Property P and Some Fixed Point Results on (𝜓,𝜙)-Weakly Contractive G-Metric Spaces

Abstract

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(𝑇^𝑛) 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,𝑎,𝑎)+𝐺(𝑎,𝑦,𝑧), for all 𝑥,𝑦,𝑧,𝑎∈𝑋 (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,𝑑_𝐺) by 𝑑_𝐺(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 (𝑥_𝑛) be a sequence of points in 𝑋. Then,(i)a point 𝑥∈𝑋 is said to be the limit of the sequence (𝑥_𝑛) if 𝐺𝑥_𝑛,𝑥_𝑚,𝑥⟶0,(as𝑛,𝑚⟶∞),(2.1) and we say that the sequence (𝑥_𝑛) is G convergent to 𝑥 (we say 𝑥→_𝑛(𝐺)𝑥),(ii)A sequence (𝑥_𝑛) is said to be G-Cauchy if 𝐺𝑥_𝑛,𝑥_𝑚,𝑥_𝑙⟶0,(as𝑛,𝑚,𝑙⟶∞),(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)𝑥→_𝑛(𝐺)𝑥, (2)𝐺(𝑥_𝑛,𝑥_𝑛,𝑥)→0(as𝑛→∞), (3)𝐺(𝑥_𝑛,𝑥,𝑥)→0(as𝑛→∞), (4)𝐺(𝑥_𝑛,𝑥_𝑚,𝑥)→0(as𝑛→∞).

Proposition 2.6. Let (𝑋,𝐺) be a G-metric space, then the following are equivalent:(1)(𝑥_𝑛) is be G-Cauchy in 𝑋,(2)𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑚)→0(as𝑛,𝑚→∞).

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=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,𝑎,𝑧)+𝐺(𝑎,𝑦,𝑧), (v)𝐺(𝑥,𝑦,𝑧)≤𝐺(𝑥,𝑎,𝑎)+𝐺(𝑦,𝑎,𝑎)+𝐺(𝑧,𝑎,𝑎).

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≤(𝑓−𝜓𝑥𝜙,𝑓m𝑦,ma𝑓xa𝑧x))𝐺𝐺(𝑥(,𝑥𝑦,𝛽𝛼,𝑦𝑧𝐺𝛽𝐺𝛼,)𝑧𝐺𝐺,(()𝑥𝑓𝐺,((,𝑥𝑥𝑓𝐺𝑓(,,𝑥𝑥𝑥𝑓𝑓(,,𝑥,𝑥𝑥𝑓𝑓,𝑓,,𝑥𝑥𝑓𝑥𝑓𝑦,,𝑥))𝑥𝑦𝑓,+)+)𝑥𝑓++)𝑥((11,)((𝐺11,−−𝐺−−(𝛼𝛽𝑦(),𝛼)𝛽𝑦𝑓((),)𝐺𝐺𝑦𝑓((𝐺𝐺,𝑦((𝑓,𝑓𝑦((𝑦𝑓,𝑦𝑓𝑦)𝑓𝑦,,𝑦,)𝑓𝑦𝑓,𝐺,,𝑦𝑓𝑦𝐺𝑓,,𝑦(𝑧𝑦𝑓𝑧,())𝑧,𝑦𝑧))𝑓)),,,))𝑧𝑓,,,𝑧𝑓,𝑧𝑓)𝑧,),,(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 {𝑥_𝑛} by 𝑥_𝑛+1=𝑓𝑥_𝑛=𝑓^𝑛𝑥₀. We may assume that 𝑥_𝑛≠𝑥_𝑛+1 for each 𝑛∈𝑁∪{0}. Since, if there exist 𝑛∈𝑁 such that 𝑥_𝑛=𝑥_𝑛+1, then 𝑥_𝑛 is a fixed point of 𝑓.
