑋 × 𝑋 → 𝑋 and𝑔 : 𝑋 → 𝑋 if 𝐹(𝑥, 𝑦) = 𝑔𝑥 and 𝐹(𝑦, 𝑥) = 𝑔𝑦.

Definition 16 (see [12]). An element(𝑥, 𝑦) ∈ 𝑋 × 𝑋 is called a

coupled common fixed point of the mappings𝐹 : 𝑋×𝑋 → 𝑋 and𝑔 : 𝑋 → 𝑋 if 𝑥 = 𝑔𝑥 = 𝐹(𝑥, 𝑦) and 𝑦 = 𝑔𝑦 = 𝐹(𝑦, 𝑥).

Definition 17 (see [12]). The mappings𝐹 : 𝑋 × 𝑋 → 𝑋 and

𝑔 : 𝑋 → 𝑋 are called commutative if

𝑔𝐹 (𝑥, 𝑦) = 𝐹 (𝑔𝑥, 𝑔𝑦) , (3) for all𝑥, 𝑦 ∈ 𝑋.

Let (𝑋, ≤) be a partially ordered set, and let 𝐺 be a 𝐺-metric on𝑋 such that (𝑋, 𝐺) is a complete 𝐺-metric space.

Choudhury and Maity [10] established some coupled fixed point theorems for the mixed monotone mapping𝐹 : 𝑋 × 𝑋 → 𝑋 under a contractive condition of the form

𝐺 (𝐹 (𝑥, 𝑦) , 𝐹 (𝑢, V) , 𝐹 (𝑤, 𝑧))

≤ 𝑘₂[𝐺 (𝑥, 𝑢, 𝑤) + 𝐺 (𝑦, V, 𝑧)] , (4) where𝑘 ∈ [0, 1).

Various authors extended and generalized the results of Choudhury and Maity [10] under different contractive conditions in G-metric spaces. For more works, one can see [26–34]. Nashine [31] generalized and extended the contractive condition (4) and thereby obtained the coupled coincidence points for a pair of commuting mappings under the following contraction:

𝐺 (𝐹 (𝑥, 𝑦) , 𝐹 (𝑢, V) , 𝐹 (𝑤, 𝑧))

≤ 𝑘 [𝐺 (𝑔𝑥, 𝑔𝑢, 𝑔𝑤) + 𝐺 (𝑔𝑦, 𝑔V, 𝑔𝑧)] , (5) where𝑘 ∈ [0, 1/2).

Mohiuddine and Alotaibi [33] further generalized the contraction (4) by considering the following more general contractive condition: 𝜑 (𝐺 (𝐹 (𝑥, 𝑦) , 𝐹 (𝑢, V) , 𝐹 (𝑤, 𝑧))) ≤ 1 2𝜑 (𝐺 (𝑥, 𝑢, 𝑤) + 𝐺 (𝑦, V, 𝑧)) − 𝜓 (𝐺 (𝑥, 𝑢, 𝑤) + 𝐺 (𝑦, V, 𝑧) 2 ) , (6)

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where𝜑, 𝜓 : [0, ∞) → [0, ∞) are functions satisfying some appropriate conditions mentioned in [33].

Interestingly, for𝜑(𝑡) = 𝑡, 𝜓(𝑡) = ((1 − 𝑘)/2)𝑡, with 0 ≤ 𝑘 < 1, condition (6) reduces to (4).

On the other hand, Karapinar et al. [32] improved various results present in the literature of coupled fixed point theory of G-metric spaces by considering a generalized 𝜙-contraction. Assigning the value𝑘𝑡 to 𝜙(𝑡) with 𝑘 ∈ [0, 1) for 𝑡 > 0, Karapinar et. al. [32] in an alternative way generalized the contraction (4) for a pair of commutative mappings as follows: 𝐺 (𝐹 (𝑥, 𝑦) , 𝐹 (𝑢, V) , 𝐹 (𝑤, 𝑧)) + 𝐺 (𝐹 (𝑥, 𝑦) , 𝐹 (𝑢, V) , 𝐹 (𝑤, 𝑧)) ≤ 𝑘 [𝐺 (𝑔𝑥, 𝑔𝑢, 𝑔𝑤) + 𝐺 (𝑔𝑦, 𝑔V, 𝑔𝑧)] , (7) where𝑘 ∈ [0, 1).

Present work extend and generalize several results present in the literature of fixed point theory of𝐺-metric spaces. Our result directly derive a result of Karapinar et. al. [32]. We give suitable examples to show how our results generalize and enrich the well-known results of Choudhury et al. [10], Nashine [31], and Mohiuddine and Alotaibi [33] by significantly weakening the involved contractive conditions. The effectiveness of the present work is shown by suitable examples and an application to the integral equations.

2. Main Results

Before proving our results, we need the following.

Denote byΦ the class of all functions 𝜑 : [0, ∞) → [0, ∞) with the following properties:

(𝜑i) 𝜑 is continuous and nondecreasing;

(𝜑ii) 𝜑(𝑡) < 𝑡 for all 𝑡 > 0;

(𝜑iii) 𝜑(𝑡 + 𝑠) ≤ 𝜑(𝑡) + 𝜑(𝑠) for all 𝑡, 𝑠 ∈ [0, ∞).

We note that(𝜑i) and (𝜑ii) imply 𝜑(𝑡) = 0 if and only if

𝑡 = 0.

Denote by Ψ the class of all functions 𝜓 : [0, ∞) → [0, ∞) with the following properties:

(𝜓i) lim𝑡 → 𝑟𝜓(𝑡) > 0 for all 𝑟 > 0;

(𝜓ii) lim𝑡 → 0+𝜓(𝑡) = 0.

Some examples of𝜑(𝑡) are 𝑘𝑡 (where 𝑘 > 0), and 𝑡/(𝑡 + 1), 𝑡/(𝑡+2) and examples of 𝜓(𝑡) are 𝑘𝑡 (where 𝑘 > 0), ln(2𝑡+1)/2. Let(𝑋, 𝐺) be a 𝐺-metric space, and let 𝐹 : 𝑋×𝑋 → 𝑋, 𝑔 : 𝑋 → 𝑋 be two mappings. We say that 𝐹 and 𝑔 are symmetric

(𝜑, 𝜓)-contractive mappings on 𝑋 if there exist 𝜑 ∈ Φ and 𝜓 ∈ Ψ such that 𝜑 ( (𝐺 (𝐹 (𝑥, 𝑦) , 𝐹 (𝑢, V) , 𝐹 (𝑤, 𝑧)) +𝐺 (𝐹 (𝑦, 𝑥) , 𝐹 (V, 𝑢) , 𝐹 (𝑧, 𝑤))) 2−1) ≤ 𝜑 (𝐺 (𝑔𝑥, 𝑔𝑢, 𝑔𝑤) + 𝐺 (𝑔𝑦, 𝑔V, 𝑔𝑧)₂ ) − 𝜓 (𝐺 (𝑔𝑥, 𝑔𝑢, 𝑔𝑤) + 𝐺 (𝑔𝑦, 𝑔V, 𝑔𝑧) 2 ) (8) for all𝑥, 𝑦, 𝑢, V, 𝑤, 𝑧 ∈ 𝑋.

Our first result is the following.

Theorem 18. Let (𝑋, ≤) be a partially ordered set, and suppose

that there exists a 𝐺-metric 𝐺 on 𝑋 such that (𝑋, 𝐺) is a

complete 𝐺-metric space. Let 𝐹 and g be symmetric (𝜑,

𝜓)-contractive mappings on𝑋 with 𝑔𝑥 ≥ 𝑔𝑢 ≥ 𝑔𝑤 and 𝑔𝑦 ≤ 𝑔V ≤

𝑔𝑧 (or 𝑔𝑥 ≤ 𝑔𝑢 ≤ 𝑔𝑤 and 𝑔𝑦 ≥ 𝑔V ≥ 𝑔𝑧) such that 𝐹 has the

mixed𝑔-monotone property. Assume that 𝐹(𝑋 × 𝑋) ⊆ 𝑔(𝑋);

both the mappings𝐹 and 𝑔 commutes and are continuous.

If there exist two elements𝑥₀, 𝑦₀ ∈ 𝑋 with 𝑔𝑥₀≤ 𝐹(𝑥₀, 𝑦₀)

and𝑔𝑦₀𝐹(𝑦₀, 𝑥₀) (or 𝑔𝑥₀ ≥ 𝐹(𝑥₀, 𝑦₀) and 𝑔𝑦₀ ≤ 𝐹(𝑦₀, 𝑥₀)),

then there exist𝑥, 𝑦 ∈ 𝑋 such that 𝑔𝑥 = 𝐹(𝑥, 𝑦) and 𝑔𝑦 =

𝐹(𝑦, 𝑥); that is, 𝐹 and 𝑔 have a coupled coincidence point in 𝑋.

Proof. Without loss of generality, assume that there exist

𝑥₀, 𝑦₀ ∈ 𝑋 such that 𝑔𝑥₀ ≤ 𝐹(𝑥₀, 𝑦₀), 𝑔𝑦₀ ≥ 𝐹(𝑦₀, 𝑥₀). Since𝐹(𝑋 × 𝑋) ⊆ 𝑔(𝑋), we can choose 𝑥₁, 𝑦₁ ∈ 𝑋 such that𝑔𝑥₁ = 𝐹(𝑥₀, 𝑦₀), 𝑔𝑦₁ = 𝐹(𝑦₀, 𝑥₀). Again we can choose 𝑥₂, 𝑦₂∈ 𝑋 such that 𝑔𝑥₂= 𝐹(𝑥₁, 𝑦₁), 𝑔𝑦₂= 𝐹(𝑦₁, 𝑥₁).

Continuing this process, we can construct sequences {𝑔𝑥_𝑛} and {𝑔𝑦_𝑛} in 𝑋 such that

𝑔𝑥_𝑛+1= 𝐹 (𝑥_𝑛, 𝑦_𝑛) , 𝑔𝑦_𝑛+1= 𝐹 (𝑦_𝑛, 𝑥_𝑛) , for all 𝑛 ≥ 0. (9) We will prove, for all𝑛 ≥ 0, that

𝑔𝑥_𝑛≤ 𝑔𝑥_𝑛+1,

𝑔𝑦_𝑛≥ 𝑔𝑦_𝑛+1. (10) Since𝑔𝑥₀ ≤ 𝐹(𝑥₀, 𝑦₀), 𝑔𝑦₀ ≥ 𝐹(𝑦₀, 𝑥₀), and 𝑔𝑥₁ = 𝐹(𝑥₀, 𝑦₀), 𝑔𝑦₁ = 𝐹(𝑦₀, 𝑥₀), we have 𝑔𝑥₀ ≤ 𝑔𝑥₁, 𝑔𝑦₀ ≥ 𝑔𝑦₁; that is, (10) holds for𝑛 = 0.

Suppose that (10) holds for some𝑛 > 0; that is, 𝑔𝑥_𝑛 ≤ 𝑔𝑥_𝑛+1, 𝑔𝑦_𝑛 ≥ 𝑔𝑦_𝑛+1. As 𝐹 has the mixed 𝑔-monotone property, using (9), we have

𝑔𝑥_𝑛+1= 𝐹 (𝑥_𝑛, 𝑦_𝑛) ≤ 𝐹 (𝑥_𝑛+1, 𝑦_𝑛) ≤ 𝐹 (𝑥𝑛+1, 𝑦𝑛+1) = 𝑔𝑥𝑛+2,

𝑔𝑦_𝑛+1= 𝐹 (𝑦_𝑛, 𝑥_𝑛) ≥ 𝐹 (𝑦_𝑛+1, 𝑥_𝑛) ≥ 𝐹 (𝑦_𝑛+1, 𝑥_𝑛+1) = 𝑔𝑦_𝑛+2.

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Then by mathematical induction, it follows that (10) holds for all𝑛 ≥ 0.

