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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012Β (2012), Article IDΒ 675094, 11 pages
http://dx.doi.org/10.1155/2012/675094
Research Article

Property P and Some Fixed Point Results on (πœ“,πœ™)-Weakly Contractive G-Metric Spaces

1Department of Mathematics, Hashemite University, P.O. Box 150459, Zarqa 13115, Jordan
2Department of Mathematics, Yarmouk University, P.O. Box 566, Irbid 21163, Jordan

Received 22 March 2012; Accepted 6 June 2012

Academic Editor: CharlesΒ Chidume

Copyright Β© 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.

Abstract

We prove some fixed point results for (πœ“,πœ™)-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 GΓ€hler [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 π‘›βˆˆπ‘. However, the converse is false. For example, consider 𝑋=[0,1] and 𝑇 defined by 𝑇π‘₯=1βˆ’π‘₯. Then 𝑇 has a unique fixed point at π‘₯=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œ‹], 𝑇π‘₯=cosπ‘₯, 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 (πœ“,πœ™)-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 πΊβˆΆπ‘‹Γ—π‘‹Γ—π‘‹β†’π‘…+ satisfy the following properties:(𝐺1)𝐺(π‘₯,𝑦,𝑧)=0 if π‘₯=𝑦=𝑧=0 (coincidence),(𝐺2)0<𝐺(π‘₯,π‘₯,𝑦), for all π‘₯,π‘¦βˆˆπ‘‹, where π‘₯≠𝑦,(𝐺3)𝐺(π‘₯,π‘₯,𝑦)≀𝐺(π‘₯,𝑦,𝑧), for all π‘₯,𝑦,π‘§βˆˆπ‘‹, with 𝑧≠𝑦,(𝐺4)𝐺(π‘₯,𝑦,𝑧)=𝐺(𝑝{π‘₯,𝑦,𝑧}), where 𝑝 is a permutation of π‘₯,𝑦,𝑧 (symmetry),(𝐺5)𝐺(π‘₯,𝑦,𝑧)≀𝐺(π‘₯,π‘Ž,π‘Ž)+𝐺(π‘Ž,𝑦,𝑧), for all π‘₯,𝑦,𝑧,π‘Žβˆˆπ‘‹ (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(π‘₯,𝑦,𝑦)=𝐺(𝑦,π‘₯,π‘₯) for all π‘₯,π‘¦βˆˆπ‘‹.

Proposition 2.3. Every G -metric space (𝑋,𝐺) will define a metric space (𝑋,𝑑𝐺) by 𝑑𝐺(π‘₯,𝑦)=𝐺(π‘₯,𝑦,𝑦)+𝐺(𝑦,π‘₯,π‘₯), for all π‘₯,π‘¦βˆˆπ‘‹.

Definition 2.4. Let (𝑋,𝐺) be a G-metric space, and (π‘₯𝑛) be a sequence of points in 𝑋. Then,(i)a point π‘₯βˆˆπ‘‹ is said to be the limit of the sequence (π‘₯𝑛) if 𝐺π‘₯𝑛,π‘₯π‘šξ€Έ,π‘₯⟢0,(as𝑛,π‘šβŸΆβˆž),(2.1) and we say that the sequence (π‘₯𝑛) is G convergent to π‘₯ (we say π‘₯𝑛(𝐺)β†’π‘₯),(ii)A sequence (π‘₯𝑛) is said to be G-Cauchy if 𝐺π‘₯𝑛,π‘₯π‘š,π‘₯π‘™ξ€ΈβŸΆ0,(as𝑛,π‘š,π‘™βŸΆβˆž),(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)π‘₯𝑛(𝐺)β†’π‘₯, (2)𝐺(π‘₯𝑛,π‘₯𝑛,π‘₯)β†’0(asπ‘›β†’βˆž), (3)𝐺(π‘₯𝑛,π‘₯,π‘₯)β†’0(asπ‘›β†’βˆž), (4)𝐺(π‘₯𝑛,π‘₯π‘š,π‘₯)β†’0(asπ‘›β†’βˆž).

Proposition 2.6. Let (𝑋,𝐺) be a G-metric space, then the following are equivalent:(1)(π‘₯𝑛) is be G-Cauchy in 𝑋,(2)𝐺(π‘₯𝑛,π‘₯π‘š,π‘₯π‘š)β†’0(as𝑛,π‘šβ†’βˆž).

