Some Fixed-Point Results for a 𝐺-Weak Contraction in 𝐺-Metric Spaces

Abstract

We introduce the concepts of a 𝐺-weak contraction mapping of types 𝐴 and 𝐵 and we establish some fixed point theorems for a 𝐺-weak contraction mapping of types 𝐴 and 𝐵 in complete 𝐺-metric spaces. Our results generalize several well-known comparable results in the literature.

1. Introduction

As it is well known, one of the most useful theorems in nonlinear analysis is the Banach contraction principle [1]. Many authors generalized this famous result in different ways. In their outstanding article [2], Mustafa and Sims introduced a new notion of generalized metric space, called 𝐺-metric space and gave a modification to the contraction principle of Banach. After then, several authors studied various fixed and common fixed point problems for adequate classes of contractive mappings in generalized metric spaces (for an up to date reference list, see [3–28]).

Despite these important advances in nonlinear analysis, our problem, imposed by theoretical and practical reasons as well, had not been studied so far. In this paper, we introduce the concepts of a 𝐺-weak contraction mapping of types 𝐴 and 𝐵 and we establish some fixed point theorems for a 𝐺-weak contraction mapping of types 𝐴 and 𝐵 in complete 𝐺-metric spaces. Our study is encouraged by its possible application, especially in discrete models for numerical analysis, where iterative processes are extensively used due to their versatility for computer simulation. These models play an essential role in applied mathematical studies of certain nonlinear processes in relation with economics, biology, numerical physics, and tribology. Also, we are motivated by several scientists, who pay attention to such kind of problems. On one hand, Berinde [29–31], Berinde and Pă‌curar [32], and Pă‌curar [33, 34] studied many existing results for almost contraction mappings in metric spaces. On other hand, Samet and Vetro [35] and Shatanawi [36] studied some existing results for almost contraction mappings in 𝐺 metric spaces.

This paper aims to establish new results for almost contraction mappings on nonlinear analysis on 𝐺-metric spaces. It is organized as follows. Next, in Section 2 our framework is introduced, in Section 3 our main results are given, while in Section 4 we introduce a nontrivial example and application to support the use ability of our results. Finally, we conclude the paper.

2. Previous Definitions and Results

To make our presentation self-contained, in this section we give basic definitions and previous results, which are used throughout the paper. This background is organized as a whole, given credit to [2] by Mustafa and Sims.

Definition 2.1. Let 𝑋 be a 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.
Then the function 𝐺 is called a generalized metric, or, more specifically, a 𝐺-metric on 𝑋, and the pair (𝑋,𝐺) is called a 𝐺-metric space.

Definition 2.2. 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 we say that the sequence {𝑥_𝑛} is 𝐺-convergent to 𝑥 or {𝑥_𝑛}𝐺-converges to 𝑥.

Thus, 𝑥_𝑛→𝑥 in a 𝐺-metric space (𝑋,𝐺) if for any 𝜀>0, there exists 𝑘∈ℕ such that 𝐺(𝑥,𝑥_𝑛,𝑥_𝑚)<𝜀 for all 𝑚,𝑛≥𝑘.

Proposition 2.3. Let (𝑋,𝐺) be a 𝐺-metric space. Then the following statements 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_𝑛,𝑥_𝑚,𝑥)→0 as 𝑛,𝑚→+∞.

Definition 2.4. Let (𝑋,𝐺) be a 𝐺-metric space, a sequence {𝑥_𝑛} is called 𝐺-Cauchy if for every 𝜀>0, there is 𝑘∈ℕ such that 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)<𝜀, for all 𝑛,𝑚,𝑙≥𝑘, that is, 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)→0 as 𝑛,𝑚,𝑙→+∞.

Proposition 2.5. Let (𝑋,𝐺) be a 𝐺-metric space. Then the following statements are equivalent.(1)The sequence {𝑥_𝑛} is 𝐺-Cauchy.(2)For every 𝜀>0, there is 𝑘∈ℕ such that 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑚)<𝜀, for all 𝑛,𝑚≥𝑘.

Definition 2.6. A 𝐺-metric space (𝑋,𝐺) is called 𝐺-complete if every 𝐺-Cauchy sequence in (𝑋,𝐺) is 𝐺-convergent in (𝑋,𝐺).

Proposition 2.7. Let (𝑋,𝐺) be a 𝐺-metric space. Then, for any 𝑥,𝑦,𝑧,𝑎∈𝑋 it follows that:(i)if 𝐺(𝑥,𝑦,𝑧)=0, then 𝑥=𝑦=𝑧;(ii)𝐺(𝑥,𝑦,𝑧)≤𝐺(𝑥,𝑥,𝑦)+𝐺(𝑥,𝑥,𝑧);(iii)𝐺(𝑥,𝑦,𝑦)≤2𝐺(𝑦,𝑥,𝑥);(iv)𝐺(𝑥,𝑦,𝑧)≤𝐺(𝑥,𝑎,𝑧)+𝐺(𝑎,𝑦,𝑧);(v)𝐺(𝑥,𝑦,𝑧)≤(2/3)[𝐺(𝑥,𝑦,𝑎)+𝐺(𝑥,𝑎,𝑧)+𝐺(𝑎,𝑦,𝑧)];(vi)𝐺(𝑥,𝑦,𝑧)≤𝐺(𝑥,𝑎,𝑎)+𝐺(𝑦,𝑎,𝑎)+𝐺(𝑧,𝑎,𝑎).

3. Main Results

We start with the following definition.

Definition 3.1 (see [37]). The function 𝜙∶[0,+∞)→[0,+∞) is called an altering distance function, if the following properties are satisfied. (1)𝜙 is continuous and nondecreasing. (2)𝜙(𝑡)=0 if and only if 𝑡=0.

