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

Abstract

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]). Let 𝑋 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 function 𝐺 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 {𝑥_𝑛} 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 the 𝐺-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→∞,(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 every 𝜀>0, there exists a positive integer 𝑁 such that 𝐺(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 function 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 {𝑥_𝑛} is 𝐺-convergent to 𝑥, {𝑓(𝑥_𝑛)} is 𝐺^′-convergent to 𝑓(𝑥).

Lemma 9 (see [1]). Let (𝑋,𝐺) be a 𝐺-metric space; then the function 𝐺(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 be 𝐺-complete (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 mapping 𝐹∶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 contractive 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 Ć‌irić‌ [12] extended the notion of mixed monotone property by introducing the notion of mixed 𝑔-monotone property in partially ordered metric spaces.

Definition 12 (see [11]). Let (𝑋,≤) be a partially ordered set and 𝐹∶𝑋×𝑋→𝑋. The mapping 𝐹 is said to have the mixed monotone property if 𝐹(𝑥,𝑦) is monotone nondecreasing in 𝑥 and monotone nonincreasing in 𝑦; that is, for any 𝑥,𝑦∈𝑋, 𝑥₁,𝑥₂∈𝑋,𝑥₁≤𝑥₂implies𝐹𝑥₁,𝑦≤𝐹𝑥₂,𝑦,𝑦₁,𝑦₂∈𝑋,𝑦₁≤𝑦₂implies𝐹𝑥,𝑦₁≥𝐹𝑥,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 mapping 𝐹 has the mixed 𝑔-monotone property if 𝐹(𝑥,𝑦) is monotone 𝑔-nondecreasing in its first argument and monotone 𝑔-nonincreasing in its second argument; that is, for any 𝑥,𝑦∈𝑋, 𝑥₁,𝑥₂∈𝑋,𝑔𝑥₁≤𝑔𝑥₂implies𝐹𝑥₁,𝑦≤𝐹𝑥₂,𝑦,𝑦₁,𝑦₂∈𝑋,𝑔𝑦₁≤𝑔𝑦₂implies𝐹𝑥,𝑦₁≥𝐹𝑥,𝑦₂. (2)

Definition 14 (see [11]). An element (𝑥,𝑦)∈𝑋×𝑋 is called a coupled fixed point of the mapping 𝐹∶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 mappings 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶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 mappings 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 if 𝑥=𝑔𝑥=𝐹(𝑥,𝑦) and 𝑦=𝑔𝑦=𝐹(𝑦,𝑥).

Definition 17 (see [12]). The mappings 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 are called commutative if 𝑔𝐹𝑥,𝑦=𝐹𝑔𝑥,𝑔𝑦, (3) for all 𝑥,𝑦∈𝑋.
Let (𝑋,≤) be a partially ordered set, and let 𝐺 be a 𝐺-metric on 𝑋 such that (𝑋,𝐺) is a complete 𝐺-metric space.

Choudhury and Maity [10] established some coupled fixed point theorems for the mixed monotone mapping 𝐹∶𝑋×𝑋→𝑋 under a contractive condition of the form 𝐺𝐹()()≤𝑘𝑥,𝑦,𝐹𝑢,𝑣,𝐹𝑤,𝑧2𝐺(𝑥,𝑢,𝑤)+𝐺𝑦,𝑣,𝑧, (4) where 𝑘∈[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: 𝐺𝐹()()𝐺,𝑥,𝑦,𝐹𝑢,𝑣,𝐹𝑤,𝑧≤𝑘𝑔𝑥,𝑔𝑢,𝑔𝑤+𝐺𝑔𝑦,𝑔𝑣,𝑔𝑧 (5) where 𝑘∈[0,1/2).

Mohiuddine and Alotaibi [33] further generalized the contraction (4) by considering the following more general contractive condition: 𝜑𝐺𝐹()()≤1𝑥,𝑦,𝐹𝑢,𝑣,𝐹𝑤,𝑧2𝜑𝐺()𝐺()𝑥,𝑢,𝑤+𝐺𝑦,𝑣,𝑧−𝜓𝑥,𝑢,𝑤+𝐺𝑦,𝑣,𝑧2, (6) where 𝜑,𝜓∶[0,∞)→[0,∞) are functions satisfying some appropriate conditions mentioned in [33].

Interestingly, for 𝜑(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 value 𝑘𝑡 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: 𝐺𝐹()()𝐹()()𝐺,𝑥,𝑦,𝐹𝑢,𝑣,𝐹𝑤,𝑧+𝐺𝑥,𝑦,𝐹𝑢,𝑣,𝐹𝑤,𝑧≤𝑘𝑔𝑥,𝑔𝑢,𝑔𝑤+𝐺𝑔𝑦,𝑔𝑣,𝑔𝑧 (7) where 𝑘∈[0,1).

Present work extend and generalize several results present in the literature of fixed point theory of 𝐺-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 by Φ the class of all functions 𝜑∶[0,∞)→[0,∞) with the following properties:(𝜑_i)𝜑 is continuous and nondecreasing;(𝜑_ii)𝜑(𝑡)<𝑡 for all 𝑡>0;(𝜑_iii)𝜑(𝑡+𝑠)≤𝜑(𝑡)+𝜑(𝑠) for all 𝑡,𝑠∈[0,∞).