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

3. Main Results

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

Definition 3.1 (see [20]). A function πœ“βˆΆ[0,∞)β†’[0,∞) is called altering distance if the following properties are satisfied:(1)πœ“ is continuous and increasing,(2)πœ“(𝑑)=0 if and only if 𝑑=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: ,πœ“(𝐺(𝑓π‘₯,𝑓𝑦,𝑓𝑧))β‰€πœ“max𝐺(π‘₯,𝑦,𝑧),𝐺(π‘₯,𝑓π‘₯,𝑓π‘₯),𝐺(𝑦,𝑓𝑦,𝑓𝑦),𝐺(𝑧,𝑓𝑧,𝑓𝑧),𝛼𝐺(𝑓π‘₯,𝑓π‘₯,𝑦)+(1βˆ’π›Ό)(𝐺(𝑓𝑦,𝑓𝑦,𝑧)),𝛽𝐺(π‘₯,𝑓π‘₯,𝑓π‘₯)+(1βˆ’π›½)(𝐺(𝑦,𝑓𝑦,𝑓𝑦)),ξƒ°ξƒͺβˆ’πœ™max𝐺(π‘₯,𝑦,𝑧),𝐺(π‘₯,𝑓π‘₯,𝑓π‘₯),𝐺(𝑦,𝑓𝑦,𝑓𝑦),𝐺(𝑧,𝑓𝑧,𝑓𝑧),𝛼𝐺(𝑓π‘₯,𝑓π‘₯,𝑦)+(1βˆ’π›Ό)(𝐺(𝑓𝑦,𝑓𝑦,𝑧)),𝛽𝐺(π‘₯,𝑓π‘₯,𝑓π‘₯)+(1βˆ’π›½)(𝐺(𝑦,𝑓𝑦,𝑓𝑦)),ξƒ°ξƒͺ(3.1) for all π‘₯,𝑦,π‘§βˆˆπ‘‹, where 0<𝛼,𝛽<1, πœ“ is an altering distance function, and πœ™βˆΆ[0,∞)β†’[0,∞) is a continuous function with πœ™(𝑑)=0 if and only if 𝑑=0. Then, 𝑓 has a unique fixed point (say 𝑒), where 𝑓 is 𝐺 continuous at 𝑒.