Let (𝑋,𝐺) be a 𝐺-metric space and 𝐹∶𝑋→𝑋 be a mapping. We set 𝑀𝐺(),𝐺𝑥,𝑦,𝑦=max𝑥,𝑦,𝑦,𝐺𝑥,𝐹𝑥,𝐹𝑥,𝐺𝑦,𝐹𝑦,𝐹𝑦𝑥,𝐹𝑦,𝐹𝑦+𝐺𝑦,𝐹𝑦,𝐹𝑦+𝐺𝑦,𝐹𝑥,𝐹𝑥3,𝑀₁𝐺,𝑁𝐺().𝑥,𝑦,𝑦=max𝑥,𝑦,𝑦,𝐺𝑦,𝐹𝑦,𝐹𝑦𝑥,𝑦,𝑦=min𝑥,𝐹𝑥,𝐹𝑥,𝐺𝑦,𝐹𝑦,𝐹𝑦,𝐺𝑦,𝐹𝑥,𝐹𝑥 (3.1)

With this setting, we introduce the following definitions.

Definition 3.2. Let (𝑋,𝐺) be a 𝐺-metric space. A mapping 𝐹∶𝑋→𝑋 is called a 𝐺-weak contraction of type 𝐴 if and only if there exist two constants 𝑎∈(0,1) and 𝐿≥0 such that 𝐺𝐹𝑥,𝐹𝑦,𝐹𝑦≤𝑎𝑀𝑥,𝑦,𝑦+𝐿𝑁𝑥,𝑦,𝑦, (3.2) for all 𝑥,𝑦∈𝑋.

Definition 3.3. Let (𝑋,𝐺) be a 𝐺-metric space and let 𝜓,𝜙,𝛾 be altering distance functions. A mapping 𝐹∶𝑋→𝑋 is called a (𝜓,𝜙,𝛾,𝐺)-weak contraction of type 𝐴 if and only if there exist a constant 𝐿≥0 such that 𝜓𝐺𝐹𝑥,𝐹𝑦,𝐹𝑦≤𝜓𝑀𝑥,𝑦,𝑦−𝜙𝑀₁𝑥,𝑦,𝑦+𝐿𝛾𝑁𝑥,𝑦,𝑦, (3.3) for all 𝑥,𝑦∈𝑋.

Now, we can introduce and prove our main result.

Theorem 3.4. Let (𝑋,𝐺) be a complete 𝐺-metric space and let 𝐹∶𝑋→𝑋 be a (𝜓,𝜙,𝐺)-weak contraction of type 𝐴. Then 𝐹 has a unique fixed point.