We note that (𝜑_i) and (𝜑_ii) 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:(𝜓_i)lim_𝑡→𝑟𝜓(𝑡)>0 for all 𝑟>0;(𝜓_ii)lim_𝑡→0+𝜓(𝑡)=0.Some examples of 𝜑(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 𝜑𝐺𝐹()(𝐹()()2𝑥,𝑦,𝐹𝑢,𝑣,𝐹𝑤,𝑧)+𝐺𝑦,𝑥,𝐹𝑣,𝑢,𝐹𝑧,𝑤⁻¹𝐺≤𝜑𝑔𝑥,𝑔𝑢,𝑔𝑤+𝐺𝑔𝑦,𝑔𝑣,𝑔𝑧2𝐺−𝜓𝑔𝑥,𝑔𝑢,𝑔𝑤+𝐺𝑔𝑦,𝑔𝑣,𝑔𝑧2 (8) for all 𝑥,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 on 𝑋 with 𝑔𝑥≥𝑔𝑢≥𝑔𝑤 and 𝑔𝑦≤𝑔𝑣≤𝑔𝑧 (or 𝑔𝑥≤𝑔𝑢≤𝑔𝑤 and 𝑔𝑦≥𝑔𝑣≥𝑔𝑧) such that 𝐹 has the mixed 𝑔-monotone property. Assume that 𝐹(𝑋×𝑋)⊆𝑔(𝑋); both the mappings 𝐹 and 𝑔 commutes and are continuous.
If there exist two elements 𝑥₀,𝑦₀∈𝑋 with 𝑔𝑥₀≤𝐹(𝑥₀,𝑦₀) and 𝑔𝑦₀𝐹(𝑦₀,𝑥₀) (or 𝑔𝑥₀≥𝐹(𝑥₀,𝑦₀) and 𝑔𝑦₀≤𝐹(𝑦₀,𝑥₀)), then there exist 𝑥,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). Since 𝐹(𝑋×𝑋)⊆𝑔(𝑋), we can choose 𝑥₁,𝑦₁∈𝑋 such that 𝑔𝑥₁=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=𝐹𝑥_𝑛,𝑦_𝑛,𝑔𝑦_𝑛+1=𝐹𝑦_𝑛,𝑥_𝑛,forall𝑛≥0. (9) We will prove, for all 𝑛≥0, that 𝑔𝑥_𝑛≤𝑔𝑥_𝑛+1,𝑔𝑦_𝑛≥𝑔𝑦_𝑛+1. (10) Since 𝑔𝑥₀≤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 for 𝑛=0.
Suppose that (10) holds for some 𝑛>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=𝐹𝑥_𝑛,𝑦_𝑛≤𝐹𝑥_𝑛+1,𝑦_𝑛≤𝐹𝑥_𝑛+1,𝑦_𝑛+1=𝑔𝑥_𝑛+2,𝑔𝑦_𝑛+1=𝐹𝑦_𝑛,𝑥_𝑛≥𝐹𝑦_𝑛+1,𝑥_𝑛≥𝐹𝑦_𝑛+1,𝑥_𝑛+1=𝑔𝑦_𝑛+2. (11) Then by mathematical induction, it follows that (10) holds for all 𝑛≥0.
If for some 𝑛≥0, we have (𝑔𝑥_𝑛+1,𝑔𝑦_𝑛+1)=(𝑔𝑥_𝑛,𝑔𝑦_𝑛), and then 𝐹(𝑥_𝑛,𝑦_𝑛)=𝑔𝑥_𝑛 and 𝐹(𝑦_𝑛,𝑥_𝑛)=𝑔𝑦_𝑛; that is, 𝐹 and 𝑔 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 either 𝑔𝑥_𝑛+1=𝐹(𝑥_𝑛,𝑦_𝑛)≠𝑔𝑥_𝑛 or 𝑔𝑦_𝑛+1=𝐹(𝑦_𝑛,𝑥_𝑛)≠𝑔𝑦_𝑛.
Using (8)–(10), we have 𝜑𝐺𝑔𝑥_𝑛+1𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+𝐺𝑔𝑦_𝑛+1,𝑔𝑦_𝑛+1𝑔𝑦_𝑛2=𝜑𝐺𝐹𝑥_𝑛,𝑦_𝑛,𝐹𝑥_𝑛,𝑦_𝑛,𝐹𝑥_𝑛−1,𝑦_𝑛−1+𝐺𝐹𝑦_𝑛,𝑥_𝑛,𝐹𝑦_𝑛,𝑥_𝑛,𝐹𝑦_𝑛−1,𝑥_𝑛−12⁻¹𝐺≤𝜑𝑔𝑥_𝑛,𝑔𝑥_𝑛,𝑔𝑥_𝑛−1+𝐺𝑔𝑦_𝑛,𝑔𝑦_𝑛,𝑔𝑦_𝑛−12𝐺−𝜓𝑔𝑥_𝑛,𝑔𝑥_𝑛,𝑔𝑥_𝑛−1+𝐺𝑔𝑦_𝑛,𝑔𝑦_𝑛,𝑔𝑦_𝑛−12. (12) Since 𝜓 is nonnegative, using (12), we get 𝜑𝐺𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+𝐺𝑔𝑦_𝑛+1,𝑔𝑦_𝑛+1,𝑔𝑦_𝑛2𝐺≤𝜑𝑔𝑥_𝑛,𝑔𝑥_𝑛,𝑔𝑥_𝑛−1+𝐺𝑔𝑦_𝑛,𝑔𝑦_𝑛,𝑔𝑦_𝑛−12. (13) By monotonicity of 𝜑, we get 𝐺𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+𝐺𝑔𝑦_𝑛+1,𝑔𝑦_𝑛+1,𝑔𝑦_𝑛2≤𝐺𝑔𝑥_𝑛,𝑔𝑥_𝑛,𝑔𝑥_𝑛−1+𝐺𝑔𝑦_𝑛,𝑔𝑦_𝑛,𝑔𝑦_𝑛−12. (14) Let 𝛿_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 some 𝛿≥0 such that Lim_𝑛→∞𝛿_𝑛=lim_𝑛→∞𝐺𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+𝐺𝑔𝑦_𝑛+1,𝑔𝑦_𝑛+1,𝑔𝑦_𝑛2=𝛿. (15) We claim that 𝛿=0.
On the contrary, suppose that 𝛿>0.
Taking limit as 𝑛→∞ on both sides of (12) and using the properties of 𝜑 and 𝜓, we have 𝜑(𝛿)=lim_𝑛→∞𝜑𝛿_𝑛≤lim_𝑛→∞𝜑𝛿_𝑛−1−𝜓𝛿_𝑛−1=𝜑(𝛿)−lim_{𝛿𝑛−1→𝛿}𝜓𝛿_𝑛−1<𝜑(𝛿),acontradiction. (16) Thus, 𝛿=0; that is, lim_𝑛→∞𝛿_𝑛=lim_𝑛→∞𝐺𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+𝐺𝑔𝑦_𝑛+1,𝑔𝑦_𝑛+1,𝑔𝑦_𝑛2=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 a 𝐺-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=𝐺𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑚(𝑘)+𝐺𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑚(𝑘)2≥𝜀. (18) Further, corresponding to 𝑚(𝑘), we can choose 𝑛(𝑘) in such a way that it is the smallest integer with 𝑛(𝑘)>𝑚(𝑘)≥𝑘 and satisfies (18). Then, 𝐺𝑔𝑥_{𝑛(𝑘)−1},𝑔𝑥_{𝑛(𝑘)−1},𝑔𝑥_𝑚(𝑘)+𝐺𝑔𝑦_{𝑛(𝑘)−1},𝑔𝑦_{𝑛(𝑘)−1},𝑔𝑦_𝑚(𝑘)2<𝜀. (19) Using (18), (19) and Lemma 7, we get 