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If for some𝑛 ≥ 0, we have (𝑔𝑥_𝑛+1, 𝑔𝑦_𝑛+1) = (𝑔𝑥_𝑛, 𝑔𝑦_𝑛), and then𝐹(𝑥_𝑛, 𝑦_𝑛) = 𝑔𝑥_𝑛 and𝐹(𝑦_𝑛, 𝑥_𝑛) = 𝑔𝑦_𝑛; that is,𝐹 and𝑔 have a coupled coincidence point. So now onwards, we suppose that(𝑔𝑥_𝑛+1, 𝑔𝑦_𝑛+1) ̸= (𝑔𝑥_𝑛, 𝑔𝑦_𝑛) for all 𝑛 ≥ 0; that is, we suppose that either𝑔𝑥_𝑛+1 = 𝐹(𝑥_𝑛, 𝑦_𝑛) ̸= 𝑔𝑥_𝑛or𝑔𝑦_𝑛+1 = 𝐹(𝑦𝑛, 𝑥𝑛) ̸= 𝑔𝑦𝑛. Using (8)–(10), we have 𝜑 (𝐺 (𝑔𝑥𝑛+1𝑔𝑥𝑛+1, 𝑔𝑥𝑛) + 𝐺 (𝑔𝑦₂ 𝑛+1, 𝑔𝑦𝑛+1𝑔𝑦𝑛)) = 𝜑 ((𝐺 (𝐹 (𝑥_𝑛, 𝑦_𝑛) , 𝐹 (𝑥_𝑛, 𝑦_𝑛) , 𝐹 (𝑥_𝑛−1, 𝑦_𝑛−1)) +𝐺 (𝐹 (𝑦_𝑛, 𝑥_𝑛) , 𝐹 (𝑦_𝑛, 𝑥_𝑛) , 𝐹 (𝑦_𝑛−1, 𝑥_𝑛−1))) 2−1) ≤ 𝜑 (𝐺 (𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑔𝑥𝑛−1) + 𝐺 (𝑔𝑦𝑛, 𝑔𝑦𝑛, 𝑔𝑦𝑛−1) 2 ) − 𝜓 (𝐺 (𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑔𝑥𝑛−1) + 𝐺 (𝑔𝑦₂ 𝑛, 𝑔𝑦𝑛, 𝑔𝑦𝑛−1)) . (12) Since𝜓 is nonnegative, using (12), we get

𝜑 (𝐺 (𝑔𝑥𝑛+1, 𝑔𝑥𝑛+1, 𝑔𝑥𝑛) + 𝐺 (𝑔𝑦𝑛+1, 𝑔𝑦𝑛+1, 𝑔𝑦𝑛) 2 ) ≤ 𝜑 (𝐺 (𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑔𝑥𝑛−1) + 𝐺 (𝑔𝑦𝑛, 𝑔𝑦𝑛, 𝑔𝑦𝑛−1) 2 ) . (13) By monotonicity of𝜑, we get 𝐺 (𝑔𝑥_𝑛+1, 𝑔𝑥_𝑛+1, 𝑔𝑥_𝑛) + 𝐺 (𝑔𝑦_𝑛+1, 𝑔𝑦_𝑛+1, 𝑔𝑦_𝑛) 2 ≤ 𝐺 (𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑔𝑥𝑛−1) + 𝐺 (𝑔𝑦𝑛, 𝑔𝑦𝑛, 𝑔𝑦𝑛−1) 2 . (14) Let𝛿_𝑛 := (𝐺(𝑔𝑥_𝑛+1, 𝑔𝑥_𝑛+1, 𝑔𝑥_𝑛) + 𝐺(𝑔𝑦_𝑛+1, 𝑔𝑦_𝑛+1, 𝑔𝑦_𝑛))/2; then{𝛿_𝑛} is a monotone decreasing sequence. Therefore, there exists some𝛿 ≥0 such that

Lim 𝑛 → ∞𝛿𝑛 = lim_{𝑛 → ∞}[𝐺 (𝑔𝑥𝑛+1, 𝑔𝑥𝑛+1, 𝑔𝑥𝑛) + 𝐺 (𝑔𝑦₂ 𝑛+1, 𝑔𝑦𝑛+1, 𝑔𝑦𝑛)] = 𝛿. (15) We claim that𝛿 = 0.

On the contrary, suppose that𝛿 > 0.

Taking limit as𝑛 → ∞ on both sides of (12) and using the properties of𝜑 and 𝜓, we have

𝜑 (𝛿) = lim_{𝑛 → ∞}𝜑 (𝛿_𝑛) ≤ lim_{𝑛 → ∞}[ 𝜑 (𝛿_𝑛−1) − 𝜓 (𝛿_𝑛−1)] = 𝜑 (𝛿) − lim

𝛿𝑛−1→ 𝛿𝜓 (𝛿𝑛−1) < 𝜑 (𝛿) , a contradiction.

(16)

Thus,𝛿 = 0; that is, lim 𝑛 → ∞𝛿𝑛 = lim_{𝑛 → ∞}[𝐺 (𝑔𝑥𝑛+1, 𝑔𝑥𝑛+1, 𝑔𝑥𝑛) + 𝐺 (𝑔𝑦𝑛+1, 𝑔𝑦𝑛+1, 𝑔𝑦𝑛) 2 ] = 0. (17) Next, we shall show that {𝑔𝑥_𝑛} and {𝑔𝑦_𝑛} are 𝐺-Cauchy sequences.

If possible, suppose that at least one of{𝑔𝑥_𝑛} and {𝑔𝑦_𝑛} is not a𝐺-Cauchy sequence. Then there exists an 𝜀 > 0 for which we can find subsequences{𝑔𝑥_𝑛(𝑘)}, {𝑔𝑥_𝑚(𝑘)} of {𝑔𝑥_𝑛} and{𝑔𝑦_𝑛(𝑘)}, {𝑔𝑦_𝑚(𝑘)} of {𝑔𝑦_𝑛} with 𝑛(𝑘) > 𝑚(𝑘) ≥ 𝑘 such that

𝑟_𝑘 =𝐺 (𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑚(𝑘)) + 𝐺 (𝑔𝑦𝑛(𝑘), 𝑔𝑦𝑛(𝑘), 𝑔𝑦𝑚(𝑘)) 2

≥ 𝜀.

(18) Further, corresponding to𝑚(𝑘), we can choose 𝑛(𝑘) in such a way that it is the smallest integer with𝑛(𝑘) > 𝑚(𝑘) ≥ 𝑘 and satisfies (18). Then,

𝐺 (𝑔𝑥_{𝑛(𝑘)−1}, 𝑔𝑥_{𝑛(𝑘)−1}, 𝑔𝑥_𝑚(𝑘)) + 𝐺 (𝑔𝑦_{𝑛(𝑘)−1}, 𝑔𝑦_{𝑛(𝑘)−1}, 𝑔𝑦_𝑚(𝑘)) 2

< 𝜀.

(19) Using (18), (19) andLemma 7, we get

𝜀 ≤ 𝑟𝑘 =𝐺 (𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑚(𝑘)) + 𝐺 (𝑔𝑦₂ 𝑛(𝑘), 𝑔𝑦𝑛(𝑘), 𝑔𝑦𝑚(𝑘)) ≤ {𝐺 (𝑔𝑥_𝑛(𝑘), 𝑔𝑥_𝑛(𝑘)𝑔𝑥_{𝑛(𝑘)−1}) + 𝐺 (𝑔𝑥_{𝑛(𝑘)−1}, 𝑔𝑥_{𝑛(𝑘)−1}, 𝑔𝑥_𝑚(𝑘)) + 𝐺 (𝑔𝑦_𝑛(𝑘), 𝑔𝑦_𝑛(𝑘), 𝑔𝑦_{𝑛(𝑘)−1}) +𝐺 (𝑔𝑦_{𝑛(𝑘)−1}, 𝑔𝑦_{𝑛(𝑘)−1}, 𝑔𝑦_𝑚(𝑘))} 2−1 < (𝐺 (𝑔𝑥_𝑛(𝑘), 𝑔𝑥_𝑛(𝑘), 𝑔𝑥_{𝑛(𝑘)−1}) +𝐺 (𝑔𝑦𝑛(𝑘), 𝑔𝑦𝑛(𝑘), 𝑔𝑦𝑛(𝑘)−1)) 2−1+ 𝜀. (20) Letting𝑘 → ∞ and using (17), we have

lim

𝑘 → ∞𝑟𝑘= lim𝑘 → ∞[ (𝐺 (𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑚(𝑘))

+𝐺 (𝑔𝑦_𝑛(𝑘), 𝑔𝑦_𝑛(𝑘), 𝑔𝑦_𝑚(𝑘))) 2−1] = 𝜀. (21)

(5)

UsingLemma 6andLemma 7, we get 𝐺 (𝑔𝑥_𝑛(𝑘), 𝑔𝑥_𝑛(𝑘), 𝑔𝑥_𝑚(𝑘)) ≤ 𝐺 (𝑔𝑥_𝑛(𝑘), 𝑔𝑥_𝑛(𝑘), 𝑔𝑥_𝑛(𝑘)+1) + 𝐺 (𝑔𝑥_𝑛(𝑘)+1, 𝑔𝑥_𝑛(𝑘)+1, 𝑔𝑥_𝑚(𝑘)) ≤ 2𝐺 (𝑔𝑥𝑛(𝑘)+1, 𝑔𝑥𝑛(𝑘)+1, 𝑔𝑥𝑛(𝑘)) + 𝐺 (𝑔𝑥_𝑛(𝑘)+1, 𝑔𝑥_𝑛(𝑘)+1, 𝑔𝑥_𝑚(𝑘)+1) + 𝐺 (𝑔𝑥_𝑚(𝑘)+1, 𝑔𝑥_𝑚(𝑘)+1, 𝑔𝑥_𝑚(𝑘)) . (22)

Similarly, we can obtain

𝐺 (𝑔𝑦_𝑛(𝑘), 𝑔𝑦_𝑛(𝑘), 𝑔𝑦_𝑚(𝑘))

≤ 2𝐺 (𝑔𝑦_𝑛(𝑘)+1, 𝑔𝑦_𝑛(𝑘)+1, 𝑔𝑦_𝑛(𝑘)) + 𝐺 (𝑔𝑦_𝑛(𝑘)+1, 𝑔𝑦_𝑛(𝑘)+1, 𝑔𝑦_𝑚(𝑘)+1) + 𝐺 (𝑔𝑦_𝑚(𝑘)+1, 𝑔𝑦_𝑚(𝑘)+1, 𝑔𝑦_𝑚(𝑘)) .

(23)

Now, for all𝑘 ≥ 0, using (22)-(23) in (18), we get

𝑟_𝑘 =𝐺 (𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑚(𝑘)) + 𝐺 (𝑔𝑦₂ 𝑛(𝑘), 𝑔𝑦𝑛(𝑘), 𝑔𝑦𝑚(𝑘)) ≤ 2𝛿𝑛(𝑘) + 𝛿𝑚(𝑘)

+ (𝐺 (𝑔𝑥_𝑛(𝑘)+1, 𝑔𝑥_𝑛(𝑘)+1, 𝑔𝑥_𝑚(𝑘)+1) + 𝐺 (𝑔𝑦_𝑛(𝑘)+1, 𝑔𝑦_𝑛(𝑘)+1, 𝑔𝑦_𝑚(𝑘)+1)) 2−1.