Proof. Fix π‘₯0βˆˆπ‘‹. Then construct a sequence {π‘₯𝑛} by π‘₯𝑛+1=𝑓π‘₯𝑛=𝑓𝑛π‘₯0. We may assume that π‘₯𝑛≠π‘₯𝑛+1 for each π‘›βˆˆπ‘βˆͺ{0}. Since, if there exist π‘›βˆˆπ‘ such that π‘₯𝑛=π‘₯𝑛+1, then π‘₯𝑛 is a fixed point of 𝑓.
From (3.1), substituting π‘₯≑π‘₯π‘›βˆ’1, 𝑦=𝑧≑π‘₯𝑛 then, for all π‘›βˆˆπ‘, πœ“ξ€·πΊξ€·π‘₯𝑛,π‘₯𝑛+1,π‘₯𝑛+1βŽ›βŽœβŽœβŽβŽ§βŽͺ⎨βŽͺβŽ©πΊξ€·π‘₯ξ€Έξ€Έβ‰€πœ“maxπ‘›βˆ’1,π‘₯𝑛,π‘₯𝑛π‘₯,πΊπ‘›βˆ’1,π‘₯𝑛,π‘₯𝑛π‘₯,𝐺𝑛,π‘₯𝑛+1,π‘₯𝑛+1ξ€Έξ€·π‘₯,𝐺𝑛,π‘₯𝑛+1,π‘₯𝑛+1ξ€Έ,ξ€·π‘₯𝛼𝐺𝑛,π‘₯𝑛,π‘₯𝑛𝐺π‘₯+(1βˆ’π›Ό)𝑛+1,π‘₯𝑛+1,π‘₯𝑛,ξ€·π‘₯ξ€Έξ€Έπ›½πΊπ‘›βˆ’1,π‘₯𝑛,π‘₯𝑛𝐺π‘₯+(1βˆ’π›½)𝑛+1,π‘₯𝑛+1,π‘₯π‘›βŽ«βŽͺ⎬βŽͺβŽ­βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽ§βŽͺ⎨βŽͺβŽ©πΊξ€·π‘₯ξ€Έξ€Έβˆ’πœ™maxπ‘›βˆ’1,π‘₯𝑛,π‘₯𝑛π‘₯,πΊπ‘›βˆ’1,π‘₯𝑛,π‘₯𝑛π‘₯,𝐺𝑛,π‘₯𝑛+1,π‘₯𝑛+1ξ€Έξ€·π‘₯,𝐺𝑛,π‘₯𝑛+1,π‘₯𝑛+1ξ€Έ,ξ€·π‘₯𝛼𝐺𝑛,π‘₯𝑛,π‘₯𝑛𝐺π‘₯+(1βˆ’π›Ό)𝑛+1,π‘₯𝑛+1,π‘₯𝑛,ξ€·π‘₯ξ€Έξ€Έπ›½πΊπ‘›βˆ’1,π‘₯𝑛,π‘₯𝑛𝐺π‘₯+(1βˆ’π›½)𝑛+1,π‘₯𝑛+1,π‘₯π‘›βŽ«βŽͺ⎬βŽͺβŽ­βŽžβŽŸβŽŸβŽ ξ‚΅ξ‚»πΊξ€·π‘₯ξ€Έξ€Έβ‰€πœ“maxπ‘›βˆ’1,π‘₯𝑛,π‘₯𝑛π‘₯,𝐺𝑛,π‘₯𝑛+1,π‘₯𝑛+1ξ€Έ,ξ€·π‘₯π›½πΊπ‘›βˆ’1,π‘₯𝑛,π‘₯𝑛𝐺π‘₯+(1βˆ’π›½)𝑛+1,π‘₯𝑛+1,π‘₯𝑛𝐺π‘₯ξ€Έξ€Έξ‚Όξ‚Άβˆ’πœ™maxπ‘›βˆ’1,π‘₯𝑛,π‘₯𝑛π‘₯,𝐺𝑛,π‘₯𝑛+1,π‘₯𝑛+1ξ€Έ,ξ€·π‘₯π›½πΊπ‘›βˆ’1,π‘₯𝑛,π‘₯𝑛𝐺π‘₯+(1βˆ’π›½)𝑛+1,π‘₯𝑛+1,π‘₯𝑛.ξ€Έξ€Έξ‚Όξ‚Ά(3.2) Let 𝑀𝑛=max{𝐺(π‘₯π‘›βˆ’1,π‘₯𝑛,π‘₯𝑛),𝐺(π‘₯𝑛,π‘₯𝑛+1,π‘₯𝑛+1)}. Then, (3.2) gives πœ“ξ€·πΊξ€·π‘₯𝑛,π‘₯𝑛+1,π‘₯𝑛+1ξ€·π‘€ξ€Έξ€Έβ‰€πœ“π‘›ξ€Έξ€·π‘€βˆ’πœ™π‘›ξ€Έ.(3.3) We have two cases, either 𝑀𝑛=𝐺(π‘₯𝑛,π‘₯𝑛+1,π‘₯𝑛+1) or 𝑀𝑛=𝐺(π‘₯π‘›βˆ’1,π‘₯𝑛,π‘₯𝑛). Suppose that, for some π‘›βˆˆπ‘0,𝑀𝑛=𝐺(π‘₯𝑛,π‘₯𝑛+1,π‘₯𝑛+1). Then, we have πœ“ξ€·πΊξ€·π‘₯𝑛,π‘₯𝑛+1,π‘₯𝑛+1𝐺π‘₯ξ€Έξ€Έβ‰€πœ“π‘›,π‘₯𝑛+1,π‘₯𝑛+1𝐺π‘₯ξ€Έξ€Έβˆ’πœ™π‘›,π‘₯𝑛+1,π‘₯𝑛+1ξ€Έξ€Έ.(3.4) Therefore, πœ“(𝐺(π‘₯𝑛,π‘₯𝑛+1,π‘₯𝑛+1))=0. Hence π‘₯𝑛=π‘₯𝑛+1. This is a contradiction since the π‘₯𝑛's are distinct.
Thus, 𝑀𝑛=𝐺(π‘₯𝑛,π‘₯𝑛+1,π‘₯𝑛+1), and (3.2) becomes πœ“ξ€·πΊξ€·π‘₯𝑛,π‘₯𝑛+1,π‘₯𝑛+1𝐺π‘₯ξ€Έξ€Έβ‰€πœ“π‘›βˆ’1,π‘₯𝑛,π‘₯𝑛𝐺π‘₯ξ€Έξ€Έβˆ’πœ™π‘›βˆ’1,π‘₯𝑛,π‘₯𝑛𝐺π‘₯ξ€Έξ€Έβ‰€πœ“π‘›βˆ’1,π‘₯𝑛,π‘₯𝑛.ξ€Έξ€Έ(3.5) But πœ“ is an increasing function. Thus, from (3.5), we get 𝐺π‘₯𝑛,π‘₯𝑛+1,π‘₯𝑛+1ξ€Έξ€·π‘₯β‰€πΊπ‘›βˆ’1,π‘₯𝑛,π‘₯𝑛,βˆ€π‘›βˆˆπ‘.(3.6) Therefore, {𝐺(π‘₯𝑛,π‘₯𝑛+1,π‘₯𝑛+1),π‘›βˆˆπ‘βˆͺ{0}} is a positive nonincreasing sequence. Hence there exists π‘Ÿβ‰₯0 such that limπ‘›β†’βˆžπΊξ€·π‘₯𝑛,π‘₯𝑛+1,π‘₯𝑛+1ξ€Έ=π‘Ÿ.(3.7) Letting π‘›β†’βˆž, and using (3.5) and the continuity of πœ“ and πœ™, we get πœ“(π‘Ÿ)β‰€πœ“(π‘Ÿ)βˆ’πœ™(π‘Ÿ).(3.8) Hence, πœ™(π‘Ÿ)=0, therefore π‘Ÿ=0, which implies that limπ‘›β†’βˆžπΊξ€·π‘₯𝑛,π‘₯𝑛+1,π‘₯𝑛+1ξ€Έ=0.(3.9) Consequently, for a given πœ€>0, there is an integer 𝑛0 such that 𝐺π‘₯𝑛,π‘₯𝑛+1,π‘₯𝑛+1ξ€Έ<πœ€2,βˆ€π‘›>𝑛0.(3.10) For π‘š,π‘›βˆˆπ‘ with π‘š>𝑛, we claim that 𝐺π‘₯𝑛,π‘₯π‘š,π‘₯π‘šξ€Έ<πœ€2,βˆ€π‘š>𝑛>𝑛0.(3.11) To show (3.11), we use induction on π‘š. Inequality (3.11) holds for π‘š=𝑛+1 from (3.10). Assume (3.11) holds for π‘š=π‘˜, that is, 𝐺π‘₯𝑛,π‘₯π‘˜,π‘₯π‘˜ξ€Έ<πœ€2,βˆ€π‘›>𝑛0.(3.12) For all 𝑛>𝑛0, take π‘š=π‘˜+1. Using (𝐺5) in Definition 2.1 and inequalities (3.10), (3.12), we get 𝐺π‘₯𝑛,π‘₯π‘˜+1,π‘₯π‘˜+1ξ€Έξ€·π‘₯≀𝐺𝑛,π‘₯𝑛+1,π‘₯𝑛+1ξ€Έξ€·π‘₯+𝐺𝑛+1,π‘₯π‘˜+1,π‘₯π‘˜+1ξ€Έξ€·π‘₯≀𝐺𝑛,π‘₯𝑛+1,π‘₯𝑛+1ξ€Έξ€·π‘₯+𝐺𝑛,π‘₯π‘˜,π‘₯π‘˜ξ€Έ<πœ€.(3.13) By induction on π‘š, we conclude that 𝐺π‘₯𝑛,π‘₯π‘š,π‘₯π‘šξ€Έ<πœ€2,βˆ€π‘š>𝑛>𝑛0.(3.14) We conclude from Proposition 2.6 that {π‘₯𝑛} is a G-Cauchy sequence in 𝑋. From the completeness of 𝑋, there exists 𝑒 in 𝑋 such that π‘₯𝑛(𝐺)→𝑒. For π‘›βˆˆπ‘, we have πœ“ξ€·πΊξ€·π‘“π‘’,𝑓𝑒,π‘₯𝑛𝐺=πœ“π‘“π‘’,𝑓𝑒,𝑓π‘₯π‘›βˆ’1βŽ›βŽœβŽœβŽβŽ§βŽͺ⎨βŽͺβŽ©πΊξ€·ξ€Έξ€Έβ‰€πœ“max𝑒,𝑒,π‘₯π‘›βˆ’1ξ€Έξ€·π‘₯,𝐺(𝑒,𝑓𝑒,𝑓𝑒),𝐺(𝑒,𝑓𝑒,𝑓𝑒),πΊπ‘›βˆ’1,π‘₯𝑛,π‘₯𝑛,𝐺𝛼𝐺(𝑓𝑒,𝑓𝑒,𝑒)+(1βˆ’π›Ό)𝑓𝑒,𝑓𝑒,π‘₯π‘›βˆ’1,⎫βŽͺ⎬βŽͺβŽ­βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽ§βŽͺ⎨βŽͺβŽ©πΊξ€·ξ€Έξ€Έπ›½πΊ(𝑒,𝑓𝑒,𝑓𝑒)+(1βˆ’π›½)(𝐺(𝑒,𝑓𝑒,𝑓𝑒))βˆ’πœ™max𝑒,𝑒,π‘₯π‘›βˆ’1ξ€Έξ€·π‘₯,𝐺(𝑒,𝑓𝑒,𝑓𝑒),𝐺(𝑒,𝑓𝑒,𝑓𝑒),πΊπ‘›βˆ’1,π‘₯𝑛,π‘₯𝑛,𝐺𝛼𝐺(𝑓𝑒,𝑓𝑒,𝑒)+(1βˆ’π›Ό)𝑓𝑒,𝑓𝑒,π‘₯π‘›βˆ’1,⎫βŽͺ⎬βŽͺβŽ­βŽžβŽŸβŽŸβŽ ξ‚΅ξ‚»πΊξ€·ξ€Έξ€Έπ›½πΊ(𝑒,𝑓𝑒,𝑓𝑒)+(1βˆ’π›½)(𝐺(𝑒,𝑓𝑒,𝑓𝑒))β‰€πœ“max𝑒,𝑒,π‘₯π‘›βˆ’1ξ€Έξ€·π‘₯,𝐺(𝑒,𝑓𝑒,𝑓𝑒),πΊπ‘›βˆ’1,π‘₯𝑛,π‘₯𝑛,𝐺𝛼𝐺(𝑓𝑒,𝑓𝑒,𝑒)+(1βˆ’π›Ό)𝑓𝑒,𝑓𝑒,π‘₯π‘›βˆ’1ξ‚΅ξ‚»πΊξ€·ξ€Έξ€Έξ‚Όξ‚Άβˆ’πœ™max𝑒,𝑒,π‘₯π‘›βˆ’1ξ€Έξ€·π‘₯,𝐺(𝑒,𝑓𝑒,𝑓𝑒),πΊπ‘›βˆ’1,π‘₯𝑛,π‘₯𝑛,𝐺𝛼𝐺(𝑓𝑒,𝑓𝑒,𝑒)+(1βˆ’π›Ό)𝑓𝑒,𝑓𝑒,π‘₯π‘›βˆ’1.ξ€Έξ€Έξ‚Όξ‚Ά(3.15) Letting π‘›β†’βˆž, and using the fact that πœ“ is continuous and 𝐺 is continuous on its variables, we get that 𝐺(𝑒,𝑓𝑒,𝑓𝑒)=0. Hence 𝑓𝑒=𝑒. So 𝑒 is a fixed point of 𝑓. Now, to show uniqueness, let 𝑣 be another fixed point of 𝑓 with 𝑣≠𝑒. Therefore, ξƒ©ξƒ―ξƒ©ξƒ―πœ“(𝐺(𝑒,𝑒,𝑣))=πœ“(𝐺(𝑓𝑒,𝑓𝑒,𝑓𝑣))β‰€πœ“max𝐺(𝑒,𝑒,𝑣),𝐺(𝑒,𝑓𝑒,𝑓𝑒),𝐺(𝑒,𝑓𝑒,𝑓𝑒),𝐺(𝑣,𝑓𝑣,𝑓𝑣),𝛼𝐺(𝑓𝑒,𝑓𝑒,𝑒)+(1βˆ’π›Ό)(𝐺(𝑓𝑒,𝑓𝑒,𝑣)),𝛽𝐺(𝑒,𝑓𝑒,𝑓𝑒)+(1βˆ’π›½)(𝐺(𝑣,𝑓𝑣,𝑓𝑣))ξƒ°ξƒͺβˆ’πœ™max𝐺(𝑒,𝑒,𝑣),𝐺(𝑒,𝑓𝑒,𝑓𝑒),𝐺(𝑒,𝑓𝑒,𝑓𝑒),𝐺(𝑣,𝑓𝑣,𝑓𝑣),𝛼𝐺(𝑓𝑒,𝑓𝑒,𝑒)+(1βˆ’π›Ό)(𝐺(𝑓𝑒,𝑓𝑒,𝑣)),𝛽𝐺(𝑒,𝑓𝑒,𝑓𝑒)+(1βˆ’π›½)(𝐺(𝑣,𝑓𝑣,𝑓𝑣))ξƒ°ξƒͺ=πœ“(max{𝐺(𝑒,𝑒,𝑣),(1βˆ’π›Ό)𝐺(𝑒,𝑒,𝑣)})βˆ’πœ™(max{𝐺(𝑒,𝑒,𝑣),(1βˆ’π›Ό)𝐺(𝑒,𝑒,𝑣)})=πœ“(𝐺(𝑒,𝑒,𝑣))βˆ’πœ™(𝐺(𝑒,𝑒,𝑣)).(3.16) Hence, πœ“(𝐺(𝑒,𝑒,𝑣))β‰€πœ“(𝐺(𝑒,𝑒,𝑣))βˆ’πœ™(𝐺(𝑒,𝑒,𝑣)).(3.17) This implies that πœ™(𝐺(𝑒,𝑒,𝑣))=0, then 𝐺(𝑒,𝑒,𝑣)=0 and 𝑒=𝑣.
Now to show that 𝑓 is 𝐺 continuous at 𝑒, let {π‘₯𝑛} be a sequence in 𝑋 with limit 𝑒(i.e.,π‘₯𝑛(𝐺)→𝑒). Using (3.1), we have πœ“ξ€·πΊξ€·π‘“π‘₯𝑛𝐺,𝑒,𝑒=πœ“π‘“π‘₯π‘›βŽ›βŽœβŽœβŽβŽ§βŽͺ⎨βŽͺβŽ©πΊξ€·π‘₯,𝑓𝑒,π‘“π‘’ξ€Έξ€Έβ‰€πœ“max𝑛π‘₯,𝑒,𝑒,𝐺𝑛,𝑓π‘₯𝑛,𝑓π‘₯𝑛,𝐺(𝑒,𝑓𝑒,𝑓𝑒),𝐺(𝑒,𝑓𝑒,𝑓𝑒),𝛼𝐺𝑓π‘₯𝑛,𝑓π‘₯𝑛π‘₯,𝑒+(1βˆ’π›Ό)(𝐺(𝑓𝑒,𝑓𝑒,𝑒)),𝛽𝐺𝑛,𝑓π‘₯𝑛,𝑓π‘₯π‘›ξ€ΈβŽ«βŽͺ⎬βŽͺβŽ­βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽ§βŽͺ⎨βŽͺβŽ©πΊξ€·π‘₯+(1βˆ’π›½)(𝐺(𝑒,𝑓𝑒,𝑓𝑒))βˆ’πœ™max𝑛π‘₯,𝑒,𝑒,𝐺𝑛,𝑓π‘₯𝑛,𝑓π‘₯𝑛,𝐺(𝑒,𝑓𝑒,𝑓𝑒),𝐺(𝑒,𝑓𝑒,𝑓𝑒),𝛼𝐺𝑓π‘₯𝑛,𝑓π‘₯𝑛π‘₯,𝑒+(1βˆ’π›Ό)(𝐺(𝑓𝑒,𝑓𝑒,𝑒)),𝛽𝐺𝑛,𝑓π‘₯𝑛,𝑓π‘₯π‘›ξ€ΈβŽ«βŽͺ⎬βŽͺβŽ­βŽžβŽŸβŽŸβŽ ξ€·ξ€½πΊξ€·π‘₯+(1βˆ’π›½)(𝐺(𝑒,𝑓𝑒,𝑓𝑒))=πœ“max𝑛,𝑒,𝑒,𝛼𝐺𝑓π‘₯𝑛,𝑓π‘₯𝑛π‘₯,𝑒,𝛽𝐺𝑛,𝑓π‘₯𝑛,𝑓π‘₯𝑛𝐺π‘₯ξ€Έξ€Ύξ€Έβˆ’πœ™max𝑛,𝑒,𝑒,𝛼𝐺𝑓π‘₯𝑛,𝑓π‘₯𝑛π‘₯,𝑒,𝛽𝐺𝑛,𝑓π‘₯𝑛,𝑓π‘₯𝑛𝐺π‘₯ξ€Έξ€Ύξ€Έβ‰€πœ“max𝑛π‘₯,𝑒,𝑒,𝛼𝐺𝑛+1,π‘₯𝑛+1ξ€Έξ€·π‘₯,𝑒,𝛽𝐺𝑛,π‘₯𝑛+1,π‘₯𝑛+1𝐺π‘₯ξ€Έξ€Ύξ€Έβ‰€πœ“max𝑛π‘₯,𝑒,𝑒,𝛼𝐺𝑛+1,π‘₯𝑛+1ξ€Έξ€·π‘₯,𝑒,𝛽𝐺𝑛,𝑒,𝑒+𝛽𝐺𝑒,π‘₯𝑛+1,π‘₯𝑛+1𝐺π‘₯ξ€Έξ€Ύξ€Έβ‰€πœ“max𝑛π‘₯,𝑒,𝑒,𝐺𝑛+1,π‘₯𝑛+1ξ€Έξ€·π‘₯,𝑒,𝛽𝐺𝑛,𝑒,𝑒+𝛽𝐺𝑒,π‘₯𝑛+1,π‘₯𝑛+1.ξ€Έξ€Ύξ€Έ(3.18) But πœ“ is an increasing function, thus from (3.18), we get 𝐺𝑓π‘₯𝑛𝐺π‘₯,𝑒,𝑒≀max𝑛π‘₯,𝑒,𝑒,𝐺𝑛+1,π‘₯𝑛+1ξ€Έξ€·π‘₯,𝑒,𝛽𝐺𝑛+1ξ€Έξ€·,𝑒,𝑒+𝛽𝐺𝑒,π‘₯𝑛+1,π‘₯𝑛+1ξ€Έξ€Ύ.(3.19) Therefore, limπ‘›β†’βˆžπΊ(𝑓π‘₯𝑛,𝑒,𝑒)=0.

