International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 675094 | https://doi.org/10.1155/2012/675094

Mona Khandaqji, Sharifa Al-Sharif, Mohammad Al-Khaleel, "Property P and Some Fixed Point Results on (𝜓,𝜙)-Weakly Contractive G-Metric Spaces", International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 675094, 11 pages, 2012. https://doi.org/10.1155/2012/675094

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

Academic Editor: Charles Chidume
Received22 Mar 2012
Accepted06 Jun 2012
Published11 Jul 2012

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 [416]. 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 𝑛𝑁. However, the converse is false. For example, consider 𝑋=[0,1] and 𝑇 defined by 𝑇𝑥=1𝑥. 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 [1719]. 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 𝐺𝑋×𝑋×𝑋𝑅+ 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,𝑦𝑋, where 𝑥𝑦,(𝐺3)𝐺(𝑥,𝑥,𝑦)𝐺(𝑥,𝑦,𝑧), for all 𝑥,𝑦,𝑧𝑋, with 𝑧𝑦,(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,𝑎𝑋 (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,𝑦𝑋.

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,𝑦𝑋.

Definition 2.4. Let (𝑋,𝐺) be a G-metric space, and (𝑥𝑛) be a sequence of points in 𝑋. Then,(i)a point 𝑥𝑋 is said to be the limit of the sequence (𝑥𝑛) if 𝐺𝑥𝑛,𝑥𝑚,𝑥0,(as𝑛,𝑚),(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 𝐺𝑥𝑛,𝑥𝑚,𝑥𝑙0,(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)𝑥𝑛(𝐺)𝑥, (2)𝐺(𝑥𝑛,𝑥𝑛,𝑥)0(as𝑛), (3)𝐺(𝑥𝑛,𝑥,𝑥)0(as𝑛), (4)𝐺(𝑥𝑛,𝑥𝑚,𝑥)0(as𝑛).

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,𝑚).

