We prove some fixed point results for (𝜓,𝜙)-weakly contractive maps in G-metric spaces, we show that these maps satisfy property P. The results presented in this paper generalize several well-known comparable results in the literature.

1. Introduction

Metric fixed point theory is an important mathematical discipline because of its applications in areas such as variational and linear inequalities, optimization, and approximation theory. Generalizations of metric spaces were proposed by Gähler [1, 2] (called 2 metricspaces). In 2005, Mustafa and Sims [3] introduced a new structure of generalized metric spaces, which are called G-metric spaces as a generalization of metric space (𝑋,𝑑), to develop and introduce a new fixed point theory for various mappings in this new structure. Many papers dealing with fixed point theorems for mappings satisfying different contractive conditions on G-metric spaces can be found in [4–16]. Let 𝑇 be a self-map of a complete metric space (𝑋,𝑑) with a nonempty fixed point set 𝐹(𝑇). Then 𝑇 is said to satisfy property 𝑃 if 𝐹(𝑇)=𝐹(𝑇𝑛) for each 𝑛∈𝑁. However, the converse is false. For example, consider 𝑋=[0,1] and 𝑇 defined by 𝑇𝑥=1−𝑥. Then 𝑇 has a unique fixed point at 𝑥=1/2, but every even iterate of 𝑇 is the identity map, which has every point of [0,1] as a fixed point. On the other hand, if 𝑋=[0,𝜋], 𝑇𝑥=cos𝑥, then every iterate of 𝑇 has the same fixed point as 𝑇 (see [17, 18]). Jeong and Rhoades [17] showed that maps satisfying many contractive conditions have property 𝑃. An interesting fact about maps satisfying property 𝑃 is that they have no nontrivial periodic points. Papers dealing with property 𝑃 are those in [17–19]. In this paper, we will prove some general fixed point theorems for (𝜓,𝜙)-weakly contractive maps in G-metric spaces, and then we show that these maps satisfy property 𝑃.

Now we give first in what follows preliminaries and basic definitions which will be used throughout the paper.

2. Preliminaries

Consistent with Mustafa and Sims [3], the following definitions and results will be needed in the sequel.

Definition 2.1. Let 𝑋 be a nonempty set and 𝐺∶𝑋×𝑋×𝑋→𝑅+ satisfy the following properties:(𝐺1)𝐺(𝑥,𝑦,𝑧)=0 if 𝑥=𝑦=𝑧=0 (coincidence),(𝐺2)0<𝐺(𝑥,𝑥,𝑦), for all 𝑥,𝑦∈𝑋, where 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(𝑝{𝑥,𝑦,𝑧}), where 𝑝 is a permutation of 𝑥,𝑦,𝑧 (symmetry),(𝐺5)𝐺(𝑥,𝑦,𝑧)≤𝐺(𝑥,ğ‘Ž,ğ‘Ž)+𝐺(ğ‘Ž,𝑦,𝑧), for all 𝑥,𝑦,𝑧,ğ‘Žâˆˆğ‘‹ (rectangle inequality).

Then, the function G is called the G-metric on 𝑋, and the pair (𝑋,𝐺) is called the G-metric space.

Definition 2.2. A G-metric is said to be symmetric if 𝐺(𝑥,𝑦,𝑦)=𝐺(𝑦,𝑥,𝑥) for all 𝑥,𝑦∈𝑋.

Proposition 2.3. Every G -metric space (𝑋,𝐺) will define a metric space (𝑋,𝑑𝐺) by 𝑑𝐺(𝑥,𝑦)=𝐺(𝑥,𝑦,𝑦)+𝐺(𝑦,𝑥,𝑥), for all 𝑥,𝑦∈𝑋.

Definition 2.4. Let (𝑋,𝐺) be a G-metric space, and (𝑥𝑛) be a sequence of points in 𝑋. Then,(i)a point 𝑥∈𝑋 is said to be the limit of the sequence (𝑥𝑛) if 𝐺𝑥𝑛,𝑥𝑚,𝑥⟶0,(as𝑛,ğ‘šâŸ¶âˆž),(2.1) and we say that the sequence (𝑥𝑛) is G convergent to 𝑥 (we say 𝑥𝑛(𝐺)→𝑥),(ii)A sequence (𝑥𝑛) is said to be G-Cauchy if 𝐺𝑥𝑛,𝑥𝑚,𝑥𝑙⟶0,(as𝑛,𝑚,ğ‘™âŸ¶âˆž),(2.2)(iii)(𝑋,𝐺) is called a complete G-metric space if every G-Cauchy sequence in X is G converge in 𝑋.

Proposition 2.5. Let (𝑋,𝐺) be a G-metric space, then the following are equivalent:(1)𝑥𝑛(𝐺)→𝑥, (2)𝐺(𝑥𝑛,𝑥𝑛,𝑥)→0(asğ‘›â†’âˆž), (3)𝐺(𝑥𝑛,𝑥,𝑥)→0(asğ‘›â†’âˆž), (4)𝐺(𝑥𝑛,𝑥𝑚,𝑥)→0(asğ‘›â†’âˆž).