𝜀≤𝑟_𝑘=𝐺𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑚(𝑘)+𝐺𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑚(𝑘)2≤𝐺𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑛(𝑘)𝑔𝑥_{𝑛(𝑘)−1}+𝐺𝑔𝑥_{𝑛(𝑘)−1},𝑔𝑥_{𝑛(𝑘)−1},𝑔𝑥_𝑚(𝑘)+𝐺𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑛(𝑘),𝑔𝑦_{𝑛(𝑘)−1}+𝐺𝑔𝑦_{𝑛(𝑘)−1},𝑔𝑦_{𝑛(𝑘)−1},𝑔𝑦_𝑚(𝑘)2⁻¹<𝐺𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑛(𝑘),𝑔𝑥_{𝑛(𝑘)−1}+𝐺𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑛(𝑘),𝑔𝑦_{𝑛(𝑘)−1}2⁻¹+𝜀. (20) Letting 𝑘→∞ and using (17), we have lim_𝑘→∞𝑟_𝑘=lim_𝑘→∞𝐺𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑚(𝑘)+𝐺𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑚(𝑘)2⁻¹=𝜀. (21) Using Lemma 6 and Lemma 7, we get 𝐺𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑚(𝑘)≤𝐺𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑛(𝑘)+1+𝐺𝑔𝑥_𝑛(𝑘)+1,𝑔𝑥_𝑛(𝑘)+1,𝑔𝑥_𝑚(𝑘)≤2𝐺𝑔𝑥_𝑛(𝑘)+1,𝑔𝑥_𝑛(𝑘)+1,𝑔𝑥_𝑛(𝑘)+𝐺𝑔𝑥_𝑛(𝑘)+1,𝑔𝑥_𝑛(𝑘)+1,𝑔𝑥_𝑚(𝑘)+1+𝐺𝑔𝑥_𝑚(𝑘)+1,𝑔𝑥_𝑚(𝑘)+1,𝑔𝑥_𝑚(𝑘). (22) Similarly, we can obtain 𝐺𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑚(𝑘)≤2𝐺𝑔𝑦_𝑛(𝑘)+1,𝑔𝑦_𝑛(𝑘)+1,𝑔𝑦_𝑛(𝑘)+𝐺𝑔𝑦_𝑛(𝑘)+1,𝑔𝑦_𝑛(𝑘)+1,𝑔𝑦_𝑚(𝑘)+1+𝐺𝑔𝑦_𝑚(𝑘)+1,𝑔𝑦_𝑚(𝑘)+1,𝑔𝑦_𝑚(𝑘). (23) Now, for all 𝑘≥0, using (22)-(23) in (18), we get 𝑟_𝑘=𝐺𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑚(𝑘)+𝐺𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑚(𝑘)2≤2𝛿_𝑛(𝑘)+𝛿_𝑚(𝑘)+𝐺𝑔𝑥_𝑛(𝑘)+1,𝑔𝑥_𝑛(𝑘)+1,𝑔𝑥_𝑚(𝑘)+1+𝐺𝑔𝑦_𝑛(𝑘)+1,𝑔𝑦_𝑛(𝑘)+1,𝑔𝑦_𝑚(𝑘)+12⁻¹. (24) By monotonicity of 𝜑 and property (𝜑_iii), we have 𝜑𝑟_𝑘≤𝜑2𝛿_𝑛(𝑘)+𝛿_𝑚(𝑘)+𝐺𝑔𝑥_𝑛(𝑘)+1,𝑔𝑥_𝑛(𝑘)+1,𝑔𝑥_𝑚(𝑘)+1+𝐺𝑔𝑦_𝑛(𝑘)+1,𝑔𝑦_𝑛(𝑘)+1,𝑔𝑦_𝑚(𝑘)+12⁻¹≤2𝜑𝛿_𝑛(𝑘)+𝜑𝛿_𝑚(𝑘)×𝐺+𝜑𝑔𝑥_𝑛(𝑘)+1,𝑔𝑥_𝑛(𝑘)+1,𝑔𝑥_𝑚(𝑘)+1+𝐺𝑔𝑦_𝑛(𝑘)+1,𝑔𝑦_𝑛(𝑘)+1,𝑔𝑦_𝑚(𝑘)+12⁻¹. (25) Also, since 𝑛(𝑘)>𝑚(𝑘), 𝑔𝑥_𝑛(𝑘)≥𝑔𝑥_𝑚(𝑘) and 𝑔𝑦_𝑛(𝑘)≤𝑔𝑦_𝑚(𝑘), using (8) and (9), we have 𝜑𝐺𝑔𝑥_𝑛(𝑘)+1,𝑔𝑥_𝑛(𝑘)+1,𝑔𝑥_𝑚(𝑘)+1+𝐺𝑔𝑦_𝑛(𝑘)+1,𝑔𝑦_𝑛(𝑘)+1,𝑔𝑦_𝑚(𝑘)+12⁻¹=𝜑𝐺𝐹𝑥_𝑛(𝑘),𝑦_𝑛(𝑘),𝐹𝑥_𝑛(𝑘),𝑦_𝑛(𝑘),𝐹𝑥_𝑚(𝑘),𝑦_𝑚(𝑘)+𝐺𝐹𝑦_𝑛(𝑘),𝑥_𝑛(𝑘),𝐹𝑦_𝑛(𝑘),𝑥_𝑛(𝑘),𝐹𝑦_𝑚(𝑘),𝑥_𝑚(𝑘)2⁻¹≤𝜑𝐺𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑚(𝑘)+𝐺𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑚(𝑘)2⁻¹−𝜓𝐺𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑚(𝑘)+𝐺𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑚(𝑘)2⁻¹=𝜑𝑟_𝑘−𝜓𝑟_𝑘. (26) Combining (25) and (26), we obtain that 𝜑𝑟_𝑘≤2𝜑𝛿_𝑛(𝑘)+𝜑𝛿_𝑚(𝑘)+𝜑𝑟_𝑘−𝜓𝑟_𝑘. (27) On letting 𝑘→∞, using (17), (21) and continuity of 𝜑, we get 𝜑(𝜀)≤2𝜑(0)+𝜑(0)+𝜑(𝜀)−lim_𝑘→∞𝜓𝑟_𝑘=2𝜑(0)+𝜑(0)+𝜑(𝜀)−lim_{𝑟𝑘→𝜀}𝜓𝑟_𝑘<𝜑(𝜀),acontradiction. (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 exist 𝑥,𝑦 in 𝑋 such that the sequences {𝑔𝑥_𝑛} and {𝑔𝑦_𝑛} are, respectively, 𝐺-convergent to 𝑥 and 𝑦, and then by Lemma 4, we have lim_𝑛→∞𝐺𝑔𝑥_𝑛,𝑔𝑥_𝑛,𝑥=lim_𝑛→∞𝐺𝑔𝑥_𝑛,𝑥,𝑥=0,lim_𝑛→∞𝐺𝑔𝑦_𝑛,𝑔𝑦_𝑛,𝑦=lim_𝑛→∞𝐺𝑔𝑦_𝑛,𝑦,𝑦=0. (29) Using the 𝐺-continuity of 𝑔, and Definition 8, we get lim_𝑛→∞𝐺𝑔𝑔𝑥_𝑛,𝑔𝑔𝑥_𝑛,𝑔𝑥=lim_𝑛→∞𝐺𝑔𝑔𝑥_𝑛,𝑔𝑥,𝑔𝑥=0,lim_𝑛→∞𝐺𝑔𝑔𝑦_𝑛,𝑔𝑔𝑦_𝑛,𝑔𝑦=lim_𝑛→∞𝐺𝑔𝑔𝑦_𝑛,𝑔𝑦,𝑔𝑦=0. (30)(31) Since 𝑔𝑥_𝑛+1=𝐹(𝑥_𝑛,𝑦_𝑛) and 𝑔𝑦_𝑛+1=𝐹(𝑦_𝑛,𝑥_𝑛), hence using commutativity of 𝐹 and 𝑔 we obtain 𝑔𝑔𝑥_𝑛+1=𝑔𝐹𝑥_𝑛,𝑦_𝑛=𝐹𝑔𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑔𝑦_𝑛+1=𝑔𝐹𝑦_𝑛,𝑥_𝑛=𝐹𝑔𝑦_𝑛,𝑔𝑥_𝑛. (32)(33) Since the mapping 𝐹 is G-continuous and the sequences {𝑔𝑥_𝑛} and {𝑔𝑦_𝑛} are, respectively, 𝐺-convergent to 𝑥 and 𝑦, hence using Definition 11, the sequence {𝐹(𝑔𝑥_𝑛,𝑔𝑦_𝑛)} is 𝐺-convergent to 𝐹(𝑥,𝑦). By uniqueness of limit and using (30), and (32) we get 𝐹(𝑥,𝑦)=𝑔𝑥. Similarly, we can show that 𝐹(𝑦,𝑥)=𝑔𝑦. Hence, (𝑥,𝑦)∈𝑋×𝑋 is a coupled coincidence point of 𝐹 and 𝑔.