Proposition 2.7. Let (𝑋,𝐺) be a G-metric space. Then, for any 𝑥,𝑦,𝑧,𝑎𝑋, 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(𝑓𝑥,𝑓𝑦,𝑓𝑧))𝜓max𝐺(𝑥,𝑦,𝑧),𝐺(𝑥,𝑓𝑥,𝑓𝑥),𝐺(𝑦,𝑓𝑦,𝑓𝑦),𝐺(𝑧,𝑓𝑧,𝑓𝑧),𝛼𝐺(𝑓𝑥,𝑓𝑥,𝑦)+(1𝛼)(𝐺(𝑓𝑦,𝑓𝑦,𝑧)),𝛽𝐺(𝑥,𝑓𝑥,𝑓𝑥)+(1𝛽)(𝐺(𝑦,𝑓𝑦,𝑓𝑦)),𝜙max𝐺(𝑥,𝑦,𝑧),𝐺(𝑥,𝑓𝑥,𝑓𝑥),𝐺(𝑦,𝑓𝑦,𝑓𝑦),𝐺(𝑧,𝑓𝑧,𝑓𝑧),𝛼𝐺(𝑓𝑥,𝑓𝑥,𝑦)+(1𝛼)(𝐺(𝑓𝑦,𝑓𝑦,𝑧)),𝛽𝐺(𝑥,𝑓𝑥,𝑓𝑥)+(1𝛽)(𝐺(𝑦,𝑓𝑦,𝑓𝑦)),(3.1) for all 𝑥,𝑦,𝑧𝑋, 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 𝑥0𝑋. Then construct a sequence {𝑥𝑛} by 𝑥𝑛+1=𝑓𝑥𝑛=𝑓𝑛𝑥0. We may assume that 𝑥𝑛𝑥𝑛+1 for each 𝑛𝑁{0}. Since, if there exist 𝑛𝑁 such that 𝑥𝑛=𝑥𝑛+1, then 𝑥𝑛 is a fixed point of 𝑓.
From (3.1), substituting 𝑥𝑥𝑛1, 𝑦=𝑧𝑥𝑛 then, for all 𝑛𝑁, 𝜓𝐺𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1𝐺𝑥𝜓max𝑛1,𝑥𝑛,𝑥𝑛𝑥,𝐺𝑛1,𝑥𝑛,𝑥𝑛𝑥,𝐺𝑛,𝑥𝑛+1,𝑥𝑛+1𝑥,𝐺𝑛,𝑥𝑛+1,𝑥𝑛+1,𝑥𝛼𝐺𝑛,𝑥𝑛,𝑥𝑛𝐺𝑥+(1𝛼)𝑛+1,𝑥𝑛+1,𝑥𝑛,𝑥𝛽𝐺𝑛1,𝑥𝑛,𝑥𝑛𝐺𝑥+(1𝛽)𝑛+1,𝑥𝑛+1,𝑥𝑛𝐺𝑥𝜙max𝑛1,𝑥𝑛,𝑥𝑛𝑥,𝐺𝑛1,𝑥𝑛,𝑥𝑛𝑥,𝐺𝑛,𝑥𝑛+1,𝑥𝑛+1𝑥,𝐺𝑛,𝑥𝑛+1,𝑥𝑛+1,𝑥𝛼𝐺𝑛,𝑥𝑛,𝑥𝑛𝐺𝑥+(1𝛼)𝑛+1,𝑥𝑛+1,𝑥𝑛,𝑥𝛽𝐺𝑛1,𝑥𝑛,𝑥𝑛𝐺𝑥+(1𝛽)𝑛+1,𝑥𝑛+1,𝑥𝑛𝐺𝑥𝜓max𝑛1,𝑥𝑛,𝑥𝑛𝑥,𝐺𝑛,𝑥𝑛+1,𝑥𝑛+1,𝑥𝛽𝐺𝑛1,𝑥𝑛,𝑥𝑛𝐺𝑥+(1𝛽)𝑛+1,𝑥𝑛+1,𝑥𝑛𝐺𝑥𝜙max𝑛1,𝑥𝑛,𝑥𝑛𝑥,𝐺𝑛,𝑥𝑛+1,𝑥𝑛+1,𝑥𝛽𝐺𝑛1,𝑥𝑛,𝑥𝑛𝐺𝑥+(1𝛽)𝑛+1,𝑥𝑛+1,𝑥𝑛.(3.2) Let 𝑀𝑛=max{𝐺(𝑥𝑛1,𝑥𝑛,𝑥𝑛),𝐺(𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1)}. Then, (3.2) gives 𝜓𝐺𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1𝑀𝜓𝑛𝑀𝜙𝑛.(3.3) We have two cases, either 𝑀𝑛=𝐺(𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1) or 𝑀𝑛=𝐺(𝑥𝑛1,𝑥𝑛,𝑥𝑛). Suppose that, for some 𝑛𝑁0,𝑀𝑛=𝐺(𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1). Then, we have 𝜓𝐺𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1𝐺𝑥𝜓𝑛,𝑥𝑛+1,𝑥𝑛+1𝐺𝑥𝜙𝑛,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 𝜓𝐺𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1𝐺𝑥𝜓𝑛1,𝑥𝑛,𝑥𝑛𝐺𝑥𝜙𝑛1,𝑥𝑛,𝑥𝑛𝐺𝑥𝜓𝑛1,𝑥𝑛,𝑥𝑛.(3.5) But 𝜓 is an increasing function. Thus, from (3.5), we get 𝐺𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1𝑥𝐺𝑛1,𝑥𝑛,𝑥𝑛,𝑛𝑁.