Some Almost Generalized (𝜓,𝜙)-Contractions in 𝐺-Metric Spaces

Abstract

In this paper, we introduce some almost generalized (𝜓,𝜙)-contractions in the setting of 𝐺-metric spaces. We prove some fixed points results for such contractions. The presented theorems improve and extend some known results in the literature. An example is also presented.

1. Introduction and Preliminaries

In 2006, a new structure of a generalized metric space was introduced by Mustafa and Sims [1] as an appropriate notion of a generalized metric space called a 𝐺-metric space. Fixed point theory in this space was initiated in [2]. Particularly, Banach contraction mapping principle was established in this work. Since then the fixed point theory in 𝐺-metric spaces has been studied and developed by many authors, see [1–29].

The following definitions and results will be needed in the sequel.

Definition 1 (see [1]). Let 𝑋 be a nonempty set, 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 𝑦≠𝑧,(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 generalized metric, or, more specially, a 𝐺-metric on 𝑋, and the pair (𝑋,𝐺) is called a 𝐺-metric space.

Every 𝐺-metric on 𝑋 defines a metric 𝑑_𝐺 on 𝑋 by 𝑑_𝐺𝑥,𝑦=𝐺𝑥,𝑦,𝑦+𝐺𝑦,𝑥,𝑥,∀𝑥,𝑦∈𝑋. (1)

Example 2. Let (𝑋,𝑑) be a metric space. The function 𝐺∶𝑋×𝑋×𝑋→[0,+∞), defined by 𝐺𝑥,𝑦,𝑧=max𝑑𝑥,𝑦,𝑑𝑦,𝑧,𝑑(𝑧,𝑥), (2) or 𝐺𝑥,𝑦,𝑧=𝑑𝑥,𝑦+𝑑𝑦,𝑧+𝑑(𝑧,𝑥), (3) for all 𝑥,𝑦,𝑧∈𝑋, is a 𝐺-metric on 𝑋.

Definition 3 (see [1]). Let (𝑋,𝐺) be a 𝐺-metric space, and let {𝑥_𝑛} be a sequence of points of 𝑋; therefore, we say that 𝑥_𝑛 is 𝐺-convergent to 𝑥∈𝑋 if lim_{𝑛,𝑚→+∞}𝐺(𝑥,𝑥_𝑛,𝑥_𝑚)=0; that is, for any 𝜀>0, there exists 𝑁∈ℕ such that 𝐺(𝑥,𝑥_𝑛,𝑥_𝑚)<𝜀, for all 𝑛,𝑚≥𝑁. We call 𝑥 the limit of the sequence and write 𝑥_𝑛→𝑥 or lim_𝑛→+∞𝑥_𝑛=𝑥.

Proposition 4 (see [1]). Let (𝑋,𝐺) be a 𝐺-metric space. 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→+∞.

Definition 5 (see [1]). Let (𝑋,𝐺) be a 𝐺-metric space. A sequence {𝑥_𝑛} is called a 𝐺-Cauchy sequence if, for any 𝜀>0, there is 𝑁∈ℕ such that 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)<𝜀 for all 𝑚,𝑛,𝑙≥𝑁; that is, 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)→0 as 𝑛,𝑚,𝑙→+∞.

Proposition 6 (see [1]). Let (𝑋,𝐺) be a 𝐺-metric space. Then the following are equivalent:(1)the sequence {𝑥_𝑛} is 𝐺-Cauchy,(2)for any 𝜀>0, there exists 𝑁∈ℕ such that 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑚)<𝜀, for all 𝑚,𝑛≥𝑁.

Definition 7 (see [1]). A 𝐺-metric space (𝑋,𝐺) is called 𝐺-complete if every 𝐺-Cauchy sequence is 𝐺-convergent in (𝑋,𝐺).

Definition 8. Let (𝑋,𝐺) be a 𝐺-metric space. A mapping 𝐹∶𝑋→𝑋 is said to be 𝐺-continuous if for any 𝐺-convergent sequence {𝑥_𝑛} to 𝑥, then {𝐹(𝑥_𝑛)} is 𝐺-convergent to 𝐹(𝑥).

Now, let ℱ denote the set of functions 𝑓∶[0,+∞)→[0,+∞) such that 𝑓(𝑡)=0 if and only if 𝑡=0. We denote by Ψ and Φ the subsets of ℱ such that Ψ=𝜓∈ℱ∶𝜓iscontinuousandnondecreasing,Φ=𝜙∈ℱ∶𝜙islowersemicontinuous. (4) There are a lot of fixed point theorems for different type contractions in the literature. In particular, Berinde [30–32] introduced the concept of an almost contraction in metric spaces and studied many interesting fixed point theorems for a Ćirić strong almost contraction. For other fixed point results on generalized almost contractions, see [33–37]. In this paper, we introduce some almost generalized (𝜓,𝜙)-contractions in the setting of 𝐺-metric spaces, and we establish some fixed points results for such contractions.

2. Main Results

Let (𝑋,𝐺) be a 𝐺-metric space. First, we consider the following expressions: 𝑀𝐺𝑥,𝑦,𝑧=max𝑥,𝑇𝑥,𝑦,𝐺𝑦,𝑇²𝑥,𝑇𝑦,𝐺𝑇𝑥,𝑇²,𝐺()𝑥,𝑇𝑦𝑥,𝑇𝑥,𝑧,𝐺𝑧,𝑇²,𝐺𝑥,𝑇𝑧𝑇𝑥,𝑇²,𝑁𝐺()(),𝐺(),𝑥,𝑇𝑧,𝐺𝑥,𝑦,𝑧𝑥,𝑦,𝑧=min𝑥,𝑇𝑥,𝑇𝑥,𝐺𝑦,𝑇𝑦,𝑇𝑦,𝐺𝑧,𝑇𝑧,𝑇𝑧𝑧,𝑇𝑥,𝑇𝑥,𝐺𝑦,𝑇𝑧,𝑇𝑧 (5) for all 𝑥,𝑦,𝑧∈𝑋.