Proposition 2.6. Let (𝑋,𝐺) be a G-metric space, then the following are equivalent:(1)(𝑥𝑛) is be G-Cauchy in 𝑋,(2)𝐺(𝑥𝑛,𝑥𝑚,𝑥𝑚)→0(as𝑛,ğ‘šâ†’âˆž).

Proposition 2.7. Let (𝑋,𝐺) be a G-metric space. Then, for any 𝑥,𝑦,𝑧,ğ‘Žâˆˆğ‘‹, it follows that:(i)If 𝐺(𝑥,𝑦,𝑧)=0, then 𝑥=𝑦=𝑧=0,(ii)𝐺(𝑥,𝑦,𝑧)≤𝐺(𝑥,𝑥,𝑦)+𝐺(𝑥,𝑥,𝑧), (iii)𝐺(𝑥,𝑥,𝑦)≤2𝐺(𝑦,𝑥,𝑥),(iv)𝐺(𝑥,𝑦,𝑧)≤𝐺(𝑥,ğ‘Ž,𝑧)+𝐺(ğ‘Ž,𝑦,𝑧), (v)𝐺(𝑥,𝑦,𝑧)≤𝐺(𝑥,ğ‘Ž,ğ‘Ž)+𝐺(𝑦,ğ‘Ž,ğ‘Ž)+𝐺(𝑧,ğ‘Ž,ğ‘Ž).

3. Main Results

Throughout the paper, 𝑁 denotes the set of all natural numbers.

Definition 3.1 (see [20]). A function 𝜓∶[0,∞)→[0,∞) is called altering distance if the following properties are satisfied:(1)𝜓 is continuous and increasing,(2)𝜓(𝑡)=0 if and only if 𝑡=0.

The altering distance functions alter the metric distance between two points and enable us to deal with relatively new classes of fixed points problems.

Theorem 3.2. Let (𝑋,𝐺) be a complete G-metric space. Let 𝑓 be a self-map on 𝑋 satisfying the following: ,𝜓(𝐺(𝑓𝑥,𝑓𝑦,𝑓𝑧))≤𝜓max𝐺(𝑥,𝑦,𝑧),𝐺(𝑥,𝑓𝑥,𝑓𝑥),𝐺(𝑦,𝑓𝑦,𝑓𝑦),𝐺(𝑧,𝑓𝑧,𝑓𝑧),𝛼𝐺(𝑓𝑥,𝑓𝑥,𝑦)+(1−𝛼)(𝐺(𝑓𝑦,𝑓𝑦,𝑧)),𝛽𝐺(𝑥,𝑓𝑥,𝑓𝑥)+(1−𝛽)(𝐺(𝑦,𝑓𝑦,𝑓𝑦)),−𝜙max𝐺(𝑥,𝑦,𝑧),𝐺(𝑥,𝑓𝑥,𝑓𝑥),𝐺(𝑦,𝑓𝑦,𝑓𝑦),𝐺(𝑧,𝑓𝑧,𝑓𝑧),𝛼𝐺(𝑓𝑥,𝑓𝑥,𝑦)+(1−𝛼)(𝐺(𝑓𝑦,𝑓𝑦,𝑧)),𝛽𝐺(𝑥,𝑓𝑥,𝑓𝑥)+(1−𝛽)(𝐺(𝑦,𝑓𝑦,𝑓𝑦)),(3.1) for all 𝑥,𝑦,𝑧∈𝑋, where 0<𝛼,𝛽<1, 𝜓 is an altering distance function, and 𝜙∶[0,∞)→[0,∞) is a continuous function with 𝜙(𝑡)=0 if and only if 𝑡=0. Then, 𝑓 has a unique fixed point (say 𝑢), where 𝑓 is 𝐺 continuous at 𝑢.