Corollary 3.3. Let 𝑇 be a self-map on a complete G-metric space 𝑋 satisfying the following for all π‘₯,𝑦,π‘§βˆˆπ‘‹βˆΆξƒ―ξƒ°πΊ(𝑓π‘₯,𝑓𝑦,𝑓𝑧)β‰€πœ†max𝐺(π‘₯,𝑦,𝑧),𝐺(π‘₯,𝑓π‘₯,𝑓π‘₯),𝐺(𝑦,𝑓𝑦,𝑓𝑦),𝐺(𝑧,𝑓𝑧,𝑓𝑧),𝛼𝐺(𝑓π‘₯,𝑓π‘₯,𝑦)+(1βˆ’π›Ό)(𝐺(𝑓𝑦,𝑓𝑦,𝑧)),𝛽𝐺(π‘₯,𝑓π‘₯,𝑓π‘₯)+(1βˆ’π›½)(𝐺(𝑦,𝑓𝑦,𝑓𝑦)),(3.20) where 0<𝛼,𝛽,πœ†<1,πœ“ is an altering distance function, and πœ™βˆΆ[0,∞)β†’[0,∞) is a continuous function with πœ™(𝑑)=0 if and only if 𝑑=0. Then 𝑓 has a unique fixed point (say 𝑒), and 𝑓 is 𝐺 continuous at 𝑒.