Proof. Consider 𝑥₀ in 𝑋 and define a sequence {𝑥_𝑛} in 𝑋 such that 𝑥_𝑛+1=𝐹𝑥_𝑛,forany𝑛∈ℕ. (3.4)
If for some 𝑛∈ℕ, 𝑥_𝑛+1=𝑥_𝑛, then 𝑥_𝑛=𝐹𝑥_𝑛, that is, 𝐹 has a fixed point. Thus, we may assume that 𝑥_𝑛+1≠𝑥_𝑛 for all 𝑛∈ℕ. Now, we finish our proof by the following steps.
Step One. We will show that lim_𝑛→0𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1=0. (3.5)
Given 𝑛∈ℕ. Since 𝐹 is (𝜓,𝜙,𝛾,𝐺)-weak contraction, we have 𝜓𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1=𝜓𝐺𝐹𝑥_𝑛−1,𝐹𝑥_𝑛,𝐹𝑥_𝑛≤𝜓𝑀𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛−𝜙𝑀₁𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛+𝐿𝛾𝑁𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛, (3.6) where 𝑀𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛=max𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛−1,𝐹𝑥_𝑛−1,𝐹𝑥_𝑛−1,𝐺𝑥_𝑛,𝐹𝑥_𝑛,𝐹𝑥_𝑛,𝐺𝑥_𝑛−1,𝐹𝑥_𝑛,𝐹𝑥_𝑛+𝐺𝑥_𝑛,𝐹𝑥_𝑛,𝐹𝑥_𝑛+𝐺𝑥_𝑛,𝐹𝑥_𝑛−1,𝐹𝑥_𝑛−13=max𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝐺𝑥_𝑛−1,𝑥_𝑛+1,𝑥_𝑛+1+𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1+𝐺𝑥_𝑛,𝑥_𝑛,𝑥_𝑛3=max𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝐺𝑥_𝑛−1,𝑥_𝑛+1,𝑥_𝑛+1+𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+13,𝑀₁𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛=max𝐺𝑥_𝑛−1,𝐹𝑥_𝑛−1,𝐹𝑥_𝑛−1,𝐺𝑥_𝑛,𝐹𝑥_𝑛,𝐹𝑥_𝑛=max𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝑁𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛=min𝐺𝑥_𝑛−1,𝐹𝑥_𝑛−1,𝐹𝑥_𝑛−1,𝐺𝑥_𝑛,𝐹𝑥_𝑛,𝐹𝑥_𝑛,𝐺𝑥_𝑛,𝐹𝑥_𝑛−1,𝐹𝑥_𝑛−1=min𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝐺𝑥_𝑛,𝑥_𝑛,𝑥_𝑛=0. (3.7) From (3.6)-(3.7), we obtain 𝜓𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1≤𝜓max𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝐺𝑥_𝑛−1,𝑥_𝑛+1,𝑥_𝑛+1+𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+13−𝜙max𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1. (3.8) By 𝐺₅, we have 𝐺𝑥_𝑛−1,𝑥_𝑛+1,𝑥_𝑛+1≤𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛+𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1. (3.9) Thus 13𝐺𝑥_𝑛−1,𝑥_𝑛+1,𝑥_𝑛+1+𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1≤max𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1. (3.10) If for some 𝑛 in ℕ, max𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1=𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1, (3.11) then 𝜓𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1≤𝜓𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1−𝜙𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1. (3.12) Thus 𝜙(𝐺(𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1))=0 and hence 𝐺(𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1)=0. Therefore, 𝑥_𝑛=𝑥_𝑛+1 which is a contradiction. So, max𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1=𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛. (3.13)
Thus, we get 𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1≤𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛∀𝑛∈ℕ,𝜓𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1≤𝜓𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛−𝜙𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛∀𝑛∈ℕ. (3.14)(3.15)
By (3.14), {𝐺(𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1)∶𝑛∈ℕ} is a nonincreasing sequence. Hence, there is 𝑟≥0 such that lim_𝑛→+∞𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1=𝑟. (3.16) Taking limit as 𝑛→+∞ in (3.15), we obtain 𝜓(𝑟)≤𝜓(𝑟)−𝜙(𝑟). (3.17) Therefore 𝜙(𝑟)=0 and hence 𝑟=0. Thus lim_𝑛→+∞𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1=0. (3.18)
Step Two. We show that {𝑥_𝑛} is a 𝐺-Cauchy sequence in 𝑋. Assume on contrary, then there exists 𝜖>0 for which we can find two subsequences (𝑥_𝑚(𝑖)) and (𝑥_𝑛(𝑖)) of (𝑥_𝑛) such that 𝑛(𝑖) is the smallest index for which 𝑛(𝑖)>𝑚(𝑖)>𝑖,𝐺𝑥_𝑚(𝑖),𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)≥𝜖. (3.19) This means that 𝐺𝑥_𝑚(𝑖),𝑥_{𝑛(𝑖)−1},𝑥_{𝑛(𝑖)−1}<𝜖. (3.20) From (3.19), (3.20), and (𝐺₅), we get 𝜖≤𝐺𝑥_𝑚(𝑖),𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)≤𝐺𝑥_𝑚(𝑖),𝑥_{𝑛(𝑖)−1},𝑥_{𝑛(𝑖)−1}+𝐺𝑥_{𝑛(𝑖)−1},𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)<𝜖+𝐺𝑥_{𝑛(𝑖)−1},𝑥_𝑛(𝑖),𝑥_𝑛(𝑖). (3.21) Taking limit as 𝑖→+∞ and using (3.18), we have lim_𝑖→+∞𝐺𝑥_𝑚(𝑖),𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)=𝜖. (3.22) Using (𝐺₅) again, we obtain 𝐺𝑥_𝑚(𝑖),𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)≤𝐺𝑥_𝑚(𝑖),𝑥_𝑚(𝑖)+1,𝑥_𝑚(𝑖)+1+𝐺𝑥_𝑚(𝑖)+1,𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖)+1+𝐺𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)≤𝐺𝑥_𝑚(𝑖),𝑥_𝑚(𝑖)+1,𝑥_𝑚(𝑖)+1+𝐺𝑥_𝑚(𝑖)+1,𝑥_𝑚(𝑖),𝑥_𝑚(𝑖)+𝐺𝑥_𝑚(𝑖),𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)+𝐺𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖)+1+𝐺𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖),𝑥_𝑛(𝑖). (3.23) Taking 