(24)

By monotonicity of𝜑 and property (𝜑iii), we have

𝜑 (𝑟_𝑘) ≤ 𝜑 (2𝛿_𝑛(𝑘)+ 𝛿_𝑚(𝑘) + (𝐺 (𝑔𝑥_𝑛(𝑘)+1, 𝑔𝑥_𝑛(𝑘)+1, 𝑔𝑥_𝑚(𝑘)+1) +𝐺 (𝑔𝑦_𝑛(𝑘)+1, 𝑔𝑦_𝑛(𝑘)+1, 𝑔𝑦_𝑚(𝑘)+1)) 2−1) ≤ 2𝜑 (𝛿_𝑛(𝑘)) + 𝜑 (𝛿_𝑚(𝑘)) + 𝜑 × ( (𝐺 (𝑔𝑥_𝑛(𝑘)+1, 𝑔𝑥_𝑛(𝑘)+1, 𝑔𝑥_𝑚(𝑘)+1) +𝐺 (𝑔𝑦_𝑛(𝑘)+1, 𝑔𝑦_𝑛(𝑘)+1, 𝑔𝑦_𝑚(𝑘)+1)) 2−1) . (25)

Also, since𝑛(𝑘) > 𝑚(𝑘), 𝑔𝑥_𝑛(𝑘)≥ 𝑔𝑥_𝑚(𝑘)and𝑔𝑦_𝑛(𝑘)≤ 𝑔𝑦_𝑚(𝑘), using (8) and (9), we have

𝜑 ( (𝐺 (𝑔𝑥_𝑛(𝑘)+1, 𝑔𝑥_𝑛(𝑘)+1, 𝑔𝑥_𝑚(𝑘)+1) +𝐺 (𝑔𝑦_𝑛(𝑘)+1, 𝑔𝑦_𝑛(𝑘)+1, 𝑔𝑦_𝑚(𝑘)+1)) 2−1) = 𝜑 ( (𝐺 (𝐹 (𝑥_𝑛(𝑘), 𝑦_𝑛(𝑘)) , 𝐹 (𝑥_𝑛(𝑘), 𝑦_𝑛(𝑘)) , 𝐹 (𝑥_𝑚(𝑘), 𝑦_𝑚(𝑘))) + 𝐺 (𝐹 (𝑦𝑛(𝑘), 𝑥𝑛(𝑘)) , 𝐹 (𝑦𝑛(𝑘), 𝑥𝑛(𝑘)) , 𝐹 (𝑦𝑚(𝑘), 𝑥𝑚(𝑘)))) 2−1) ≤ 𝜑 ( (𝐺 (𝑔𝑥_𝑛(𝑘), 𝑔𝑥_𝑛(𝑘), 𝑔𝑥_𝑚(𝑘)) +𝐺 (𝑔𝑦_𝑛(𝑘), 𝑔𝑦_𝑛(𝑘), 𝑔𝑦_𝑚(𝑘))) 2−1) − 𝜓 ( (𝐺 (𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑚(𝑘)) +𝐺 (𝑔𝑦_𝑛(𝑘), 𝑔𝑦_𝑛(𝑘), 𝑔𝑦_𝑚(𝑘))) 2−1) = 𝜑 (𝑟_𝑘) − 𝜓 (𝑟_𝑘) . (26) Combining (25) and (26), we obtain that

𝜑 (𝑟_𝑘) ≤ 2𝜑 (𝛿_𝑛(𝑘)) + 𝜑 (𝛿_𝑚(𝑘)) + 𝜑 (𝑟_𝑘) − 𝜓 (𝑟_𝑘) . (27) On letting𝑘 → ∞, using (17), (21) and continuity of𝜑, we get 𝜑 (𝜀) ≤ 2𝜑 (0) + 𝜑 (0) + 𝜑 (𝜀) − lim 𝑘 → ∞𝜓 (𝑟𝑘) = 2𝜑 (0) + 𝜑 (0) + 𝜑 (𝜀) − lim_𝑟𝑘_{→ 𝜀}𝜓 (𝑟_𝑘) < 𝜑 (𝜀) , a contradiction. (28)

Therefore both{𝑔𝑥_𝑛} and {𝑔𝑦_𝑛} are G-Cauchy sequences in 𝑋. Now, since the G-metric space (𝑋, 𝐺) is G-complete, there exist𝑥, 𝑦 in 𝑋 such that the sequences {𝑔𝑥_𝑛} and {𝑔𝑦_𝑛} are, respectively,𝐺-convergent to 𝑥 and 𝑦, and then byLemma 4, we have lim 𝑛 → ∞𝐺 (𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑥) = lim𝑛 → ∞𝐺 (𝑔𝑥𝑛, 𝑥, 𝑥) = 0, lim 𝑛 → ∞𝐺 (𝑔𝑦𝑛, 𝑔𝑦𝑛, 𝑦) = lim𝑛 → ∞𝐺 (𝑔𝑦𝑛, 𝑦, 𝑦) = 0. (29)

Using the𝐺-continuity of 𝑔, andDefinition 8, we get lim

𝑛 → ∞𝐺 (𝑔𝑔𝑥𝑛, 𝑔𝑔𝑥𝑛, 𝑔𝑥) = lim𝑛 → ∞𝐺 (𝑔𝑔𝑥𝑛, 𝑔𝑥, 𝑔𝑥) = 0,

(30) lim

𝑛 → ∞𝐺 (𝑔𝑔𝑦𝑛, 𝑔𝑔𝑦𝑛, 𝑔𝑦) = lim𝑛 → ∞𝐺 (𝑔𝑔𝑦𝑛, 𝑔𝑦, 𝑔𝑦) = 0. (31)

Since𝑔𝑥_𝑛+1 = 𝐹(𝑥_𝑛, 𝑦_𝑛) and 𝑔𝑦_𝑛+1 = 𝐹(𝑦_𝑛, 𝑥_𝑛), hence using commutativity of𝐹 and 𝑔 we obtain

𝑔𝑔𝑥_𝑛+1= 𝑔𝐹 (𝑥_𝑛, 𝑦_𝑛) = 𝐹 (𝑔𝑥_𝑛, 𝑔𝑦_𝑛) , (32) 𝑔𝑔𝑦_𝑛+1= 𝑔𝐹 (𝑦_𝑛, 𝑥_𝑛) = 𝐹 (𝑔𝑦_𝑛, 𝑔𝑥_𝑛) . (33)

(6)

Since the mapping 𝐹 is G-continuous and the sequences {𝑔𝑥_𝑛} and {𝑔𝑦_𝑛} are, respectively, 𝐺-convergent to 𝑥 and 𝑦, hence usingDefinition 11, the sequence{𝐹(𝑔𝑥_𝑛, 𝑔𝑦_𝑛)} is 𝐺-convergent to𝐹(𝑥, 𝑦). By uniqueness of limit and using (30), and (32) we get𝐹(𝑥, 𝑦) = 𝑔𝑥. Similarly, we can show that 𝐹(𝑦, 𝑥) = 𝑔𝑦. Hence, (𝑥, 𝑦) ∈ 𝑋 × 𝑋 is a coupled coincidence point of𝐹 and 𝑔.

In the next theorem, we omit the continuity hypotheses of the mapping𝐹 along with the commutativity of mappings 𝐹 and 𝑔. We need the following definition.

Definition 19. Let (𝑋, ≤) be a partially ordered set, and

suppose that there exists a 𝐺-metric 𝐺 on 𝑋. We say that (𝑋, 𝐺, ≤) is regular if the following conditions hold:

(i) if a nondecreasing sequence{𝑥_𝑛} ⊆ 𝑋 such that 𝑥_𝑛 → 𝑥, then 𝑥_𝑛≤ 𝑥 for all 𝑛,

(ii) if a nonincreasing sequence{𝑦_𝑛} ⊆ 𝑋 such that 𝑦_𝑛 → 𝑦, then 𝑦 ≤ 𝑦_𝑛for all𝑛.

Theorem 20. Let (𝑋, ≤) be a partially ordered set, and suppose

that there exists a𝐺-metric 𝐺 on 𝑋. Let 𝐹 and 𝑔 be symmetric

(𝜑, 𝜓)-contractive mappings on 𝑋 with 𝑔𝑥 ≥ 𝑔𝑢 ≥ 𝑔𝑤 and 𝑔𝑦 ≤ 𝑔V ≤ 𝑔𝑧 (or 𝑔𝑥 ≤ 𝑔𝑢 ≤ 𝑔𝑤 and 𝑔𝑦 ≥ 𝑔V ≥ 𝑔𝑧)

such that𝐹 has the mixed 𝑔-monotone property. Assume that

(𝑋, 𝐺, ≤) is regular. Suppose that (𝑔(𝑋), 𝐺) is 𝐺-complete and 𝐹(𝑋 × 𝑋) ⊆ 𝑔(𝑋). Suppose that there exist 𝑥₀, 𝑦₀ ∈ 𝑋 with 𝑔𝑥₀≤ 𝐹(𝑥₀, 𝑦₀) and 𝑔𝑦₀≥ 𝐹(𝑦₀, 𝑥₀) (or 𝑔𝑥₀≥ 𝐹(𝑥₀, 𝑦₀) and 𝑔𝑦₀ ≤ 𝐹(𝑦₀, 𝑥₀)); then 𝐹 and 𝑔 have a coupled coincidence

point in𝑋; that is, there exist 𝑥, 𝑦 ∈ 𝑋 such that 𝑔𝑥 = 𝐹(𝑥, 𝑦)

and𝑔𝑦 = 𝐹(𝑦, 𝑥).

Proof. Proceeding exactly as in Theorem 18, we have that

{𝑔𝑥𝑛} and {𝑔𝑦𝑛} are Cauchy sequences in the complete

G-metric space(𝑔(𝑋), 𝐺). Then there exist 𝑥, 𝑦 ∈ 𝑋 such that 𝑔𝑥𝑛 → 𝑔𝑥 and 𝑔𝑦𝑛 → 𝑔𝑦; that is,

Lim

𝑛 → ∞𝐺 (𝑔𝑥𝑛, 𝑔𝑥, 𝑔𝑥) = lim𝑛 → ∞𝐺 (𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑔𝑥) = 0,

lim

𝑛 → ∞𝐺 (𝑔𝑦𝑛, 𝑔𝑦, 𝑔𝑦) = lim𝑛 → ∞𝐺 (𝑔𝑦𝑛, 𝑔𝑦𝑛, 𝑔𝑦) = 0.

(34)

Since {𝑔𝑥_𝑛} is nondecreasing and {𝑔𝑦_𝑛} is nonincreasing, using the regularity of (𝑋, 𝐺, ≤), we have 𝑔𝑥_𝑛 ≤ 𝑔𝑥 and 𝑔𝑦 ≤ 𝑔𝑦_𝑛for all𝑛 ≥ 0. Using (8), we get

𝜑 ( (𝐺 (𝐹 (𝑥, 𝑦) , 𝑔𝑥_𝑛+1, 𝑔𝑥_𝑛+1) +𝐺 (𝐹 (𝑦, 𝑥) , 𝑔𝑦_𝑛+1, 𝑔𝑦_𝑛+1)) 2−1) = 𝜑 ( (𝐺 (𝐹 (𝑥, 𝑦) , 𝐹 (𝑥_𝑛, 𝑦_𝑛) , 𝐹 (𝑥_𝑛, 𝑦_𝑛)) +𝐺 (𝐹 (𝑦, 𝑥) , 𝐹 (𝑦𝑛, 𝑥𝑛) , 𝐹 (𝑦𝑛, 𝑥𝑛))) 2−1) ≤ 𝜑 (𝐺 (𝑔𝑥, 𝑔𝑥𝑛, 𝑔𝑥𝑛) + 𝐺 (𝑔𝑦, 𝑔𝑦₂ 𝑛, 𝑔𝑦𝑛)) − 𝜓 (𝐺 (𝑔𝑥, 𝑔𝑥𝑛, 𝑔𝑥𝑛) + 𝐺 (𝑔𝑦, 𝑔𝑦𝑛, 𝑔𝑦𝑛) 2 ) . (35)