In the next theorem, we omit the continuity hypotheses of the mapping 𝐹 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_𝑛 for all 𝑛.

Theorem 20. Let (𝑋,≤) be a partially ordered set, and suppose that there exists a 𝐺-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 that 𝐹 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)  (or𝑔𝑥₀≥𝐹(𝑥₀,𝑦₀) and 𝑔𝑦₀≤𝐹(𝑦₀,𝑥₀)); then 𝐹 and 𝑔 have a coupled coincidence point in 𝑋; that is, there exist 𝑥,𝑦∈𝑋 such that 𝑔𝑥=𝐹(𝑥,𝑦) and 𝑔𝑦=𝐹(𝑦,𝑥).

Proof. Proceeding exactly as in Theorem 18, we have that {𝑔𝑥_𝑛} and {𝑔𝑦_𝑛} are G-Cauchy sequences in the complete G-metric space (𝑔(𝑋),𝐺). Then there exist 𝑥,𝑦∈𝑋 such that 𝑔𝑥_𝑛→𝑔𝑥 and 𝑔𝑦_𝑛→𝑔𝑦; that is, Lim_𝑛→∞𝐺𝑔𝑥_𝑛,𝑔𝑥,𝑔𝑥=lim_𝑛→∞𝐺𝑔𝑥_𝑛,𝑔𝑥_𝑛,𝑔𝑥=0,lim_𝑛→∞𝐺𝑔𝑦_𝑛,𝑔𝑦,𝑔𝑦=lim_𝑛→∞𝐺𝑔𝑦_𝑛,𝑔𝑦_𝑛,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_𝑛 for all 𝑛≥0. Using (8), we get 𝜑𝐺𝐹𝑥,𝑦,𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+1+𝐺𝐹𝑦,𝑥,𝑔𝑦_𝑛+1,𝑔𝑦_𝑛+12⁻¹=𝜑𝐺𝐹𝑥,𝑦,𝐹𝑥_𝑛,𝑦_𝑛,𝐹𝑥_𝑛,𝑦_𝑛+𝐺𝐹𝑦,𝑥,𝐹𝑦_𝑛,𝑥_𝑛,𝐹𝑦_𝑛,𝑥_𝑛2⁻¹𝐺≤𝜑𝑔𝑥,𝑔𝑥_𝑛,𝑔𝑥_𝑛+𝐺𝑔𝑦,𝑔𝑦_𝑛,𝑔𝑦_𝑛2𝐺−𝜓𝑔𝑥,𝑔𝑥_𝑛,𝑔𝑥_𝑛+𝐺𝑔𝑦,𝑔𝑦_𝑛,𝑔𝑦_𝑛2. (35) On letting 𝑛→∞ and using (34) and the properties of 𝜑 and 𝜓, we obtain that 𝜑lim_𝑛→∞𝐺𝐹𝑥,𝑦,𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+1+𝐺𝐹𝑦,𝑥,𝑔𝑦_𝑛+1,𝑔𝑦_𝑛+12⁻¹≤𝜑lim_𝑛→∞𝐺𝑔𝑥,𝑔𝑥_𝑛,𝑔𝑥_𝑛+𝐺𝑔𝑦,𝑔𝑦_𝑛,𝑔𝑦_𝑛2−lim_𝑛→∞𝜓𝐺𝑔𝑥,𝑔𝑥_𝑛,𝑔𝑥_𝑛+𝐺𝑔𝑦,𝑔𝑦_𝑛,𝑔𝑦_𝑛2=0, (36) which yields lim_𝑛→∞𝐺𝐹𝑥,𝑦,𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+1+𝐺𝐹𝑦,𝑥,𝑔𝑦_𝑛+1,𝑔𝑦_𝑛+12⁻¹=0. (37) Hence we can obtain that lim_𝑛→∞𝐺𝐹𝑥,𝑦,𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+1=0=lim_𝑛→∞𝐺𝐹𝑦,𝑥,𝑔𝑦_𝑛+1,𝑔𝑦_𝑛+1. (38) On the other hand, by condition (G5), we have 𝐺𝐹𝐹≤𝐺𝐹𝑥,𝑦,𝑔𝑥,𝑔𝑥+𝐺𝑦,𝑥,𝑔𝑦,𝑔𝑦𝑥,𝑦,𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+1+𝐺𝑔𝑥_𝑛+1+𝐺𝐹,𝑔𝑥,𝑔𝑥𝑦,𝑥,𝑔𝑦_𝑛+1,𝑔𝑦_𝑛+1+𝐺𝑔𝑦_𝑛+1,𝑔𝑦,𝑔𝑦. (39) Letting 𝑛→∞ in (39) and using (34)–(38), we have 𝐺𝐹𝑥,𝑦,𝑔𝑥,𝑔𝑥+𝐺𝐹𝑦,𝑥,𝑔𝑦,𝑔𝑦=0. (40) Thus 𝐹(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 of Theorem 18 that shows that Theorem 18 is more general than Theorem 3.1 in [31], since the contractive condition (8) is more general than (5).

Example 21. Let 𝑋=ℝ. Then (𝑋,≤) is a partially ordered set with the natural ordering of real numbers. Let 𝐺∶𝑋×𝑋×𝑋→𝑅⁺ be defined by 𝐺𝑥,𝑦,𝑧=𝑥−𝑦+𝑦−𝑧+|𝑧−𝑥|for𝑥,𝑦,𝑧∈𝑋. (41) Then (𝑋,𝐺) is a complete 𝐺-metric space.
Define 𝐹∶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.
Clearly 𝐹(𝑋×𝑋)⊆𝑔(𝑋), 𝐹 and 𝑔 are continuous, 𝐹 has the mixed 𝑔-monotone property, and the pair (𝐹,𝑔) is commutative and satisfies condition (8) but does not satisfy the condition (5). Assume, to the contrary, that there exists some 𝑘∈[0,1/2) such that (5) holds. Then, we must have 𝑥−2𝑦8−𝑢−2𝑣8+𝑢−2𝑣8−𝑤−2𝑧8+𝑤−2𝑧8−𝑥−2𝑦8𝑥≤𝑘2−𝑢2+𝑢2−𝑤2+𝑤2−𝑥2+𝑦2−𝑣2+𝑣2−𝑧2+𝑧2−𝑦2=𝑘2|+||+|++||+(|𝑥−𝑢𝑢−𝑤𝑤−𝑥|)𝑦−𝑣𝑣−𝑧𝑧−𝑦 (42) for all 𝑥≥𝑢≥𝑤 and 𝑦≤𝑣≤𝑧. Taking 𝑥=𝑢=𝑤, 𝑣<𝑧 in the last inequality and setting 𝜌∶=|𝑣−𝑧|/2+|𝑦−𝑣|/2+|𝑧−𝑦|/2, we obtain 𝜌2≤𝑘𝜌,𝜌>0, (43) which implies that 1/2≤𝑘, a contradiction since 𝑘∈[0,1/2). Hence 𝐹 and 𝑔 do not satisfy (5).
Indeed, for 𝑥≥𝑢≥𝑤 and 𝑦≤𝑣≤𝑧, we have 𝑥−2𝑦8−𝑢−2𝑣8≤18||+1𝑥−𝑢4𝑦−𝑣,𝑢−2𝑣8−𝑤−2𝑧8≤18||+1𝑢−𝑤4|𝑣−𝑧|,𝑤−2𝑧8−𝑥−2𝑦8≤18||+1𝑤−𝑥4𝑧−𝑦,𝑦−2𝑥8−𝑣−2𝑢8≤18+1𝑦−𝑣4|𝑥−𝑢|,𝑣−2𝑢8−𝑧−2𝑤8≤18||+1𝑣−𝑧4|𝑢−𝑤|,𝑧−2𝑤8−𝑦−2𝑥8≤18+1𝑧−𝑦4|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 of Theorem 18; by Theorem 18, we obtain that 𝐹 and 𝑔 have a coupled coincidence point (0,0), but Theorem 3.1 in [31] cannot be applied to 𝐹 and 𝑔 in this example.
Now, putting 𝑔=𝐼_𝑋 (the identity map of 𝑋) in the previous results, we obtain the following.