𝑖→+∞ in the above inequality and using (3.18) and (3.22), we arrive at lim_𝑖→+∞𝐺𝑥_𝑚(𝑖)+1,𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖)+1=𝜖. (3.24) Again applying (𝐺₅), we have 𝐺𝑥_𝑚(𝑖),𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖)+1≤𝐺𝑥_𝑚(𝑖),𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)+𝐺𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖)+1≤𝐺𝑥_𝑚(𝑖),𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖)+1+𝐺𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)+𝐺𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖)+1. (3.25) By using (3.18), (3.22) and on taking limit as 𝑖→+∞ in the above inequality, we get lim_𝑖→+∞𝐺𝑥_𝑚(𝑖),𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖)+1=𝜖. (3.26) Also, 𝐺𝑥_𝑚(𝑖)+1,𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)≤𝐺𝑥_𝑚(𝑖)+1,𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖)+1+𝐺𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)≤𝐺𝑥_𝑚(𝑖)+1,𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)+𝐺𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖)+1+𝐺𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖),𝑥_𝑛(𝑖). (3.27) By taking the limit in the above inequalities and using (3.18), (3.24), we obtain lim_𝑖→+∞𝐺𝑥_𝑚(𝑖)+1,𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)=𝜖. (3.28) Now, we have 𝜓𝐺𝑥_𝑚(𝑖)+1,𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖)+1=𝜓𝐺𝐹𝑥_𝑚(𝑖),𝐹𝑥_𝑛(𝑖),𝐹𝑥_𝑛(𝑖)≤𝜓max𝐺𝑥_𝑚(𝑖),𝑥_𝑛(𝑖),𝑥_𝑛(𝑖),𝐺𝑥_𝑚(𝑖),𝐹𝑥_𝑚(𝑖),𝐹𝑥_𝑚(𝑖),𝐺𝑥_𝑛(𝑖),𝐹𝑥_𝑛(𝑖),𝐹𝑥_𝑛(𝑖),13𝐺𝑥_𝑚(𝑖),𝐹𝑥_𝑛(𝑖),𝐹𝑥_𝑛(𝑖)+𝐺𝑥_𝑛(𝑖),𝐹𝑥_𝑛(𝑖),𝐹𝑥_𝑛(𝑖)+𝐺𝑥_𝑛(𝑖),𝐹𝑥_𝑚(𝑖),𝐹𝑥_𝑚(𝑖)−𝜙max𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘),𝐺𝑥_𝑛(𝑘),𝑇𝑥_𝑛(𝑘),𝑇𝑥_𝑛(𝑘)+𝐿𝛾min𝐺𝑥_𝑚(𝑖),𝑥_𝑛(𝑖),𝑥_𝑛(𝑖),𝐺𝑥_𝑛(𝑖),𝐹𝑥_𝑛(𝑖),𝐹𝑥_𝑛(𝑖),𝐺𝑥_𝑛(𝑖),𝐹𝑥_𝑚(𝑖),𝐹𝑥_𝑚(𝑖)≤𝜓max𝐺𝑥_𝑚(𝑖),𝑥_𝑛(𝑖),𝑥_𝑛(𝑖),𝐺𝑥_𝑚(𝑖),𝑥_𝑚(𝑖)+1,𝑥_𝑚(𝑖)+1,𝐺𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖)+1,13𝐺𝑥_𝑚(𝑖),𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖)+1+𝐺𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖)+1+𝐺𝑥_𝑛(𝑖),𝑥_𝑚(𝑖)+1,𝑥_𝑚(𝑖)+1−𝜙max𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘),𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1+𝐿𝛾min𝐺𝑥_𝑚(𝑖),𝑥_𝑛(𝑖),𝑥_𝑛(𝑖),𝐺𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖)+1,𝐺𝑥_𝑛(𝑖),𝑥_𝑚(𝑖)+1,𝑥_𝑚(𝑖)+1≤𝜓max𝐺𝑥_𝑚(𝑖),𝑥_𝑛(𝑖),𝑥_𝑛(𝑖),𝐺𝑥_𝑚(𝑖),𝑥_𝑚(𝑖)+1,𝑥_𝑚(𝑖)+1,𝐺𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖)+1,13𝐺𝑥_𝑚(𝑖),𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖)+1+𝐺𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖)+1+2𝐺𝑥_𝑛(𝑖),𝑥_𝑛(𝑖),𝑥_𝑚(𝑖)+1−𝜙max𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘),𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1+𝐿𝛾min𝐺𝑥_𝑚(𝑖),𝑥_𝑛(𝑖),𝑥_𝑛(𝑖),𝐺𝑥_𝑛(𝑖),𝑥_𝑛(𝑖)+1,𝑥_𝑛(𝑖)+1,𝐺𝑥_𝑛(𝑖),𝑥_𝑚(𝑖)+1,𝑥_𝑚(𝑖)+1. (3.29) Letting 𝑖→+∞, using (3.18), (3.22), (3.24), (3.26), (3.28), and the continuity of 𝜓, 𝜙, and 𝛾, we have 𝜓(𝜖)≤𝜓(𝜖)−𝜙(𝜖). (3.30) Thus 𝜓(𝜖)=0 and hence 𝜖=0, a contradiction. Thus {𝑥_𝑛} is a 𝐺-Cauchy sequence in 𝑋.
Step Three. We show that 𝐹 has a fixed point.
Since {𝑥_𝑛} is a 𝐺-Cauchy sequence in the complete 𝐺-metric space 𝑋, there is 𝑢∈𝑋 such that lim_𝑛→+∞𝐺𝑥_𝑛,𝑢,𝑢=lim_𝑛→+∞𝐺𝑥_𝑛,𝑥_𝑛,𝑢=0. (3.31) Since 𝐹 is (𝜓,𝜙,𝛾,𝐺)-weak contraction, we have 𝜓𝐺𝑥_𝑛+1,𝐹𝑢,𝐹𝑢=𝜓𝐺𝐹𝑥_𝑛𝑀𝑥,𝐹𝑢,𝐹𝑢≤𝜓_𝑛,𝑢,𝑢−𝜙𝑀₁𝑥_𝑛,𝑢,𝑢+𝐿𝛾𝑁𝑥_𝑛,𝑢,𝑢, (3.32) where 𝑀𝑥_𝑛,𝑢,𝑢=max𝐺𝑥_𝑛,𝑢,𝑢,𝐺𝑥_𝑛,𝐹𝑥_𝑛,𝐹𝑥_𝑛(),1,𝐺𝑢,𝐹𝑢,𝐹𝑢3𝐺𝑥_𝑛,𝐹𝑢,𝐹𝑢+𝐺(𝑢,𝐹𝑢,𝐹𝑢)+𝐺𝑢,𝐹𝑥_𝑛,𝐹𝑥_𝑛=max𝐺𝑥_𝑛,𝑢,𝑢,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1(),1,𝐺𝑢,𝐹𝑢,𝐹𝑢3𝐺𝑥_𝑛,𝐹𝑢,𝐹𝑢+𝐺(𝑢,𝐹𝑢,𝐹𝑢)+𝐺𝑢,𝑥_𝑛+1,𝑥_𝑛+1,𝑀₁𝑥_𝑛,𝑢,𝑢=max𝐺𝑥_𝑛(,𝑁𝑥,𝑢,𝑢,𝐺𝑢,𝑇𝑢,𝑇𝑢)_𝑛,𝑢,𝑢=min𝐺𝑥_𝑛,𝐹𝑥_𝑛,𝐹𝑥_𝑛,𝐺(𝑢,𝐹𝑢,𝐹𝑢),𝐺𝑢,𝐹𝑥_𝑛,𝐹𝑥_𝑛=min𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝐺(𝑢,𝐹𝑢,𝐹𝑢),𝐺𝑢,𝑥_𝑛+1,𝑥_𝑛+1. (3.33) Letting 𝑛→+∞ in (3.33) we get 𝑀𝑥_𝑛(),𝑀,𝑢,𝑢→𝐺𝑢,𝐹𝑢,𝐹𝑢₁𝑥_𝑛(),𝑁𝑥,𝑢,𝑢→𝐺𝑢,𝐹𝑢,𝐹𝑢_𝑛,𝑢,𝑢→0. (3.34) On letting 𝑛→+∞ in (3.32) and using the continuity of 𝜓, 𝜙, and 𝛾, we get 𝜓(𝐺(𝑢,𝐹𝑢,𝐹𝑢))≤𝜓(𝐺(𝑢,𝐹𝑢,𝐹𝑢))−𝜙(𝐺(𝑢,𝐹𝑢,𝐹𝑢)). (3.35) Therefore 𝜓(𝐺(𝑢,𝐹𝑢,𝐹𝑢))=0 and hence 𝑢=𝐹𝑢. Thus 𝑢 is a fixed point of 𝐹.
Step Four. We prove the uniqueness of the fixed point.
In this respect, we proceed by reductio ad absurdum. Suppose that there are two fixed points of 𝐹, say 𝑢,𝑣∈𝑋 such that 𝑢≠𝑣. Since 𝐹 is (𝜓,𝜙,𝛾,𝐺)-weak contraction, we have 𝜓(𝐺(𝐹𝑢,𝐹𝑣,𝐹𝑣))≤𝜓(𝑀(𝑢,𝑣,𝑣))−𝜙𝑀₁(𝑢,𝑣,𝑣)+𝐿𝛾(𝑁(𝑢,𝑣,𝑣)), (3.36) where 𝑀()𝐺()()(),𝐺()()()𝑢,𝑣,𝑣=max𝑢,𝑣,𝑣,𝐺𝑢,𝐹𝑢,𝐹𝑢,𝐺𝑣,𝐹𝑣,𝐹𝑣𝑢,𝐹𝑣,𝐹𝑣+𝐺𝑣,𝐹𝑣,𝐹𝑣+𝐺𝑣,𝐹𝑢,𝐹𝑢3𝐺(),𝐺()()=max𝑢,𝑣,𝑣𝑢,𝑣,𝑣+𝐺𝑣,𝑢,𝑢3𝐺(),𝐺()()≤max𝑢,𝑣,𝑣𝑢,𝑣,𝑣+2𝐺𝑣,𝑣,𝑢3(),𝑀=𝐺𝑢,𝑣,𝑣₁(){𝐺()((),𝑁(){𝐺()()({𝐺()()(𝑢,𝑣,𝑣=max𝑢,𝑣,𝑣,𝐺𝑣,𝐹𝑣,𝐹𝑣)}=𝐺𝑢,𝑣,𝑣𝑢,𝑣,𝑣=min𝑢,𝐹𝑢,𝐹𝑢,𝐺𝑣,𝐹𝑣,𝐹𝑣,𝐺𝑣,𝐹𝑢,𝐹𝑢)}=min𝑢,𝑢,𝑢,𝐺𝑣,𝑣,𝑣,𝐺𝑣,𝑢,𝑢)}=0. (3.37) By (3.36)-(3.37), we obtain 𝐺(𝑢,𝑣,𝑣)=𝐺(𝐹𝑢,𝐹𝑣,𝐹𝑣)≤𝜙(𝐺(𝑢,𝑣,𝑣))−𝜙(𝐺(𝑢,𝑣,𝑣))<𝜓(𝐺(𝑢,𝑣,𝑣)), (3.38) a contradiction. Thus, 𝜙(𝐺(𝑢,𝑣,𝑣))=0, and hence 𝑢=𝑣. Therefore the fixed point of 𝐹 is unique.