On letting𝑛 → ∞ and using (34) and the properties of𝜑 and𝜓, we obtain that

𝜑 ( lim_{𝑛 → ∞}((𝐺 (𝐹 (𝑥, 𝑦) , 𝑔𝑥_𝑛+1, 𝑔𝑥_𝑛+1) +𝐺 (𝐹 (𝑦, 𝑥) , 𝑔𝑦_𝑛+1, 𝑔𝑦_𝑛+1)) 2−1) ) ≤ 𝜑 ( lim_{𝑛 → ∞}(𝐺 (𝑔𝑥, 𝑔𝑥𝑛, 𝑔𝑥𝑛) + 𝐺 (𝑔𝑦, 𝑔𝑦𝑛, 𝑔𝑦𝑛) 2 )) − lim_{𝑛 → ∞}𝜓 (𝐺 (𝑔𝑥, 𝑔𝑥𝑛, 𝑔𝑥𝑛) + 𝐺 (𝑔𝑦, 𝑔𝑦₂ 𝑛, 𝑔𝑦𝑛)) = 0, (36) which yields lim 𝑛 → ∞( (𝐺 (𝐹 (𝑥, 𝑦) , 𝑔𝑥𝑛+1, 𝑔𝑥𝑛+1) +𝐺 (𝐹 (𝑦, 𝑥) , 𝑔𝑦_𝑛+1, 𝑔𝑦_𝑛+1)) 2−1) = 0. (37) Hence we can obtain that

lim

𝑛 → ∞𝐺 (𝐹 (𝑥, 𝑦) , 𝑔𝑥𝑛+1, 𝑔𝑥𝑛+1)

= 0 = lim_{𝑛 → ∞}𝐺 (𝐹 (𝑦, 𝑥) , 𝑔𝑦_𝑛+1, 𝑔𝑦_𝑛+1) . (38) On the other hand, by condition (G5), we have

𝐺 (𝐹 (𝑥, 𝑦) , 𝑔𝑥, 𝑔𝑥) + 𝐺 (𝐹 (𝑦, 𝑥) , 𝑔𝑦, 𝑔𝑦)

≤ {𝐺 (𝐹 (𝑥, 𝑦) , 𝑔𝑥_𝑛+1, 𝑔𝑥_𝑛+1) + 𝐺 (𝑔𝑥_𝑛+1, 𝑔𝑥, 𝑔𝑥)} + {𝐺 (𝐹 (𝑦, 𝑥) , 𝑔𝑦_𝑛+1, 𝑔𝑦_𝑛+1) + 𝐺 (𝑔𝑦_𝑛+1, 𝑔𝑦, 𝑔𝑦)} .

(39) Letting𝑛 → ∞ in (39) and using (34)–(38), we have

𝐺 (𝐹 (𝑥, 𝑦) , 𝑔𝑥, 𝑔𝑥) + 𝐺 (𝐹 (𝑦, 𝑥) , 𝑔𝑦, 𝑔𝑦) = 0. (40) Thus𝐹(𝑥, 𝑦) = 𝑔𝑥 and 𝑔𝑦 = 𝐹(𝑦, 𝑥). Therefore, we proved that(𝑥, 𝑦) is a coupled coincidence point of 𝐹 and 𝑔.

Next we give an example in support ofTheorem 18that shows thatTheorem 18is more general than Theorem 3.1 in [31], since the contractive condition (8) is more general than (5).

Example 21. Let𝑋 = R. Then (𝑋, ≤) is a partially ordered set

with the natural ordering of real numbers. Let𝐺 : 𝑋 × 𝑋 × 𝑋 → 𝑅+_{be defined by}

𝐺 (𝑥, 𝑦, 𝑧) = 󵄨󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑦 − 𝑧󵄨󵄨󵄨󵄨 + |𝑧 − 𝑥|

for𝑥, 𝑦, 𝑧 ∈ 𝑋. (41) Then(𝑋, 𝐺) is a complete 𝐺-metric space.

Define𝐹 : 𝑋 × 𝑋 → 𝑋 by 𝐹(𝑥, 𝑦) = (𝑥 − 2𝑦)/8, (𝑥, 𝑦) ∈ 𝑋 × 𝑋 and 𝑔 : 𝑋 → 𝑋 by 𝑔(𝑥) = 𝑥/2, 𝑥 ∈ 𝑋.

Clearly𝐹(𝑋 × 𝑋) ⊆ 𝑔(𝑋), 𝐹 and 𝑔 are continuous, 𝐹 has the mixed𝑔-monotone property, and the pair (𝐹, 𝑔) is

(7)

commutative and satisfies condition (8) but does not satisfy the condition (5). Assume, to the contrary, that there exists some𝑘 ∈ [0, 1/2) such that (5) holds. Then, we must have

(󵄨󵄨󵄨󵄨_󵄨󵄨󵄨𝑥 − 2𝑦 8 − 𝑢 − 2V 8 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝑢 − 2V 8 − 𝑤 − 2𝑧 8 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨_󵄨󵄨󵄨𝑤 − 2𝑧 8 − 𝑥 − 2𝑦 8 󵄨󵄨󵄨󵄨󵄨󵄨󵄨) ≤ 𝑘 [(󵄨󵄨󵄨󵄨_󵄨󵄨󵄨𝑥 2− 𝑢 2󵄨󵄨󵄨󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝑢 2− 𝑤 2󵄨󵄨󵄨󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝑤 2 − 𝑥 2󵄨󵄨󵄨󵄨󵄨󵄨󵄨) + (󵄨󵄨󵄨󵄨_󵄨󵄨󵄨𝑦 2 − V 2󵄨󵄨󵄨󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨󵄨󵄨󵄨 V 2− 𝑧 2󵄨󵄨󵄨󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝑧 2 − 𝑦 2󵄨󵄨󵄨󵄨󵄨󵄨󵄨)] =𝑘₂[ (|𝑥 − 𝑢| + |𝑢 − 𝑤| + |𝑤 − 𝑥|) + (󵄨󵄨󵄨󵄨𝑦 − V󵄨󵄨󵄨󵄨 + |V − 𝑧| + 󵄨󵄨󵄨󵄨𝑧 − 𝑦󵄨󵄨󵄨󵄨)] (42)

for all𝑥 ≥ 𝑢 ≥ 𝑤 and 𝑦 ≤ V ≤ 𝑧. Taking 𝑥 = 𝑢 = 𝑤, V < 𝑧 in the last inequality and setting𝜌 := |V−𝑧|/2+|𝑦−V|/2+|𝑧−𝑦|/2, we obtain

𝜌

2 ≤ 𝑘𝜌, 𝜌 > 0, (43) which implies that1/2 ≤ 𝑘, a contradiction since 𝑘 ∈ [0, 1/2). Hence𝐹 and 𝑔 do not satisfy (5).

Indeed, for𝑥 ≥ 𝑢 ≥ 𝑤 and 𝑦 ≤ V ≤ 𝑧, we have 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑥 − 2𝑦8 − 𝑢 − 2V 8 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ 1 8|𝑥 − 𝑢| + 1 4󵄨󵄨󵄨󵄨𝑦 − V󵄨󵄨󵄨󵄨, 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑢 − 2V8 − 𝑤 − 2𝑧 8 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ 1 8|𝑢 − 𝑤| + 1 4|V − 𝑧| , 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑤 − 2𝑧8 − 𝑥 − 2𝑦 8 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ 1 8|𝑤 − 𝑥| + 1 4󵄨󵄨󵄨󵄨𝑧 − 𝑦󵄨󵄨󵄨󵄨, 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑦 − 2𝑥8 − V − 2𝑢 8 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ 1 8󵄨󵄨󵄨󵄨𝑦 − V󵄨󵄨󵄨󵄨 + 14|𝑥 − 𝑢| , 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨V − 2𝑢8 − 𝑧 − 2𝑤 8 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ 1 8|V − 𝑧| + 1 4|𝑢 − 𝑤| , 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑧 − 2𝑤8 − 𝑦 − 2𝑥 8 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ 1 8󵄨󵄨󵄨󵄨𝑧 − 𝑦󵄨󵄨󵄨󵄨 + 14|𝑤 − 𝑥| . (44)

By summing up the above six inequalities, we get exactly (8) with 𝜑(𝑡) = (1/3)𝑡, 𝜓(𝑡) = (1/12)𝑡. Also, 𝑥₀ = −1, 𝑦₀= 1 are the two points in 𝑋 such that 𝑔𝑥₀≤ 𝐹(𝑥₀, 𝑦₀) and 𝑔𝑦₀≥ 𝐹(𝑦₀, 𝑥₀). Now 𝐹, 𝑔, 𝜑, and 𝜓 satisfy all the conditions

ofTheorem 18; byTheorem 18, we obtain that𝐹 and 𝑔 have

a coupled coincidence point(0, 0), but Theorem 3.1 in [31] cannot be applied to𝐹 and 𝑔 in this example.

Now, putting 𝑔 = 𝐼_𝑋 (the identity map of𝑋) in the previous results, we obtain the following.

Corollary 22. Let (𝑋, ≤) be a partially ordered set, and let 𝐺 be a𝐺-metric on 𝑋. Let 𝐹 : 𝑋 × 𝑋 → 𝑋 be a mapping satisfying

(8) (with𝑔 = 𝐼_𝑋); that is,

𝜑 ( (𝐺 (𝐹 (𝑥, 𝑦) , 𝐹 (𝑢, V) , 𝐹 (𝑤, 𝑧)) +𝐺 (𝐹 (𝑦, 𝑥) , 𝐹 (V, 𝑢) , 𝐹 (𝑧, 𝑤))) 2−1) ≤ 𝜑 (𝐺 (𝑥, 𝑢, 𝑤) + 𝐺 (𝑦, V, 𝑧)₂ ) − 𝜓 (𝐺 (𝑥, 𝑢, 𝑤) + 𝐺 (𝑦, V, 𝑧) 2 ) (45)

for all𝑥, 𝑦, 𝑢, V, 𝑤, 𝑧 ∈ 𝑋 with 𝑥 ≥ 𝑢 ≥ 𝑤 and 𝑦 ≤ V ≤ 𝑧 (or

𝑥 ≤ 𝑢 ≤ 𝑤 and 𝑦 ≥ V ≥ 𝑧). Assume that (𝑋, 𝐺) is complete and 𝐹 has the mixed monotone property. Also suppose that either

(a)𝐹 is continuous, or (b)(𝑋, ≤, 𝐺) is regular.

If there exist two elements𝑥₀, 𝑦₀∈ 𝑋 with 𝑥₀ ≤ 𝐹(𝑥₀, 𝑦₀) and

𝑦₀ ≥ 𝐹(𝑦₀, 𝑥₀) (or 𝑥₀ ≥ 𝐹(𝑥₀, 𝑦₀) and 𝑦₀ ≤ 𝐹(𝑦₀, 𝑥₀)), then

there exist𝑥, 𝑦 ∈ 𝑋 such that 𝑥 = 𝐹(𝑥, 𝑦) and 𝑦 = 𝐹(𝑦, 𝑥);

that is,𝐹 has a coupled fixed point in 𝑋.

The following example shows that the contractive condition

(45) is more general than the contractive conditions (4) and (6).

Example 23. Let𝑋 = R. Then (𝑋, ≤) is a partially ordered set

with the natural ordering of real numbers. Let𝐺 : 𝑋 × 𝑋 × 𝑋 → 𝑅+be defined by

𝐺 (𝑥, 𝑦, 𝑧) = 󵄨󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑦 − 𝑧󵄨󵄨󵄨󵄨 + |𝑧 − 𝑥|

for𝑥, 𝑦, 𝑧 ∈ 𝑋. (46) Then(𝑋, 𝐺) is a complete 𝐺-metric space.

Define𝐹 : 𝑋 × 𝑋 → 𝑋 by 𝐹(𝑥, 𝑦) = (𝑥 − 5𝑦)/10, (𝑥, 𝑦) ∈ 𝑋 × 𝑋.

Then𝐹 is continuous and satisfies the mixed monotone property. We note that𝐹 satisfies condition (45) but does not satisfy the conditions (4) and (6). Indeed, assume that there exists𝑘 ∈ [0, 1), such that (4) holds. Then, we must have

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑥 − 5𝑦10 − 𝑢 − 5V 10 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝑢 − 5V 10 − 𝑤 − 5𝑧 10 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨_󵄨󵄨󵄨𝑤 − 5𝑧₁₀ −𝑥 − 5𝑦₁₀ 󵄨󵄨󵄨󵄨_󵄨󵄨󵄨 ≤ 𝑘 2{ (|𝑥 − 𝑢| + |𝑢 − 𝑤| + |𝑤 − 𝑥|) + (󵄨󵄨󵄨󵄨𝑦 − V󵄨󵄨󵄨󵄨 + |V − 𝑧| + 󵄨󵄨󵄨󵄨𝑧 − 𝑦󵄨󵄨󵄨󵄨)} 𝑥 ≥ 𝑢 ≥ 𝑤, 𝑦 ≤ V ≤ 𝑧, (47)

by which, for𝑥 = 𝑢 = 𝑤, we get 󵄨󵄨󵄨󵄨𝑦 − V󵄨󵄨󵄨󵄨 + |V − 𝑧| + 󵄨󵄨󵄨󵄨𝑧 − 𝑦󵄨󵄨󵄨󵄨

(8)

which forV < 𝑧 implies 1 ≤ 𝑘, a contradiction, since 𝑘 ∈ [0, 1). Hence 𝐹 does not satisfy (4).