Proof. We get the result by taking πœ“(𝑑)=𝑑 and πœ™(𝑑)=π‘‘βˆ’πœ†π‘‘, 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: ⎧βŽͺ⎨βŽͺ⎩1𝐺(𝑓π‘₯,𝑓𝑦,𝑓𝑧)β‰€πœ†max𝐺(π‘₯,𝑦,𝑧),𝐺(π‘₯,𝑓π‘₯,𝑓π‘₯),𝐺(𝑦,𝑓𝑦,𝑓𝑦),𝐺(𝑧,𝑓𝑧,𝑓𝑧),21(𝐺(𝑓π‘₯,𝑓π‘₯,𝑦)+(𝐺(𝑓𝑦,𝑓𝑦,𝑧))),2⎫βŽͺ⎬βŽͺ⎭(𝐺(π‘₯,𝑓π‘₯,𝑓π‘₯)+𝐺(𝑦,𝑓𝑦,𝑓𝑦)),(3.21) for all π‘₯,𝑦,π‘§βˆˆπ‘‹ where 0<πœ†<1,πœ“ is an altering distance function and, πœ™βˆΆ[0,∞)β†’[0,∞) is a continuous function with πœ™(𝑑)=0 if and only if 𝑑=0. Then 𝑓 has a unique fixed point (say 𝑒), and 𝑓 is 𝐺 continuous at 𝑒.