Proof. Fix 𝑥0∈𝑋. Then construct a sequence {𝑥𝑛} by 𝑥𝑛+1=𝑓𝑥𝑛=𝑓𝑛𝑥0. We may assume that 𝑥𝑛≠𝑥𝑛+1 for each 𝑛∈𝑁∪{0}. Since, if there exist 𝑛∈𝑁 such that 𝑥𝑛=𝑥𝑛+1, then 𝑥𝑛 is a fixed point of 𝑓.
From (3.1), substituting 𝑥≡𝑥𝑛−1, 𝑦=𝑧≡𝑥𝑛 then, for all 𝑛∈𝑁, 𝜓𝐺𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1âŽ›âŽœâŽœâŽâŽ§âŽªâŽ¨âŽªâŽ©ğºî€·ğ‘¥î€¸î€¸â‰¤ğœ“max𝑛−1,𝑥𝑛,𝑥𝑛𝑥,𝐺𝑛−1,𝑥𝑛,𝑥𝑛𝑥,𝐺𝑛,𝑥𝑛+1,𝑥𝑛+1𝑥,𝐺𝑛,𝑥𝑛+1,𝑥𝑛+1,𝑥𝛼𝐺𝑛,𝑥𝑛,𝑥𝑛𝐺𝑥+(1−𝛼)𝑛+1,𝑥𝑛+1,𝑥𝑛,𝑥𝛽𝐺𝑛−1,𝑥𝑛,𝑥𝑛𝐺𝑥+(1−𝛽)𝑛+1,𝑥𝑛+1,ğ‘¥ğ‘›âŽ«âŽªâŽ¬âŽªâŽ­âŽžâŽŸâŽŸâŽ âŽ›âŽœâŽœâŽâŽ§âŽªâŽ¨âŽªâŽ©ğºî€·ğ‘¥î€¸î€¸âˆ’ğœ™max𝑛−1,𝑥𝑛,𝑥𝑛𝑥,𝐺𝑛−1,𝑥𝑛,𝑥𝑛𝑥,𝐺𝑛,𝑥𝑛+1,𝑥𝑛+1𝑥,𝐺𝑛,𝑥𝑛+1,𝑥𝑛+1,𝑥𝛼𝐺𝑛,𝑥𝑛,𝑥𝑛𝐺𝑥+(1−𝛼)𝑛+1,𝑥𝑛+1,𝑥𝑛,𝑥𝛽𝐺𝑛−1,𝑥𝑛,𝑥𝑛𝐺𝑥+(1−𝛽)𝑛+1,𝑥𝑛+1,ğ‘¥ğ‘›âŽ«âŽªâŽ¬âŽªâŽ­âŽžâŽŸâŽŸâŽ î‚µî‚»ğºî€·ğ‘¥î€¸î€¸â‰¤ğœ“max𝑛−1,𝑥𝑛,𝑥𝑛𝑥,𝐺𝑛,𝑥𝑛+1,𝑥𝑛+1,𝑥𝛽𝐺𝑛−1,𝑥𝑛,𝑥𝑛𝐺𝑥+(1−𝛽)𝑛+1,𝑥𝑛+1,𝑥𝑛𝐺𝑥−𝜙max𝑛−1,𝑥𝑛,𝑥𝑛𝑥,𝐺𝑛,𝑥𝑛+1,𝑥𝑛+1,𝑥𝛽𝐺𝑛−1,𝑥𝑛,𝑥𝑛𝐺𝑥+(1−𝛽)𝑛+1,𝑥𝑛+1,𝑥𝑛.(3.2) Let 𝑀𝑛=max{𝐺(𝑥𝑛−1,𝑥𝑛,𝑥𝑛),𝐺(𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1)}. Then, (3.2) gives 𝜓𝐺𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1𝑀≤𝜓𝑛𝑀−𝜙𝑛.(3.3) We have two cases, either 𝑀𝑛=𝐺(𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1) or 𝑀𝑛=𝐺(𝑥𝑛−1,𝑥𝑛,𝑥𝑛). Suppose that, for some 𝑛∈𝑁0,𝑀𝑛=𝐺(𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1). Then, we have 𝜓𝐺𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1𝐺𝑥≤𝜓𝑛,𝑥𝑛+1,𝑥𝑛+1𝐺𝑥−𝜙𝑛,𝑥𝑛+1,𝑥𝑛+1.(3.4) Therefore, 𝜓(𝐺(𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1))=0. Hence 𝑥𝑛=𝑥𝑛+1. This is a contradiction since the 𝑥𝑛's are distinct.
Thus, 𝑀𝑛=𝐺(𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1), and (3.2) becomes 𝜓𝐺𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1𝐺𝑥≤𝜓𝑛−1,𝑥𝑛,𝑥𝑛𝐺𝑥−𝜙𝑛−1,𝑥𝑛,𝑥𝑛𝐺𝑥≤𝜓𝑛−1,𝑥𝑛,𝑥𝑛.(3.5) But 𝜓 is an increasing function. Thus, from (3.5), we get 𝐺𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1𝑥≤𝐺𝑛−1,𝑥𝑛,𝑥𝑛,∀𝑛∈𝑁.(3.6) Therefore, {𝐺(𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1),𝑛∈𝑁∪{0}} is a positive nonincreasing sequence. Hence there exists 𝑟≥0 such that limğ‘›â†’âˆžğºî€·ğ‘¥ğ‘›,𝑥𝑛+1,𝑥𝑛+1=𝑟.(3.7) Letting ğ‘›â†’âˆž, and using (3.5) and the continuity of 𝜓 and 𝜙, we get 𝜓(𝑟)≤𝜓(𝑟)−𝜙(𝑟).(3.8) Hence, 𝜙(𝑟)=0, therefore 𝑟=0, which implies that limğ‘›â†’âˆžğºî€·ğ‘¥ğ‘›,𝑥𝑛+1,𝑥𝑛+1=0.(3.9) Consequently, for a given 𝜀>0, there is an integer 𝑛0 such that 𝐺𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1<𝜀2,∀𝑛>𝑛0.(3.10) For 𝑚,𝑛∈𝑁 with 𝑚>𝑛, we claim that 𝐺𝑥𝑛,𝑥𝑚,𝑥𝑚<𝜀2,∀𝑚>𝑛>𝑛0.(3.11) To show (3.11), we use induction on 𝑚. Inequality (3.11) holds for 𝑚=𝑛+1 from (3.10). Assume (3.11) holds for 𝑚=𝑘, that is, 𝐺𝑥𝑛,𝑥𝑘,𝑥𝑘<𝜀2,∀𝑛>𝑛0.(3.12) For all 𝑛>𝑛0, take 𝑚=𝑘+1. Using (𝐺5) in Definition 2.1 and inequalities (3.10), (3.12), we get 𝐺𝑥𝑛,𝑥𝑘+1,𝑥𝑘+1𝑥≤𝐺𝑛,𝑥𝑛+1,𝑥𝑛+1𝑥+𝐺𝑛+1,𝑥𝑘+1,𝑥𝑘+1𝑥≤𝐺𝑛,𝑥𝑛+1,𝑥𝑛+1𝑥+𝐺𝑛,𝑥𝑘,𝑥𝑘<𝜀.