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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 907951, 5 pages
http://dx.doi.org/10.1155/2012/907951
Review Article

Quasi-Contractive Mappings in Modular Metric Spaces

1Department of Mathematics Education and RINS, Gyeongsang National University, jinju 660-701, Republic of Korea
2Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
3Department of Mathematics and Computer Sciences, Sabzevar Tarbiat Moallem University, Sabzevar, Iran

Received 6 October 2011; Accepted 30 November 2011

Academic Editor: Rudong Chen

Copyright © 2012 Yeol J. E. Cho 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 the existence of fixed point and uniqueness of quasi-contractive mappings in modular metric spaces which was introduced by Ćirić

1. Introduction and Preliminaries

In this paper, we prove the existence and uniqueness of fixed points of quasi-contractive mappings in modular metric spaces which develop the theory of metric spaces generated by modulars. Throughout the paper 𝔛 is a nonempty set and 𝜆>0. The notion of a metric modular was introduced by Chistyakov [1] as follows.

Definition 1.1. A function 𝜔(0,)×𝔛×𝔛[0,] is said to be a metric modular on 𝔛 (or, simply, a modular if no ambiguity arises) if it satisfies three axioms:(i)for any 𝑥,𝑦𝔛, 𝜔𝜆(𝑥,𝑦)=0 for all 𝜆>0 if and only if 𝑥=𝑦;(ii)𝜔𝜆(𝑥,𝑦)=𝜔𝜆(𝑦,𝑥) for all 𝜆>0, and 𝑥,𝑦𝔛;(iii)𝜔𝜆+𝜇(𝑥,𝑦)𝜔𝜆(𝑥,𝑧)+𝜔𝜇(𝑧,𝑦) for all 𝜆,𝜇>0 and 𝑥,𝑦𝔛.

Definition 1.2. Let (𝔛,𝜔) be a metric modular space.(1)A sequence {𝑥𝑛} in 𝔛𝜔 is said to be 𝜔-convergent to a point 𝑥𝔛 if, for all 𝜆>0, 𝜔𝜆𝑥𝑛,𝑥0(1.1) as 𝑛.(2)A subset of 𝔛𝜔 is said to be 𝜔-closed if the 𝜔-limit of a 𝜔-convergent sequence of always belongs to .(3)A subset of 𝔛𝜔 is said to be 𝜔-complete if every 𝜔-Cauchy sequence in is 𝜔-convergent and its 𝜔-limit is in .

Definition 1.3. The metric modular 𝜔 is said to have the Fatou property if 𝜔𝜆(𝑥,𝑦)liminf𝑛𝜔𝑥𝑛,𝑦(1.2) for all 𝑦𝔛𝜔 and 𝜆(0,), where {𝑥𝑛}𝜔-converges to 𝑥.

2. Main Results

Definition 2.1. Let (𝔛,𝜔) be a metric modular space, and let be a nonempty subset of 𝔛𝜔. The self-mapping 𝑇 is said to be quasi-contraction if there exists 0<𝑘<1 such that 𝜔𝜆𝜔(𝑇(𝑥),𝑇(𝑦))𝑘max𝜆(𝑥,𝑦),𝜔𝜆(𝑥,𝑇(𝑥)),𝜔𝜆(𝑦,𝑇(𝑦)),𝜔𝜆(𝑥,𝑇(𝑦)),𝜔𝜆(𝑇(𝑥),𝑦)(2.1) for any 𝑥,𝑦𝔛 and 𝜆(0,).

Let 𝑇 be a mapping, and let be a nonempty subset of 𝔛𝜔. For any 𝑥, define the orbit𝒪(𝑥)=𝑥,𝑇(𝑥),𝑇2(𝑥),(2.2) and its 𝜔-diameter by𝛿𝜔𝜔(𝑥)=diam(𝒪(𝑥))=sup𝜆(𝑇𝑛(𝑥),𝑇𝑚(𝑥))𝑛,𝑚).(2.3)

Lemma 2.2. Let (𝔛,𝜔) be a metric modular space, and let be a nonempty subset of 𝔛𝜔. Let 𝑇 be a quasi-contractive mapping, and let 𝑥 be such that 𝛿𝜔(𝑥)<. Then, for any 𝑛1, one has 𝛿𝜔(𝑇(𝑥))𝑘𝑛𝛿𝜔(𝑥),(2.4) where 𝑘 is the constant associated with the mapping of 𝑇. Moreover, one has 𝜔𝜆𝑇𝑛(𝑥),𝑇𝑛+𝑚(𝑥)𝑘𝑛𝛿𝜔(𝑥)(2.5) for any 𝑛,𝑚1 and 𝜆(0,).