From (3.1), substituting 𝑥≡𝑥_𝑛−1, 𝑦=𝑧≡𝑥_𝑛 then, for all 𝑛∈𝑁, 𝜓𝐺≤𝑥𝜓_𝑛,⎛⎜⎜⎝𝑥max_𝑛+1,𝑥⎧⎪⎨⎪⎩𝐺_𝑛+1𝑥_𝑛−1,𝑥_𝑛,𝑥𝛼_𝑛𝐺,𝐺𝑥_𝑛𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝛽𝐺𝑥,_𝑛−1𝑥_𝑛,𝑥_𝑛+(1−𝛼)𝐺𝑥_𝑛+1,𝑥_𝑛+1,𝑥_𝑛,−𝜙⎛⎜⎜⎝max⎧⎪⎨⎪⎩𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛+(1−𝛽)𝐺𝑥_𝑛+1,𝑥_𝑛+1,𝑥_𝑛⎫⎪⎬⎪⎭⎞⎟⎟⎠,𝑥_𝑛,𝑥𝛼_𝑛𝐺,𝐺𝑥_𝑛𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝛽𝐺𝑥,_𝑛−1𝑥_𝑛,𝑥_𝑛+(1−𝛼)𝐺𝑥_𝑛+1,𝑥_𝑛+1,𝑥_𝑛,≤𝜓max𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛+(1−𝛽)𝐺𝑥_𝑛+1,𝑥_𝑛+1,𝑥_𝑛⎫⎪⎬⎪⎭⎞⎟⎟⎠𝛽𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,−𝜙max𝐺,𝑥𝑥_𝑛,𝑥_𝑛𝑛−1+(1−𝛽)𝐺𝑥_𝑛+1,𝑥_𝑛+1,𝑥_𝑛𝛽𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,,𝑥_𝑛,𝑥_𝑛+(1−𝛽)𝐺𝑥_𝑛+1,𝑥_𝑛+1,𝑥_𝑛.(3.2) Let 𝑀_𝑛=max{𝐺(𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛),𝐺(𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1)}. Then, (3.2) gives 𝜓𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1≤𝜓𝑀_𝑛−𝜙𝑀_𝑛.(3.3) We have two cases, either 𝑀_𝑛=𝐺(𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1) or 𝑀_𝑛=𝐺(𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛). Suppose that, for some 𝑛∈𝑁₀,𝑀_𝑛=𝐺(𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1). Then, we have 𝜓𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1≤𝜓𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1−𝜙𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1.(3.4) Therefore, 𝜓(𝐺(𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1))=0. Hence 𝑥_𝑛=𝑥_𝑛+1. This is a contradiction since the 𝑥_𝑛's are distinct.