Corollary 3.5. Let (𝑋,𝐺) be a complete 𝐺-metric space and let 𝐹∶𝑋→𝑋 be a 𝐺-weak contraction of type 𝐴. Then 𝐹 has a unique fixed point.

Proof. Since 𝐹 is a 𝐺-weak contraction, there are two constant 𝑎∈[0,1) and 𝐿≥0 such that 𝐺𝐹𝑥,𝐹𝑦,𝐹𝑦≤𝑎𝑀𝑥,𝑦,𝑦+𝐿𝑁𝑥,𝑦,𝑦. (3.39) Define altering distance functions 𝜓,𝜙,𝛾∶[0,+∞)→[0,+∞) by 𝜓(𝑡)=𝛾(𝑡)=𝑡 and 𝜙(𝑡)=(1−𝑎)𝑡. Then 𝜓𝐺𝑀𝑀𝜓𝑁𝑀𝑀𝐹𝑥,𝐹𝑦,𝐹𝑦≤𝜓𝑥,𝑦,𝑦−𝜙𝑥,𝑦,𝑦+𝐿𝑥,𝑦,𝑦≤𝜓𝑥,𝑦,𝑦−𝜙₁𝑥,𝑦,𝑦+𝐿𝜓𝑁𝑥,𝑦,𝑦. (3.40) Thus 𝐹 is (𝜓,𝜙,𝛾,𝐺)-weak contraction. Hence by Theorem 3.4, we conclude that 𝐹 has a unique fixed point.

Corollary 3.6. Let (𝑋,𝐺) be a complete 𝐺-metric space and 𝐹∶𝑋→𝑋 a mapping. Suppose there exists 𝑎∈[0,1) such that 𝐺𝐺(),𝐺𝐹𝑥,𝐹𝑦,𝐹𝑦≤𝑎max𝑥,𝑦,𝑦,𝐺𝑥,𝐹𝑥,𝐹𝑥,𝐺𝑦,𝐹𝑦,𝐹𝑦𝑥,𝐹𝑦,𝐹𝑦+𝐺𝑦,𝐹𝑦,𝐹𝑦+𝐺𝑦,𝐹𝑥,𝐹𝑥3, (3.41) for all 𝑥,𝑦∈𝑋. Then 𝐹 has a unique fixed point.

Corollary 3.7. Let (𝑋,𝐺) be a complete 𝐺-metric space and let 𝐹∶𝑋→𝑋 be a mapping. Suppose there exist nonnegative real numbers 𝑎,𝑏,𝑐,and𝑑 with 𝑎+𝑏+𝑐+𝑑<1 such that 𝐺()+𝑑𝐹𝑥,𝐹𝑦,𝐹𝑦≤𝑎𝐺𝑥,𝑦,𝑦+𝑏𝐺𝑥,𝐹𝑥,𝐹𝑥+𝑐𝐺𝑦,𝐹𝑦,𝐹𝑦3𝐺𝑥,𝐹𝑦,𝐹𝑦+𝐺𝑦,𝐹𝑦,𝐹𝑦+𝐺𝑦,𝐹𝑥,𝐹𝑥, (3.42) for all 𝑥,𝑦∈𝑋. Then 𝐹 has a unique fixed point.

Proof. Follows from Corollary 3.6, by noting that ()+𝑑𝑎𝐺𝑥,𝑦,𝑦+𝑏𝐺𝑥,𝐹𝑥,𝐹𝑥+𝑐𝐺𝑦,𝐹𝑦,𝐹𝑦3𝐺≤()𝐺(),𝐺𝑥,𝐹𝑦,𝐹𝑦+𝐺𝑦,𝐹𝑦,𝐹𝑦+𝐺𝑦,𝐹𝑥,𝐹𝑥𝑎+𝑏+𝑐+𝑑max𝑥,𝑦,𝑦,𝐺𝑥,𝐹𝑥,𝐹𝑥,𝐺𝑦,𝐹𝑦,𝐹𝑦𝑥,𝐹𝑦,𝐹𝑦+𝐺𝑦,𝐹𝑦,𝐹𝑦+𝐺𝑦,𝐹𝑥,𝐹𝑥3. (3.43)

Consider again (𝑋,𝐺) be a 𝐺-metric space, and F∶𝑋→𝑋 a mapping. We set 𝑚𝐺(),𝐺𝑥,𝑥,𝑦=max𝑥,𝑥,𝑦,𝐺𝑥,𝑥,𝐹𝑥,𝐺𝑦,𝑦,𝐹𝑦𝑥,𝑥,𝐹𝑦+𝐺𝑦,𝑦,𝐹𝑦+𝐺𝑦,𝑦,𝐹𝑥3,𝑚₁𝐺,𝑛𝐺().𝑥,𝑥,𝑦=max𝑥,𝑥,𝑦,𝐺𝑦,𝑦,𝐹𝑦𝑥,𝑥,𝑦=min𝑥,𝑥,𝐹𝑥,𝐺𝑦,𝑦,𝐹𝑦,𝐺𝑦,𝑦,𝐹𝑥 (3.44)

Now, we introduce the following definitions.