Corollary 22. Let (𝑋,≤) be a partially ordered set, and let 𝐺 be a 𝐺-metric on 𝑋. Let 𝐹∶𝑋×𝑋→𝑋 be a mapping satisfying (8) (with 𝑔=𝐼_𝑋); that is, 𝜑𝐺𝐹()(𝐹()()2𝑥,𝑦,𝐹𝑢,𝑣,𝐹𝑤,𝑧)+𝐺𝑦,𝑥,𝐹𝑣,𝑢,𝐹𝑧,𝑤⁻¹𝐺()≤𝜑𝑥,𝑢,𝑤+𝐺𝑦,𝑣,𝑧2𝐺()−𝜓𝑥,𝑢,𝑤+𝐺𝑦,𝑣,𝑧2 (45) for all 𝑥,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 elements 𝑥₀,𝑦₀∈𝑋 with 𝑥₀≤𝐹(𝑥₀,𝑦₀) and 𝑦₀≥𝐹(𝑦₀,𝑥₀)  (or 𝑥₀≥𝐹(𝑥₀,𝑦₀) and 𝑦₀≤𝐹(𝑦₀,𝑥₀)), then there exist 𝑥,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. Let 𝑋=ℝ. Then (𝑋,≤) is a partially ordered set with the natural ordering of real numbers. Let 𝐺∶𝑋×𝑋×𝑋→𝑅⁺ be defined by 𝐺𝑥,𝑦,𝑧=𝑥−𝑦+𝑦−𝑧+|𝑧−𝑥|for𝑥,𝑦,𝑧∈𝑋. (46) Then (𝑋,𝐺) is a complete 𝐺-metric space.
Define 𝐹∶𝑋×𝑋→𝑋 by 𝐹(𝑥,𝑦)=(𝑥−5𝑦)/10, (𝑥,𝑦)∈𝑋×𝑋.
Then 𝐹 is continuous and satisfies the mixed monotone property. We note that 𝐹 satisfies condition (45) but does not satisfy the conditions (4) and (6). Indeed, assume that there exists 𝑘∈[0,1), such that (4) holds. Then, we must have 𝑥−5𝑦−10𝑢−5𝑣+10𝑢−5𝑣−10𝑤−5𝑧+10𝑤−5𝑧−10𝑥−5𝑦≤𝑘102|+||+|++||+(|𝑥−𝑢𝑢−𝑤𝑤−𝑥|)𝑦−𝑣𝑣−𝑧𝑧−𝑦𝑥≥𝑢≥𝑤,𝑦≤𝑣≤𝑧, (47) by which, for 𝑥=𝑢=𝑤, we get +||++||+,𝑦−𝑣𝑣−𝑧𝑧−𝑦≤𝑘𝑦−𝑣𝑣−𝑧𝑧−𝑦 (48) which for 𝑣<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 functions 𝜑 and 𝜓 satisfying appropriate conditions as in [33] such that (6) holds. This means that 𝜑𝑥−5𝑦−10𝑢−5𝑣+10𝑢−5𝑣−10𝑤−5𝑧+10𝑤−5𝑧−10𝑥−5𝑦≤1102𝜑||+||+||++||+||+||+||++||+2𝑥−𝑢𝑢−𝑤𝑤−𝑥𝑦−𝑣𝑣−𝑧𝑧−𝑦−𝜓𝑥−𝑢𝑢−𝑤𝑤−𝑥𝑦−𝑣𝑣−𝑧𝑧−𝑦⁻¹, (49) for all 𝑥≥𝑢≥𝑤 and 𝑦≤𝑣≤𝑧. Taking 𝑥=𝑢=𝑤, 𝑣<𝑧 in the previous inequality and setting 𝛼∶=(|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 all 𝛼>0, 𝜓(𝛼)≤0; that is, 𝜓(𝛼)=0, which contradicts the definition of 𝜓.
This shows that 𝐹 does not satisfy (6).
Finally, we prove that (45) holds. Indeed, for 𝑥≥𝑢≥𝑤 and 𝑦≤𝑣≤𝑧, we have 𝑥−5𝑦−10𝑢−5𝑣≤110||+110𝑥−𝑢2𝑦−𝑣,𝑢−5𝑣−10𝑤−5𝑧≤110||+110𝑢−𝑤2|𝑣−𝑧|,𝑤−5𝑧−10𝑥−5𝑦≤110||+110𝑤−𝑥2𝑧−𝑦,𝑦−5𝑥−10𝑣−5𝑢≤110+110𝑦−𝑣2|𝑥−𝑢|,𝑣−5𝑢−10𝑧−5𝑤≤110||+110𝑣−𝑧2|𝑢−𝑤|,𝑧−5𝑤−10𝑦−5𝑥≤110+110𝑧−𝑦2|𝑤−𝑥|. (51) By summing up the above six inequalities, we get exactly (45) with 𝜑(𝑡)=(1/2)𝑡, 𝜓(𝑡)=(1/5)𝑡. Also, 𝑥₀=−1, 𝑦₀=1 are the two points in 𝑋 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 to 𝐹 in this example.

Corollary 24. Let (𝑋,≤) be a partially ordered set, and let 𝐺 be a G-metric on 𝑋. Let 𝐹∶𝑋×𝑋→𝑋 be a mapping having mixed monotone property on 𝑋. Suppose there exists 𝜓∈Ψ such that 𝐺𝐹()()𝐹()()≤𝐺()𝐺()𝑥,𝑦,𝐹𝑢,𝑣,𝐹𝑤,𝑧+𝐺𝑦,𝑥,𝐹𝑣,𝑢,𝐹𝑧,𝑤𝑥,𝑢,𝑤+𝐺𝑦,𝑣𝑧−2𝜓𝑥,𝑢,𝑤+𝐺𝑦,𝑣,𝑧2 (52) for all 𝑥,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 also suppose that either(i)𝐹 is continuous, or(ii)(𝑋,≤,𝐺) is regular.If there exist two elements 𝑥₀,𝑦₀∈𝑋 with 𝑥₀≤𝐹(𝑥₀,𝑦₀) and 𝑦₀≥𝐹(𝑦₀,𝑥₀) (or 𝑥₀≥𝐹(𝑥₀,𝑦₀) and 𝑦₀≤𝐹(𝑦₀,𝑥₀)), then there exist 𝑥,𝑦∈𝑋 such that 𝑥=𝐹(𝑥,𝑦) and 𝑦=𝐹(𝑦,𝑥); that is, 𝐹 has a coupled fixed point in 𝑋.