(3.6) Therefore, {𝐺(𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1),𝑛𝑁{0}} is a positive nonincreasing sequence. Hence there exists 𝑟0 such that lim𝑛𝐺𝑥𝑛,𝑥𝑛+1,𝑥𝑛+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 lim𝑛𝐺𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1=0.(3.9) Consequently, for a given 𝜀>0, there is an integer 𝑛0 such that 𝐺𝑥𝑛,𝑥𝑛+1,𝑥𝑛+1<𝜀2,𝑛>𝑛0.(3.10) For 𝑚,𝑛𝑁 with 𝑚>𝑛, we claim that 𝐺𝑥𝑛,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, 𝐺𝑥𝑛,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 𝐺𝑥𝑛,𝑥𝑘+1,𝑥𝑘+1𝑥𝐺𝑛,𝑥𝑛+1,𝑥𝑛+1𝑥+𝐺𝑛+1,𝑥𝑘+1,𝑥𝑘+1𝑥𝐺𝑛,𝑥𝑛+1,𝑥𝑛+1𝑥+𝐺𝑛,𝑥𝑘,𝑥𝑘<𝜀.(3.13) By induction on 𝑚, we conclude that 𝐺𝑥𝑛,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 𝑛𝑁, we have 𝜓𝐺𝑓𝑢,𝑓𝑢,𝑥𝑛𝐺=𝜓𝑓𝑢,𝑓𝑢,𝑓𝑥𝑛1𝐺𝜓max𝑢,𝑢,𝑥𝑛1𝑥,𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺𝑛1,𝑥𝑛,𝑥𝑛,𝐺𝛼𝐺(𝑓𝑢,𝑓𝑢,𝑢)+(1𝛼)𝑓𝑢,𝑓𝑢,𝑥𝑛1,𝐺𝛽𝐺(𝑢,𝑓𝑢,𝑓𝑢)+(1𝛽)(𝐺(𝑢,𝑓𝑢,𝑓𝑢))𝜙max𝑢,𝑢,𝑥𝑛1𝑥,𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺𝑛1,𝑥𝑛,𝑥𝑛,𝐺𝛼𝐺(𝑓𝑢,𝑓𝑢,𝑢)+(1𝛼)𝑓𝑢,𝑓𝑢,𝑥𝑛1,𝐺𝛽𝐺(𝑢,𝑓𝑢,𝑓𝑢)+(1𝛽)(𝐺(𝑢,𝑓𝑢,𝑓𝑢))𝜓max𝑢,𝑢,𝑥𝑛1𝑥,𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺𝑛1,𝑥𝑛,𝑥𝑛,𝐺𝛼𝐺(𝑓𝑢,𝑓𝑢,𝑢)+(1𝛼)𝑓𝑢,𝑓𝑢,𝑥𝑛1𝐺𝜙max𝑢,𝑢,𝑥𝑛1𝑥,𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺𝑛1,𝑥𝑛,𝑥𝑛,𝐺𝛼𝐺(𝑓𝑢,𝑓𝑢,𝑢)+(1𝛼)𝑓𝑢,𝑓𝑢,𝑥𝑛1.(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 𝑣𝑢. 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)+(1𝛼)(𝐺(𝑓𝑢,𝑓𝑢,𝑣)),𝛽𝐺(𝑢,𝑓𝑢,𝑓𝑢)+(1𝛽)(𝐺(𝑣,𝑓𝑣,𝑓𝑣))𝜙max𝐺(𝑢,𝑢,𝑣),𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺(𝑣,𝑓𝑣,𝑓𝑣),𝛼𝐺(𝑓𝑢,𝑓𝑢,𝑢)+(1𝛼)(𝐺(𝑓𝑢,𝑓𝑢,𝑣)),𝛽𝐺(𝑢,𝑓𝑢,𝑓𝑢)+(1𝛽)(𝐺(𝑣,𝑓𝑣,𝑓𝑣))=𝜓(max{𝐺(𝑢,𝑢,𝑣),(1𝛼)𝐺(𝑢,𝑢,𝑣)})𝜙(max{𝐺(𝑢,𝑢,𝑣),(1𝛼)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 𝜓𝐺𝑓𝑥𝑛𝐺,𝑢,𝑢=𝜓𝑓𝑥𝑛𝐺𝑥,𝑓𝑢,𝑓𝑢𝜓max𝑛𝑥,𝑢,𝑢,𝐺𝑛,𝑓𝑥𝑛,𝑓𝑥𝑛,𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝛼𝐺𝑓𝑥𝑛,𝑓𝑥𝑛𝑥,𝑢+(1𝛼)(𝐺(𝑓𝑢,𝑓𝑢,𝑢)),𝛽𝐺𝑛,𝑓𝑥𝑛,𝑓𝑥𝑛𝐺𝑥+(1𝛽)(𝐺(𝑢,𝑓𝑢,𝑓𝑢))𝜙max𝑛𝑥,𝑢,𝑢,𝐺𝑛,𝑓𝑥𝑛,𝑓𝑥𝑛,𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝛼𝐺𝑓𝑥𝑛,𝑓𝑥𝑛𝑥,𝑢+(1𝛼)(𝐺(𝑓𝑢,𝑓𝑢,𝑢)),𝛽𝐺𝑛,𝑓𝑥𝑛,𝑓𝑥𝑛𝐺𝑥+(1𝛽)(𝐺(𝑢,𝑓𝑢,𝑓𝑢))=𝜓max𝑛,𝑢,𝑢,𝛼𝐺𝑓𝑥𝑛,𝑓𝑥𝑛𝑥,𝑢,𝛽𝐺𝑛,𝑓𝑥𝑛,𝑓𝑥𝑛𝐺𝑥𝜙max𝑛,𝑢,𝑢,𝛼𝐺𝑓𝑥𝑛,𝑓𝑥𝑛𝑥,𝑢,𝛽𝐺𝑛,𝑓𝑥𝑛,𝑓𝑥𝑛𝐺𝑥𝜓max𝑛𝑥,𝑢,𝑢,𝛼𝐺𝑛+1,𝑥𝑛+1𝑥,𝑢,𝛽𝐺𝑛,𝑥𝑛+1,𝑥𝑛+1𝐺𝑥𝜓max𝑛𝑥,𝑢,𝑢,𝛼𝐺𝑛+1,𝑥𝑛+1𝑥,𝑢,𝛽𝐺𝑛,𝑢,𝑢+𝛽𝐺𝑢,𝑥𝑛+1,𝑥𝑛+1𝐺𝑥𝜓max𝑛𝑥,𝑢,𝑢,𝐺𝑛+1,𝑥𝑛+1𝑥,𝑢,𝛽𝐺𝑛,𝑢,𝑢+𝛽𝐺𝑢,𝑥𝑛+1,𝑥𝑛+1.