Our first result for almost generalized (𝜓,𝜙)-contractions is the following.

Theorem 9. Let (𝑋,𝐺) be a complete 𝐺-metric space. Let 𝑇∶𝑋→𝑋 be a self-mapping. Suppose there exist 𝜓∈Ψ, 𝜙∈Φ, and 𝐿≥0 such that for all 𝑥,𝑦,𝑧∈𝑋, 𝜓𝐺𝑀𝑀.𝑇𝑥,𝑇𝑦,𝑇𝑧≤𝜓𝑥,𝑦,𝑧−𝜙𝑥,𝑦,𝑧+𝐿𝑁𝑥,𝑦,𝑧 (6) Then 𝑇 has a unique fixed point; say 𝑢∈𝑋.

Proof. Let 𝑥₀∈𝑋, and define a sequence {𝑥_𝑛} in 𝑋 such that 𝑥_𝑛+1=𝑇𝑥_𝑛,forany𝑛∈ℕ. (7) If for some 𝑛∈ℕ,𝑥_𝑛+1=𝑥_𝑛, then 𝑥_𝑛=𝑇𝑥_𝑛, and the proof is completed. Thus, we may assume that 𝑥_𝑛+1≠𝑥_𝑛 for all 𝑛∈ℕ. By (6) we have 𝜓𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1=𝜓𝐺𝑇𝑥_𝑛−1,𝑇𝑥_𝑛,𝑇𝑥_𝑛≤𝜓𝑀𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛−𝜙𝑀𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛+𝐿𝑁𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛, (8) where 𝑀𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛=max𝐺𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝑥_𝑛,𝐺𝑥_𝑛,𝑇²𝑥_𝑛−1,𝑇𝑥_𝑛,𝐺𝑇𝑥_𝑛−1,𝑇²𝑥_𝑛−1,𝑇𝑥_𝑛,𝐺𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝑥_𝑛,𝐺𝑥_𝑛,𝑇²𝑥_𝑛−1,𝑇𝑥_𝑛,𝐺𝑇𝑥_𝑛−1,𝑇²𝑥_𝑛−1,𝑇𝑥_𝑛,𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛=max𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_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,𝑇𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝐺𝑥_𝑛,𝑇𝑥_𝑛,𝑇𝑥_𝑛,𝐺𝑥_𝑛,𝑇𝑥_𝑛,𝑇𝑥_𝑛,𝐺𝑥_𝑛,𝑇𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝐺𝑥_𝑛,𝑇𝑥_𝑛,𝑇𝑥_𝑛=min𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝐺𝑥_𝑛,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1=min𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,0=0. (9) From (8) and (9), we get 𝜓𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1≤𝜓max𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1−𝜙max𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1. (10) If max𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1=𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1, (11) then by (10) we have 𝜓𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1≤𝜓𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1−𝜙𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1. (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. (13) Therefore, we get 𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1≤𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛∀𝑛∈ℕ, (14) and (10) becomes 𝜓𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1≤𝜓𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛−𝜙𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛∀𝑛∈ℕ. (15) Thus, by (14), the sequence {𝐺(𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1)} is monotone nonincreasing. It follows that 𝐺(𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1)→𝛼 as 𝑛→+∞ for some 𝛼≥0. Next we claim that 𝛼=0. On taking limit as 𝑛→+∞ in (15), we obtain 𝜓(𝛼)≤𝜓(𝛼)−𝜙(𝛼). (16) Hence 𝜙(𝛼)=0 and we get 𝛼=0. Hence lim_𝑛→+∞𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1=0. (17) Next, we show that {𝑥_𝑛} is a 𝐺-Cauchy sequence. On contrary, assume that {𝑥_𝑛} is not a 𝐺-Cauchy sequence. Then, there is an 𝜀>0 for which we can find subsequences {𝑥_𝑛(𝑘)},{𝑥_𝑚(𝑘)} of {𝑥_𝑛} with 𝑚(𝑘)>𝑛(𝑘)≥𝑘 such that 𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)𝐺𝑥≥𝜀,_{𝑚(𝑘)−1},𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)<𝜀. (18) Using (18) and (G5), we have 𝜀≤𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)≤𝐺𝑥_𝑚(𝑘),𝑥_{𝑚(𝑘)−1},𝑥_{𝑚(𝑘)−1}+𝐺𝑥_{𝑚(𝑘)−1},𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)<𝜀+2𝐺𝑥_{𝑚(𝑘)−1},𝑥_𝑚(𝑘),𝑥_𝑚(𝑘). (19) Taking limit as 𝑘→+∞ and using (17), we have lim_𝑘→+∞𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)=𝜀. (20) By (G5), we get 𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)≤𝐺𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1+𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1+𝐺𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)≤𝐺𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1+𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1+2𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1≤𝐺𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1+𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1+2𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1≤3𝐺𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1+𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1+2𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1. (21) On taking limit as 𝑘→+∞ in the above inequality and using (17) and (20), we obtain lim_𝑘→+∞𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1=𝜀. (22) Again, by (G5), we have 𝐺𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+2,𝑥_𝑛(𝑘)+1≤𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1+𝐺𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1,𝑥_𝑚(𝑘)+2≤𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1+𝐺𝑥_𝑚(𝑘)+2,𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1+𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1≤𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1+2𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+2,𝑥_𝑚(𝑘)+2+𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+2,𝑥_𝑚(𝑘)+2+𝐺𝑥_𝑚(𝑘)+2,𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1≤𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1+2𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+2,𝑥_𝑚(𝑘)+2+𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+2,𝑥_𝑚(𝑘)+2+𝐺𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑚(𝑘)+2≤𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1+2𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+2,𝑥_𝑚(𝑘)+2+𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+2,𝑥_𝑚(𝑘)+2+2𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1+𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑚(𝑘)+2. (23) On taking limit as 𝑘→+∞ in the above inequality and using (17) and (22), we obtain lim_𝑘→+∞𝐺𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+2,𝑥_𝑛(𝑘)+1=𝜀. (24) We have also, by (G5), 𝐺𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)≤𝐺𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1+𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘),𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)≤𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+𝐺𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘). (25) On taking limit as 𝑘→+∞ in the above inequality and using (17) and (G3), we obtain lim_𝑘→+∞𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)=lim_𝑘→+∞𝐺𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘). (26) We have also, by (G3) and (G5), 𝐺𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1≤𝐺𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+1≤𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1+𝐺𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1,𝑥_𝑚(k)+1,𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)≤𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+1. (27) Now, on taking limit as 𝑘→+∞ in (27), and using (17), (20), and (26), we obtain lim_𝑘→+∞𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)=lim_𝑘→+∞𝐺𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)=𝜀. (28) Furthermore, by (G3) and (G5), we get 𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1≤𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+2,𝑥_𝑛(𝑘)+1≤𝐺𝑥_𝑚(𝑘)+2,𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1+𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+1≤2𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+2,x_𝑚(𝑘)+2+𝐺𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+𝐺𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1≤2𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+2,𝑥_𝑚(𝑘)+2+2𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1+𝐺𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1. (29) On