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Journal of Applied Mathematics
VolumeΒ 2012, Article IDΒ 856974, 7 pages
http://dx.doi.org/10.1155/2012/856974
Research Article

Periodic Points and Fixed Points for the Weaker (πœ™,πœ‘)-Contractive Mappings in Complete Generalized Metric Spaces

Department of Applied Mathematics, National Hsinchu University of Education, No. 521 Nanda Road, Hsinchu City 300, Taiwan

Received 19 November 2011; Accepted 14 December 2011

Academic Editor: SongΒ Cen

Copyright Β© 2012 Chi-Ming Chen and W. Y. Sun. 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 introduce the notion of weaker (πœ™,πœ‘)-contractive mapping in complete metric spaces and prove the periodic points and fixed points for this type of contraction. Our results generalize or improve many recent fixed point theorems in the literature.

1. Introduction and Preliminaries

Let (𝑋,𝑑) be a metric space, 𝐷 a subset of 𝑋 and π‘“βˆΆπ·β†’π‘‹ a map. We say 𝑓 is contractive if there exists π›Όβˆˆ[0,1) such that, for all π‘₯,π‘¦βˆˆπ·, 𝑑(𝑓π‘₯,𝑓𝑦)≀𝛼⋅𝑑(π‘₯,𝑦).(1.1) The well-known Banach’s fixed point theorem asserts that if 𝐷=𝑋, 𝑓 is contractive and (𝑋,𝑑) is complete, then 𝑓 has a unique fixed point in 𝑋. It is well known that the Banach contraction principle [1] is a very useful and classical tool in nonlinear analysis. In 1969, Boyd and Wong [2] introduced the notion of Ξ¦-contraction. A mapping π‘“βˆΆπ‘‹β†’π‘‹ on a metric space is called Ξ¦-contraction if there exists an upper semicontinuous function Φ∢[0,∞)β†’[0,∞) such that𝑑(𝑓π‘₯,𝑓𝑦)≀Φ(𝑑(π‘₯,𝑦))βˆ€π‘₯,π‘¦βˆˆπ‘‹.(1.2)

In 2000, Branciari [3] introduced the following notion of a generalized metric space where the triangle inequality of a metric space had been replaced by an inequality involving three terms instead of two. Later, many authors worked on this interesting space (e.g., [4–9]).

Let (𝑋,𝑑) be a generalized metric space. For 𝛾>0 and π‘₯βˆˆπ‘‹, we define 𝐡𝛾(π‘₯)∢={π‘¦βˆˆπ‘‹βˆ£π‘‘(π‘₯,𝑦)<𝛾}.(1.3) Branciari [3] also claimed that {𝐡𝛾(π‘₯)βˆΆπ›Ύ>0,π‘₯βˆˆπ‘‹} is a basis for a topology on 𝑋, 𝑑 is continuous in each of the coordinates and a generalized metric space is a Hausdorff space. We recall some definitions of a generalized metric space, as follows.

Definition 1.1 (see [3]). Let 𝑋 be a nonempty set and π‘‘βˆΆπ‘‹Γ—π‘‹β†’[0,∞) a mapping such that for all π‘₯,π‘¦βˆˆπ‘‹ and for all distinct point 𝑒,π‘£βˆˆπ‘‹ each of them different from π‘₯ and 𝑦, one has(i)𝑑(π‘₯,𝑦)=0 if and only if π‘₯=𝑦;(ii)𝑑(π‘₯,𝑦)=𝑑(𝑦,π‘₯);(iii)𝑑(π‘₯,𝑦)≀𝑑(π‘₯,𝑒)+𝑑(𝑒,𝑣)+𝑑(𝑣,𝑦) (rectangular inequality). Then (𝑋,𝑑) is called a generalized metric space (or shortly g.m.s).

Definition 1.2 (see [3]). Let (𝑋,𝑑) be a g.m.s, {π‘₯𝑛} a sequence in 𝑋, and π‘₯βˆˆπ‘‹. We say that {π‘₯𝑛} is g.m.s convergent to π‘₯ if and only if 𝑑(π‘₯𝑛,π‘₯)β†’0 as π‘›β†’βˆž. We denote by π‘₯𝑛→π‘₯ as π‘›β†’βˆž.

Definition 1.3 (see [3]). Let (𝑋,𝑑) be a g.m.s, {π‘₯𝑛} a sequence in 𝑋, and π‘₯βˆˆπ‘‹. We say that {π‘₯𝑛} is g.m.s Cauchy sequence if and only if for each πœ€>0, there exists 𝑛0βˆˆβ„• such that 𝑑(π‘₯π‘š,π‘₯𝑛)<πœ€ for all 𝑛>π‘š>𝑛0.