(3.18) But 𝜓 is an increasing function, thus from (3.18), we get 𝐺𝑓𝑥𝑛𝐺𝑥,𝑢,𝑢max𝑛𝑥,𝑢,𝑢,𝐺𝑛+1,𝑥𝑛+1𝑥,𝑢,𝛽𝐺𝑛+1,𝑢,𝑢+𝛽𝐺𝑢,𝑥𝑛+1,𝑥𝑛+1.(3.19) Therefore, lim𝑛𝐺(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,𝑧𝑋𝐺(𝑓𝑥,𝑓𝑦,𝑓𝑧)𝜆max𝐺(𝑥,𝑦,𝑧),𝐺(𝑥,𝑓𝑥,𝑓𝑥),𝐺(𝑦,𝑓𝑦,𝑓𝑦),𝐺(𝑧,𝑓𝑧,𝑓𝑧),𝛼𝐺(𝑓𝑥,𝑓𝑥,𝑦)+(1𝛼)(𝐺(𝑓𝑦,𝑓𝑦,𝑧)),𝛽𝐺(𝑥,𝑓𝑥,𝑓𝑥)+(1𝛽)(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)=𝑡𝜆𝑡, 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,𝑧𝑋 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)=𝑡𝜆𝑡, 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)+3𝐺(𝑓𝑦,𝑓𝑦,𝑦),(3.22) for all 𝑥,𝑦,𝑧𝑋, 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)=𝑡𝜆𝑡, 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 𝑛𝑁. Fix 𝑛>1, and assume that 𝑢𝐹(𝑓𝑛). We claim that 𝑢𝐹(𝑓). To prove the claim, suppose that 𝑢𝑓𝑢. Using (3.1), we have 𝜓𝐺𝑓(𝐺(𝑢,𝑓𝑢,𝑓𝑢))=𝜓𝑛𝑢,𝑓𝑛+1𝑢,𝑓𝑛+1𝑢𝐺=𝜓𝑓𝑓𝑛1𝑢,𝑓𝑓𝑛𝑢,𝑓𝑓𝑛𝑢𝐺𝑓𝜓max𝑛1𝑓𝑢,𝑢,𝑢,𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝛼𝐺(𝑢,𝑢,𝑢)+(1𝛼)(𝐺(𝑓𝑢,𝑓𝑢,𝑢))𝛽𝐺𝑛1𝐺𝑓𝑢,𝑢,𝑢+(1𝛽)(𝐺(𝑢,𝑓𝑢,𝑓𝑢))𝜙max𝑛1𝑓𝑢,𝑢,𝑢,𝐺(𝑢,𝑓𝑢,𝑓𝑢),𝛼𝐺(𝑢,𝑢,𝑢)+(1𝛼)(𝐺(𝑓𝑢,𝑓𝑢,𝑢))𝛽𝐺𝑛1𝐺𝑓𝑢,𝑢,𝑢+(1𝛽)(𝐺(𝑢,𝑓𝑢,𝑓𝑢))=𝜓max𝑛1𝐺𝑓𝑢,𝑢,𝑢,𝐺(𝑢,𝑓𝑢,𝑓𝑢)𝜙max𝑛1.𝑢,𝑢,𝑢,𝐺(𝑢,𝑓𝑢,𝑓𝑢)(3.23) Letting 𝑀=max{𝐺(𝑓𝑛1𝑢,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(𝑓𝑛1𝑢,𝑢,𝑢), then (3.1) gives that 𝜓𝐺𝑓𝑛𝑢,𝑓𝑛+1𝑢,𝑓𝑛+1𝑢𝐺𝑓=𝜓(𝐺(𝑢,𝑓𝑢,𝑓𝑢))𝜓𝑛1𝐺𝑓𝑢,𝑢,𝑢𝜙𝑛1𝐺𝑓𝑢,𝑢,𝑢=𝜓𝑛1𝑢,𝑓𝑛𝑢,𝑓𝑛𝑢𝐺𝑓𝜙𝑛1𝑢,𝑓𝑛𝑢,𝑓𝑛𝑢𝐺𝑓𝜓𝑛2𝑢,𝑓𝑛1𝑢,𝑓𝑛1𝑢𝐺𝑓𝜙𝑛2𝑢,𝑓𝑛1𝑢,𝑓𝑛1𝑢𝐺𝑓𝜙𝑛1𝑢,𝑓𝑛𝑢,𝑓𝑛𝑢𝜓(𝐺(𝑢,𝑓𝑢,𝑓𝑢))𝑛1𝑘=0𝜙𝐺𝑓𝑛𝑘1𝑢,𝑓𝑛𝑘𝑢,𝑓𝑛𝑘𝑢.(3.26) Therefore, 𝑛1𝑘=0𝜙𝐺𝑓𝑛𝑘1𝑢,𝑓𝑛𝑘𝑢,𝑓𝑛𝑘𝑢=0,(3.27) which implies that 𝜙(𝐺(𝑓𝑛𝑘1𝑢,𝑓𝑛𝑘𝑢,𝑓𝑛𝑘𝑢))=0, for all (0𝑘𝑛1). Thus, 𝜙(𝐺(𝑢,𝑓𝑢,𝑓𝑢))=0, and by a property of 𝜙, we have 𝑢=𝑓𝑢. This is a contradiction.
Therefore, 𝑢𝐹(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)+(1𝛼)(𝐺(𝑓𝑦,𝑓𝑦,𝑧)),𝛽𝐺(𝑥,𝑓𝑥,𝑓𝑥)+(1𝛽)(𝐺(𝑦,𝑓𝑦,𝑓𝑦)),,(3.28) where 𝛼,𝛽(0,1].