(3.13) By induction on 𝑚, we conclude that 𝐺𝑥𝑛,𝑥𝑚,𝑥𝑚<𝜀2,∀𝑚>𝑛>𝑛0.(3.14) We conclude from Proposition 2.6 that {𝑥𝑛} is a G-Cauchy sequence in 𝑋. From the completeness of 𝑋, there exists 𝑢 in 𝑋 such that 𝑥𝑛(𝐺)→𝑢. For 𝑛∈𝑁, we have 𝜓𝐺𝑓𝑢,𝑓𝑢,𝑥𝑛𝐺=𝜓𝑓𝑢,𝑓𝑢,𝑓𝑥𝑛−1âŽ›âŽœâŽœâŽâŽ§âŽªâŽ¨âŽªâŽ©ğºî€·î€¸î€¸â‰¤ğœ“max𝑢,𝑢,𝑥𝑛−1𝑥,𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺𝑛−1,𝑥𝑛,𝑥𝑛,𝐺𝛼𝐺(𝑓𝑢,𝑓𝑢,𝑢)+(1−𝛼)𝑓𝑢,𝑓𝑢,𝑥𝑛−1,âŽ«âŽªâŽ¬âŽªâŽ­âŽžâŽŸâŽŸâŽ âŽ›âŽœâŽœâŽâŽ§âŽªâŽ¨âŽªâŽ©ğºî€·î€¸î€¸ğ›½ğº(𝑢,𝑓𝑢,𝑓𝑢)+(1−𝛽)(𝐺(𝑢,𝑓𝑢,𝑓𝑢))−𝜙max𝑢,𝑢,𝑥𝑛−1𝑥,𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺𝑛−1,𝑥𝑛,𝑥𝑛,𝐺𝛼𝐺(𝑓𝑢,𝑓𝑢,𝑢)+(1−𝛼)𝑓𝑢,𝑓𝑢,𝑥𝑛−1,âŽ«âŽªâŽ¬âŽªâŽ­âŽžâŽŸâŽŸâŽ î‚µî‚»ğºî€·î€¸î€¸ğ›½ğº(𝑢,𝑓𝑢,𝑓𝑢)+(1−𝛽)(𝐺(𝑢,𝑓𝑢,𝑓𝑢))≤𝜓max𝑢,𝑢,𝑥𝑛−1𝑥,𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺𝑛−1,𝑥𝑛,𝑥𝑛,𝐺𝛼𝐺(𝑓𝑢,𝑓𝑢,𝑢)+(1−𝛼)𝑓𝑢,𝑓𝑢,𝑥𝑛−1𝐺−𝜙max𝑢,𝑢,𝑥𝑛−1𝑥,𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺𝑛−1,𝑥𝑛,𝑥𝑛,𝐺𝛼𝐺(𝑓𝑢,𝑓𝑢,𝑢)+(1−𝛼)𝑓𝑢,𝑓𝑢,𝑥𝑛−1.(3.15) Letting ğ‘›â†’âˆž, and using the fact that 𝜓 is continuous and 𝐺 is continuous on its variables, we get that 𝐺(𝑢,𝑓𝑢,𝑓𝑢)=0. Hence 𝑓𝑢=𝑢. So 𝑢 is a fixed point of 𝑓. Now, to show uniqueness, let 𝑣 be another fixed point of 𝑓 with 𝑣≠𝑢. Therefore, 𝜓(𝐺(𝑢,𝑢,𝑣))=𝜓(𝐺(𝑓𝑢,𝑓𝑢,𝑓𝑣))≤𝜓max𝐺(𝑢,𝑢,𝑣),𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺(𝑣,𝑓𝑣,𝑓𝑣),𝛼𝐺(𝑓𝑢,𝑓𝑢,𝑢)+(1−𝛼)(𝐺(𝑓𝑢,𝑓𝑢,𝑣)),𝛽𝐺(𝑢,𝑓𝑢,𝑓𝑢)+(1−𝛽)(𝐺(𝑣,𝑓𝑣,𝑓𝑣))−𝜙max𝐺(𝑢,𝑢,𝑣),𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺(𝑣,𝑓𝑣,𝑓𝑣),𝛼𝐺(𝑓𝑢,𝑓𝑢,𝑢)+(1−𝛼)(𝐺(𝑓𝑢,𝑓𝑢,𝑣)),𝛽𝐺(𝑢,𝑓𝑢,𝑓𝑢)+(1−𝛽)(𝐺(𝑣,𝑓𝑣,𝑓𝑣))=𝜓(max{𝐺(𝑢,𝑢,𝑣),(1−𝛼)𝐺(𝑢,𝑢,𝑣)})−𝜙(max{𝐺(𝑢,𝑢,𝑣),(1−𝛼)𝐺(𝑢,𝑢,𝑣)})=𝜓(𝐺(𝑢,𝑢,𝑣))−𝜙(𝐺(𝑢,𝑢,𝑣)).(3.16) Hence, 𝜓(𝐺(𝑢,𝑢,𝑣))≤𝜓(𝐺(𝑢,𝑢,𝑣))−𝜙(𝐺(𝑢,𝑢,𝑣)).(3.17) This implies that 𝜙(𝐺(𝑢,𝑢,𝑣))=0, then 𝐺(𝑢,𝑢,𝑣)=0 and 𝑢=𝑣.
Now to show that 𝑓 is 𝐺 continuous at 𝑢, let {𝑥𝑛} be a sequence in 𝑋 with limit 𝑢(i.e.,𝑥𝑛(𝐺)→𝑢). Using (3.1), we have 𝜓𝐺𝑓𝑥𝑛𝐺,𝑢,𝑢=ğœ“ğ‘“ğ‘¥ğ‘›âŽ›âŽœâŽœâŽâŽ§âŽªâŽ¨âŽªâŽ©ğºî€·ğ‘¥,𝑓𝑢,𝑓𝑢≤𝜓max𝑛𝑥,𝑢,𝑢,𝐺𝑛,𝑓𝑥𝑛,𝑓𝑥𝑛,𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝛼𝐺𝑓𝑥𝑛,𝑓𝑥𝑛𝑥,𝑢+(1−𝛼)(𝐺(𝑓𝑢,𝑓𝑢,𝑢)),𝛽𝐺𝑛,𝑓𝑥𝑛,ğ‘“ğ‘¥ğ‘›î€¸âŽ«âŽªâŽ¬âŽªâŽ­âŽžâŽŸâŽŸâŽ âŽ›âŽœâŽœâŽâŽ§âŽªâŽ¨âŽªâŽ©ğºî€·ğ‘¥+(1−𝛽)(𝐺(𝑢,𝑓𝑢,𝑓𝑢))−𝜙max𝑛𝑥,𝑢,𝑢,𝐺𝑛,𝑓𝑥𝑛,𝑓𝑥𝑛,𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝛼𝐺𝑓𝑥𝑛,𝑓𝑥𝑛𝑥,𝑢+(1−𝛼)(𝐺(𝑓𝑢,𝑓𝑢,𝑢)),𝛽𝐺𝑛,𝑓𝑥𝑛,ğ‘“ğ‘¥ğ‘›î€¸âŽ«âŽªâŽ¬âŽªâŽ­âŽžâŽŸâŽŸâŽ î€·î€½ğºî€·ğ‘¥+(1−𝛽)(𝐺(𝑢,𝑓𝑢,𝑓𝑢))=𝜓max𝑛,𝑢,𝑢,𝛼𝐺𝑓𝑥𝑛,𝑓𝑥𝑛𝑥,𝑢,𝛽𝐺𝑛,𝑓𝑥𝑛,𝑓𝑥𝑛𝐺𝑥−𝜙max𝑛,𝑢,𝑢,𝛼𝐺𝑓𝑥𝑛,𝑓𝑥𝑛𝑥,𝑢,𝛽𝐺𝑛,𝑓𝑥𝑛,𝑓𝑥𝑛𝐺𝑥≤𝜓max𝑛𝑥,𝑢,𝑢,𝛼𝐺𝑛+1,𝑥𝑛+1𝑥,𝑢,𝛽𝐺𝑛,𝑥𝑛+1,𝑥𝑛+1𝐺𝑥≤𝜓max𝑛𝑥,𝑢,𝑢,𝛼𝐺𝑛+1,𝑥𝑛+1𝑥,𝑢,𝛽𝐺𝑛,𝑢,𝑢+𝛽𝐺𝑢,𝑥𝑛+1,𝑥𝑛+1𝐺𝑥≤𝜓max𝑛𝑥,𝑢,𝑢,𝐺𝑛+1,𝑥𝑛+1𝑥,𝑢,𝛽𝐺𝑛,𝑢,𝑢+𝛽𝐺𝑢,𝑥𝑛+1,𝑥𝑛+1.(3.18) But 𝜓 is an increasing function, thus from (3.18), we get 𝐺𝑓𝑥𝑛𝐺𝑥,𝑢,𝑢≤max𝑛𝑥,𝑢,𝑢,𝐺𝑛+1,𝑥𝑛+1𝑥,𝑢,𝛽𝐺𝑛+1,𝑢,𝑢+𝛽𝐺𝑢,𝑥𝑛+1,𝑥𝑛+1.(3.19) Therefore, limğ‘›â†’âˆžğº(𝑓𝑥𝑛,𝑢,𝑢)=0.