taking limit as 𝑘→+∞ in the above inequality, and using (17), (20), (22), and (28), we obtain lim_𝑘→+∞𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+2,𝑥_𝑛(𝑘)+1=𝜀. (30) Now, we have 𝜓𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1=𝜓𝐺𝑇𝑥_𝑚(𝑘),𝑇𝑥_𝑛(𝑘),𝑇𝑥_𝑛(𝑘)≤𝜓𝑀𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)−𝜙𝑀𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+𝐿𝑁𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)≤𝜓max𝐺𝑥_𝑚(𝑘),𝑇𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝐺𝑥_𝑛(𝑘),𝑇²𝑥_𝑚(𝑘),𝑇𝑥_𝑛(𝑘),𝐺𝑇𝑥_𝑚(𝑘),𝑇²𝑥_𝑚(𝑘),𝑇𝑥_𝑛(𝑘),𝐺𝑥_𝑚(𝑘),𝑇𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝐺𝑥_𝑛(𝑘),𝑇²𝑥_𝑚(𝑘),𝑇𝑥_𝑛(𝑘),𝐺𝑇𝑥_𝑚(𝑘),𝑇²𝑥_𝑚(𝑘),𝑇𝑥_𝑛(𝑘),𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)−𝜙max𝐺𝑥_𝑚(𝑘),𝑇𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝐺𝑥_𝑛(𝑘),𝑇²𝑥_𝑚(𝑘),𝑇𝑥_𝑛(𝑘),𝐺𝑇𝑥_𝑚(𝑘),𝑇²𝑥_𝑚(𝑘),𝑇𝑥_𝑛(𝑘),𝐺𝑥_𝑚(𝑘),𝑇𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝐺𝑥_𝑛(𝑘),𝑇²𝑥_𝑚(𝑘),𝑇𝑥_𝑛(𝑘),𝐺𝑇𝑥_𝑚(𝑘),𝑇²𝑥_𝑚(𝑘),𝑇𝑥_𝑛(𝑘),𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+𝐿min𝐺𝑥_𝑚(𝑘),𝑇𝑥_𝑚(𝑘),𝑇𝑥_𝑚(𝑘),𝐺𝑥_𝑛(𝑘),𝑇𝑥_𝑛(𝑘),𝑇𝑥_𝑛(𝑘),𝐺𝑥_𝑛(𝑘),𝑇𝑥_𝑛(𝑘),𝑇𝑥_𝑛(𝑘),𝐺𝑥_𝑛(𝑘),𝑇𝑥_𝑚(𝑘),𝑇𝑥_𝑚(𝑘),𝐺𝑥_𝑛(𝑘),𝑇𝑥_𝑛(𝑘),𝑇𝑥_𝑛(𝑘)≤𝜓max𝐺𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘),𝐺𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+2,𝑥_𝑛(𝑘)+1,𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+2,𝑥_𝑛(𝑘)+1,𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)−𝜙max𝐺𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘),𝐺𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+2,𝑥_𝑛(𝑘)+1,𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+2,𝑥_𝑛(𝑘)+1,𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+𝐿min𝐺𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1,𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1,𝐺𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1. (31) Letting 𝑘→+∞, and using (17), (20), (22), (24), (28), and (30), and the properties of 𝜙 and 𝜓, we have 𝜓(𝜀)≤𝜓(𝜀)−𝜙(𝜀). (32) Thus 𝜙(𝜀)=0 and hence 𝜀=0, a contradiction. Thus {𝑥_𝑛} is a 𝐺-Cauchy sequence in 𝑋.
Now, since (𝑋,𝐺) is 𝐺-complete, there are 𝑥∈𝑋 such that {𝑥_𝑛} is 𝐺-convergent to 𝑥; that is lim_𝑛→+∞𝐺𝑥_𝑛,𝑥_𝑛,𝑥=lim_𝑛→+∞𝐺𝑥_𝑛,𝑥,𝑥=0. (33) By (6), we get 𝜓𝐺𝑥_𝑛+1,𝑥_𝑛+1𝐺,𝑇𝑥=𝜓𝑇𝑥_𝑛,𝑇𝑥_𝑛𝑀𝑥,𝑇𝑥≤𝜓_𝑛,𝑥_𝑛,𝑥−𝜙𝑀𝑥_𝑛,𝑥_𝑛,𝑥+𝐿𝑁𝑥_𝑛,𝑥_𝑛,𝑥, (34) where 𝑀𝑥_𝑛,𝑥_𝑛,𝑥=max𝐺𝑥_𝑛,𝑇𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑇²𝑥_𝑛,𝑇𝑥_𝑛,𝐺𝑇𝑥_𝑛,𝑇²𝑥_𝑛,𝑇𝑥_𝑛,𝐺𝑥_𝑛,𝑇𝑥_𝑛,𝐺,𝑥𝑥,𝑇²𝑥_𝑛,𝐺,𝑇𝑥𝑇𝑥_𝑛,𝑇²𝑥_𝑛,𝑇𝑥,𝐺𝑥_𝑛,𝑥_𝑛𝐺𝑥,𝑥=max_𝑛,𝑥_𝑛+1,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+2,𝑥_𝑛+1,𝐺𝑥_𝑛+1,𝑥_𝑛+2,𝑥_𝑛+1,𝐺𝑥_𝑛,𝑥_𝑛+1,𝐺,𝑥𝑥,𝑥_𝑛+2,𝑇𝑥,𝐺𝑥_𝑛+1,𝑥_𝑛+2,𝐺𝑥,𝑇𝑥_𝑛,𝑥_𝑛,𝑁𝑥,𝑥_𝑛,𝑥_𝑛,𝑥=min𝐺𝑥_𝑛,𝑇𝑥_𝑛,𝑇𝑥_𝑛,𝐺𝑥_𝑛,𝑇𝑥_𝑛,𝑇𝑥_𝑛,𝐺(𝑥,𝑇𝑥,𝑇𝑥),𝐺𝑥,𝑇𝑥_𝑛,𝑇𝑥_𝑛,𝐺𝑥_𝑛𝐺𝑥,𝑇𝑥,𝑇𝑥=min_𝑛,𝑥_𝑛+1,𝑥_𝑛+1(),𝐺,𝐺𝑥,𝑇𝑥,𝑇𝑥𝑥,𝑥_𝑛+1,𝑥_𝑛+1,𝐺𝑥_𝑛,𝑇𝑥,𝑇𝑥. (35) Letting 𝑛→+∞ in (35), we get lim_𝑛→+∞𝑀𝑥_𝑛,𝑥_𝑛,𝑥=𝐺(𝑥,𝑥,𝑇𝑥),lim_𝑛→+∞𝑁𝑥_𝑛,𝑥_𝑛,𝑥=0. (36) On letting 𝑛→+∞ in (34), and using the properties of 𝜓 and 𝜙 and (36), we obtain 𝜓(𝐺(𝑥,𝑥,𝑇𝑥))≤𝜓(𝐺(𝑥,𝑥,𝑇𝑥))−𝜙(𝐺(𝑥,𝑥,𝑇𝑥)). (37) Therefore 𝐺(𝑥,𝑥,𝑇𝑥)=0 and hence 𝑥=𝑇𝑥. Thus 𝑥 is a fixed point of 𝑇.
Now our purpose is to check that such point is unique. Suppose that there are two fixed points of 𝑇; say 𝑥,𝑦∈𝑋 such that 𝑥≠𝑦. By (6), we have 𝜓𝐺𝐺𝑀𝑀,𝑥,𝑥,𝑦=𝜓𝑇𝑥,𝑇𝑥,𝑇𝑦≤𝜓𝑥,𝑥,𝑦−𝜙𝑥,𝑥,𝑦+𝐿𝑁𝑥,𝑥,𝑦 (38) where 𝑀𝑥,𝑥,𝑦=max𝐺(𝑥,𝑇𝑥,𝑥),𝐺𝑥,𝑇²,𝐺𝑥,𝑇𝑥𝑇𝑥,𝑇²,𝐺𝑥,𝑇𝑥,𝐺𝑥,𝑇𝑥,𝑦𝑦,𝑇²,𝐺𝑥,𝑇𝑦𝑇𝑥,𝑇²𝐺,𝑁{𝐺()(),𝐺,𝐺𝑥,𝑇𝑦,𝐺𝑥,𝑥,𝑦=max𝑥,𝑥,𝑦,𝐺𝑦,𝑥,𝑦𝑥,𝑥,𝑦=min𝑥,𝑇𝑥,𝑇𝑥,𝐺𝑥,𝑇𝑥,𝑇𝑥𝑦,𝑇𝑦,𝑇𝑦,𝐺𝑦,𝑇𝑥,𝑇𝑥𝑥,𝑇𝑦,𝑇𝑦=0. (39) Similarly, we can prove that 𝜓𝐺𝐺𝑀𝑀,𝑥,𝑦,𝑦=𝜓𝑇𝑥,𝑇𝑦,𝑇𝑦≤𝜓𝑥,𝑦,𝑦−𝜙𝑥,𝑦,𝑦+𝐿𝑁𝑥,𝑦,𝑦 (40) where 𝑀𝐺,𝑁𝑥,𝑦,𝑦=max𝑥,𝑥,𝑦,𝐺𝑦,𝑥,𝑦𝑥,𝑦,𝑦=0. (41) If max𝐺𝑥,𝑥,𝑦,𝐺𝑦,𝑥,𝑦=𝐺𝑥,𝑥,𝑦. (42) By (38) and (39), we get 𝜓𝐺𝑥,𝑥,𝑦≤𝜓𝐺𝑥,𝑥,𝑦−𝜙𝐺𝑥,𝑥,𝑦, (43) a contradiction. Then max𝐺𝑥,𝑥,𝑦,𝐺𝑦,𝑥,𝑦=𝐺𝑥,𝑦,𝑦. (44) By (40) and (41), we get 𝜓𝐺𝑥,𝑦,𝑦≤𝜓𝐺𝑥,𝑦,𝑦−𝜙𝐺𝑥,𝑦,𝑦, (45) a contradiction. Thus, 𝑥=𝑦, and hence the fixed point of 𝑇 is unique.