Definition 3.8. Let (𝑋,𝐺) be a 𝐺-metric space. A mapping 𝐹∶𝑋→𝑋 is called a 𝐺-weak contraction of type 𝐵 if and only if there exist two constants 𝑎∈(0,1) and 𝐿≥0 such that 𝐺𝐹𝑥,𝐹𝑥,𝐹𝑦≤𝑎𝑚𝑥,𝑥,𝑦+𝐿𝑛𝑥,𝑥,𝑦, (3.45) for all 𝑥,𝑦∈𝑋.

Definition 3.9. Let (𝑋,𝐺) be a 𝐺-metric space and let 𝜓,𝜙,𝛾 be altering distance functions. A mapping 𝐹∶𝑋→𝑋 is called a (𝜓,𝜙,𝛾,𝐺)-weak contraction of type 𝐵 if and only if there exist a constant 𝐿≥0 such that 𝜓𝐺𝐹𝑥,𝐹𝑥,𝐹𝑦≤𝜓𝑚𝑥,𝑥,𝑦−𝜙𝑚₁𝑥,𝑥,𝑦+𝐿𝛾𝑛𝑥,𝑥,𝑦 (3.46) for all 𝑥,𝑦∈𝑋.

Following the same arguments as those in the proof of Theorem 3.4, we obtain the following.

Theorem 3.10. Let (𝑋,𝐺) be a complete 𝐺-metric space and let 𝜓,𝜙,𝛾∶[0,+∞)→[0,+∞) be altering distance function. If 𝐹∶𝑋→𝑋 be a (𝜓,𝜙,𝛾,𝐺)-weak contraction of type 𝐵, then 𝐹 has a unique fixed point.

The following results are direct consequences of Theorem 3.10.

Corollary 3.11. Let (𝑋,𝐺) be a complete 𝐺-metric space and let 𝐹∶𝑋→𝑋 be a 𝐺-weak contraction of type 𝐵. Then 𝐹 has a unique fixed point.

Corollary 3.12. Let (𝑋,𝐺) be a complete 𝐺-metric space and let 𝐹∶𝑋→𝑋 be a mapping. Suppose there exists 𝑎∈[0,1) such that 𝐺𝐹𝑥,𝐹x𝐺(),𝐺,𝐹𝑦≤𝑎max𝑥,𝑥,𝑦,𝐺𝑥,𝑥,𝐹𝑥,𝐺𝑦,𝑦,𝐹𝑦𝑥,𝑥,𝐹𝑦+𝐺𝑦,𝑦,𝐹𝑦+𝐺𝑦,𝑦,𝐹𝑥3, (3.47) for all 𝑥,𝑦∈𝑋. Then 𝐹 has a unique fixed point.

Corollary 3.13. Let (𝑋,𝐺) be a complete 𝐺-metric space and let 𝐹∶𝑋→𝑋 be a mapping. Suppose there exist nonnegative real numbers 𝑎,𝑏,𝑐,and𝑑 with 𝑎+𝑏+𝑐+𝑑<1, such that 𝐺()+𝑑𝐹𝑥,𝐹𝑥,𝐹𝑦≤𝑎𝐺𝑥,𝑥,𝑦+𝑏𝐺𝑥,𝑥,𝐹𝑥+𝑐𝐺𝑦,𝑦,𝐹𝑦3𝐺𝑥,𝑥,𝐹𝑦+𝐺𝑦,𝑦,𝐹𝑦+𝐺𝑦,𝑦,𝐹𝑥, (3.48) for all 𝑥,𝑦∈𝑋. Then 𝐹 has a unique fixed point.

4. Example and Application

In this section, we introduce an example and application to support the validity of our results.

Example 4.1. Let 𝑋={0,1,2,3,…}. Define 𝐺∶𝑋×𝑋×𝑋→𝑋 by 𝐺𝑥,𝑦,𝑧=𝑥+𝑦+𝑧,if𝑥,𝑦,𝑧arealldistinctanddifferentfromzero,𝑥+𝑧,if𝑥=𝑦≠𝑧andallaredifferentfromzero,𝑦+𝑧+1,if𝑥=0,𝑦≠𝑧and𝑦,𝑧aredifferentfromzero,𝑦+2,if𝑥=0,𝑧=𝑦≠0,𝑧+1,if𝑥=0,𝑦=0,𝑧≠0,0,if𝑥=𝑦=𝑥, (4.1) and 𝐹∶𝑋→𝑋 by 𝐹𝑥=0,if𝑥=0,𝑥−1,if𝑥≥1. (4.2) Also, define 𝜓,𝜙,𝛾∶[0,+∞)→[0,+∞) by 𝜓(𝑡)=𝑡² and 𝜙(𝑡)=𝛾(𝑡)=𝑡. Then (a)(𝑋,𝐺,≤) is a complete nonsymmetric 𝐺-metric space, (b)for all 𝑥,𝑦∈𝑋 and 𝐿≥0, we have 𝜓𝐺𝐹𝑥,𝐹𝑦,𝐹𝑦≤𝜓𝑀𝑥,𝑦,𝑦−𝜙𝑀₁𝑥,𝑦,𝑦+𝐿𝛾𝑁𝑥,𝑦,𝑦, (4.3)
that is, 𝐹 is (𝜓,𝜙,𝛾,𝐺)-weak contraction of type 𝐴.