Proof. Note that if 𝜓∈Ψ, then for all 𝑟>0, 𝑟𝜓∈Ψ. Now divide (52) by 4 and take 𝜑(𝑡)=(1/2)𝑡, 𝑡∈[0,∞); then condition (52) reduces to (8) with 𝜓₁=(1/2)𝜓 and 𝑔(𝑥)=𝑥; and hence by Theorem 18 and Theorem 20, we obtain Corollary 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 a 𝐺-metric 𝐺 on 𝑋 such that (𝑋,𝐺) is a complete 𝐺-metric space. Let 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶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 number 𝑘∈[0,1) such that 𝐺𝐹()()𝐹()()𝐺𝑥,𝑦,𝐹𝑢,𝑣,𝐹𝑤,𝑧+𝐺𝑦,𝑥,𝐹𝑣,𝑢,𝐹𝑧,𝑤≤𝑘𝑔𝑥,𝑔𝑢,𝑔𝑤+𝐺𝑔𝑦,𝑔𝑣,𝑔𝑧 (53) for all 𝑥,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 with 𝐹, or(ii)(𝑔(𝑋),𝐺) is complete and (𝑋,≤,𝐺) is regular.If there exist two elements 𝑥₀,𝑦₀∈𝑋 with 𝑔𝑥₀≤𝐹(𝑥₀,𝑦₀) and 𝑔𝑦₀≥𝐹(𝑦₀,𝑥₀) (or 𝑔𝑥₀≥𝐹(𝑥₀,𝑦₀) and 𝑔𝑦₀≤𝐹(𝑦₀,𝑥₀)), then there exist 𝑥,𝑦∈𝑋 such that 𝑔𝑥=𝐹(𝑥,𝑦) and 𝑔𝑦=𝐹(𝑦,𝑥).

Proof. Define 𝜑,𝜓:[0,∞)→[0,∞), 𝜑(𝑡)=𝑡/2 and 𝜓(𝑡)=(1−𝑘)(𝑡/2), 0≤𝑘<1. Then (53) holds. Hence the result follows from Theorem 18 or Theorem 20.

Remark 26. The choice of functions 𝐹 and 𝑔 in Example 21 shows that Theorem 25 is 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 functions 𝐹 and 𝑔, but (53) holds exactly for 𝑘=3/4 with 𝑥₀=−1 and 𝑦₀=1 and yields (0,0) as the coupled coincidence point of 𝐹 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 a 𝐺-metric on 𝑋 such that (𝑋,𝐺) is a complete G-metric space. Let 𝐹∶𝑋×𝑋→𝑋 be a mapping having mixed monotone property on 𝑋. Suppose there exists a real number 𝑘∈[0,1) such that 𝐺𝐹()()𝐹()()𝐺()𝑥,𝑦,𝐹𝑢,𝑣,𝐹𝑤,𝑧+𝐺𝑦,𝑥,𝐹𝑣,𝑢,𝐹𝑧,𝑤≤𝑘𝑥,𝑢,𝑤+𝐺𝑦,𝑣,𝑧 (54) for all 𝑥,𝑦,𝑢,𝑣,𝑤,𝑧∈𝑋 with 𝑥≥𝑢≥𝑤 and 𝑦≤𝑣≤𝑧 (or 𝑥≤𝑢≤𝑤 and 𝑦≥𝑣≥𝑧). Suppose that either(i)𝐹 is continuous, or(ii)(𝑋,≤,𝐺) is regular.If there exist two elements 𝑥₀,𝑦₀∈𝑋 with 𝑥₀≤𝐹(𝑥₀,𝑦₀) and 𝑦₀≥𝐹(𝑦₀,𝑥₀) (or 𝑥₀≥𝐹(𝑥₀,𝑦₀) and 𝑦₀≤𝐹(𝑦₀,𝑥₀)), then there exist 𝑥,𝑦∈𝑋 such that 𝑥=𝐹(𝑥,𝑦) and 𝑦=𝐹(𝑦,𝑥); that is, 𝐹 has a coupled fixed point in 𝑋.

Remark 28. The choice of function 𝐹 in Example 23 shows 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 for 𝑘=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. From Theorem 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, if 𝑔(𝑥)=𝐹(𝑥,𝑦), 𝑔(𝑦)=𝐹(𝑦,𝑥) and 𝑔(𝑥^∗)=𝐹(𝑥^∗,𝑦^∗), 𝑔(𝑦^∗)=𝐹(𝑦^∗,𝑥^∗), then 𝑔(𝑥)=𝑔𝑥^∗,𝑔𝑦=𝑔𝑦^∗. (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, set 𝑥₀=𝑥, 𝑦₀=𝑦, 𝑥^∗₀=𝑥^∗, and 𝑦^∗₀=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=𝐹𝑥_𝑛,𝑦_𝑛,𝑔𝑦_𝑛+1=𝐹𝑦_𝑛,𝑥_𝑛,𝑔𝑥^∗_𝑛+1=𝐹𝑥^∗_𝑛,𝑦^∗_𝑛,𝑔𝑦^∗_𝑛+1=𝐹𝑦^∗_𝑛,𝑥^∗_𝑛,forall𝑛≥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 𝜑𝐺𝑔𝑢_𝑛+1,𝑔𝑢_𝑛+1,𝑔𝑥+𝐺𝑔𝑣_𝑛+1,𝑔𝑣_𝑛+1,𝑔𝑦2=𝜑𝐺𝐹𝑢_𝑛,𝑣_𝑛,𝐹𝑢_𝑛,𝑣_𝑛𝐹𝑣,𝐹𝑥,𝑦+𝐺_𝑛,𝑢_𝑛,𝐹𝑣_𝑛,𝑢_𝑛,𝐹𝑦,𝑥2⁻¹𝐺≤𝜑𝑔𝑢_𝑛,𝑔𝑢_𝑛,𝑔𝑥+𝐺𝑔𝑣_𝑛,𝑔𝑣_𝑛,𝑔𝑦2𝐺−𝜓𝑔𝑢_𝑛,𝑔𝑢_𝑛,𝑔𝑥+𝐺𝑔𝑣_𝑛,𝑔𝑣_𝑛,𝑔𝑦2. (57) Since 𝜓 is nonnegative, we have 𝜑𝐺𝑔𝑢_𝑛+1,𝑔𝑢_𝑛+1,𝑔𝑥+𝐺𝑔𝑣_𝑛+1,𝑔𝑣_𝑛+1,𝑔𝑦2𝐺≤𝜑𝑔𝑢_𝑛,𝑔𝑢_𝑛,𝑔𝑥+𝐺𝑔𝑣_𝑛,𝑔𝑣_𝑛,𝑔𝑦2. (58) By monotonicity of 𝜑, we have 𝐺𝑔𝑢_𝑛+1,𝑔𝑢_𝑛+1,𝑔𝑥+𝐺𝑔𝑣_𝑛+1,𝑔𝑣_𝑛+1,𝑔𝑦2≤𝐺𝑔𝑢_𝑛,𝑔𝑢_𝑛,𝑔𝑥+𝐺𝑔𝑣_𝑛,𝑔𝑣_𝑛,𝑔𝑦2. (59) Thus, the sequence {𝛾_𝑛} defined by 𝛾_𝑛=(𝐺(𝑔𝑢_𝑛,𝑔𝑢_𝑛,𝑔𝑥)+𝐺(𝑔𝑣_𝑛,𝑔𝑣_𝑛,𝑔𝑦))/2 is monotonically decreasing, so there exists some 𝛾≥0 such that lim_𝑛→∞𝛾_𝑛=𝛾.
We shall show that 𝛾=0. Suppose, to the contrary, that 𝛾>0. Then taking limit as 𝑛→∞ in (57) and using the continuity of 𝜑, we have 𝜑𝛾≤𝜑𝛾−lim_{𝛾𝑛→𝛾}𝜓𝛾_𝑛<𝜑𝛾,acontradiction. (60) Thus, 𝛾=0; that is, lim_𝑛→∞𝛾_𝑛=0.
Hence, it follows that 𝑔𝑢_𝑛→𝑔𝑥, 𝑔𝑣_𝑛→𝑔𝑦.
Similarly, we can show that 𝑔𝑢_𝑛→𝑔𝑥^∗, 𝑔𝑣_𝑛→𝑔𝑦^∗.
By uniqueness of limit, it follows that 𝑔𝑥=𝑔𝑥^∗ and 𝑔𝑦=𝑔𝑦^∗. Thus, we proved (55).
Since 𝑔𝑥=𝐹(𝑥,𝑦), 𝑔𝑦=𝐹(𝑦,𝑥), and the pair (𝐹,𝑔) is commuting, it follows that ,.𝑔𝑔𝑥=𝑔𝐹𝑥,𝑦=𝐹𝑔𝑥,𝑔𝑦𝑔𝑔𝑦=𝑔𝐹𝑦,𝑥=𝐹𝑔𝑦,𝑔𝑥 (61) Denote 𝑔𝑥=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) with 𝑥^∗=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 have 𝑝=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, 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. 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)+^𝑇₀𝑆(𝑓((𝑠(𝑠)𝑦(𝑡)(𝑡)+𝑡,𝑠)𝑠,𝑥))+𝑘𝑠,𝑦𝑑𝑠,=𝑝^𝑇₀𝑆(𝑡,𝑠)𝑓𝑠,𝑦(𝑠)+𝑘(𝑠,𝑥(𝑠))𝑑𝑠. (65)