Corollary 3.3. Let 𝑇 be a self-map on a complete G-metric space 𝑋 satisfying the following for all 𝑥,𝑦,𝑧∈𝑋∶𝐺(𝑓𝑥,𝑓𝑦,𝑓𝑧)≤𝜆max𝐺(𝑥,𝑦,𝑧),𝐺(𝑥,𝑓𝑥,𝑓𝑥),𝐺(𝑦,𝑓𝑦,𝑓𝑦),𝐺(𝑧,𝑓𝑧,𝑓𝑧),𝛼𝐺(𝑓𝑥,𝑓𝑥,𝑦)+(1−𝛼)(𝐺(𝑓𝑦,𝑓𝑦,𝑧)),𝛽𝐺(𝑥,𝑓𝑥,𝑓𝑥)+(1−𝛽)(𝐺(𝑦,𝑓𝑦,𝑓𝑦)),(3.20) where 0<𝛼,𝛽,𝜆<1,𝜓 is an altering distance function, and 𝜙∶[0,∞)→[0,∞) is a continuous function with 𝜙(𝑡)=0 if and only if 𝑡=0. Then 𝑓 has a unique fixed point (say 𝑢), and 𝑓 is 𝐺 continuous at 𝑢.

Proof. We get the result by taking 𝜓(𝑡)=𝑡 and 𝜙(𝑡)=𝑡−𝜆𝑡, then apply Theorem 3.2.