Thus, 𝑀_𝑛=𝐺(𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1), and (3.2) becomes 𝜓𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1≤≤𝜓𝜓𝐺𝐺𝑥𝑥_{𝑛−1𝑛−1},𝑥_𝑛,𝑥_𝑛−𝜙𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,𝑥_𝑛,𝑥_𝑛.(3.5) But 𝜓 is an increasing function. Thus, from (3.5), we get 𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1≤𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,∀𝑛∈𝑁.(3.6) Therefore, {𝐺(𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1),𝑛∈𝑁∪{0}} is a positive nonincreasing sequence. Hence there exists 𝑟≥0 such that lim_𝑛→∞𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1=𝑟.(3.7) Letting 𝑛→∞, 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 lim_𝑛→∞𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1=0.(3.9) Consequently, for a given 𝜀>0, there is an integer 𝑛₀ such that 𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1<𝜀2,∀𝑛>𝑛⁰.(3.10) For 𝑚,𝑛∈𝑁 with 𝑚>𝑛, we claim that 𝐺𝑥_𝑛,𝑥_𝑚,𝑥_𝑚<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𝑥_𝑛,𝑥_𝑘,𝑥_𝑘<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𝑥_𝑛,𝑥_𝑘+1,𝑥_𝑘+1≤≤𝐺𝐺𝑥𝑥_𝑛𝑛,𝑥_𝑛+1,𝑥_𝑛+1+𝐺𝑥_𝑛+1,𝑥_𝑘+1,𝑥_𝑘+1,𝑥_𝑛+1,𝑥_𝑛+1+𝐺𝑥_𝑛,𝑥_𝑘,𝑥_𝑘<𝜀.(3.13) By induction on 𝑚, we conclude that 𝐺𝑥_𝑛,𝑥_𝑚,𝑥_𝑚<𝜀2,∀𝑚>𝑛>𝑛⁰.