Proof. We get the result by taking πœ“(𝑑)=𝑑 and πœ™(𝑑)=π‘‘βˆ’πœ†π‘‘, 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: ⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩1𝐺(𝑓π‘₯,𝑓𝑦,𝑓𝑧)β‰€πœ†max𝐺(π‘₯,𝑦,𝑧),𝐺(π‘₯,𝑓π‘₯,𝑓π‘₯),𝐺(𝑦,𝑓𝑦,𝑓𝑦),𝐺(𝑧,𝑓𝑧,𝑓𝑧),32𝐺(𝑓π‘₯,𝑓π‘₯,𝑦)+31𝐺(𝑓𝑦,𝑓𝑦,𝑧),32𝐺(𝑓π‘₯,𝑓π‘₯,π‘₯)+3⎫βŽͺβŽͺ⎬βŽͺβŽͺ⎭𝐺(𝑓𝑦,𝑓𝑦,𝑦),(3.22) for all π‘₯,𝑦,π‘§βˆˆπ‘‹, where 0<πœ†<1,πœ“ is an altering distance function, and πœ™βˆΆ[0,∞)β†’[0,∞) is a continuous function with πœ™(𝑑)=0 if and only if 𝑑=0. Then 𝑓 has a unique fixed point (say 𝑒) and 𝑓 is 𝐺 continuous at 𝑒.