As consequence of Theorem 9, we present the following corollaries.

Corollary 10. Let (𝑋,𝐺) be a complete 𝐺-metric space. Let 𝑇∶𝑋→𝑋 be a self mapping. Suppose there exist 𝑘∈[0,1) and 𝐿≥0 such that for all 𝑥,𝑦,𝑧∈𝑋𝐺𝑇𝑥,𝑇𝑦,𝑇𝑧≤𝑘𝑀𝑥,𝑦,𝑧+𝐿𝑁𝑥,𝑦,𝑧. (46) Then 𝑇 has a unique fixed point; say 𝑢∈𝑋.

Proof. It suffices to take 𝜓(𝑡)=𝑡 and 𝜙(𝑡)=(1−𝑘)(𝑡) in Theorem 9.

Corollary 11. Let (𝑋,𝐺) be a complete 𝐺-metric space. Let 𝑇∶𝑋→𝑋 be a self mapping. Suppose there exist 𝑘∈[0,1) and 𝐿≥0 such that for all 𝑥,𝑦,𝑧∈𝑋𝐺𝑇𝑥,𝑇𝑦,𝑇𝑧≤𝑘𝐺𝑥,𝑦,𝑧+𝐿𝑁𝑥,𝑦,𝑧. (47) Then 𝑇 has a unique fixed point; say 𝑢∈𝑋.

Proof. Note that 𝐺(𝑥,𝑦,𝑧)≤𝑀(𝑥,𝑦,𝑧) in Corollary 10.

Remark 12. Corollary 11 is a generalization of Mustafa’s result [2].

Our second main result is given as follows.

Theorem 13. Let (𝑋,𝐺) be a complete 𝐺-metric space. Let 𝑇∶𝑋→𝑋 be a self mapping. Suppose there exist 𝜓∈Ψ,  𝜙∈Φ and 𝐿≥0 such that for all 𝑥,𝑦∈𝑋: 𝜓𝐺𝑇𝑥,𝑇𝑦,𝑇²𝑥≤𝜓𝑀^∗𝑥,𝑦,𝑥−𝜙𝑀^∗𝑥,𝑦,𝑥+𝐿𝑁^∗𝑥,𝑦,𝑥, (48) where 𝑀^∗𝐺𝑥,𝑦,𝑥=max𝑥,𝑇𝑥,𝑦,𝐺𝑦,𝑇²𝑥,𝑇𝑦,𝐺𝑇𝑥,𝑇²,𝐺()𝑥,𝑇𝑦𝑥,𝑇𝑥,𝑇𝑥,𝐺𝑇𝑥,𝑇²𝑥,𝑇²𝑥,𝑁,𝐺𝑥,𝑦,𝑇𝑥_∗𝑥,𝑦,x=min𝐺(𝑥,𝑇𝑥,𝑇𝑥),𝐺𝑦,𝑇𝑦,𝑇𝑦,𝐺𝑦,𝑇𝑥,𝑇𝑥 (49) for all 𝑥,𝑦∈𝑋. Then 𝑇 has a unique fixed point; say 𝑢∈𝑋.