(3.14) We conclude from Proposition 2.6 that {𝑥_𝑛} is a G-Cauchy sequence in 𝑋. From the completeness of 𝑋, there exists 𝑢 in 𝑋 such that 𝑥→_𝑛(𝐺)𝑢. For 𝑛∈𝑁, we have 𝜓𝐺≤𝑓𝜓𝑢⎛⎜⎜⎝,m𝑓𝑢a,x𝑥_𝑛⎧⎪⎨⎪⎩𝐺=𝑢𝜓,𝑢,𝐺𝑥𝑓_𝑛−1𝑢,𝑓𝑢,𝑓𝑥_𝑛−1𝛼𝐺(𝑓𝑢,,𝐺𝑓(𝑢𝑢,,𝑢𝑓)𝑢+,𝑓(1𝑢)−,𝐺𝛼)(𝑢,𝐺𝑓𝑢𝑓,𝑓𝑢𝑢,𝑓),𝑢𝐺,𝑥𝑥_{𝑛−1𝑛−1},𝑥_𝑛,𝑥_𝑛,−𝜙⎛⎜⎜⎝max⎧⎪⎨⎪⎩𝐺𝑢,𝑢,𝑥𝛽𝐺_𝑛−1(𝑢,𝑓𝑢,𝑓𝑢)+(1−𝛽)(𝐺(𝑢,𝑓𝑢,𝑓𝑢)),⎫⎪⎬⎪⎭⎞⎟⎟⎠𝛼𝐺(𝑓𝑢,,𝐺𝑓(𝑢𝑢,,𝑢𝑓)𝑢+,𝑓(1𝑢)−,𝐺𝛼)(𝑢,𝐺𝑓𝑢𝑓,𝑓𝑢𝑢,𝑓),𝑢𝐺,𝑥𝑥_{𝑛−1𝑛−1},𝑥_𝑛,𝑥_𝑛,≤𝜓max𝐺𝑢,𝑢,𝑥𝛽_𝑛−1𝐺(𝑢,𝑓𝑢,𝑓𝑢)+(1−𝛽)(𝐺(𝑢,𝑓𝑢,𝑓𝑢)),⎫⎪⎬⎪⎭⎞⎟⎟⎠𝛼𝐺(𝑓𝑢,𝑓𝑢,𝑢,)𝐺+((𝑢1,𝑓−𝑢,𝛼𝑓)𝑢)𝐺,𝐺𝑓𝑢𝑥,𝑓𝑢_𝑛−1,,𝑥𝑥_𝑛,_𝑛−1𝑥_𝑛,−𝜙max𝐺𝑢,𝑢,𝑥_𝑛−1𝛼𝐺(𝑓𝑢,𝑓𝑢,𝑢,)𝐺+((𝑢1,𝑓−𝑢,𝛼𝑓)𝑢)𝐺,𝐺𝑓𝑢𝑥,𝑓𝑢_𝑛−1,,𝑥𝑥_𝑛,_𝑛−1𝑥_𝑛,.(3.15) Letting 𝑛→∞, 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))≤===𝜓𝜓𝜓𝜓−−(((𝜙𝜙𝐺m𝐺m(a((mx𝑢a𝑓mx,a𝑢𝑢{xa,,𝑓𝐺x𝑣{𝑢)(𝐺,𝐺)𝑢𝑓,−((𝑣𝐺𝑢𝑢𝑢),𝜙,,)(𝑣𝑢𝑢𝑢(),,𝛼𝐺,𝛽,𝑣𝑣𝑢𝐺𝐺())(,𝛼1𝛽𝑢,,𝑣(𝐺(𝐺(,−𝑓𝐺)𝑢1𝑢𝑢,,((,𝛼(−𝑓,𝐺𝑓𝑢𝑢𝑣𝑓)𝑢𝑢,,)𝛼(𝐺𝑢,𝑓,)𝑓𝑢𝑓,)𝑓.𝑢𝑢,(𝑢𝐺𝑢𝑢,𝑓,𝑢),𝑓)𝑓,𝑢(𝑢+𝑢+𝑢𝑢,𝑢)),𝑓),(+𝑣(+,𝑢1𝑢1)𝐺,)}(−𝑣(−,1)1()𝐺𝛼𝑢}−𝛽−,))()𝑓𝛼𝑢(𝛽(𝐺𝑢,𝐺))𝑓,(((𝑓𝐺𝑢(𝐺𝑓𝑣𝑢,𝑢,(𝑓)(𝑓,𝑓𝑣,𝑢𝑓𝑢𝐺𝑣,)𝑢,𝑓,,𝑓,(𝑓𝐺𝑣𝑣𝑣𝑢𝑣,),,(𝑓))𝑓𝑣𝑣)𝑣,𝑣),))𝑓,),𝑓𝑣𝑣,𝑓),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 {𝑥_𝑛} be a sequence in 𝑋 with limit 𝑢(i.e.,𝑥→_𝑛(𝐺)𝑢). Using (3.1), we have 𝜓𝐺𝑓𝑥_𝑛,𝑢,𝑢≤=𝜓𝜓⎛⎜⎜⎝m𝐺ax𝑓𝑥⎧⎪⎨⎪⎩_𝑛𝐺,𝑓𝑢𝑥,𝑓_𝑛𝑢,𝑢𝛼,𝑢𝐺,𝑓𝐺𝑥_𝑛𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝛽𝐺𝑥_𝑛,𝑓𝑥_𝑛,𝑢+(1−𝛼)(𝐺(𝑓𝑢,𝑓𝑢,𝑢)),−𝜙⎛⎜⎜⎝max⎧⎪⎨⎪⎩𝐺𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑥_𝑛+(1−𝛽)(𝐺(𝑢,𝑓𝑢,𝑓𝑢))⎫⎪⎬⎪⎭⎞⎟⎟⎠,𝑢𝛼,𝑢𝐺,𝑓𝐺𝑥_𝑛𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝛽𝐺𝑥_𝑛,𝑓𝑥_𝑛,𝑢+(1−𝛼)(𝐺(𝑓𝑢,𝑓𝑢,𝑢)),=𝜓max𝐺𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑥_𝑛+(1−𝛽)(𝐺(𝑢,𝑓𝑢,𝑓𝑢))⎫⎪⎬⎪⎭⎞⎟⎟⎠−𝜙max𝐺,𝑥𝑢_𝑛,𝑢,𝛼𝐺𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝑢,𝛽𝐺𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑥_𝑛≤𝜓max𝐺𝑥_𝑛,𝑢,𝑢,𝛼𝐺𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝑢,𝛽𝐺𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑥_𝑛≤𝜓max𝐺𝑥_𝑛,𝑢,𝑢,𝛼𝐺𝑥_𝑛+1,𝑥_𝑛+1,𝑢,𝛽𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1≤𝜓max𝐺𝑥_𝑛,𝑢,𝑢,𝛼𝐺𝑥_𝑛+1,𝑥_𝑛+1,𝑢,𝛽𝐺𝑥_𝑛,𝑢,𝑢+𝛽𝐺𝑢,𝑥_𝑛+1,𝑥_𝑛+1,𝑢,𝑢,𝐺𝑥_𝑛+1,𝑥_𝑛+1,𝑢,𝛽𝐺𝑥_𝑛,𝑢,𝑢+𝛽𝐺𝑢,𝑥_𝑛+1,𝑥_𝑛+1.(3.18) But 𝜓 is an increasing function, thus from (3.18), we get 𝐺𝑓𝑥_𝑛,𝑢,𝑢≤max𝐺𝑥_𝑛,𝑢,𝑢,𝐺𝑥_𝑛+1,𝑥_𝑛+1,𝑢,𝛽𝐺𝑥_𝑛+1,𝑢,𝑢+𝛽𝐺𝑢,𝑥_𝑛+1,𝑥_𝑛+1.(3.19) Therefore, lim_𝑛→∞𝐺(𝑓𝑥_𝑛,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)≤𝜆max𝐺(𝑥,𝑦𝛼,𝛽𝑧𝐺𝐺),((𝑓𝐺𝑥𝑥,(,𝑓𝑥𝑓𝑥,𝑥𝑓,,𝑓𝑥𝑦𝑥,)𝑓)+𝑥+)((1,1𝐺−−(𝛼𝑦𝛽),)𝑓((𝐺𝑦𝐺,(𝑓(𝑓𝑦𝑦𝑦,),𝑓,𝑓𝑦𝐺𝑦,,𝑓(𝑧𝑧𝑦),))𝑓),𝑧,𝑓𝑧),,(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: 𝐺(𝑓𝑥,𝑓𝑦,𝑓𝑧)≤𝜆max⎧⎪⎨⎪⎩𝐺(𝑥,𝑦,𝑧)12,1𝐺(𝐺(𝑥(𝑓,𝑥𝑓,𝑥𝑓,𝑥𝑓,𝑥𝑦)),+𝐺((𝐺𝑦,(𝑓𝑓𝑦𝑦,,𝑓𝑓𝑦𝑦),,𝑧𝐺))()𝑧,,𝑓𝑧,𝑓𝑧),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: 𝐺(𝑓𝑥,𝑓𝑦,𝑓𝑧)≤𝜆max⎧⎪⎪⎨⎪⎪⎩𝐺(𝑥,𝑦,𝑧),1𝐺(𝑥,𝑓𝑥,𝑓𝑥),𝐺(𝑦,𝑓𝑦,𝑓𝑦),𝐺(𝑧,𝑓𝑧,𝑓𝑧),31𝐺(𝑓𝑥,𝑓𝑥,𝑦)+32𝐺(𝑓𝑦,𝑓𝑦,𝑧),3𝐺(𝑓𝑥,𝑓𝑥,𝑥)+23𝐺(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(𝑓^𝑛)≠𝜑 for each 𝑛∈𝑁. Fix 𝑛>1, and assume that 𝑢∈𝐹(𝑓^𝑛). We