Next, we prove that (6) is not satisfied, either. Assume, to the contrary, that there exist functions𝜑 and 𝜓 satisfying appropriate conditions as in [33] such that (6) holds. This means that 𝜑 (󵄨󵄨󵄨󵄨_󵄨󵄨󵄨𝑥 − 5𝑦 10 − 𝑢 − 5V 10 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝑢 − 5V 10 − 𝑤 − 5𝑧 10 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨_󵄨󵄨󵄨𝑤 − 5𝑧₁₀ −𝑥 − 5𝑦₁₀ 󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨)} ≤ 1 2𝜑 ( |𝑥 − 𝑢| + |𝑢 − 𝑤| + |𝑤 − 𝑥| + 󵄨󵄨󵄨󵄨𝑦 − V󵄨󵄨󵄨󵄨 + |V − 𝑧| + 󵄨󵄨󵄨󵄨𝑧 − 𝑦󵄨󵄨󵄨󵄨) − 𝜓 ( (|𝑥 − 𝑢| + |𝑢 − 𝑤| + |𝑤 − 𝑥| + 󵄨󵄨󵄨󵄨𝑦 − V󵄨󵄨󵄨󵄨 + |V − 𝑧| + 󵄨󵄨󵄨󵄨𝑧 − 𝑦󵄨󵄨󵄨󵄨)2−1_{) ,} (49)

for all𝑥 ≥ 𝑢 ≥ 𝑤 and 𝑦 ≤ V ≤ 𝑧. Taking 𝑥 = 𝑢 = 𝑤, V < 𝑧 in the previous inequality and setting𝛼 := (|𝑦 − V| + |V − 𝑧| + |𝑧 − 𝑦|)/2, we obtain

𝜑 (𝛼) ≤ 1₂𝜑 (2𝛼) − 𝜓 (𝛼) , 𝛼 > 0. (50) Since 𝜑 satisfy the subadditive property, we have (1/2)𝜑(2𝛼) ≤ 𝜑(𝛼), and therefore, we deduce that, for all𝛼 > 0, 𝜓(𝛼) ≤ 0; that is, 𝜓(𝛼) = 0, which contradicts the definition of𝜓.

This shows that𝐹 does not satisfy (6).

Finally, we prove that (45) holds. Indeed, for𝑥 ≥ 𝑢 ≥ 𝑤 and𝑦 ≤ V ≤ 𝑧, we have 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑥 − 5𝑦10 − 𝑢 − 5V 10 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ 1 10|𝑥 − 𝑢| + 1 2󵄨󵄨󵄨󵄨𝑦 − V󵄨󵄨󵄨󵄨, 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑢 − 5V10 − 𝑤 − 5𝑧 10 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ 1 10|𝑢 − 𝑤| + 1 2|V − 𝑧| , 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑤 − 5𝑧10 − 𝑥 − 5𝑦 10 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ 1 10|𝑤 − 𝑥| + 1 2󵄨󵄨󵄨󵄨𝑧 − 𝑦󵄨󵄨󵄨󵄨, 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑦 − 5𝑥10 − V − 5𝑢 10 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ 1 10󵄨󵄨󵄨󵄨𝑦 − V󵄨󵄨󵄨󵄨 + 12|𝑥 − 𝑢| , 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨V − 5𝑢10 − 𝑧 − 5𝑤 10 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ 1 10|V − 𝑧| + 1 2|𝑢 − 𝑤| , 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑧 − 5𝑤10 − 𝑦 − 5𝑥 10 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ 1 10󵄨󵄨󵄨󵄨𝑧 − 𝑦󵄨󵄨󵄨󵄨 + 12|𝑤 − 𝑥| . (51)

By summing up the above six inequalities, we get exactly (45) with𝜑(𝑡) = (1/2)𝑡, 𝜓(𝑡) = (1/5)𝑡. Also, 𝑥₀ = −1, 𝑦₀ = 1 are the two points in𝑋 such that 𝑥₀ ≤ 𝐹(𝑥₀, 𝑦₀) and 𝑦₀ ≥ 𝐹(𝑦₀, 𝑥₀).

By Corollary 22, we obtain that 𝐹 has a coupled fixed

point(0, 0), but Theorems 3.1 and 3.2 in [10] and Theorem 3.1 in [33] cannot be applied to𝐹 in this example.

Corollary 24. Let (𝑋, ≤) be a partially ordered set, and let 𝐺

be a G-metric on𝑋. Let 𝐹 : 𝑋 × 𝑋 → 𝑋 be a mapping having

mixed monotone property on𝑋. Suppose there exists 𝜓 ∈ Ψ

such that [𝐺 (𝐹 (𝑥, 𝑦) , 𝐹 (𝑢, V) , 𝐹 (𝑤, 𝑧)) +𝐺 (𝐹 (𝑦, 𝑥) , 𝐹 (V, 𝑢) , 𝐹 (𝑧, 𝑤))] ≤ [ 𝐺 (𝑥, 𝑢, 𝑤) + 𝐺 (𝑦, V𝑧)] − 2𝜓 (𝐺 (𝑥, 𝑢, 𝑤) + 𝐺 (𝑦, V, 𝑧) 2 ) (52)

𝑥 ≤ 𝑢 ≤ 𝑤 and 𝑦 ≥ V ≥ 𝑧). Assume that (𝑋, 𝐺) is complete,

and also suppose that either

(i)𝐹 is continuous, or (ii)(𝑋, ≤, 𝐺) is regular.

𝑦0 ≥ 𝐹(𝑦0, 𝑥0) (or 𝑥0 ≥ 𝐹(𝑥0, 𝑦0) and 𝑦0 ≤ 𝐹(𝑦0, 𝑥0)), then

Proof. Note that if 𝜓 ∈ Ψ, then for all 𝑟 > 0, 𝑟𝜓 ∈ Ψ.

Now divide (52) by 4 and take𝜑(𝑡) = (1/2)𝑡, 𝑡 ∈ [0, ∞); then condition (52) reduces to (8) with 𝜓₁ = (1/2)𝜓 and 𝑔(𝑥) = 𝑥; and hence byTheorem 18 and Theorem 20, we obtainCorollary 24.

The following result provides us the recent result of Karapinar et. al. [32, Corollary 2.5].

Theorem 25 (see [32]). Let(𝑋, ≤) be a partially ordered set,

and suppose that there exists a𝐺-metric 𝐺 on 𝑋 such that

(𝑋, 𝐺) is a complete 𝐺-metric space. Let 𝐹 : 𝑋 × 𝑋 → 𝑋

and𝑔 : 𝑋 → 𝑋 be two mappings such that 𝐹 has the mixed

𝑔-monotone property on 𝑋 and 𝐹(𝑋 × 𝑋) ⊆ 𝑔(𝑋). Suppose

that there exists a real number𝑘 ∈ [0, 1) such that

𝐺 (𝐹 (𝑥, 𝑦) , 𝐹 (𝑢, V) , 𝐹 (𝑤, 𝑧)) + 𝐺 (𝐹 (𝑦, 𝑥) , 𝐹 (V, 𝑢) , 𝐹 (𝑧, 𝑤)) ≤ 𝑘 [𝐺 (𝑔𝑥, 𝑔𝑢, 𝑔𝑤) + 𝐺 (𝑔𝑦, 𝑔V, 𝑔𝑧)]

(53)

for all𝑥, 𝑦, 𝑢, V, 𝑤, 𝑧 ∈ 𝑋 with 𝑔𝑥 ≥ 𝑔𝑢 ≥ 𝑔𝑤 and 𝑔𝑦 ≤ 𝑔V ≤

𝑔𝑧 (or 𝑔𝑥 ≤ 𝑔𝑢 ≤ 𝑔𝑤 and 𝑔𝑦 ≥ 𝑔V ≥ 𝑔𝑧). Also suppose that

either

(i)𝐹 and 𝑔 are continuous, (𝑋, 𝐺) is complete, and 𝑔

commutes with𝐹, or

(ii)(𝑔(𝑋), 𝐺) is complete and (𝑋, ≤, 𝐺) is regular.

If there exist two elements𝑥₀, 𝑦₀∈ 𝑋 with 𝑔𝑥₀≤ 𝐹(𝑥₀, 𝑦₀) and

𝑔𝑦₀ ≥ 𝐹(𝑦₀, 𝑥₀) (or 𝑔𝑥₀ ≥ 𝐹(𝑥₀, 𝑦₀) and 𝑔𝑦₀ ≤ 𝐹(𝑦₀, 𝑥₀)),

then there exist𝑥, 𝑦 ∈ 𝑋 such that 𝑔𝑥 = 𝐹(𝑥, 𝑦) and 𝑔𝑦 =

𝐹(𝑦, 𝑥).

Proof. Define𝜑, 𝜓:[0, ∞) → [0, ∞), 𝜑(𝑡) = 𝑡/2 and 𝜓(𝑡) =

(1 − 𝑘)(𝑡/2), 0 ≤ 𝑘 < 1. Then (53) holds. Hence the result follows fromTheorem 18orTheorem 20.

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Remark 26. The choice of functions𝐹 and 𝑔 inExample 21

shows thatTheorem 25is more general than Theorem 3.1 in [31], since the contractive condition (53) is more general than (5). Indeed, the contractive condition (5) does not hold for the choice of functions𝐹 and 𝑔, but (53) holds exactly for𝑘 = 3/4 with𝑥₀ = −1 and 𝑦₀ = 1 and yields (0, 0) as the coupled coincidence point of𝐹 and 𝑔.

Now, putting 𝑔 = 𝐼_𝑋 (the identity map of 𝑋) in

Theorem 25, we obtain the following.

Corollary 27. Let (𝑋, ≤) be a partially ordered set, and let 𝐺 be a𝐺-metric on 𝑋 such that (𝑋, 𝐺) is a complete G-metric space. Let𝐹 : 𝑋 × 𝑋 → 𝑋 be a mapping having mixed monotone

property on𝑋. Suppose there exists a real number 𝑘 ∈ [0, 1)

such that

𝐺 (𝐹 (𝑥, 𝑦) , 𝐹 (𝑢, V) , 𝐹 (𝑤, 𝑧)) + 𝐺 (𝐹 (𝑦, 𝑥) , 𝐹 (V, 𝑢) , 𝐹 (𝑧, 𝑤)) ≤ 𝑘 [ 𝐺 (𝑥, 𝑢, 𝑤) + 𝐺 (𝑦, V, 𝑧)]

(54)

𝑥 ≤ 𝑢 ≤ 𝑤 and 𝑦 ≥ V ≥ 𝑧). Suppose that either (i)𝐹 is continuous, or

(ii)(𝑋, ≤, 𝐺) is regular.

𝑦₀ ≥ 𝐹(𝑦₀, 𝑥₀) (or 𝑥₀ ≥ 𝐹(𝑥₀, 𝑦₀) and 𝑦₀ ≤ 𝐹(𝑦₀, 𝑥₀)), then

Remark 28. The choice of function𝐹 inExample 23shows

that Corollary 27 is more general than Theorem 3.1 and

Theorem 3.2 in [10], since the contractive condition (54) is more general than (4). Indeed the contractive condition (4) does not hold for the choice of function 𝐹, but (54) holds exactly for𝑘 = 3/5 with 𝑥₀= −1, 𝑦₀= 1.

Next we prove the existence and uniqueness of the coupled common fixed point for our main result.