Proof. Let 𝑥₀ be an arbitrary point in 𝑋 and define a sequence {𝑥_𝑛} in 𝑋 such that 𝑥_𝑛+1=𝑇𝑥_𝑛,forany𝑛∈ℕ. (50) If for some 𝑛∈ℕ,𝑥_𝑛+1=𝑥_𝑛, then 𝑥_𝑛=𝑇𝑥_𝑛 and the proof is completed. Thus, we may assume that 𝑥_𝑛+1≠𝑥_𝑛 for all 𝑛∈ℕ. By (48), we have 𝜓𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1=𝜓𝐺𝑇𝑥_𝑛−1,𝑇𝑥_𝑛,𝑇²𝑥_𝑛−1≤𝜓𝑀^∗𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛−1−𝜙𝑀^∗𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛−1+𝐿𝑁^∗𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛−1, (51) where 𝑀^∗𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛−1=max𝐺𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝑥_𝑛,𝐺𝑥_𝑛,𝑇²𝑥_𝑛−1,𝑇𝑥_𝑛,𝐺𝑇𝑥_𝑛−1,𝑇²𝑥_𝑛−1,𝑇𝑥_𝑛,𝐺𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝐺𝑇𝑥_𝑛−1,𝑇²𝑥_𝑛−1,𝑇²𝑥_𝑛−1,𝐺𝑥_𝑛−1,𝑥_𝑛,𝑇𝑥_𝑛−1=max𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛=max𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝑁^∗𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛−1=min𝐺𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝐺𝑥_𝑛,𝑇𝑥_𝑛,𝑇𝑥_𝑛,𝐺𝑥_𝑛,𝑇𝑥_𝑛−1,𝑇𝑥_𝑛−1=min𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝐺𝑥_𝑛,𝑥_𝑛,𝑥_𝑛=0. (52) From (51) and (52), we get 𝜓𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1≤𝜓max𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1−𝜙max𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1. (53) If max𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1=𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1, (54) then by (53) we have 𝜓𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1≤𝜓𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1−𝜙𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1. (55) Thus 𝜓(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. (56) Therefore, we get 𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1≤𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛∀𝑛∈ℕ, (57) and (53) becomes 𝜓𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1≤𝜓𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛−𝜙𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛∀𝑛∈ℕ. (58) Thus, by (57), the sequence {𝐺(𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1)} is monotone nonincreasing. It follows that 𝐺(𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1)→𝑔 as 𝑛→+∞ for some 𝑔≥0. Next we claim that 𝑔=0. On taking limit as 𝑛→+∞ in (58), we obtain 𝜓𝑔≤𝜓𝑔−𝜙𝑔. (59) Hence 𝜙(𝑔)=0 and we get 𝑔=0. Then lim_𝑛→+∞𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1=0. (60) Next, we show that {𝑥_𝑛} is a 𝐺-Cauchy sequence. On contrary, assume that {𝑥_𝑛} is not a 𝐺-Cauchy sequence. Then, there is an 𝜀>0 for which we can find subsequences {𝑥_𝑛(𝑘)},{𝑥_𝑚(𝑘)} of {𝑥_𝑛} with 𝑚(𝑘)>𝑛(𝑘)≥𝑘 such that 𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)𝐺𝑥≥𝜀,_{𝑚(𝑘)−1},𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)<𝜀. (61) Using (61) and (G5), we have 𝜀≤𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)≤𝐺𝑥_𝑚(𝑘),𝑥_{𝑚(𝑘)−1},𝑥_{𝑚(𝑘)−1}+𝐺𝑥_{𝑚(𝑘)−1},𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)<𝜀+2𝐺𝑥_{𝑚(𝑘)−1},𝑥_𝑚(𝑘),𝑥_𝑚(𝑘). (62) Taking limit as 𝑘→+∞ and using (60), we have lim_𝑘→+∞𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)=𝜀. (63) By (G5), and using (60) and (63), we obtain, as stated in the proof of Theorem 9, lim_𝑘→+∞𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1=𝜀. (64) Again, by (G5), and using (60) and (64), similarly as in the proof of Theorem 9, we obtain lim_𝑘→+∞𝐺𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+2,𝑥_𝑛(𝑘)+1=𝜀. (65) Furthermore, by (G3) and (G5), we get 𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1≤𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+2,𝑥_𝑛(𝑘)+1≤𝐺𝑥_𝑚(𝑘)+2,𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1+𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+1≤2𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+2,𝑥_𝑚(𝑘)+2+𝐺𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+𝐺𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1≤2𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+2,𝑥_𝑚(𝑘)+2+2𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1+𝐺𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1. (66) By the proof of Theorem 9, we have stated that lim_𝑘→+∞𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)=𝜀. (67) So, on taking limit as 𝑘→+∞ in (66), and using (60), (64), and (67), we obtain lim_𝑘→+∞𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+2,𝑥_𝑛(𝑘)+1=𝜀. (68) Furthermore, by (G3) and (G5), we get 