Definition 1.4 (see [3]). Let (𝑋,𝑑) be a g.m.s. Then 𝑋 is called complete g.m.s if every g.m.s Cauchy sequence is g.m.s convergent in 𝑋.

In this paper, we also recall the notion of Meir-Keeler function (see [10]). A function πœ™βˆΆ[0,∞)β†’[0,∞) is said to be a Meir-Keeler function if for each πœ‚>0, there exists 𝛿>0 such that for π‘‘βˆˆ[0,∞) with πœ‚β‰€π‘‘<πœ‚+𝛿, we have πœ™(𝑑)<πœ‚. Generalization of the above function has been a heavily investigated branch research. Particularly, in [11, 12], the authors proved the existence and uniqueness of fixed points for various Meir-Keeler-type contractive functions. We now introduce the notion of weaker Meir-Keeler function πœ™βˆΆ[0,∞)β†’[0,∞), as follows.

Definition 1.5. We call πœ™βˆΆ[0,∞)β†’[0,∞) a weaker Meir-Keeler function if for each πœ‚>0, there exists 𝛿>0 such that for π‘‘βˆˆ[0,∞) with πœ‚β‰€π‘‘<πœ‚+𝛿, there exists 𝑛0βˆˆβ„• such that πœ™π‘›0(𝑑)<πœ‚.

2. Main Results

In the paper, we denote by Ξ¦ the class of functions πœ™βˆΆ[0,∞)β†’[0,∞) satisfying the following conditions:

(πœ™1) πœ™βˆΆ[0,∞)β†’[0,∞) is a weaker Meir-Keeler function;

(πœ™2) πœ™(𝑑)>0 for 𝑑>0, πœ™(0)=0;

(πœ™3) for all π‘‘βˆˆ(0,∞), {πœ™π‘›(𝑑)}π‘›βˆˆβ„• is decreasing;

(πœ™4) if limπ‘›β†’βˆžπ‘‘π‘›=𝛾, then limπ‘›β†’βˆžπœ™(𝑑𝑛)≀𝛾.

And we denote by Θ the class of functions πœ‘βˆΆ[0,∞)β†’[0,∞) satisfying the following conditions:

(πœ‘1) πœ‘ is continuous;

(πœ‘2) πœ‘(𝑑)>0 for 𝑑>0 and πœ‘(0)=0.

Our main result is the following.

Theorem 2.1. Let (𝑋,𝑑) be a Hausdorff and complete g.m.s, and let π‘“βˆΆπ‘‹β†’π‘‹ be a function satisfying 𝑑(𝑓π‘₯,𝑓𝑦)β‰€πœ™(𝑑(π‘₯,𝑦))βˆ’πœ‘(𝑑(π‘₯,𝑦))(2.1) for all π‘₯,π‘¦βˆˆπ‘‹ and πœ™βˆˆΞ¦, πœ‘βˆˆΞ˜. Then 𝑓 has a periodic point πœ‡ in 𝑋, that is, there exists πœ‡βˆˆπ‘‹ such that πœ‡=π‘“π‘πœ‡ for some π‘βˆˆβ„•.