Theorem 29. In addition to the hypotheses of Theorem 18,

suppose that for every(𝑥, 𝑦), (𝑥∗, 𝑦∗) ∈ 𝑋 × 𝑋, there exists

a (𝑢, V) ∈ 𝑋 × 𝑋 such that (𝐹(𝑢, V), 𝐹(V, 𝑢)) is comparable to(𝐹(𝑥, 𝑦), 𝐹(𝑦, 𝑥)) and (𝐹(𝑥∗, 𝑦∗), 𝐹(𝑦∗, 𝑥∗)). Then 𝐹 and 𝑔 have a unique coupled common fixed point; that is, there exists

a unique(𝑥, 𝑦) ∈ 𝑋 × 𝑋 such that 𝑥 = 𝑔(𝑥) = 𝐹(𝑥, 𝑦) and

𝑦 = 𝑔(𝑦) = 𝐹(𝑦, 𝑥).

Proof. FromTheorem 18, the set of coupled coincidences is

nonempty. In order to prove the theorem, we shall first show that if(𝑥, 𝑦) and (𝑥∗, 𝑦∗) are coupled coincidence points, that is, if𝑔(𝑥) = 𝐹(𝑥, 𝑦), 𝑔(𝑦) = 𝐹(𝑦, 𝑥) and 𝑔(𝑥∗) = 𝐹(𝑥∗, 𝑦∗), 𝑔(𝑦∗_{) = 𝐹(𝑦}∗_{, 𝑥}∗_{), then}

𝑔 (𝑥) = 𝑔 (𝑥∗_{) ,} _{𝑔 (𝑦) = 𝑔 (𝑦}∗_{) .} ₍₅₅₎

By assumption, there is (𝑢, V) ∈ 𝑋 × 𝑋 such that (𝐹(𝑢, V), 𝐹(V, 𝑢)) is comparable with (𝐹(𝑥, 𝑦), 𝐹(𝑦, 𝑥)) and (𝐹(𝑥∗, 𝑦∗), 𝐹(𝑦∗, 𝑥∗)). Put 𝑢0= 𝑢, V0= V and choose 𝑢1, V1∈

𝑋, so that 𝑔𝑢1= 𝐹(𝑢0, V0), 𝑔V1= 𝐹(V0, 𝑢0).

Then, as in the proof of Theorem (8), we can inductively define the sequences {𝑔𝑢_𝑛} and {𝑔V_𝑛} such that 𝑔𝑢_𝑛+1 = 𝐹(𝑢_𝑛, V_𝑛) and 𝑔V_𝑛+1= 𝐹(V_𝑛, 𝑢_𝑛).

Further, set𝑥₀ = 𝑥, 𝑦₀ = 𝑦, 𝑥∗₀ = 𝑥∗, and𝑦₀∗ = 𝑦∗, and on the same way define the sequences{𝑔𝑥_𝑛}, {𝑔𝑦_𝑛} and {𝑔𝑥∗_𝑛}, {𝑔𝑦∗_𝑛}. Then, it is easy to show that

𝑔𝑥_𝑛+1= 𝐹 (𝑥_𝑛, 𝑦_𝑛) , 𝑔𝑦_𝑛+1= 𝐹 (𝑦_𝑛, 𝑥_𝑛) , 𝑔𝑥∗_𝑛+1= 𝐹 (𝑥∗_𝑛, 𝑦_𝑛∗) , 𝑔𝑦∗_𝑛+1= 𝐹 (𝑦∗_𝑛, 𝑥∗_𝑛) ,

for all𝑛 ≥ 0.

(56)

Since(𝐹(𝑢, V), 𝐹(V, 𝑢)) = (𝑔𝑢₁, 𝑔V₁) and (𝐹(𝑥, 𝑦), 𝐹(𝑦, 𝑥)) = (𝑔𝑥₁, 𝑔𝑦₁) = (𝑔𝑥, 𝑔𝑦) are comparable, then 𝑔𝑢₁ ≥ 𝑔𝑥 and 𝑔V₁ ≤ 𝑔𝑦. It is easy to show that (𝑔𝑢_𝑛, 𝑔V_𝑛) and (𝑔𝑥, 𝑔𝑦) are comparable; that is,𝑔𝑢_𝑛 ≥ 𝑔𝑥 and 𝑔V_𝑛 ≤ 𝑔𝑦 for all 𝑛 ≥ 1. Thus from (8), we have

𝜑 (𝐺 (𝑔𝑢𝑛+1, 𝑔𝑢𝑛+1, 𝑔𝑥) + 𝐺 (𝑔V₂ 𝑛+1, 𝑔V𝑛+1, 𝑔𝑦)) = 𝜑 ((𝐺 (𝐹 (𝑢_𝑛, V_𝑛) , 𝐹 (𝑢_𝑛, V_𝑛) , 𝐹 (𝑥, 𝑦)) +𝐺 (𝐹 (V_𝑛, 𝑢_𝑛) , 𝐹 (V_𝑛, 𝑢_𝑛) , 𝐹 (𝑦, 𝑥))) 2−1) ≤ 𝜑 (𝐺 (𝑔𝑢𝑛, 𝑔𝑢𝑛, 𝑔𝑥) + 𝐺 (𝑔V𝑛, 𝑔V𝑛, 𝑔𝑦) 2 ) − 𝜓 (𝐺 (𝑔𝑢𝑛, 𝑔𝑢𝑛, 𝑔𝑥) + 𝐺 (𝑔V𝑛, 𝑔V𝑛, 𝑔𝑦) 2 ) . (57)

Since𝜓 is nonnegative, we have

𝜑 (𝐺 (𝑔𝑢𝑛+1, 𝑔𝑢𝑛+1, 𝑔𝑥) + 𝐺 (𝑔V𝑛+1, 𝑔V𝑛+1, 𝑔𝑦) 2 ) ≤ 𝜑 (𝐺 (𝑔𝑢𝑛, 𝑔𝑢𝑛, 𝑔𝑥) + 𝐺 (𝑔V𝑛, 𝑔V𝑛, 𝑔𝑦) 2 ) . (58) By monotonicity of𝜑, we have 𝐺 (𝑔𝑢𝑛+1, 𝑔𝑢𝑛+1, 𝑔𝑥) + 𝐺 (𝑔V𝑛+1, 𝑔V𝑛+1, 𝑔𝑦) 2 ≤ 𝐺 (𝑔𝑢𝑛, 𝑔𝑢𝑛, 𝑔𝑥) + 𝐺 (𝑔V𝑛, 𝑔V𝑛, 𝑔𝑦) 2 . (59)

Thus, the sequence{𝛾_𝑛} defined by 𝛾_𝑛 = (𝐺(𝑔𝑢_𝑛, 𝑔𝑢_𝑛, 𝑔𝑥) + 𝐺(𝑔V𝑛, 𝑔V𝑛, 𝑔𝑦))/2 is monotonically decreasing, so there

exists some𝛾 ≥ 0 such that lim_{𝑛 → ∞}𝛾_𝑛 = 𝛾.

We shall show that𝛾 = 0. Suppose, to the contrary, that 𝛾 > 0. Then taking limit as 𝑛 → ∞ in (57) and using the continuity of𝜑, we have

𝜑 (𝛾) ≤ 𝜑 (𝛾) − lim_𝛾

𝑛→ 𝛾𝜓 (𝛾𝑛) < 𝜑 (𝛾) , a contradiction.

(60) Thus,𝛾 = 0; that is, lim_{𝑛 → ∞}𝛾_𝑛= 0.

Hence, it follows that𝑔𝑢_𝑛 → 𝑔𝑥, 𝑔V_𝑛 → 𝑔𝑦. Similarly, we can show that𝑔𝑢_𝑛 → 𝑔𝑥∗,𝑔V_𝑛 → 𝑔𝑦∗.

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By uniqueness of limit, it follows that𝑔𝑥 = 𝑔𝑥∗and𝑔𝑦 = 𝑔𝑦∗_{. Thus, we proved (}₅₅_).

Since𝑔𝑥 = 𝐹(𝑥, 𝑦), 𝑔𝑦 = 𝐹(𝑦, 𝑥), and the pair (𝐹, 𝑔) is commuting, it follows that

𝑔𝑔𝑥 = 𝑔𝐹 (𝑥, 𝑦) = 𝐹 (𝑔𝑥, 𝑔𝑦) ,

𝑔𝑔𝑦 = 𝑔𝐹 (𝑦, 𝑥) = 𝐹 (𝑔𝑦, 𝑔𝑥) . (61) Denote𝑔𝑥 = 𝑧, 𝑔𝑦 = 𝑤. Then from (61), we have

𝑔𝑧 = 𝐹 (𝑧, 𝑤) , 𝑔𝑤 = 𝐹 (𝑤, 𝑧) . (62) Thus(𝑧, 𝑤) is a coupled coincidence point.

Then from (55) with𝑥∗ = 𝑧 and 𝑦∗ = 𝑤, it follows that 𝑔𝑧 = 𝑔𝑥 and 𝑔𝑤 = 𝑔𝑦; that is,

𝑔𝑧 = 𝑧, 𝑔𝑤 = 𝑤. (63) Combining (62) and (63), we obtain

𝑧 = 𝑔𝑧 = 𝐹 (𝑧, 𝑤) , 𝑤 = 𝑔𝑤 = 𝐹 (𝑤, 𝑧) . (64) Therefore,(𝑧, 𝑤) is the coupled common fixed point of 𝐹 and 𝑔.

To prove the uniqueness, assume that (𝑝, 𝑞) is another coupled common fixed point. Then by (55), we have𝑝 = 𝑔𝑝 = 𝑔𝑧 = 𝑧 and 𝑞 = 𝑔𝑞 = 𝑔𝑤 = 𝑤.

Theorem 30. In addition to the hypotheses of Theorem 20,

suppose that, for every(𝑥, 𝑦), (𝑥∗, 𝑦∗) ∈ 𝑋 × 𝑋, there exists

a (𝑢, V) ∈ 𝑋 × 𝑋 such that (𝐹(𝑢, V), 𝐹(V, 𝑢)) is comparable to (𝐹(𝑥, 𝑦), 𝐹(𝑦, 𝑥)) and (𝐹(𝑥∗, 𝑦∗), 𝐹(𝑦∗, 𝑥∗)). If 𝐹 and 𝑔

commute, then𝐹 and 𝑔 have a unique coupled common fixed

point; that is, there exists a unique(𝑥, 𝑦) ∈ 𝑋 × 𝑋 such that

𝑥 = 𝑔(𝑥) = 𝐹(𝑥, 𝑦) and 𝑦 = 𝑔(𝑦) = 𝐹(𝑦, 𝑥).

Proof. Proceeding exactly as in Theorem 29 result follows

immediately.

3. Applications to Integral Equations

Motivated by the works of Aydi et al. [24] and Luong and Thuan [35], in this section, we study the existence of solutions to nonlinear integral equations using some of our main results.

Consider the integral equations in the following system:

𝑥 (𝑡) = 𝑝 (𝑡) + ∫𝑇

0 𝑆 (𝑡, 𝑠) [𝑓 (𝑠, 𝑥 (𝑠)) + 𝑘 (𝑠, 𝑦 (𝑠))] 𝑑𝑠,

𝑦 (𝑡) = 𝑝 (𝑡) + ∫𝑇

0 𝑆 (𝑡, 𝑠) [ 𝑓 (𝑠, 𝑦 (𝑠)) + 𝑘 (𝑠, 𝑥 (𝑠))] 𝑑𝑠.

(65)

LetΘ denote the class of functions 𝜃 : [0, ∞) → [0, ∞) which satisfies the following conditions:

(i)𝜃 is increasing;

(ii) there exists𝜓 ∈ Ψ such that 𝜃(𝑟) = (𝑟/2) − 𝜓(𝑟/2) for all𝑟 ∈ [0, ∞).