Corollary 3.4. Let (𝑋,𝐺) be a complete G-metric space. Let 𝑓 be a self-map on 𝑋 satisfying the following: ⎧⎪⎨⎪⎩1𝐺(𝑓𝑥,𝑓𝑦,𝑓𝑧)≤𝜆max𝐺(𝑥,𝑦,𝑧),𝐺(𝑥,𝑓𝑥,𝑓𝑥),𝐺(𝑦,𝑓𝑦,𝑓𝑦),𝐺(𝑧,𝑓𝑧,𝑓𝑧),21(𝐺(𝑓𝑥,𝑓𝑥,𝑦)+(𝐺(𝑓𝑦,𝑓𝑦,𝑧))),2⎫⎪⎬⎪⎭(𝐺(𝑥,𝑓𝑥,𝑓𝑥)+𝐺(𝑦,𝑓𝑦,𝑓𝑦)),(3.21) for all 𝑥,𝑦,𝑧∈𝑋 where 0<𝜆<1,𝜓 is an altering distance function and, 𝜙∶[0,∞)→[0,∞) is a continuous function with 𝜙(𝑡)=0 if and only if 𝑡=0. Then 𝑓 has a unique fixed point (say 𝑢), and 𝑓 is 𝐺 continuous at 𝑢.

Proof. We get the result by taking 𝜓(𝑡)=𝑡 and 𝜙(𝑡)=𝑡−𝜆𝑡, 𝛼=𝛽=1/2 in Theorem 3.2.

Corollary 3.5. Let (𝑋,𝐺) be a complete G-metric space. Let 𝑓 be a self-map on 𝑋 satisfying the following: ⎧⎪⎪⎨⎪⎪⎩1𝐺(𝑓𝑥,𝑓𝑦,𝑓𝑧)≤𝜆max𝐺(𝑥,𝑦,𝑧),𝐺(𝑥,𝑓𝑥,𝑓𝑥),𝐺(𝑦,𝑓𝑦,𝑓𝑦),𝐺(𝑧,𝑓𝑧,𝑓𝑧),32𝐺(𝑓𝑥,𝑓𝑥,𝑦)+31𝐺(𝑓𝑦,𝑓𝑦,𝑧),32𝐺(𝑓𝑥,𝑓𝑥,𝑥)+3⎫⎪⎪⎬⎪⎪⎭𝐺(𝑓𝑦,𝑓𝑦,𝑦),(3.22) for all 𝑥,𝑦,𝑧∈𝑋, where 0<𝜆<1,𝜓 is an altering distance function, and 𝜙∶[0,∞)→[0,∞) is a continuous function with 𝜙(𝑡)=0 if and only if 𝑡=0. Then 𝑓 has a unique fixed point (say 𝑢) and 𝑓 is 𝐺 continuous at 𝑢.