claim that 𝑢∈𝐹(𝑓). To prove the claim, suppose that 𝑢≠𝑓𝑢. Using (3.1), we have 𝜓(𝐺(𝑢,𝑓𝑢,𝑓𝑢))≤=𝜓𝜓⎛⎜⎜⎝m𝐺ax𝑓^𝑛⎧⎪⎨⎪⎩𝑢,𝑓^𝑛+1𝐺𝑢,𝑓𝑓^𝑛+1_𝑛−1𝑢=𝜓𝐺𝑓𝑓^𝑛−1𝑢,𝑓𝑓^𝑛𝑢,𝑓𝑓^𝑛𝑢𝛽𝐺𝛼𝐺𝑓(𝑢,_𝑛−1𝑢,𝑢)𝑢+,𝑢(,1𝑢−,𝛼𝐺)((𝐺𝑢,(𝑓𝑓𝑢𝑢,,𝑓𝑓𝑢𝑢),,𝑢))−𝜙⎛⎜⎜⎝max⎧⎪⎨⎪⎩𝐺𝑢,𝑓𝑢,𝑢_𝑛−1+(1−𝛽)(𝐺(𝑢,𝑓𝑢,𝑓𝑢))⎫⎪⎬⎪⎭⎞⎟⎟⎠𝛽𝐺𝛼𝐺𝑓(𝑢,_𝑛−1𝑢,𝑢)𝑢+,𝑢(,1𝑢−,𝛼𝐺)((𝐺𝑢,(𝑓𝑓𝑢𝑢,,𝑓𝑓𝑢𝑢),,𝑢))=𝜓max𝐺𝑓_𝑛−1𝑢,𝑢,𝑢+(1−𝛽)(𝐺(𝑢,𝑓𝑢,𝑓𝑢))⎫⎪⎬⎪⎭⎞⎟⎟⎠−𝜙max𝐺𝑓𝑢,_𝑛−1𝑢,𝑢,𝐺(𝑢,𝑓𝑢,𝑓𝑢)𝑢,𝑢,𝑢,𝐺(𝑢,𝑓𝑢,𝑓𝑢).(3.23) Letting 𝑀=max{𝐺(𝑓^𝑛−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(𝑓^𝑛−1𝑢,𝑢,𝑢), then (3.1) gives that 𝜓𝐺𝑓^𝑛𝑢,𝑓^𝑛+1𝑢,𝑓^𝑛+1𝑢≤=𝜓𝜓(𝐺𝐺(𝑢𝑓,𝑓𝑢_𝑛−1,𝑓𝑢))=𝜓𝐺𝑓_𝑛−1𝑢,𝑢,𝑢−𝜙𝐺𝑓^𝑛−1𝑢,𝑢,𝑢≤𝜓𝐺𝑓_𝑛−2𝑢,𝑓^𝑛𝑢,𝑓^𝑛𝑢−𝜙𝐺𝑓^𝑛−1𝑢,𝑓^𝑛𝑢,𝑓^𝑛𝑢−𝜙𝐺𝑓𝑢,_𝑛−2𝑓^𝑛−1𝑢,𝑓^𝑛−1𝑢≤⋯≤𝜓(𝐺(𝑢,𝑢𝑓,𝑢𝑓,𝑓𝑢^𝑛−1)𝑢),−𝑓^𝑛−1_𝑛−1𝑢−𝜙𝐺𝑓^𝑛−1𝑢,𝑓^𝑛𝑢,𝑓^𝑛𝑢_𝑘=0𝜙𝐺𝑓^{𝑛−𝑘−1}𝑢,𝑓^𝑛−𝑘𝑢,𝑓^𝑛−𝑘𝑢.(3.26) Therefore, ^𝑛−1_𝑘=0𝜙𝐺𝑓^{𝑛−𝑘−1}𝑢,𝑓^𝑛−𝑘𝑢,𝑓^𝑛−𝑘𝑢=0,(3.27) which implies that 𝜙(𝐺(𝑓^{𝑛−𝑘−1}𝑢,𝑓^𝑛−𝑘𝑢,𝑓^𝑛−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. Let𝑀_𝛼,𝛽(𝑥,𝑦,𝑧)=max𝐺(𝑥,𝑦𝛽𝛼,𝑧𝐺𝐺),((𝑥𝑓𝐺,𝑥𝑓(,𝑥𝑥𝑓,,𝑥𝑓𝑓,𝑥𝑥𝑦,))𝑓++𝑥)((11,𝐺−−(𝛼𝛽𝑦),)𝑓((𝐺𝐺𝑦,((𝑓𝑓𝑦𝑦,𝑦)𝑓,,𝑓𝑦𝐺,𝑦𝑓,(𝑧𝑦𝑧)),))𝑓,,𝑧,𝑓𝑧),,(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𝑀_𝛼,𝛽(𝑥,𝑦,𝑧)−𝜙𝑀_𝛼,𝛽(𝑥,𝑦,𝑧)=18𝑀^𝛼,𝛽(𝑥,𝑦,𝑧).(3.29) We have 𝐺(𝑓𝑥,𝑓𝑦,𝑓𝑧)=max|||𝑥=18−8𝑦|||,|||8𝑦−8𝑧|||,|||8𝑧−𝑥8|||=81max||𝑥−𝑦||,||𝑦−𝑧||,|𝑧−𝑥|≤81𝐺(𝑥,𝑦,𝑧)8𝑀^𝛼,𝛽=𝜓𝑀(_𝛼,𝛽𝑥,𝑦,𝑧)(𝑥,𝑦,𝑧)−𝜙𝑀_𝛼,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.

Acknowledgment

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