For example,𝜃₁(𝑥) = 𝛼𝑥 (where 0 ≤ 𝛼 ≤ 1/2), 𝜃₂(𝑥) = 𝑥2_{/2(𝑥 + 1) are some members of Θ.}

We shall analyze the system (65) under the following assumptions:

(i)𝑓, 𝑘 : [0, 𝑇] × R → R are continuous; (ii)𝑝 : [0, 𝑇] → R is continuous;

(iii)𝑆 : [0, 𝑇] × R → [0, ∞) is continuous;

(iv) there exist𝜆 > 0 and 𝜃 ∈ Θ such that for all 𝑥, 𝑦 ∈ R, 𝑦 ≥ 𝑥, 0 ≤ 𝑓 (𝑠, 𝑦) − 𝑓 (𝑠, 𝑥) ≤ 𝜆𝜃 (𝑦 − 𝑥) , 0 ≤ 𝑘 (𝑠, 𝑥) − 𝑘 (𝑠, 𝑦) ≤ 𝜆𝜃 (𝑦 − 𝑥) ; (66) (v) we suppose that 3𝜆sup_{𝑡∈[0,𝑇]}∫𝑇 0 𝑆 (𝑡, 𝑠) 𝑑𝑠 ≤ 1 2. (67) (vi) there exist continuous functions𝛼, 𝛽 : [0, 𝑇] → R

such that 𝛼 (𝑡) ≤ 𝑝 (𝑡) + ∫𝑇 0 𝑆 (𝑡, 𝑠) (𝑓 (𝑠, 𝛼 (𝑠)) + 𝑘 (𝑠, 𝛽 (𝑠))) 𝑑𝑠, 𝛽 (𝑡) ≥ 𝑝 (𝑡) + ∫𝑇 0 𝑆 (𝑡, 𝑠) (𝑓 (𝑠, 𝛽 (𝑠)) + 𝑘 (𝑠, 𝛼 (𝑠))) 𝑑𝑠. (68)

Consider the space𝑋 = 𝐶([0, 𝑇], R) of continuous functions defined on[0, 𝑇] endowed with the (G-complete) G-metric given by

𝐺 (𝑢, V, 𝑤)

= sup_{𝑡∈[0,𝑇]}|𝑢 (𝑡) − V (𝑡)| + sup_{𝑡∈[0,𝑇]}|V (𝑡) − 𝑤 (𝑡)| + sup_{𝑡∈[0,𝑇]}|𝑤 (𝑡) − 𝑢 (𝑡)| ∀𝑢, V, 𝑤 ∈ 𝑋.

(69)

Endow𝑋 with the partial order ≤ given by 𝑥, 𝑦 ∈ 𝑋, 𝑥 ≤ 𝑦 ⇐⇒ 𝑥(𝑡) ≤ 𝑦(𝑡) for all 𝑡 ∈ [0, 𝑇]. Also, we may adjust as in [18] to prove that(𝑋, 𝐺, ≤) is regular.

Theorem 31. Under assumptions (i)–(vi), the system (65) has

a solution in𝑋2= (𝐶([0, 𝑇], R))2.

Proof. Consider the operator𝐹 : 𝑋 × 𝑋 → 𝑋 defined by

𝐹 (𝑥, 𝑦) (𝑡) = 𝑝 (𝑡) + ∫𝑇

0 𝑆 (𝑡, 𝑠) [𝑓 (𝑠, 𝑥 (𝑠)) + 𝑘 (𝑠, 𝑦 (𝑠))] 𝑑𝑠

for 𝑡 ∈ [0, 𝑇] , 𝑥, 𝑦 ∈ 𝑋. (70)

First, we shall prove that𝐹 has the mixed monotone property. In fact, for𝑥₁≤ 𝑥₂and𝑡 ∈ [0, 𝑇], we have

𝐹 (𝑥₂, 𝑦) (𝑡) − 𝐹 (𝑥1, 𝑦) (𝑡)

= ∫𝑇

0 𝑆 (𝑡, 𝑠) [ 𝑓 (𝑠, 𝑥2(𝑠)) − 𝑓 (𝑠, 𝑥1(𝑠))] 𝑑𝑠.

(11)

Taking into account that𝑥₁(𝑡) ≤ 𝑥₂(𝑡) for all 𝑡 ∈ [0, 𝑇], so by (iv),𝑓(𝑠, 𝑥₂(𝑠)) ≥ 𝑓(𝑠, 𝑥₁(𝑠)). Then 𝐹(𝑥₂, 𝑦)(𝑡) ≥ 𝐹(𝑥₁, 𝑦)(𝑡) for all𝑡 ∈ [0, 𝑇]; that is,

𝐹 (𝑥₁, 𝑦) ≤ 𝐹 (𝑥₂, 𝑦) . (72) Similarly, for𝑦₁≤ 𝑦₂and𝑡 ∈ [0, 𝑇], we have

𝐹 (𝑥, 𝑦₁) (𝑡) − 𝐹 (𝑥, 𝑦2) (𝑡)

= ∫𝑇

0 𝑆 (𝑡, 𝑠) [ 𝑘 (𝑠, 𝑦1(𝑠)) − 𝑘 (𝑠, 𝑦2(𝑠))] 𝑑𝑠.

(73)

Having𝑦₁(𝑡) ≤ 𝑦₂(𝑡), so by (iv), 𝑘(𝑠, 𝑦₁(𝑠)) ≥ 𝑘(𝑠, 𝑦₂(𝑠)). Then 𝐹(𝑥, 𝑦₁)(𝑡) ≥ 𝐹(𝑥, 𝑦₂)(𝑡) for all 𝑡 ∈ [0, 𝑇]; that is,

𝐹 (𝑥, 𝑦₁) ≥ 𝐹 (𝑥, 𝑦₂) . (74) Therefore,𝐹 has the mixed monotone property.

First we estimate the quantity𝐺(𝐹(𝑥, 𝑦), 𝐹(𝑢, V), 𝐹(𝑤, 𝑧)) for all𝑥, 𝑦, 𝑢, V, 𝑤, 𝑧 ∈ 𝑋 with 𝑥 ≥ 𝑢 ≥ 𝑤 and 𝑦 ≤ V ≤ 𝑧. Since𝐹 has the mixed monotone property, we have

𝐹 (𝑤, 𝑧) ≤ 𝐹 (𝑢, V) ≤ 𝐹 (𝑥, 𝑦) . (75) We obtain 𝐺 (𝐹 (𝑥, 𝑦) , 𝐹 (𝑢, V) , 𝐹 (𝑤, 𝑧)) = sup_{𝑡∈[0,𝑇]}󵄨󵄨󵄨󵄨𝐹(𝑥,𝑦)(𝑡) − 𝐹(𝑢,V)(𝑡)󵄨󵄨󵄨󵄨 + sup_{𝑡∈[0,𝑇]}|𝐹 (𝑢, V) (𝑡) − 𝐹 (𝑤, 𝑧) (𝑡)| + sup_{𝑡∈[0,𝑇]}󵄨󵄨󵄨󵄨𝐹(𝑤,𝑧)(𝑡) − 𝐹(𝑥,𝑦)(𝑡)󵄨󵄨󵄨󵄨 = sup_{𝑡∈[0,𝑇]}(𝐹 (𝑥, 𝑦) (𝑡) − 𝐹 (𝑢, V) (𝑡)) + sup_{𝑡∈[0,𝑇]}(𝐹 (𝑢, V) (𝑡) − 𝐹 (𝑤, 𝑧) (𝑡)) + sup_{𝑡∈[0,𝑇]}(𝐹 (𝑥, 𝑦) (𝑡) − 𝐹 (𝑤, 𝑧) (𝑡)) . (76)

Also for all𝑡 ∈ [0, 𝑇], from (iv), we have 𝐹 (𝑥, 𝑦) − 𝐹 (𝑢, V) = ∫𝑇 0 𝑆 (𝑡, 𝑠) [𝑓 (𝑠, 𝑥 (𝑠)) − 𝑓 (𝑠, 𝑢 (𝑠))] 𝑑𝑠 + ∫𝑇 0 𝑆 (𝑡, 𝑠) [ 𝑘 (𝑠, 𝑦 (𝑠)) − 𝑘 (𝑠, V (𝑠))] 𝑑𝑠 ≤ 𝜆 ∫𝑇 0 𝑆 (𝑡, 𝑠) [𝜃 (𝑥 (𝑠) − 𝑢 (𝑠)) + 𝜃 (V (𝑠) − 𝑦 (𝑠))] 𝑑𝑠. (77) Since the function𝜃 is increasing 𝑥 ≥ 𝑢 ≥ 𝑤, and 𝑦 ≤ V ≤ 𝑧, we have 𝜃 (𝑥 (𝑠) − 𝑢 (𝑠)) ≤ 𝜃 (sup𝑡∈𝐼|𝑥 (𝑡) − 𝑢 (𝑡)|) , 𝜃 (V (𝑠) − 𝑦 (𝑠)) ≤ 𝜃 (sup_𝑡∈𝐼󵄨󵄨󵄨󵄨V(𝑡) − 𝑦(𝑡)󵄨󵄨󵄨󵄨). (78) Hence by (77), we obtain 󵄨󵄨󵄨󵄨𝐹(𝑥,𝑦) − 𝐹(𝑢,V)󵄨󵄨󵄨󵄨 ≤ 𝜆 ∫𝑇 0 𝑆 (𝑡, 𝑠) [ 𝜃 (sup𝑡∈𝐼|𝑥 (𝑡) − 𝑢 (𝑡)|) +𝜃 (sup_𝑡∈𝐼󵄨󵄨󵄨󵄨V(𝑡) − 𝑦(𝑡)󵄨󵄨󵄨󵄨)]𝑑𝑠, (79)

as all the quantities on the right hand side of (77) are nonnegative, so (79) is justified.

Summing (79), (80), and (81) and then taking the supremum with respect to𝑡, we get

𝐺 (𝐹 (𝑥, 𝑦) , 𝐹 (𝑢, V) , 𝐹 (𝑤, 𝑧)) ≤ 𝜆sup_{𝑡∈[0,𝑇]}∫𝑇 0 𝑆 (𝑡, 𝑠) 𝑑𝑠 ⋅ [ 𝜃 (sup_𝑡∈𝐼|𝑥 (𝑡) − 𝑢 (𝑡)|) +𝜃 (sup_𝑡∈𝐼|𝑥 (𝑡) − 𝑤 (𝑡)|) + 𝜃 (sup_𝑡∈𝐼|𝑢 (𝑡) − 𝑤 (𝑡)|)] + 𝜆sup_{𝑡∈[0,𝑇]}∫𝑇 0 𝑆 (𝑡, 𝑠) 𝑑𝑠 ⋅ [ 𝜃 (sup_𝑡∈𝐼󵄨󵄨󵄨󵄨V(𝑡) − 𝑦(𝑡)󵄨󵄨󵄨󵄨) +𝜃 (sup_𝑡∈𝐼󵄨󵄨󵄨󵄨𝑧(𝑡) − 𝑦(𝑡)󵄨󵄨󵄨󵄨) + 𝜃(sup_𝑡∈𝐼|𝑧 (𝑡) − V (𝑡)|)] . (82) Further, since𝜃 is increasing, so that we have

(12)

Then by (82), we have 𝐺 (𝐹 (𝑥, 𝑦) , 𝐹 (𝑢, V) , 𝐹 (𝑤, 𝑧)) ≤ 𝜆sup_{𝑡∈[0,𝑇]}∫𝑇 0 𝑆 (𝑡, 𝑠) 𝑑𝑠 ⋅ 3𝜃 (𝐺 (𝑥, 𝑢, 𝑤)) + 𝜆 sup 𝑡∈[0,𝑇] × ∫𝑇 0 𝑆 (𝑡, 𝑠) 𝑑𝑠 ⋅ 3𝜃 (𝐺 (𝑦, V, 𝑧)) = 3𝜆sup_{𝑡∈[0,𝑇]} × ∫𝑇 0 𝑆 (𝑡, 𝑠) 𝑑𝑠 ⋅ (𝜃 (𝐺 (𝑥, 𝑢, 𝑤)) + 𝜃 (𝐺 (𝑦, V, 𝑧))) . (85)

Similarly, we can obtain that 𝐺 (𝐹 (𝑦, 𝑥) , 𝐹 (V, 𝑢) , 𝐹 (𝑧, 𝑤))

≤ 3𝜆sup_{𝑡∈[0,𝑇]} × ∫𝑇

0 𝑆 (𝑡, 𝑠) 𝑑𝑠 ⋅ (𝜃 (𝐺 (𝑥, 𝑢, 𝑤)) + 𝜃 (𝐺 (𝑦, V, 𝑧))) .