Proof. Given π‘₯0, define a sequence {π‘₯𝑛} in 𝑋 by π‘₯𝑛+1=𝑓π‘₯𝑛forπ‘›βˆˆβ„•βˆͺ{0}.(2.2)Step 1. We will prove that limπ‘›β†’βˆžπ‘‘ξ€·π‘₯𝑛,π‘₯𝑛+1ξ€Έ=0,limπ‘›β†’βˆžπ‘‘ξ€·π‘₯𝑛,π‘₯𝑛+2ξ€Έ=0.(2.3) Using inequality (2.1), we have that for each π‘›βˆˆβ„•βˆͺ{0}𝑑π‘₯𝑛,π‘₯𝑛+1ξ€Έξ€·=𝑑𝑓π‘₯π‘›βˆ’1,𝑓π‘₯𝑛𝑑π‘₯β‰€πœ™π‘›βˆ’1,π‘₯𝑛𝑑π‘₯ξ€Έξ€Έβˆ’πœ‘π‘›βˆ’1,π‘₯𝑛𝑑π‘₯ξ€Έξ€Έβ‰€πœ™π‘›βˆ’1,π‘₯𝑛,ξ€Έξ€Έ(2.4) and so 𝑑π‘₯𝑛,π‘₯𝑛+1𝑑π‘₯β‰€πœ™π‘›βˆ’1,π‘₯π‘›ξ€·πœ™ξ€·π‘‘ξ€·π‘₯ξ€Έξ€Έβ‰€πœ™π‘›βˆ’2,π‘₯π‘›βˆ’1ξ€Έξ€Έξ€Έ=πœ™2𝑑π‘₯π‘›βˆ’2,π‘₯π‘›βˆ’1ξ€Έξ€Έβ‰€β‹―β‰€πœ™π‘›ξ€·π‘‘ξ€·π‘₯0,π‘₯1.ξ€Έξ€Έ(2.5) Since {πœ™π‘›(𝑑(π‘₯0,π‘₯1))}π‘›βˆˆβ„• is decreasing, it must converge to some πœ‚β‰₯0. We claim that πœ‚=0. On the contrary, assume that πœ‚>0. Then by the definition of weaker Meir-Keeler function πœ™, there exists 𝛿>0 such that for π‘₯0,π‘₯1βˆˆπ‘‹ with πœ‚β‰€π‘‘(π‘₯0,π‘₯1)<𝛿+πœ‚, there exists 𝑛0βˆˆβ„• such that πœ™π‘›0(𝑑(π‘₯0,π‘₯1))<πœ‚. Since limπ‘›β†’βˆžπœ™π‘›(𝑑(π‘₯0,π‘₯1))=πœ‚, there exists 𝑝0βˆˆβ„• such that πœ‚β‰€πœ™π‘(𝑑(π‘₯0,π‘₯1))<𝛿+πœ‚, for all 𝑝β‰₯𝑝0. Thus, we conclude that πœ™π‘0+𝑛0(𝑑(π‘₯0,π‘₯1))<πœ‚. So we get a contradiction. Therefore, limπ‘›β†’βˆžπœ™π‘›(𝑑(π‘₯0,π‘₯1))=0, that is, limπ‘›β†’βˆžπ‘‘ξ€·π‘₯𝑛,π‘₯𝑛+1ξ€Έ=0.(2.6) Using inequality (2.1), we also have that for each π‘›βˆˆβ„•π‘‘ξ€·π‘₯𝑛,π‘₯𝑛+2ξ€Έξ€·=𝑑𝑓π‘₯π‘›βˆ’1,𝑓π‘₯𝑛+1𝑑π‘₯β‰€πœ™π‘›βˆ’1,π‘₯𝑛+1𝑑π‘₯ξ€Έξ€Έβˆ’πœ‘π‘›βˆ’1,π‘₯𝑛+1𝑑π‘₯ξ€Έξ€Έβ‰€πœ™π‘›βˆ’1,π‘₯𝑛+1,ξ€Έξ€Έ(2.7) and so 𝑑π‘₯𝑛,π‘₯𝑛+2𝑑π‘₯β‰€πœ™π‘›βˆ’1,π‘₯𝑛+1ξ€·πœ™ξ€·π‘‘ξ€·π‘₯ξ€Έξ€Έβ‰€πœ™π‘›βˆ’2,π‘₯𝑛=πœ™2𝑑π‘₯π‘›βˆ’2,π‘₯π‘›ξ€Έξ€Έβ‰€β‹―β‰€πœ™π‘›ξ€·π‘‘ξ€·π‘₯0,π‘₯2.ξ€Έξ€Έ(2.8) Since {πœ™π‘›(𝑑(π‘₯0,π‘₯2))}π‘›βˆˆβ„• is decreasing, by the same proof process, we also conclude limπ‘›β†’βˆžπ‘‘ξ€·π‘₯𝑛,π‘₯𝑛+2ξ€Έ=0.(2.9)
Next, we claim that {π‘₯𝑛} is g.m.s Cauchy. We claim that the following result holds.
Step 2. Claim that for every πœ€>0, there exists π‘›βˆˆβ„• such that if 𝑝,π‘žβ‰₯𝑛 then 𝑑(π‘₯𝑝,π‘₯π‘ž)<πœ€.