Proof. We get the result by taking 𝜓(𝑡)=𝑡 and 𝜙(𝑡)=𝑡−𝜆𝑡, 𝛼=𝛽=1/3 in Theorem 3.2.

Theorem 3.6. Under the condition of Theorem 3.2, 𝑓 has property 𝑃.

Proof. From Theorem 3.2, 𝑓 has a fixed point. Therefore 𝐹(𝑓𝑛)≠𝜑 for each 𝑛∈𝑁. Fix 𝑛>1, and assume that 𝑢∈𝐹(𝑓𝑛). We claim that 𝑢∈𝐹(𝑓). To prove the claim, suppose that 𝑢≠𝑓𝑢. Using (3.1), we have 𝜓𝐺𝑓(𝐺(𝑢,𝑓𝑢,𝑓𝑢))=𝜓𝑛𝑢,𝑓𝑛+1𝑢,𝑓𝑛+1𝑢𝐺=𝜓𝑓𝑓𝑛−1𝑢,𝑓𝑓𝑛𝑢,ğ‘“ğ‘“ğ‘›ğ‘¢âŽ›âŽœâŽœâŽâŽ§âŽªâŽ¨âŽªâŽ©ğºî€·ğ‘“î€¸î€¸â‰¤ğœ“max𝑛−1𝑓𝑢,𝑢,𝑢,𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝛼𝐺(𝑢,𝑢,𝑢)+(1−𝛼)(𝐺(𝑓𝑢,𝑓𝑢,𝑢))𝛽𝐺𝑛−1î€¸âŽ«âŽªâŽ¬âŽªâŽ­âŽžâŽŸâŽŸâŽ âŽ›âŽœâŽœâŽâŽ§âŽªâŽ¨âŽªâŽ©ğºî€·ğ‘“ğ‘¢,𝑢,𝑢+(1−𝛽)(𝐺(𝑢,𝑓𝑢,𝑓𝑢))−𝜙max𝑛−1𝑓𝑢,𝑢,𝑢,𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝛼𝐺(𝑢,𝑢,𝑢)+(1−𝛼)(𝐺(𝑓𝑢,𝑓𝑢,𝑢))𝛽𝐺𝑛−1î€¸âŽ«âŽªâŽ¬âŽªâŽ­âŽžâŽŸâŽŸâŽ î€·î€½ğºî€·ğ‘“ğ‘¢,𝑢,𝑢+(1−𝛽)(𝐺(𝑢,𝑓𝑢,𝑓𝑢))=𝜓max𝑛−1𝐺𝑓𝑢,𝑢,𝑢,𝐺(𝑢,𝑓𝑢,𝑓𝑢)−𝜙max𝑛−1.𝑢,𝑢,𝑢,𝐺(𝑢,𝑓𝑢,𝑓𝑢)(3.23) Letting 𝑀=max{𝐺(𝑓𝑛−1𝑢,𝑢,𝑢),𝐺(𝑢,𝑓𝑢,𝑓𝑢)}, we deduce form (3.23), 𝜓(𝐺(𝑢,𝑓𝑢,𝑓𝑢))≤𝜓(𝑀)−𝜙(𝑀).(3.24) If 𝑀=𝐺(𝑢,𝑓𝑢,𝑓𝑢), then 𝜓(𝐺(𝑢,𝑓𝑢,𝑓𝑢))≤𝜓(𝐺(𝑢,𝑓𝑢,𝑓𝑢))−𝜙(𝐺(𝑢,𝑓𝑢,𝑓𝑢)),(3.25) hence, 𝜙(𝐺(𝑢,𝑓𝑢,𝑓𝑢))=0. By a property of 𝜙, we deduce that 𝐺(𝑢,𝑓𝑢,𝑓𝑢)=0, therefore, 𝑢=𝑓𝑢. This is a contradiction. On the other hand, if 𝑀=𝐺(𝑓𝑛−1𝑢,𝑢,𝑢), then (3.1) gives that 𝜓𝐺𝑓𝑛𝑢,𝑓𝑛+1𝑢,𝑓𝑛+1𝑢𝐺𝑓=𝜓(𝐺(𝑢,𝑓𝑢,𝑓𝑢))≤𝜓𝑛−1𝐺𝑓𝑢,𝑢,𝑢−𝜙𝑛−1𝐺𝑓𝑢,𝑢,𝑢=𝜓𝑛−1𝑢,𝑓𝑛𝑢,𝑓𝑛𝑢𝐺𝑓−𝜙𝑛−1𝑢,𝑓𝑛𝑢,𝑓𝑛𝑢𝐺𝑓≤𝜓𝑛−2𝑢,𝑓𝑛−1𝑢,𝑓𝑛−1𝑢𝐺𝑓−𝜙𝑛−2𝑢,𝑓𝑛−1𝑢,𝑓𝑛−1𝑢𝐺𝑓−𝜙𝑛−1𝑢,𝑓𝑛𝑢,𝑓𝑛𝑢≤⋯≤𝜓(𝐺(𝑢,𝑓𝑢,𝑓𝑢))−𝑛−1𝑘=0𝜙𝐺𝑓𝑛−𝑘−1𝑢,𝑓𝑛−𝑘𝑢,𝑓𝑛−𝑘𝑢.(3.26) Therefore, 𝑛−1𝑘=0𝜙𝐺𝑓𝑛−𝑘−1𝑢,𝑓𝑛−𝑘𝑢,𝑓𝑛−𝑘𝑢=0,(3.27) which implies that 𝜙(𝐺(𝑓𝑛−𝑘−1𝑢,𝑓𝑛−𝑘𝑢,𝑓𝑛−𝑘𝑢))=0, for all (0≤𝑘≤𝑛−1). Thus, 𝜙(𝐺(𝑢,𝑓𝑢,𝑓𝑢))=0, and by a property of 𝜙, we have 𝑢=𝑓𝑢. This is a contradiction.
Therefore, 𝑢∈𝐹(𝑓), and 𝑓 has property 𝑃. Let𝑀𝛼,𝛽(𝑥,𝑦,𝑧)=max𝐺(𝑥,𝑦,𝑧),𝐺(𝑥,𝑓𝑥,𝑓𝑥),𝐺(𝑦,𝑓𝑦,𝑓𝑦),𝐺(𝑧,𝑓𝑧,𝑓𝑧),𝛼𝐺(𝑓𝑥,𝑓𝑥,𝑦)+(1−𝛼)(𝐺(𝑓𝑦,𝑓𝑦,𝑧)),𝛽𝐺(𝑥,𝑓𝑥,𝑓𝑥)+(1−𝛽)(𝐺(𝑦,𝑓𝑦,𝑓𝑦)),,(3.28) where 𝛼,𝛽∈(0,1].