Example 3.7. Let 𝑋=[0,1] and 𝐺(𝑥,𝑦,𝑧)=max{|𝑥𝑦|,|𝑦𝑧|,|𝑧𝑥|} be a G-metric on 𝑋. Define 𝑓𝑋𝑋 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 𝜓𝑀𝛼,𝛽𝑀(𝑥,𝑦,𝑧)𝜙𝛼,𝛽=1(𝑥,𝑦,𝑧)8𝑀𝛼,𝛽(𝑥,𝑦,𝑧).(3.29) We have |||𝑥𝐺(𝑓𝑥,𝑓𝑦,𝑓𝑧)=max8𝑦8|||,|||𝑦8𝑧8|||,|||𝑧8𝑥8|||=18||||,||||=1max𝑥𝑦𝑦𝑧,|𝑧𝑥|81𝐺(𝑥,𝑦,𝑧)8𝑀𝛼,𝛽𝑀(𝑥,𝑦,𝑧)=𝜓𝛼,𝛽𝑀(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)𝑑𝑠𝑀𝛼,𝛽0(𝑥,𝑦,𝑧)𝜆(𝑠)𝑑𝑠𝑀𝛼,𝛽0(𝑥,𝑦,𝑧)𝜇(𝑠)𝑑𝑠,(4.1) where 𝜆, 𝜇Λ 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.