Proof. The proof of part (a) follows from [12]. Now, given 𝑥,𝑦∈𝑋. We divide the proof of part (b) into the following cases.
Case  1 (𝑥=𝑦). Here, we have 𝜓(𝐺(𝐹𝑥,𝐹𝑦,𝐹𝑦))=0 and hence 𝜓𝐺𝐹𝑥,𝐹𝑦,𝐹𝑦≤𝜓𝑀𝑥,𝑦,𝑦−𝜙𝑀₁𝑥,𝑦,𝑦+𝐿𝛾𝑁𝑥,𝑦,𝑦. (4.4)
Case  2 (𝑥>𝑦=1). Here 𝐹𝑥=𝑥−1 and 𝐹𝑦=0. Thus 𝐺()𝑀()𝐺()()(),1𝐹𝑥,𝐹𝑦,𝐹𝑦=𝐺𝑥−1,0,0=𝑥,𝑥,1,1=max𝑥,1,1,𝐺𝑥,𝑥−1,𝑥−1,𝐺1,0,03(𝐺()()(1𝑥,0,0+𝐺1,0,0+𝐺1,𝑥−1,𝑥−1))=max𝑥+1,2𝑥−1,2,3(1𝑥+1+2+𝑥)=max𝑥+1,2𝑥−1,3(𝑀2𝑥+3)=2𝑥−1,₁(){𝐺()({}𝑥,1,1=max𝑥,1,1,𝐺1,0,0)}=max𝑥+1,2=𝑥+1. (4.5) Since 𝑥²≤(2𝑥−1)²−(𝑥+1), (4.6) we have 𝜓𝐺𝐹𝑥,𝐹𝑦,𝐹𝑦≤𝜓𝑀𝑥,𝑦,𝑦−𝜙𝑀₁𝑥,𝑦,𝑦. (4.7) Hence 𝜓𝐺𝐹𝑥,𝐹𝑦,𝐹𝑦≤𝜓𝑀𝑥,𝑦,𝑦−𝜙𝑀₁𝑥,𝑦,𝑦+𝐿𝛾𝑁𝑥,𝑦,𝑦. (4.8)
Case  3 (𝑥>𝑦≥2). Here 𝐹𝑥=𝑥−1 and 𝐹𝑦=𝑦−1.
Subcase 3.1 (𝑥=𝑦+1). Here 𝐺𝐹𝑀𝐺,1𝑦+1,𝐹𝑦,𝐹𝑦=𝐺𝑦,𝑦−1,𝑦−1=2𝑦−1,𝑦+1,𝑦,𝑦=max𝑦+1,𝑦,𝑦,𝐺𝑦+1,𝑦,𝑦,𝐺𝑦,𝑦−1,𝑦−13𝐺1𝑦+1,𝑦−1,𝑦−1+𝐺𝑦,𝑦−1,𝑦−1+𝐺𝑦,𝑦,𝑦=max2𝑦+1,2𝑦−1,3𝑀2𝑦+2𝑦−1+0=2𝑦+1,₁𝑦+1,𝑦,𝑦=max𝐺𝑦+1,𝑦,𝑦,𝐺𝑦,𝑦−1,𝑦−1=max2𝑦+1,2𝑦−1=2𝑦+1. (4.9) Since 2𝑦−1²≤2𝑦+1²−2𝑦+1, (4.10) we have 𝜓𝐺𝐹𝑥,𝐹𝑦,𝐹𝑦≤𝜓𝑀𝑥,𝑦,𝑦−𝜙𝑀₁𝑥,𝑦,𝑦. (4.11) Hence 𝜓𝐺𝐹𝑥,𝐹𝑦,𝐹𝑦≤𝜓𝑀𝑥,𝑦,𝑦−𝜙𝑀₁𝑥,𝑦,𝑦+𝐿𝛾𝑁𝑥,𝑦,𝑦. (4.12)
Subcase 3.2 ( 𝑥>𝑦+1). Here 𝐺𝑀𝐺(),1𝐹𝑥,𝐹𝑦,𝐹𝑦=𝐺𝑥−1,𝑦−1,𝑦−1=𝑥+𝑦−2,𝑥,𝑦,𝑦=max𝑥,𝑦,𝑦,𝐺𝑥,𝑥−1,𝑥−1,𝐺𝑦,𝑦−1,𝑦−13𝐺1𝑥,𝑦−1,𝑦−1+𝐺𝑦,𝑦−1,𝑦−1+𝐺𝑦,𝑥−1,𝑥−1=max𝑥+𝑦,2𝑥−1,2𝑦−1,31𝑥+𝑦−1+2𝑦−1+𝑦+𝑥−1=max𝑥+𝑦,2𝑥−1,2𝑦−1,3𝑀2𝑥+4𝑦−3=2𝑥−1,₁𝑥,𝑦,𝑦=max𝐺𝑥,𝑦,𝑦,𝐺𝑦,𝑦−1,𝑦−1=max𝑥+𝑦,2𝑦−1=𝑥+𝑦. (4.13) Since 𝑥+𝑦−2²≤(2𝑥−3)²≤(2𝑥−1)²−𝑥+𝑦, (4.14) we have 𝜓𝐺𝐹𝑥,𝐹𝑦,𝐹𝑦≤𝜓𝑀𝑥,𝑦,𝑦−𝜙𝑀₁𝑥,𝑦,𝑦. (4.15) Hence 𝜓𝐺𝐹𝑥,𝐹𝑦,𝐹𝑦≤𝜓𝑀𝑥,𝑦,𝑦−𝜙𝑀₁𝑥,𝑦,𝑦+𝐿𝛾𝑁𝑥,𝑦,𝑦. (4.16)
Case  4 (𝑥>𝑦=0). Here, we have 𝐹𝑥=𝑥−1 and 𝐹𝑦=0.
Subcase 4.1 ( 𝑥=1). Here, we have 𝐺𝐹𝑥,𝐹𝑦,𝐹𝑦=𝐺(0,0,0)=0. (4.17) Hence 𝜓𝐺𝐹𝑥,𝐹𝑦,𝐹𝑦≤𝜓𝑀𝑥,𝑦,𝑦−𝜙𝑀₁𝑥,𝑦,𝑦+𝐿𝛾𝑁𝑥,𝑦,𝑦. (4.18)
Subcase 4.2 (𝑥≥2). Here, 𝐺()𝑀()𝐺()()(),1𝐹𝑥,𝐹𝑦,𝐹𝑦=𝐺𝑥−1,0,0=𝑥,𝑥,0,0=max𝑥,0,0,𝐺𝑥,𝑥−1,𝑥−1,𝐺0,0,03(𝐺()()(1𝑥,0,0+𝐺0,0,0+𝐺0,𝑥−1,𝑥−1))=max𝑥+1,2𝑥−1,0,3(1𝑥+1+0+𝑥−1+2)=max𝑥+1,2𝑥−1,0,3(𝑀2𝑥+2)=2𝑥−1,₁(𝑥,0,0)=max{𝐺(𝑥,0,0),𝐺(0,0,0)}=𝑥+1. (4.19) Since 𝑥²≤(2𝑥−1)²−(𝑥+1), (4.20) we have 𝜓𝐺𝐹𝑥,𝐹𝑦,𝐹𝑦≤𝜓𝑀𝑥,𝑦,𝑦−𝜙𝑀₁𝑥,𝑦,𝑦. (4.21) Hence 𝜓𝐺𝐹𝑥,𝐹𝑦,𝐹𝑦≤𝜓𝑀𝑥,𝑦,𝑦−𝜙𝑀₁𝑥,𝑦,𝑦+𝐿𝛾𝑁𝑥,𝑦,𝑦. (4.22)
Thus 𝐹 satisfies all the hypotheses of Theorem 3.4. Hence 𝐹 has a unique fixed point. Here 0 is the unique fixed point of 𝐹.