Let Θ denote the class of functions 𝜃∶[0,∞)→[0,∞) which satisfies the following conditions:(i)𝜃 is increasing;(ii)there exists 𝜓∈Ψ such that 𝜃(𝑟)=(𝑟/2)−𝜓(𝑟/2) for all 𝑟∈[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 exist 𝜆>0 and 𝜃∈Θ such that for all 𝑥,𝑦∈ℝ, 𝑦≥𝑥, (),();0≤𝑓𝑠,𝑦−𝑓𝑠,𝑥≤𝜆𝜃𝑦−𝑥0≤𝑘𝑠,𝑥−𝑘𝑠,𝑦≤𝜆𝜃𝑦−𝑥 (66)(v)we suppose that 3𝜆sup_{𝑡∈[0,𝑇]}^𝑇₀𝑆()1𝑡,𝑠𝑑𝑠≤2. (67)(vi)there exist continuous functions 𝛼,𝛽∶[0,𝑇]→ℝ such that 𝛼(𝑡)≤𝑝(𝑡)+^𝑇₀𝑆()𝑓((𝑠(𝑠)𝛽(𝑡)(𝑡)+𝑡,𝑠𝑠,𝛼))+𝑘𝑠,𝛽𝑑𝑠,≥𝑝^𝑇₀𝑆(𝑡,𝑠)𝑓𝑠,𝛽(𝑠)+𝑘(𝑠,𝛼(𝑠))𝑑𝑠. (68)Consider the space 𝑋=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) Endow 𝑋 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 in 𝑋²=(𝐶([0,𝑇],ℝ))².