References

  1. S. Gähler, “G-metrische Räume und ihre topologische Struktur,” Mathematische Nachrichten, vol. 26, pp. 115–148, 1963. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  2. S. Gähler, “Zur geometric 2-metriche raume,” Revue Roumaine de Mathématiques Pures et Appliquées, vol. 40, pp. 664–669, 1966. View at: Google Scholar
  3. Z. Mustafa and B. Sims, “A new A pproach to Generalize Metric Spaces,” Journal of Nonlinear and Convex Analysis, vol. 7, no. 2, pp. 289–297, 2006. View at: Google Scholar
  4. Z. Mustafa, M. Khandagji, and W. Shatanawi, “Fixed point results on complete G-metric spaces,” Studia Scientiarum Mathematicarum Hungarica, vol. 48, no. 3, pp. 304–319, 2011. View at: Publisher Site | Google Scholar
  5. Z. Mustafa and B. Sims, “Fixed point theorems for contractive mappings in complete G-metric spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 917175, 10 pages, 2009. View at: Publisher Site | Google Scholar
  6. W. Shatanawi, “Fixed point theory for contractive mappings satisfying Φ-maps in G-metric spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 181650, 9 pages, 2010. View at: Publisher Site | Google Scholar
  7. H. Aydi, B. Damjanović, B. Samet, and W. Shatanawi, “Coupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces,” Mathematical and Computer Modelling, vol. 54, no. 9-10, pp. 2443–2450, 2011. View at: Publisher Site | Google Scholar
  8. B. S. Choudhury, “Unique fixed point theorem for weakly C-contractive mappings,” Kathmandu University Journal of Science, Engineering and Technology, vol. 5, no. 1, pp. 6–13, 2009. View at: Google Scholar
  9. W. Shatanawi, “Fixed point theorems for nonlinear weakly C-contractive mappings in metric spaces,” Mathematical and Computer Modelling, vol. 54, no. 11-12, pp. 2816–2826, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  10. M. Abbas, T. Nazir, and S. Radenović, “Common fixed points of four maps in partially ordered metric spaces,” Applied Mathematics Letters, vol. 24, no. 9, pp. 1520–1526, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  11. M. Abbas, T. Nazir, and S. Radenović, “Some periodic point results in generalized metric spaces,” Applied Mathematics and Computation, vol. 217, no. 8, pp. 4094–4099, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  12. Z. Mustafa, H. Aydi, and E. Karapinar, “On common fixed points in G-metric spaces using (E, A) property,” Computers & Mathematics with Applications. In press. View at: Publisher Site | Google Scholar
  13. N. Tahat, H. Aydi, E. Karapinar, and W. Shatanawi, “Common fixed point for single-valued and multi-valued maps satisfying a generalized contraction in G-metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 48, 2012. View at: Publisher Site | Google Scholar
  14. M. Abbas, T. Nazir, and D. Đorić, “Common fixed point of mappings satisfying (E, A) property in generalized metric spaces,” Applied Mathematics and Computation, vol. 218, no. 14, pp. 7665–7670, 2012. View at: Publisher Site | Google Scholar
  15. M. Abbas, S. H. Khan, and T. Nazir, “Common fixed points of R-weakly commuting maps in generalized metric space,” Fixed Point Theory and Applications, vol. 2011, article 41, 2011. View at: Publisher Site | Google Scholar
  16. L. Gajić and Z. L. Crvenković, “A fixed point result for mappings with contractive iterate at a point in G-metric spaces,” Filomat, vol. 25, no. 2, pp. 53–58, 2011. View at: Publisher Site | Google Scholar
  17. G. S. Jeong and B. E. Rhoades, “Maps for which F(T)=F(Tn),” in Fixed Point Theory and Applications, vol. 6, pp. 71–105, Nova Science Publishers, New York, NY, USA, 2007. View at: Google Scholar | Zentralblatt MATH
  18. G. S. Jeong and B. E. Rhoades, “More maps for which F(T)=F(Tn),” Demonstratio Mathematica, vol. 40, no. 3, pp. 671–680, 2007. View at: Google Scholar | Zentralblatt MATH
  19. B. E. Rhoades and M. Abbas, “Maps satisfying generalized contractive conditions of integral type for which F(T)=F(Tn),” International Journal of Pure and Applied Mathematics, vol. 45, no. 2, pp. 225–231, 2008. View at: Google Scholar | Zentralblatt MATH
  20. M. S. Khan, M. Swaleh, and S. Sessa, “Fixed point theorems by altering distances between the points,” Bulletin of the Australian Mathematical Society, vol. 30, no. 1, pp. 1–9, 1984. View at: Publisher Site | Google Scholar | Zentralblatt MATH

Copyright © 2012 Mona Khandaqji et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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