(86)

Summing (85) and (86), dividing by 2, and using (v), we get (𝐺 (𝐹 (𝑥, 𝑦) , 𝐹 (𝑢, V) , 𝐹 (𝑤, 𝑧)) +𝐺 (𝐹 (𝑦, 𝑥) , 𝐹 (V, 𝑢) , 𝐹 (𝑧, 𝑤))) 2−1 ≤ 3𝜆sup_{𝑡∈[0,𝑇]} × ∫𝑇 0 𝑆 (𝑡, 𝑠) 𝑑𝑠 ⋅ (𝜃 (𝐺 (𝑥, 𝑢, 𝑤)) + 𝜃 (𝐺 (𝑦, V, 𝑧))) . ≤ (𝜃 (𝐺 (𝑥, 𝑢, 𝑤)) + 𝜃 (𝐺 (𝑦, V, 𝑧))) 2 . (87) Since𝜃 is increasing, we have

𝜃 (𝐺 (𝑥, 𝑢, 𝑤)) ≤ 𝜃 (𝐺 (𝑥, 𝑢, 𝑤) + 𝐺 (𝑦, V, 𝑧)) , 𝜃 (𝐺 (𝑦, V, 𝑧)) ≤ 𝜃 (𝐺 (𝑥, 𝑢, 𝑤) + 𝐺 (𝑦, V, 𝑧)) (88) and so (𝜃 (𝐺 (𝑥, 𝑢, 𝑤)) + 𝜃 (𝐺 (𝑦, V, 𝑧))) 2 ≤ 𝜃 (𝐺 (𝑥, 𝑢, 𝑤) + 𝐺 (𝑦, V, 𝑧)) = 𝐺 (𝑥, 𝑢, 𝑤) + 𝐺 (𝑦, V, 𝑧)₂ − 𝜓 (𝐺 (𝑥, 𝑢, 𝑤) + 𝐺 (𝑦, V, 𝑧)₂ ) (89)

by definition of𝜃. Thus, using (87) and (89), we finally get (𝐺 (𝐹 (𝑥, 𝑦) , 𝐹 (𝑢, V) , 𝐹 (𝑤, 𝑧)) +𝐺 (𝐹 (𝑦, 𝑥) , 𝐹 (V, 𝑢) , 𝐹 (𝑧, 𝑤))) 2−1 ≤ 𝐺 (𝑥, 𝑢, 𝑤) + 𝐺 (𝑦, V, 𝑧) 2 − 𝜓 ( 𝐺 (𝑥, 𝑢, 𝑤) + 𝐺 (𝑦, V, 𝑧) 2 ) , (90) or 𝐺 (𝐹 (𝑥, 𝑦) , 𝐹 (𝑢, V) , 𝐹 (𝑤, 𝑧)) + 𝐺 (𝐹 (𝑦, 𝑥) , 𝐹 (V, 𝑢) , 𝐹 (𝑧, 𝑤)) ≤ 𝐺 (𝑥, 𝑢, 𝑤) + 𝐺 (𝑦, V, 𝑧) − 2𝜓 (𝐺 (𝑥, 𝑢, 𝑤) + 𝐺 (𝑦, V, 𝑧)₂ ) , (91)

which is just the contractive condition (52) inCorollary 24. Let𝛼, 𝛽 be the functions appearing in assumption (vi); we get

𝛼 ≤ 𝐹 (𝛼, 𝛽) , 𝛽 ≥ 𝐹 (𝛽, 𝛼) . (92) ApplyingCorollary 24, we deduce the existence of𝑥, 𝑦 ∈ 𝑋 such that

𝑥 = 𝐹 (𝑥, 𝑦) , 𝑦 = 𝐹 (𝑦, 𝑥) ; (93) that is,(𝑥, 𝑦) is a solution of the system (65).

4. Conclusion

In the setup of ordered 𝐺-metric spaces, we established some coupled coincidence and common coupled fixed point theorems for the mixed 𝑔-monotone mappings satisfying symmetric (𝜑, 𝜓)-contractive conditions. We accompanied our theoretical results by an applied example and an applica-tion to integral equaapplica-tions. Contractive condiapplica-tions presented in this paper extend, complement, and unify the contractions in [10,31,33,34] as well as several other contractions as in relevant items from the reference section of this paper and in the literature in general.

References

[1] Z. Mustafa and B. Sims, “A new approach to generalized metric spaces,” Journal of Nonlinear and Convex Analysis, vol. 7, no. 2, pp. 289–297, 2006.

[2] Z. Mustafa, H. Obiedat, and F. Awawdeh, “Some fixed point theorem for mapping on complete G-metric spaces,” Fixed Point

Theory and Applications, vol. 2008, Article ID 189870, 2008.

[3] Z. Mustafa, W. Shatanawi, and M. Bataineh, “Fixed point theo-rems on uncomplete G-metric spaces,” Journal of Mathematics

and Statistics, vol. 4, no. 4, pp. 196–201, 2008.

[4] Z. Mustafa, W. Shatanawi, and M. Bataineh, “Existence of fixed point results in G-metric spaces,” International Journal of

Mathematics and Mathematical Sciences, vol. 2009, Article ID

(13)

[5] Z. Mustafa and B. Sims, “Fixed point theorems for contractive mappings in complete G-Metric spaces,” Fixed Point Theory and

Applications, vol. 2009, Article ID 917175, 2009.

[6] M. Abbas and B. E. Rhoades, “Common fixed point results for noncommuting mappings without continuity in generalized metric spaces,” Applied Mathematics and Computation, vol. 215, no. 1, pp. 262–269, 2009.

[7] R. Saadati, S. M. Vaezpour, P. Vetro, and B. E. Rhoades, “Fixed point theorems in generalized partially ordered G-metric spaces,” Mathematical and Computer Modelling, vol. 52, no. 5-6, pp. 797–801, 2010.

[8] R. Chugh, T. Kadian, A. Rani, and B. E. Rhoades, “Property P in G-metric spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 401684, 2010.

[9] M. Abbas, T. Nazir, and S. Radenovi´c, “Some periodic point results in generalized metric spaces,” Applied Mathematics and

Computation, vol. 217, no. 8, pp. 4094–4099, 2010.

[10] B. S. Choudhury and P. Maity, “Coupled fixed point results in generalized metric spaces,” Mathematical and Computer

Modelling, vol. 54, no. 1-2, pp. 73–79, 2011.

[11] T. G. Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,” Nonlinear

Analysis: Theory, Methods and Applications, vol. 65, no. 7, pp.

1379–1393, 2006.

[12] V. Lakshmikantham and L. ´Ciri´c, “Coupled fixed point the-orems for nonlinear contractions in partially ordered metric spaces,” Nonlinear Analysis: Theory, Methods and Applications, vol. 70, no. 12, pp. 4341–4349, 2009.

[13] B. S. Choudhury and A. Kundu, “A coupled coincidence point result in partially ordered metric spaces for compatible map-pings,” Nonlinear Analysis: Theory, Methods and Applications, vol. 73, no. 8, pp. 2524–2531, 2010.

[14] J. Harjani, B. Lpez, and K. Sadarangani, “Fixed point theorems for mixed monotone operators and applications to integral equations,” Nonlinear Analysis: Theory, Methods and

Applica-tions, vol. 74, no. 5, pp. 1749–1760, 2011.

[15] N. V. Luong and N. X. Thuan, “Coupled fixed points in partially ordered metric spaces and application,” Nonlinear Analysis:

Theory, Methods and Applications, vol. 74, no. 3, pp. 983–992,

2011.

[16] B. Samet, “Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces,”

Nonlinear Analysis: Theory, Methods and Applications, vol. 72,

no. 12, pp. 4508–4517, 2010.

[17] A. C. M. Ran and M. C. B. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,” Proceedings of the American Mathematical Society, vol. 132, no. 5, pp. 1435–1443, 2004.

[18] J. J. Nieto and R. Rodr´ıguez-L´opez, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, vol. 22, no. 3, pp. 223–239, 2005. [19] J. J. Nieto and R. Rodr´ıguez-L´opez, “Existence and uniqueness

of fixed point in partially ordered sets and applications to ordi-nary differential equations,” Acta Mathematica Sinica, English

Series, vol. 23, no. 12, pp. 2205–2212, 2007.

[20] L. Ćirić, N. Cakić, M. Rajović, and J. S. Ume, “Monotone generalized nonlinear contractions in partially ordered metric spaces,” Fixed Point Theory and Applications, vol. 2008, Article ID 131294, 11 pages, 2008.

[21] J. Harjani and K. Sadarangani, “Fixed point theorems for weakly contractive mappings in partially ordered sets,” Nonlinear

Anal-ysis: Theory, Methods and Applications, vol. 71, no. 7-8, pp. 3403–

3410, 2009.

[22] J. Harjani and K. Sadarangani, “Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations,” Nonlinear Analysis: Theory, Methods and

Applications, vol. 72, no. 3-4, pp. 1188–1197, 2010.

[23] E. Karapinar, “Couple fixed point theorems for nonlinear con-tractions in cone metric spaces,” Computers and Mathematics

with Applications, vol. 59, no. 12, pp. 3656–3668, 2010.

[24] H. Aydi, E. Karapınar, and W. Shatanawi, “Tripled coincidence point results for generalized contractions in ordered generalized metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 101, 2012.

[25] M. Jain, K. Tas, S. Kumar, and N. Gupta, “Coupled common fixed points involving a(𝜙, 𝜓)—contractive condition for mixed g-monotone operators in partially ordered metric spaces,”

Journal of Inequalities and Applications, vol. 2012, article 285,

2012.

[26] M. Abbas, A. R. Khan, and T. Nazir, “Coupled common fixed point results in two generalized metric spaces,” Applied

Mathematics and Computation, vol. 217, no. 13, pp. 6328–6336,

2011.

[27] H. Aydi, B. Damjanovi´c Bosko, B. Samet, and W. Shatanawi, “Coupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces,” Mathematical and Computer

[28] W. Shatanawi, M. Abbas, and B. Samet, “Common coupled fixed points for mapping satisfying(𝜓, 𝜙)-weakly contractive condi-tion in generalized metric spaces,” submitted for publicacondi-tion. In press.

[29] N. V. Luong and N. X. Thuan, “Coupled fixed point theorems in partially ordered G-metric spaces,” Mathematical and Computer

[30] H. Aydi, M. Postolache, and W. Shatanawi, “Coupled fixed point results for (𝜓, 𝜑)-weakly contractive mappings in ordered G-metric spaces,” Computers and Mathematics with Applications, vol. 63, no. 1, pp. 298–309, 2012.

[31] H. K. Nashine, “Coupled common fixed point results in ordered G-metric spaces,” Journal of Nonlinear Science and Its

Applica-tions, vol. 5, no. 1, pp. 1–13, 2012.

[32] E. Karapinar, B. Kaymakcalan, and K. Tas, “On coupled fixed point theorems on partially ordered G-metric spaces,” Journal

of Inequalities and Applications, vol. 2012, article 200, 2012.

[33] S. A. Mohiuddine and A. Alotaibi, “On coupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 897198, 15 pages, 2012.

[34] H. S. Ding and E. Karapinar, “A note on some coupled fixed-point theorems on G-metric spaces,” Journal of Inequalities and

Applications, vol. 2012, article 170, 2012.

[35] N. V. Luong and N. X. Thuan, “Coupled fixed points in partially ordered metric spaces and application,” Nonlinear Analysis:

Theory, Methods and Applications, vol. 74, no. 3, pp. 983–992,