Proof. Consider the operator 𝐹∶𝑋×𝑋→𝑋 defined by 𝐹(𝑡)(𝑡)+𝑥,𝑦=𝑝^𝑇₀𝑆(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠))+𝑘𝑠,𝑦(𝑠)𝑑𝑠for𝑡∈[0,𝑇],𝑥,𝑦∈𝑋. (70) First, we shall prove that 𝐹 has the mixed monotone property.
In fact, for 𝑥₁≤𝑥₂ and 𝑡∈[0,𝑇], we have 𝐹𝑥₂,𝑦(𝑡)−𝐹𝑥₁(𝑡)=,𝑦_𝑇0𝑆(𝑡,𝑠)𝑓𝑠,𝑥₂(𝑠)−𝑓𝑠,𝑥₁(𝑠)𝑑𝑠. (71) Taking into account that 𝑥₁(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 all 𝑡∈[0,𝑇]; that is, 𝐹𝑥₁,𝑦≤𝐹𝑥₂,𝑦. (72) Similarly, for 𝑦₁≤𝑦₂ and 𝑡∈[0,𝑇], we have 𝐹𝑥,𝑦₁(𝑡)−𝐹𝑥,𝑦₂(𝑡)=^𝑇₀𝑆(𝑡,𝑠)𝑘𝑠,𝑦₁(𝑠)−𝑘𝑠,𝑦₂(𝑠)𝑑𝑠. (73) Having 𝑦₁(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, 𝐹𝑥,𝑦₁≥𝐹𝑥,𝑦₂. (74) Therefore, 𝐹 has the mixed monotone property.
First we estimate the quantity 𝐺(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣),𝐹(𝑤,𝑧)) for all 𝑥,𝑦,𝑢,𝑣,𝑤,𝑧∈𝑋 with 𝑥≥𝑢≥𝑤 and 𝑦≤𝑣≤𝑧. Since 𝐹 has the mixed monotone property, we have 𝐹(𝑤,𝑧)≤𝐹(𝑢,𝑣)≤𝐹𝑥,𝑦. (75) We obtain 𝐺𝐹𝑥,𝑦,𝐹(𝑢,𝑣),𝐹(𝑤,𝑧)=sup_{𝑡∈[0,𝑇]}𝐹𝑥,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)−𝐹𝑥,𝑦(𝑡)=sup_{𝑡∈[0,𝑇]}𝐹𝑥,𝑦(𝑡)−𝐹(𝑢,𝑣)(𝑡)+sup_{𝑡∈[0,𝑇]}(𝐹(𝑢,𝑣)(𝑡)−𝐹(𝑤,𝑧)(𝑡))+sup_{𝑡∈[0,𝑇]}𝐹𝑥,𝑦(𝑡)−𝐹(𝑤,𝑧)(𝑡). (76) Also for all 𝑡∈[0,𝑇], from (iv), we have 𝐹()=𝑥,𝑦−𝐹𝑢,𝑣_𝑇0𝑆(𝑓((𝑠((𝑠+𝑡,𝑠)𝑠,𝑥))−𝑓𝑠,𝑢))𝑑𝑠_𝑇0𝑆(𝑡,𝑠)𝑘𝑠,𝑦(𝑠)−𝑘(𝑠,𝑣(𝑠))𝑑𝑠≤𝜆^𝑇₀𝑆(𝑡,𝑠)𝜃(𝑥(𝑠)−𝑢(𝑠))+𝜃𝑣(𝑠)−𝑦(𝑠)𝑑𝑠. (77) Since the function 𝜃 is increasing 𝑥≥𝑢≥𝑤, and 𝑦≤𝑣≤𝑧, we have 𝜃(𝑥(𝑠)−𝑢(𝑠))≤𝜃sup_𝑡∈𝐼|𝑥(𝑡)(𝑡,𝜃𝑣(𝑠)(𝑠)−𝑢)|−𝑦≤𝜃sup_𝑡∈𝐼𝑣(𝑡)−𝑦(𝑡). (78) Hence by (77), we obtain 𝐹𝑥,𝑦−𝐹(𝑢,𝑣)≤𝜆^𝑇₀𝑆(𝑡,𝑠)𝜃sup_𝑡∈𝐼|𝑥(𝑡)−𝑢(𝑡)|+𝜃sup_𝑡∈𝐼𝑣(𝑡)−𝑦(𝑡)𝑑𝑠, (79) as all the quantities on the right hand side of (77) are nonnegative, so (79) is justified.
Similarly, we can obtain that 𝐹𝑥,𝑦−𝐹(𝑤,𝑧)≤𝜆^𝑇₀𝑆(𝑡,𝑠)𝜃sup_𝑡∈𝐼|𝑥(𝑡)−𝑤(𝑡)|+𝜃sup_𝑡∈𝐼𝑧(𝑡)−𝑦(𝑡)𝑑𝑠,|𝐹(𝑢,𝑣)−𝐹(𝑤,𝑧)|≤𝜆^𝑇₀𝑆(𝑡,𝑠)𝜃sup_𝑡∈𝐼|𝑢(𝑡)−𝑤(𝑡)|+𝜃sup_𝑡∈𝐼|𝑧(𝑡)−𝑣(𝑡)|𝑑𝑠. (80)(81) Summing (79), (80), and (81) and then taking the supremum with respect to 𝑡, we get 𝐺𝐹𝑥,𝑦,𝐹(𝑢,𝑣),𝐹(𝑤,𝑧)≤𝜆sup_{𝑡∈[0,𝑇]}^𝑇₀𝑆()⋅𝜃𝑡,𝑠𝑑𝑠sup_𝑡∈𝐼|𝑥(𝑡)−𝑢(𝑡)|+𝜃sup_𝑡∈𝐼|𝑥(𝑡)−𝑤(𝑡)|+𝜃sup_𝑡∈𝐼|𝑢(𝑡)−𝑤(𝑡)|+𝜆sup_{𝑡∈[0,𝑇]}^𝑇₀𝑆()⋅𝜃𝑡,𝑠𝑑𝑠sup_𝑡∈𝐼𝑣(𝑡)−𝑦(𝑡)+𝜃sup_𝑡∈𝐼𝑧(𝑡)−𝑦(𝑡)+𝜃sup_𝑡∈𝐼|𝑧(𝑡)−𝑣(𝑡)|. (82) Further, since 𝜃 is increasing, so that we have 𝜃sup_𝑡∈𝐼|𝑥(𝑡)(𝑡(𝐺(,𝜃−𝑢)|≤𝜃𝑥,𝑢,𝑤))sup_𝑡∈𝐼|𝑥(𝑡)(𝑡(𝐺(,𝜃−𝑤)|≤𝜃𝑥,𝑢,𝑤))sup_𝑡∈𝐼|𝑢(𝑡)−𝑤(𝑡)|≤𝜃(𝐺(𝑥,𝑢,𝑤)). (83) Similarly, we have 𝜃sup_𝑡∈𝐼𝑣(𝑡)(𝑡𝐺,𝜃−𝑦)≤𝜃𝑦,𝑣,𝑧sup_𝑡∈𝐼𝑧(𝑡)(𝑡𝐺,𝜃−𝑦)≤𝜃𝑦,𝑣,𝑧sup_𝑡∈𝐼|𝑧(𝑡)−𝑣(𝑡)|≤𝜃𝐺𝑦,𝑣,𝑧. (84) Then by (82), we have 𝐺𝐹𝑥,𝑦,𝐹(𝑢,𝑣),𝐹(𝑤,𝑧)≤𝜆sup_{𝑡∈[0,𝑇]}^𝑇₀𝑆()(𝐺(𝑡,𝑠𝑑𝑠⋅3𝜃𝑥,𝑢,𝑤))+𝜆sup_{[]𝑡∈0,𝑇}×^𝑇₀𝑆(𝑡,𝑠)𝑑𝑠⋅3𝜃𝐺𝑦,𝑣,𝑧=3𝜆sup_{𝑡∈[0,𝑇]}×^𝑇₀𝑆(𝑡,𝑠)𝑑𝑠⋅𝜃(𝐺(𝑥,𝑢,𝑤))+𝜃𝐺𝑦,𝑣,𝑧. (85) Similarly, we can obtain that 𝐺𝐹𝑦,𝑥,𝐹(𝑣,𝑢),𝐹(𝑧,𝑤)≤3𝜆sup_{𝑡∈[0,𝑇]}×^𝑇₀𝑆(𝑡,𝑠)𝑑𝑠⋅𝜃(𝐺(𝑥,𝑢,𝑤))+𝜃𝐺𝑦,𝑣,𝑧. (86) Summing (85) and (86), dividing by 2, and using (v), we get 𝐺𝐹()()𝐹()()2𝑥,𝑦,𝐹𝑢,𝑣,𝐹𝑤,𝑧+𝐺𝑦,𝑥,𝐹𝑣,𝑢,𝐹𝑧,𝑤⁻¹≤3𝜆sup_{𝑡∈[0,𝑇]}×^𝑇₀𝑆()𝜃(𝐺(𝐺.≤𝜃(𝐺(𝐺𝑡,𝑠𝑑𝑠⋅𝑥,𝑢,𝑤))+𝜃𝑦,𝑣,𝑧𝑥,𝑢,𝑤))+𝜃𝑦,𝑣,𝑧2. (87) Since 𝜃 is increasing, we have 𝜃(𝐺(𝐺(),𝜃𝐺𝐺()𝑥,𝑢,𝑤))≤𝜃𝑥,𝑢,𝑤+𝐺𝑦,𝑣,𝑧𝑦,𝑣,𝑧≤𝜃𝑥,𝑢,𝑤+𝐺𝑦,𝑣,𝑧 (88) and so 𝜃(𝐺(𝑥,𝑢,𝑤))+𝜃𝐺𝑦,𝑣,𝑧2𝐺()=𝐺()≤𝜃𝑥,𝑢,𝑤+𝐺𝑦,𝑣,𝑧𝑥,𝑢,𝑤+𝐺𝑦,𝑣,𝑧2𝐺()−𝜓𝑥,𝑢,𝑤+𝐺𝑦,𝑣,𝑧2 (89) by definition of 𝜃. Thus, using (87) and (89), we finally get 𝐺𝐹()()𝐹()()2𝑥,𝑦,𝐹𝑢,𝑣,𝐹𝑤,𝑧+𝐺𝑦,𝑥,𝐹𝑣,𝑢,𝐹𝑧,𝑤⁻¹≤𝐺(𝑥,𝑢,𝑤)+𝐺𝑦,𝑣,𝑧2𝐺()−𝜓𝑥,𝑢,𝑤+𝐺𝑦,𝑣,𝑧2, (90) or 𝐺𝐹()()𝐹()()()𝐺()𝑥,𝑦,𝐹𝑢,𝑣,𝐹𝑤,𝑧+𝐺𝑦,𝑥,𝐹𝑣,𝑢,𝐹𝑧,𝑤≤𝐺𝑥,𝑢,𝑤+𝐺𝑦,𝑣,𝑧−2𝜓𝑥,𝑢,𝑤+𝐺𝑦,𝑣,𝑧2, (91) which is just the contractive condition (52) in Corollary 24.
Let 𝛼, 𝛽 be the functions appearing in assumption (vi); we get 𝛼≤𝐹𝛼,𝛽,𝛽≥𝐹𝛽,𝛼. (92) Applying Corollary 24, we deduce the existence of 𝑥,𝑦∈𝑋 such that 𝑥=𝐹𝑥,𝑦,𝑦=𝐹𝑦,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 application to integral equations. Contractive conditions 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.