Suppose the above statement is false. Then there exists πœ–>0 such that for any π‘›βˆˆβ„•, there are 𝑝𝑛,π‘žπ‘›βˆˆβ„• with 𝑝𝑛>π‘žπ‘›β‰₯𝑛 satisfying 𝑑π‘₯π‘žπ‘›,π‘₯𝑝𝑛β‰₯πœ–.(2.10) Further, corresponding to π‘žπ‘›β‰₯𝑛, we can choose 𝑝𝑛 in such a way that it the smallest integer with 𝑝𝑛>π‘žπ‘›β‰₯𝑛 and 𝑑(π‘₯π‘žπ‘›,π‘₯𝑝𝑛)β‰₯πœ–. Therefore, 𝑑(π‘₯π‘žπ‘›,π‘₯π‘π‘›βˆ’1)<πœ–. By the rectangular inequality and (2.3), we have ξ€·π‘₯πœ–β‰€π‘‘π‘π‘›,π‘₯π‘žπ‘›ξ€Έξ€·π‘₯≀𝑑𝑝𝑛,π‘₯π‘π‘›βˆ’2ξ€Έξ€·π‘₯+π‘‘π‘π‘›βˆ’2,π‘₯π‘π‘›βˆ’1ξ€Έξ€·π‘₯+π‘‘π‘π‘›βˆ’1,π‘₯π‘žπ‘›ξ€Έξ€·π‘₯<𝑑𝑝𝑛,π‘₯π‘π‘›βˆ’2ξ€Έξ€·π‘₯+π‘‘π‘π‘›βˆ’2,π‘₯π‘π‘›βˆ’1ξ€Έ+πœ–.(2.11) Let π‘›β†’βˆž. Then we get limπ‘›β†’βˆžπ‘‘ξ€·π‘₯𝑝𝑛,π‘₯π‘žπ‘›ξ€Έ=πœ–.(2.12) On the other hand, we have 𝑑π‘₯𝑝𝑛,π‘₯π‘žπ‘›ξ€Έξ€·π‘₯≀𝑑𝑝𝑛,π‘₯π‘π‘›βˆ’1ξ€Έξ€·π‘₯+π‘‘π‘π‘›βˆ’1,π‘₯π‘žπ‘›βˆ’1ξ€Έξ€·π‘₯+π‘‘π‘žπ‘›βˆ’1,π‘₯π‘žπ‘›ξ€Έ,𝑑π‘₯π‘π‘›βˆ’1,π‘₯π‘žπ‘›βˆ’1ξ€Έξ€·π‘₯β‰€π‘‘π‘π‘›βˆ’1,π‘₯𝑝𝑛π‘₯+𝑑𝑝𝑛,π‘₯π‘žπ‘›ξ€Έξ€·π‘₯+π‘‘π‘žπ‘›,π‘₯π‘žπ‘›βˆ’1ξ€Έ.(2.13) Let π‘›β†’βˆž. Then we get limπ‘›β†’βˆžπ‘‘ξ€·π‘₯π‘π‘›βˆ’1,π‘₯π‘žπ‘›βˆ’1ξ€Έ=πœ–.(2.14) Using inequality (2.1), we have 𝑑π‘₯𝑝𝑛,π‘₯π‘žπ‘›ξ€Έξ€·=𝑑𝑓π‘₯π‘π‘›βˆ’1,𝑓π‘₯π‘žπ‘›βˆ’1𝑑π‘₯β‰€πœ™π‘π‘›βˆ’1,π‘₯π‘žπ‘›βˆ’1𝑑π‘₯ξ€Έξ€Έβˆ’πœ‘π‘π‘›βˆ’1,π‘₯π‘žπ‘›βˆ’1.ξ€Έξ€Έ(2.15) Letting π‘›β†’βˆž, using the definitions of the functions πœ™ and πœ‘, we have πœ–β‰€πœ–βˆ’πœ‘(πœ–),(2.16) which implies that πœ‘(πœ–)=0. By the definition of the function πœ‘, we have πœ–=0. So we get a contradiction. Therefore {π‘₯𝑛} is g.m.s Cauchy.
Step 3. We claim that 𝑓 has a periodic point in 𝑋.
Suppose, on contrary, 𝑓 has no periodic point. Then {π‘₯𝑛} is a sequence of distinct points, that is, π‘₯𝑝≠π‘₯π‘ž for all 𝑝,π‘žβˆˆβ„• with π‘β‰ π‘ž. By Step 2, since 𝑋 is complete g.m.s, there exists πœˆβˆˆπ‘‹ such that π‘₯π‘›β†’πœˆ. Using inequality (2.1), we have 𝑑𝑓π‘₯𝑛𝑑π‘₯,π‘“πœˆβ‰€πœ™π‘›ξ€·π‘‘ξ€·π‘₯,πœˆξ€Έξ€Έβˆ’πœ‘π‘›.,πœˆξ€Έξ€Έ(2.17) Letting π‘›β†’βˆž, we have 𝑑𝑓π‘₯𝑛,π‘“πœˆβŸΆ0,asπ‘›βŸΆβˆž,(2.18) that is, π‘₯𝑛+1=𝑓π‘₯π‘›βŸΆπ‘“πœˆ,asπ‘›βŸΆβˆž.(2.19) As (𝑋,𝑑) is Hausdorff, we have 𝜈=π‘“πœˆ, a contradiction with our assumption that 𝑓 has no periodic point. Therefore, there exists πœˆβˆˆπ‘‹ such that 𝜈=𝑓𝑝(𝜈) for some π‘βˆˆβ„•. So 𝑓 has a periodic point in 𝑋.