As an application of our results, we introduce some fixed-point theorems of integral type.

Denote by Φ the set of functions 𝜙∶[0,+∞)→[0,+∞) satisfying the following hypotheses: (1)𝜙 is a Lebesgue integrable function on each compact subset of [0,+∞), (2)for every 𝜖>0, we have ∫^𝜖₀𝜙(𝑠)𝑑𝑠>0.

It is an easy matter, to see that the mapping 𝜓∶[0,+∞)→[0,+∞) defined by 𝜓(𝑡)=^𝑡₀𝜙(𝑠)𝑑𝑠 (4.23) is an altering distance function. Now, we have the following result.

Corollary 4.2. Let (𝑋,𝐺) be a complete 𝐺-metric space. Let 𝐹∶𝑋→𝑋 be a mapping. Suppose that there exist 𝐿≥0 such that for 𝜙,𝜆,𝜇∈Φ, we have ₀^{𝐺(𝐹𝑥,𝐹𝑦,𝐹𝑦)}𝜙(𝑠)𝑑𝑠≤₀^{max{𝐺(𝑥,𝑦,𝑦),𝐺(𝑥,𝐹𝑥,𝐹𝑥),𝐺(𝑦,𝐹𝑦,𝐹𝑦),(1/3)(𝐺(𝑥,𝐹𝑦,𝐹𝑦)+𝐺(𝑦,𝐹𝑦,𝐹𝑦)+𝐺(𝑦,𝐹𝑥,𝐹𝑥))}}𝜙(𝑠)−𝑑𝑠_{0max{𝐺(𝑥,𝑦,𝑦),𝐺(𝑦,𝐹𝑦,𝐹𝑦)}}𝜆(𝑠)𝑑𝑠+𝐿₀^{min{𝐺(𝑥,𝐹𝑥,𝐹𝑥),𝐺(𝑦,𝐹𝑦,𝐹𝑦),𝐺(𝑦,𝐹𝑥,𝐹𝑥)}}𝜇(𝑠)𝑑𝑠, (4.24) for 𝑥,𝑦∈𝑋. Then 𝐹 has a unique fixed point.

Proof. Follows from Theorem 3.4 by taking 𝜓(𝑡)=^𝑡₀𝜙(𝑠)𝑑𝑠,𝜙(𝑡)=^𝑡₀𝜆(𝑠)𝑑𝑠,𝛾(𝑡)=^𝑡₀𝜇(𝑠)𝑑𝑠. (4.25)

Corollary 4.3. Let (𝑋,𝐺) be a complete 𝐺-metric space. Let 𝐹∶𝑋→𝑋 be a mapping. Suppose that there exist 𝐿≥0 such that for 𝜙,𝜆,𝜇∈Φ, we have ₀^{𝐺(𝐹𝑥,𝐹𝑥,𝐹𝑦)}𝜙(𝑠)𝑑𝑠≤₀^{max{𝐺(𝑥,𝑥,𝑦),𝐺(𝑥,𝑥,𝐹𝑥),𝐺(𝑦,𝑦,𝐹𝑦),(1/3)(𝐺(𝑥,𝑥,𝐹𝑦)+𝐺(𝑦,𝑦,𝐹𝑦)+𝐺(𝑦,𝑦,𝐹𝑥))}}𝜙(𝑠)−𝑑𝑠_{0max{𝐺(𝑥,𝑥,𝑦),𝐺(𝑦,𝑦,𝐹𝑦)}}𝜆(𝑠)𝑑𝑠+𝐿^{min{𝐺(𝑥,𝑥,F}₀^{𝑥),𝐺(𝑦,𝑦,𝐹𝑦),𝐺(𝑦,𝑦,𝐹𝑥)}}𝜇(𝑠)𝑑𝑠, (4.26) for 𝑥,𝑦∈𝑋. Then 𝐹 has a unique fixed point.

Proof. Follows from Theorem 3.10 by taking 𝜓(𝑡)=^𝑡₀𝜙(𝑠)𝑑𝑠,𝜙(𝑡)=^𝑡₀𝜆(𝑠)𝑑𝑠,𝛾(𝑡)=^𝑡₀𝜇(𝑠)𝑑𝑠. (4.27)

5. Conclusion

In this paper we introduced contractive conditions of two kinds, independent of the existing ones, to establish new results on nonlinear analysis on 𝐺-metric spaces. More accurately, we established fixed point results for two kinds of 𝐺-weak contraction mappings, in complete 𝐺-metric spaces. We illustrated our theory with nontrivial example. As applications of our main theorems, we introduced fixed point results for mappings satisfying some contractive conditions of integral type in 𝐺-metric spaces.