𝐺𝑥_{𝑛(𝑘)−1},𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+1≤𝐺𝑥_{𝑛(𝑘)−1},𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘),x_𝑚(𝑘)+1≤𝐺𝑥_{𝑛(𝑘)−1},𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)≤𝐺𝑥_{𝑛(𝑘)−1},𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+2𝐺𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1+𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘),𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(k)≤𝐺𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1+𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)≤𝐺𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1+𝐺𝑥_𝑛(𝑘),𝑥_{𝑛(𝑘)−1},𝑥_{𝑛(𝑘)−1}+𝐺𝑥_{𝑛(𝑘)−1},𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+1. (69) On taking limit as 𝑘→+∞ in above inequalities, and using (60) and (63), we have lim_𝑘→+∞𝐺𝑥_{𝑛(𝑘)−1},𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+1=𝜀. (70) Also, by (G3) and (G5), we get 𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)≤𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)≤2𝐺𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1+𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘),𝐺𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)≤𝐺𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+1+𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘),𝑥_𝑛(𝑘). (71) On taking limit as 𝑘→+∞ in above inequalities, and using (60), (63), and (70), we obtain lim_𝑘→+∞𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)=𝜀. (72) Now, we have 𝜓𝐺𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+2,𝑥_𝑛(𝑘)+1=𝜓𝐺𝑇𝑥_{𝑛(𝑘)−1},𝑇𝑥_𝑚(𝑘)+1,𝑇²𝑥_{𝑛(𝑘)−1}≤𝜓𝑀^∗𝑥_{𝑛(𝑘)−1},𝑥_𝑚(𝑘)+1,𝑥_{𝑛(𝑘)−1}−𝜙𝑀^∗𝑥_{𝑛(𝑘)−1},𝑥_𝑚(𝑘)+1,𝑥_{𝑛(𝑘)−1}+𝐿𝑁^∗𝑥_{𝑛(𝑘)−1},𝑥_𝑚(𝑘)+1,𝑥_{𝑛(𝑘)−1}≤𝜓max𝐺𝑥_{𝑛(𝑘)−1},𝑇𝑥_{𝑛(𝑘)−1},𝑥_𝑚(𝑘)+1,𝐺𝑥_𝑚(𝑘)+1,𝑇²𝑥_{𝑛(𝑘)−1},𝑇𝑥_𝑚(𝑘)+1,𝐺𝑇𝑥_{𝑛(𝑘)−1},𝑇²𝑥_{𝑛(𝑘)−1},𝑇𝑥_𝑚(𝑘)+1,𝐺𝑥_{𝑛(𝑘)−1},𝑇𝑥_{𝑛(𝑘)−1},𝑇𝑥_{𝑛(𝑘)−1},𝐺𝑇𝑥_{𝑛(𝑘)−1},𝑇²𝑥_{𝑛(𝑘)−1},𝑇²𝑥_{𝑛(𝑘)−1},𝐺𝑥_{𝑛(𝑘)−1},𝑥_𝑚(𝑘)+1,𝑇𝑥_{𝑛(𝑘)−1}−𝜙max𝐺𝑥_{𝑛(𝑘)−1},𝑇𝑥_{𝑛(𝑘)−1},𝑥_𝑚(𝑘)+1,𝐺𝑥_𝑚(𝑘)+1,𝑇²𝑥_{𝑛(𝑘)−1},𝑇𝑥_𝑚(𝑘)+1,𝐺𝑇𝑥_{𝑛(𝑘)−1},𝑇²𝑥_{𝑛(𝑘)−1},𝑇𝑥_𝑚(𝑘)+1,𝐺𝑥_{𝑛(𝑘)−1},𝑇𝑥_{𝑛(𝑘)−1},𝑇𝑥_{𝑛(𝑘)−1},𝐺𝑇𝑥_{𝑛(𝑘)−1},𝑇²𝑥_{𝑛(𝑘)−1},𝑇²𝑥_{𝑛(𝑘)−1},𝐺𝑥_{𝑛(𝑘)−1},𝑥_𝑚(𝑘)+1,𝑇𝑥_{𝑛(𝑘)−1}+𝐿min𝐺𝑥_{𝑛(𝑘)−1},𝑇𝑥_{𝑛(𝑘)−1},𝑇𝑥_{𝑛(𝑘)−1},𝐺𝑥_𝑚(𝑘)+1,𝑇𝑥_𝑚(𝑘)+1,𝑇𝑥_𝑚(𝑘)+1,𝐺𝑥_𝑚(𝑘)+1,𝑇𝑥_{𝑛(𝑘)−1},𝑇𝑥_{𝑛(𝑘)−1}≤𝜓max𝐺𝑥_{𝑛(𝑘)−1},𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+1,𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+1,𝑥_𝑚(𝑘)+2,𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑚(𝑘)+2,𝐺𝑥_{𝑛(𝑘)−1},𝑥_𝑛(𝑘),𝑥_𝑛(𝑘),𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1−𝜙max𝐺𝑥_{𝑛(𝑘)−1},𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+1,𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+1,𝑥_𝑚(𝑘)+2,𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑚(𝑘)+2,𝐺𝑥_{𝑛(𝑘)−1},𝑥_𝑛(𝑘),𝑥_𝑛(𝑘),𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1+𝐿min𝐺𝑥_{𝑛(𝑘)−1},𝑥_𝑛(𝑘),𝑥_𝑛(𝑘),𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+2,𝑥_𝑚(𝑘)+2,𝐺𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘),𝑥_𝑛(𝑘). (73) Letting 𝑘→+∞, and using (60), (65), (68), (70), and (72), and the properties of 𝜙 and 𝜓, we have 𝜓(𝜀)≤𝜓(𝜀)−𝜙(𝜀). (74) Thus 𝜙(𝜀)=0 and hence 𝜀=0, a contradiction. Thus {𝑥_𝑛} is a 𝐺-Cauchy sequence in 𝑋.
Now, since (𝑋,𝐺) is 𝐺-complete, there is 𝑥∈𝑋 such that {𝑥_𝑛} is 𝐺-convergent to 𝑥; that is lim_𝑛→+∞𝐺𝑥_𝑛,𝑥_𝑛,𝑥=lim_𝑛→+∞𝐺𝑥_𝑛,𝑥,𝑥=0. (75) By (48), we get 𝜓𝐺𝑥_𝑛,𝑇𝑥,𝑥_𝑛+1=𝜓𝐺𝑇𝑥_𝑛−1,𝑇𝑥,𝑇²𝑥_𝑛−1≤𝜓𝑀^∗𝑥_𝑛−1,𝑥,𝑥_𝑛−1−𝜙𝑀^∗𝑥_𝑛−1,𝑥,𝑥_𝑛−1+𝐿𝑁^∗𝑥_𝑛−1,𝑥,𝑥_𝑛−1, (76) where 𝑀^∗𝑥_𝑛−1,𝑥,𝑥_𝑛−1=max𝐺𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝑥,𝐺𝑥,𝑇²𝑥_𝑛−1,𝐺,𝑇𝑥𝑇𝑥_𝑛−1,𝑇²𝑥_𝑛−1,𝐺𝑥,𝑇𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝐺𝑇𝑥_𝑛−1,𝑇²𝑥_𝑛−1,𝑇²𝑥_𝑛−1,𝐺𝑥_𝑛−1,𝑥,𝑇𝑥_𝑛−1=max𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥,𝐺𝑥,𝑥_𝑛+1,𝐺𝑥,𝑇𝑥_𝑛,𝑥_𝑛+1,𝑇𝑥,𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝐺𝑥_𝑛−1,𝑥,𝑥_𝑛,𝑁^∗𝑥_𝑛−1,𝑥,𝑥_𝑛−1=min𝐺𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝐺(𝑥,𝑇𝑥,𝑇𝑥),𝐺𝑥,𝑇𝑥_𝑛−1,𝑇𝑥_𝑛−1=min𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺(𝑥,𝑇𝑥,𝑇𝑥),𝐺𝑥,𝑥_𝑛,𝑥_𝑛. (77) Letting 𝑛→+∞ in (77), we get lim_𝑛→+∞𝑀^∗𝑥_𝑛−1,𝑥,𝑥_𝑛−1=𝐺(𝑥,𝑥,𝑇𝑥),lim_𝑛→+∞𝑁^∗𝑥_𝑛−1,𝑥,𝑥_𝑛−1=0. (78) On letting 𝑛→+∞ in (76), and using the properties of 𝜓 and 𝜙 and (78), we obtain 𝜓(𝐺(𝑥,𝑥,𝑇𝑥))≤𝜓(𝐺(𝑥,𝑥,𝑇𝑥))−𝜙(𝐺(𝑥,𝑥,𝑇𝑥)). (79) Therefore 𝐺(𝑥,𝑥,𝑇𝑥)=0 and hence 𝑥=𝑇𝑥. Thus 𝑥 is a fixed point of 𝑇.
Now our purpose is to check that such point is unique. Suppose that there are two fixed points of 𝑇; say 𝑥,𝑦∈𝑋 such that 𝑥≠𝑦. By (48), we have 𝜓𝐺𝑥,𝑦,𝑥=𝜓𝐺𝑇𝑥,𝑇y,𝑇²𝑥≤𝜓𝑀^∗𝑥,𝑦,𝑥−𝜙𝑀^∗𝑥,𝑦,𝑥+𝐿𝑁^∗𝑥,𝑦,𝑥, (80) where 𝑀^∗𝐺𝑥,𝑦,𝑥=max𝑥,𝑇𝑥,𝑦,𝐺𝑦,𝑇²𝑥,𝑇𝑦,𝐺𝑇𝑥,𝑇²,𝐺()𝑥,𝑇𝑦𝑥,𝑇𝑥,𝑇𝑥,𝐺𝑇𝑥,𝑇²𝑥,𝑇²𝑥𝐺,𝑁,𝐺𝑥,𝑦,𝑇𝑥=max𝑥,𝑥,𝑦,𝐺𝑦,𝑥,𝑦_∗𝐺()𝑥,𝑦,𝑥=min𝑥,𝑇𝑥,𝑇𝑥,𝐺𝑦,𝑇𝑦,𝑇𝑦,𝐺𝑦,𝑇𝑥,𝑇𝑥=0. (81) Similarly, we can prove that 𝜓𝐺𝐺𝑦,𝑥,𝑦=𝜓𝑇𝑦,𝑇𝑥,𝑇²𝑦≤𝜓𝑀^∗𝑦,𝑥,𝑦−𝜙𝑀^∗𝑦,𝑥,𝑦+𝐿𝑁^∗𝑦,𝑥,𝑦, (82) where 𝑀^∗𝐺,𝑁𝑦,𝑥,𝑦=max𝑥,𝑥,𝑦,𝐺𝑦,𝑥,𝑦_∗𝑦,𝑥,𝑦=0. (83) If max𝐺𝑥,𝑥,𝑦,𝐺𝑦,𝑥,𝑦=𝐺𝑥,𝑥,𝑦. (84) By (80) and (81), we get 𝜓𝐺𝑥,𝑥,𝑦≤𝜓𝐺𝑥,𝑥,𝑦−𝜙𝐺𝑥,𝑥,𝑦, (85) a contradiction. Then max𝐺𝑥,𝑥,𝑦,𝐺𝑦,𝑥,𝑦=𝐺𝑥,𝑦,𝑦. (86) By (82) and (83), we get 𝜓𝐺𝑥,𝑦,𝑦≤𝜓𝐺𝑥,𝑦,𝑦−𝜙𝐺𝑥,𝑦,𝑦, (87) a contradiction. Thus, 𝑥=𝑦, and hence the fixed point of 𝑇 is unique.