Following Theorem 2.1, it is easy to get the below fixed point result.

Theorem 2.2. Let (𝑋,𝑑) be a Hausdorff and complete g.m.s, and let π‘“βˆΆπ‘‹β†’π‘‹ be a function satisfying 𝑑(𝑓π‘₯,𝑓𝑦)β‰€πœ™(𝑑(π‘₯,𝑦))βˆ’πœ‘(𝑑(π‘₯,𝑦))(2.20) for all π‘₯,π‘¦βˆˆπ‘‹, where πœ™βˆˆΞ¦ with 0<πœ™(𝑑)<𝑑 for all 𝑑>0, and πœ‘βˆˆΞ˜. Then 𝑓 has a unique fixed point in 𝑋.

Proof. From Theorem 2.1, we conclude that 𝑓 has a periodic point πœˆβˆˆπ‘‹, that is, there exists πœˆβˆˆπ‘‹ such that 𝜈=𝑓𝑝(𝜈) for some π‘βˆˆβ„•. If 𝑝=1, then we complete the proof, that is, 𝜈 is a fixed point of 𝑓. If 𝑝>1, then we will show that πœ‡=π‘“π‘βˆ’1𝜈 is a fixed point of 𝑓. Suppose that it is not the case, that is, π‘“π‘βˆ’1πœˆβ‰ π‘“π‘πœˆ. Then Using inequality (2.1), we have 𝑑𝑓(𝜈,π‘“πœˆ)=π‘‘π‘πœˆ,𝑓𝑝+1πœˆξ€Έξ€·π‘‘ξ€·π‘“β‰€πœ™π‘βˆ’1𝜈,π‘“π‘πœˆξ€·π‘‘ξ€·π‘“ξ€Έξ€Έβˆ’πœ‘π‘βˆ’1𝜈,π‘“π‘πœˆξ€·π‘‘ξ€·π‘“ξ€Έξ€Έ<πœ™π‘βˆ’1𝜈,π‘“π‘πœˆξ€·π‘“ξ€Έξ€Έβ‰€π‘‘π‘βˆ’1𝜈,π‘“π‘πœˆξ€Έ.(2.21) Using inequality (2.1), we also have π‘‘ξ€·π‘“π‘βˆ’1𝜈,π‘“π‘πœˆξ€Έξ€·π‘‘ξ€·π‘“β‰€πœ™π‘βˆ’2𝜈,π‘“π‘βˆ’1πœˆξ€·π‘‘ξ€·π‘“ξ€Έξ€Έβˆ’πœ‘π‘βˆ’2𝜈,π‘“π‘βˆ’1πœˆξ€·π‘‘ξ€·π‘“ξ€Έξ€Έβ‰€πœ™π‘βˆ’2𝜈,π‘“π‘βˆ’1πœˆξ€·π‘“ξ€Έξ€Έβ‰€π‘‘π‘βˆ’2𝜈,π‘“π‘βˆ’1πœˆξ€Έ.(2.22) Continuing this process, we conclude that 𝑑𝑓(𝜈,π‘“πœˆ)<π‘‘π‘βˆ’1𝜈,π‘“π‘πœˆξ€Έξ€·π‘“β‰€π‘‘π‘βˆ’2𝜈,π‘“π‘βˆ’1πœˆξ€Έβ‰€β‹―β‰€π‘‘(𝜈,π‘“πœˆ),(2.23) which implies a contradiction. Thus, πœ‡=π‘“π‘βˆ’1𝜈 is a fixed point of 𝑓.
Finally, to prove the uniqueness of the fixed point, suppose πœ‡,𝜈 are fixed points of 𝑓. Then, 𝑑(πœ‡,𝜈)=𝑑(π‘“πœ‡,π‘“πœˆ)β‰€πœ™(𝑑(πœ‡,𝜈))βˆ’πœ‘(𝑑(πœ‡,𝜈)),(2.24) which implies that 𝑑(πœ‡,𝜈)=0, that is, πœ‡=𝜈. So we complete the proof.

Acknowledgment

This research was supported by the National Science Council of the Republic of China.

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