Example 3.7. Let 𝑋=[0,1] and 𝐺(𝑥,𝑦,𝑧)=max{|𝑥−𝑦|,|𝑦−𝑧|,|𝑧−𝑥|} be a G-metric on 𝑋. Define 𝑓∶𝑋→𝑋 by 𝑓(𝑥)=𝑥/8. We take 𝜓(𝑡)=𝑡 and 𝜙(𝑡)=7/8𝑡, for 𝑡∈[0,∞) and 𝛼,𝛽∈(0,1]. So that 𝜓𝑀𝛼,𝛽𝑀(𝑥,𝑦,𝑧)−𝜙𝛼,𝛽=1(𝑥,𝑦,𝑧)8𝑀𝛼,𝛽(𝑥,𝑦,𝑧).(3.29) We have |||𝑥𝐺(𝑓𝑥,𝑓𝑦,𝑓𝑧)=max8−𝑦8|||,|||𝑦8−𝑧8|||,|||𝑧8−𝑥8|||=18||||,||||=1max𝑥−𝑦𝑦−𝑧,|𝑧−𝑥|8≤1𝐺(𝑥,𝑦,𝑧)8𝑀𝛼,𝛽𝑀(𝑥,𝑦,𝑧)=𝜓𝛼,𝛽𝑀(𝑥,𝑦,𝑧)−𝜙𝛼,𝛽.(𝑥,𝑦,𝑧)(3.30)

4. Applications

Denote by Λ the set of functions 𝜆∶[0,∞)→[0,∞) satisfying the following hypotheses.(1)𝜆 is a Lebesgue integral mapping on each compact of [0,∞).(2)For every 𝜀>0, we have ∫𝑡0𝜆(𝑠)𝑑𝑠>0.It is an easy matter to see that the mapping 𝜓∶[0,∞)→[0,∞), defined by ∫𝜓(𝑡)=𝑡0𝜆(𝑠)𝑑𝑠, is an altering distance function. Now, we have the following result.

Theorem 4.1. Let (𝑋,𝐺) be a complete G-metric space. Let 𝑓 be a self-map on 𝑋 satisfying the following: 0𝐺(𝑓𝑥,𝑓𝑦,𝑓𝑧)𝜆(𝑠)𝑑𝑠≤𝑀𝛼,𝛽0(𝑥,𝑦,𝑧)𝜆(𝑠)𝑑𝑠−𝑀𝛼,𝛽0(𝑥,𝑦,𝑧)𝜇(𝑠)𝑑𝑠,(4.1) where 𝜆, 𝜇∈Λ and 𝛼,𝛽∈(0,1]. Then 𝑓 has a unique fixed point.

Proof. It follows from Theorem by taking ∫𝜓(𝑡)=𝑡0𝜆(𝑠)𝑑𝑠 and ∫𝜙(𝑡)=𝑡0𝜇(𝑠)𝑑𝑠.


The authors thank the referees for their valuable comments and suggestions.