Finally, we provide the following example.

Example 14. Let 𝑋=[0,1] and 𝐺∶𝑋×𝑋×𝑋→ℝ be defined by 𝐺=0𝑥,𝑦,𝑧if𝑥=𝑦=𝑧,max𝑥,𝑦,𝑧otherwise. (88) Then (𝑋,𝐺) is a 𝐺-complete 𝐺-metric space. Let 𝑇∶𝑋→𝑋 be defined by 1𝑇𝑥=4𝑥if10≤𝑥<3,18𝑥⁴if13≤𝑥≤1. (89) Take 𝜓(𝑡)=(1/2)𝑡 and 𝜙(𝑡)=(1/4)𝑡 for all 𝑡∈[0,+∞[. We examine the following cases.
(i) Let 0≤𝑥,𝑦<1/3. Then 𝐺𝑇𝑥,𝑇²1𝑥,𝑇𝑦=max41𝑥,116𝑥,4𝑦=141max𝑥,4≤1𝑥,𝑦4max𝑥,𝑦=𝜓𝑀^∗𝑥,𝑦,𝑥−𝜙𝑀^∗𝑥,𝑦,𝑥, (90) where 𝑀^∗(𝑥,𝑦,𝑥)=max{𝑥,𝑦}.
(ii) Let 1/3≤𝑥,𝑦<1. Then 𝐺𝑇𝑥,𝑇²1𝑥,𝑇𝑦=max8𝑥⁴,1𝑥64¹⁶,18𝑦⁴≤141max𝑥,8𝑥⁴≤1,𝑦4max𝑥,𝑦=𝜓𝑀^∗𝑥,𝑦,𝑥−𝜙𝑀^∗𝑥,𝑦,𝑥, (91) where 𝑀^∗(𝑥,𝑦,𝑧)=max{𝑥,𝑦}.
(iii) Let 0≤𝑥<1/3≤𝑦<1. Then 𝐺𝑇𝑥,𝑇²1𝑥,𝑇𝑦=max41𝑥,116𝑥,8𝑦⁴≤141max𝑥,4≤1𝑥,𝑦4𝑦=𝜓𝑀^∗𝑥,𝑦,𝑥−𝜙𝑀^∗𝑥,𝑦,𝑥, (92) where 𝑀^∗(𝑥,𝑦,𝑥)=𝑦.
(iv) Let 0≤𝑦<1/3≤𝑥<1. Then 𝐺𝑇𝑥,𝑇²1𝑥,𝑇𝑦=max8𝑥⁴,1𝑥64¹⁶,14𝑦≤141max𝑥,8𝑥⁴≤1,𝑦4𝑥=𝜓𝑀^∗𝑥,𝑦,𝑥−𝜙𝑀^∗𝑥,𝑦,𝑥, (93) where 𝑀^∗(𝑥,𝑦,𝑥)=𝑥. Then 𝐺𝑇𝑥,𝑇²𝑥,𝑇𝑦≤𝜓𝑀^∗𝑥,𝑦,𝑥−𝜙𝑀^∗𝑥,𝑦,𝑥+𝐿𝑁^∗𝑥,𝑦,𝑥 (94) for all 𝐿≥0. Then the conditions of Theorem 13, hold and 𝑇 has a unique fixed point. Notice that 𝑢=0 is the desired fixed point of 𝑇.