Proof. We get the result by taking πœ“(𝑑)=𝑑 and πœ™(𝑑)=π‘‘βˆ’πœ†π‘‘, 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)β‰ πœ‘ for each π‘›βˆˆπ‘. Fix 𝑛>1, and assume that π‘’βˆˆπΉ(𝑓𝑛). We claim that π‘’βˆˆπΉ(𝑓). To prove the claim, suppose that 𝑒≠𝑓𝑒. Using (3.1), we have πœ“ξ€·πΊξ€·π‘“(𝐺(𝑒,𝑓𝑒,𝑓𝑒))=πœ“π‘›π‘’,𝑓𝑛+1𝑒,𝑓𝑛+1𝑒𝐺=πœ“π‘“π‘“π‘›βˆ’1𝑒,𝑓𝑓𝑛𝑒,π‘“π‘“π‘›π‘’βŽ›βŽœβŽœβŽβŽ§βŽͺ⎨βŽͺβŽ©πΊξ€·π‘“ξ€Έξ€Έβ‰€πœ“maxπ‘›βˆ’1𝑓𝑒,𝑒,𝑒,𝐺(𝑒,𝑓𝑒,𝑓𝑒),𝛼𝐺(𝑒,𝑒,𝑒)+(1βˆ’π›Ό)(𝐺(𝑓𝑒,𝑓𝑒,𝑒))π›½πΊπ‘›βˆ’1ξ€ΈβŽ«βŽͺ⎬βŽͺβŽ­βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽ§βŽͺ⎨βŽͺβŽ©πΊξ€·π‘“π‘’,𝑒,𝑒+(1βˆ’π›½)(𝐺(𝑒,𝑓𝑒,𝑓𝑒))βˆ’πœ™maxπ‘›βˆ’1𝑓𝑒,𝑒,𝑒,𝐺(𝑒,𝑓𝑒,𝑓𝑒),𝛼𝐺(𝑒,𝑒,𝑒)+(1βˆ’π›Ό)(𝐺(𝑓𝑒,𝑓𝑒,𝑒))π›½πΊπ‘›βˆ’1ξ€ΈβŽ«βŽͺ⎬βŽͺβŽ­βŽžβŽŸβŽŸβŽ ξ€·ξ€½πΊξ€·π‘“π‘’,𝑒,𝑒+(1βˆ’π›½)(𝐺(𝑒,𝑓𝑒,𝑓𝑒))=πœ“maxπ‘›βˆ’1𝐺𝑓𝑒,𝑒,𝑒,𝐺(𝑒,𝑓𝑒,𝑓𝑒)ξ€Ύξ€Έβˆ’πœ™maxπ‘›βˆ’1ξ€Έ.𝑒,𝑒,𝑒,𝐺(𝑒,𝑓𝑒,𝑓𝑒)ξ€Ύξ€Έ(3.23) Letting 𝑀=max{𝐺(π‘“π‘›βˆ’1𝑒,𝑒,𝑒),𝐺(𝑒,𝑓𝑒,𝑓𝑒)}, we deduce form (3.23), πœ“(𝐺(𝑒,𝑓𝑒,𝑓𝑒))β‰€πœ“(𝑀)βˆ’πœ™(𝑀).(3.24) If 𝑀=𝐺(𝑒,𝑓𝑒,𝑓𝑒), then πœ“(𝐺(𝑒,𝑓𝑒,𝑓𝑒))β‰€πœ“(𝐺(𝑒,𝑓𝑒,𝑓𝑒))βˆ’πœ™(𝐺(𝑒,𝑓𝑒,𝑓𝑒)),(3.25) hence, πœ™(𝐺(𝑒,𝑓𝑒,𝑓𝑒))=0. By a property of πœ™, we deduce that 𝐺(𝑒,𝑓𝑒,𝑓𝑒)=0, therefore, 𝑒=𝑓𝑒. This is a contradiction. On the other hand, if 𝑀=𝐺(π‘“π‘›βˆ’1𝑒,𝑒,𝑒), then (3.1) gives that πœ“ξ€·πΊξ€·π‘“π‘›π‘’,𝑓𝑛+1𝑒,𝑓𝑛+1𝑒𝐺𝑓=πœ“(𝐺(𝑒,𝑓𝑒,𝑓𝑒))β‰€πœ“π‘›βˆ’1𝐺𝑓𝑒,𝑒,π‘’ξ€Έξ€Έβˆ’πœ™π‘›βˆ’1𝐺𝑓𝑒,𝑒,𝑒=πœ“π‘›βˆ’1𝑒,𝑓𝑛𝑒,π‘“π‘›π‘’ξ€·πΊξ€·π‘“ξ€Έξ€Έβˆ’πœ™π‘›βˆ’1𝑒,𝑓𝑛𝑒,π‘“π‘›π‘’ξ€·πΊξ€·π‘“ξ€Έξ€Έβ‰€πœ“π‘›βˆ’2𝑒,π‘“π‘›βˆ’1𝑒,π‘“π‘›βˆ’1π‘’ξ€·πΊξ€·π‘“ξ€Έξ€Έβˆ’πœ™π‘›βˆ’2𝑒,π‘“π‘›βˆ’1𝑒,π‘“π‘›βˆ’1π‘’ξ€·πΊξ€·π‘“ξ€Έξ€Έβˆ’πœ™π‘›βˆ’1𝑒,𝑓𝑛𝑒,π‘“π‘›π‘’ξ€Έξ€Έβ‰€β‹―β‰€πœ“(𝐺(𝑒,𝑓𝑒,𝑓𝑒))βˆ’π‘›βˆ’1ξ“π‘˜=0πœ™ξ€·πΊξ€·π‘“π‘›βˆ’π‘˜βˆ’1𝑒,π‘“π‘›βˆ’π‘˜π‘’,π‘“π‘›βˆ’π‘˜π‘’.ξ€Έξ€Έ(3.26) Therefore, π‘›βˆ’1ξ“π‘˜=0πœ™ξ€·πΊξ€·π‘“π‘›βˆ’π‘˜βˆ’1𝑒,π‘“π‘›βˆ’π‘˜π‘’,π‘“π‘›βˆ’π‘˜π‘’ξ€Έξ€Έ=0,(3.27) which implies that πœ™(𝐺(π‘“π‘›βˆ’π‘˜βˆ’1𝑒,π‘“π‘›βˆ’π‘˜π‘’,π‘“π‘›βˆ’π‘˜π‘’))=0, for all (0β‰€π‘˜β‰€π‘›βˆ’1). Thus, πœ™(𝐺(𝑒,𝑓𝑒,𝑓𝑒))=0, and by a property of πœ™, we have 𝑒=𝑓𝑒. This is a contradiction.
Therefore, π‘’βˆˆπΉ(𝑓), and 𝑓 has property 𝑃. Let𝑀𝛼,𝛽(π‘₯,𝑦,𝑧)=max𝐺(π‘₯,𝑦,𝑧),𝐺(π‘₯,𝑓π‘₯,𝑓π‘₯),𝐺(𝑦,𝑓𝑦,𝑓𝑦),𝐺(𝑧,𝑓𝑧,𝑓𝑧),𝛼𝐺(𝑓π‘₯,𝑓π‘₯,𝑦)+(1βˆ’π›Ό)(𝐺(𝑓𝑦,𝑓𝑦,𝑧)),𝛽𝐺(π‘₯,𝑓π‘₯,𝑓π‘₯)+(1βˆ’π›½)(𝐺(𝑦,𝑓𝑦,𝑓𝑦)),,(3.28) where 𝛼,π›½βˆˆ(0,1].

Example 3.7. Let 𝑋=[0,1] and 𝐺(π‘₯,𝑦,𝑧)=max{|π‘₯βˆ’π‘¦|,|π‘¦βˆ’π‘§|,|π‘§βˆ’π‘₯|} be a G-metric on 𝑋. Define π‘“βˆΆπ‘‹β†’π‘‹ by 𝑓(π‘₯)=π‘₯/8. We take πœ“(𝑑)=𝑑 and πœ™(𝑑)=7/8𝑑, for π‘‘βˆˆ[0,∞) and 𝛼,π›½βˆˆ(0,1]. So that πœ“ξ€·π‘€π›Ό,𝛽𝑀(π‘₯,𝑦,𝑧)βˆ’πœ™π›Ό,𝛽=1(π‘₯,𝑦,𝑧)8𝑀𝛼,𝛽(π‘₯,𝑦,𝑧).(3.29) We have |||π‘₯𝐺(𝑓π‘₯,𝑓𝑦,𝑓𝑧)=max8βˆ’π‘¦8|||,|||𝑦8βˆ’π‘§8|||,|||𝑧8βˆ’π‘₯8|||=18ξ€½||||,||||ξ€Ύ=1maxπ‘₯βˆ’π‘¦π‘¦βˆ’π‘§,|π‘§βˆ’π‘₯|8≀1𝐺(π‘₯,𝑦,𝑧)8𝑀𝛼,𝛽𝑀(π‘₯,𝑦,𝑧)=πœ“π›Ό,𝛽𝑀(π‘₯,𝑦,𝑧)βˆ’πœ™π›Ό,𝛽.(π‘₯,𝑦,𝑧)(3.30)

4. Applications

Denote by Ξ› the set of functions πœ†βˆΆ[0,∞)β†’[0,∞) satisfying the following hypotheses.(1)πœ† is a Lebesgue integral mapping on each compact of [0,∞).(2)For every πœ€>0, we have βˆ«π‘‘0πœ†(𝑠)𝑑𝑠>0.It is an easy matter to see that the mapping πœ“βˆΆ[0,∞)β†’[0,∞), defined by βˆ«πœ“(𝑑)=𝑑0πœ†(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: ξ€œ0𝐺(𝑓π‘₯,𝑓𝑦,𝑓𝑧)ξ€œπœ†(𝑠)𝑑𝑠≀𝑀𝛼,𝛽0(π‘₯,𝑦,𝑧)ξ€œπœ†(𝑠)π‘‘π‘ βˆ’π‘€π›Ό,𝛽0(π‘₯,𝑦,𝑧)πœ‡(𝑠)𝑑𝑠,(4.1) where πœ†, πœ‡βˆˆΞ› and 𝛼,π›½βˆˆ(0,1]. Then 𝑓 has a unique fixed point.

Proof. It follows from Theorem by taking βˆ«πœ“(𝑑)=𝑑0πœ†(𝑠)𝑑𝑠 and βˆ«πœ™(𝑑)=𝑑0πœ‡(𝑠)𝑑𝑠.

Acknowledgment

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

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