Proof. For each 𝑛,𝑚1, we have 𝜔𝜆(𝑇𝑛(𝑥),𝑇𝑚𝜔(𝑦))𝑘max𝜆𝑇𝑛1(𝑥),𝑇𝑚1(𝑦),𝜔𝜆𝑇𝑛1(𝑥),𝑇𝑛,𝜔(𝑥)𝜆𝑇𝑚1(𝑦),𝑇𝑚(𝑦),𝜔𝜆𝑇𝑛1(𝑥),𝑇𝑚(𝑦),𝜔𝜆𝑇𝑛(𝑥),𝑇𝑚1(𝑦)(2.6) for any 𝑥,𝑦 and 𝜆(0,). This obviously implies that 𝛿𝜔(𝑇𝑛(𝑥))𝑘𝛿𝜔𝑇𝑛1(𝑥)(2.7) for any 𝑛1. Hence, for any 𝑛1, we have 𝛿𝜔(𝑇𝑛(𝑥))𝑘𝑛𝛿𝜔(𝑥).(2.8) Moreover, for any 𝑛,𝑚1, we have 𝜔𝜆(𝑇𝑛(𝑥),𝑇𝑛+𝑚(𝑥)𝛿𝜔(𝑇𝑛(𝑥))𝑘𝑛𝛿𝜔(𝑥).(2.9) This completes the proof.

The next lemma is helpful to prove the main result in this paper.

Lemma 2.3. Let (𝔛,𝜔) be a modular metric space, and let be a 𝜔-complete nonempty subset of 𝔛𝜔. Let 𝑇 be quasi-contractive mapping, and let 𝑥 be such that 𝛿𝜔(𝑥)<. Then {𝑇𝑛(𝑥)}𝜔-converges to a point 𝜈. Moreover, one has 𝜔𝜆(𝑇𝑛(𝑋)𝜈)𝑘𝑛𝛿𝜔(𝑥)(2.10) for all 𝑛1 and 𝜆(0,).

Proof. From Lemma 2.2, we know that {𝑇𝑛(𝑥)} is a 𝜔-Cauchy sequence in . Since is 𝜔-complete, then there exists 𝜈 such that {𝑇𝑛(𝑥)}𝜔-converges to 𝜈. Since 𝜔𝜆𝑇𝑛(𝑥),𝑇𝑛+𝑚(𝑥)𝑘𝑛𝛿𝜔(𝑥)(2.11) for any 𝑛,𝑚1 and 𝜔 satisfies the Fatou property, and letting 𝑚, we have 𝜔𝜆(𝑇𝑛(𝑥),𝜈)liminf𝑚𝜔𝜆𝑇𝑛(𝑥),𝑇𝑛+𝑚(𝑥)𝑘𝑛𝛿𝜔(𝑥).(2.12) This completes the proof.

Next, we prove that 𝜈 is, in fact, a fixed point of 𝑇 and it is unique provided some extra assumptions.

Theorem 2.4. Let 𝑇,, and 𝑥 be as in Lemma 2.3. Suppose that 𝜔𝜆(𝜈,𝑇(𝜈))< and 𝜔𝜆(𝑥,𝑇(𝑥))< for all 𝜆(0,). Then the 𝜔-limit of {𝑇𝑛(𝑥)} is a fixed point of 𝑇, that is, 𝑇(𝜈)=𝜈. Moreover, if 𝜈 is any fixed point of 𝑇 in such that 𝜔𝜆(𝜈,𝜈)< for all 𝜆(0,), then one has 𝜈=𝜈.

Proof. We have 𝜔𝜆𝜔(𝑇(𝑥),𝑇(𝜈))𝑘max𝜆(𝑥,𝜈),𝜔𝜆(𝑥,𝑇(𝑥)),𝜔𝜆(𝜈,𝑇(𝜈)),𝜔𝜆(𝑥,𝑇(𝜈)),𝜔𝜆(𝑇(𝑥),𝜈).(2.13) From Lemma 2.3, it follows that 𝜔𝜆𝛿(𝑇(𝑥),𝑇(𝜈))𝑘max𝜔(𝑥),𝜔𝜆(𝜈,𝑇(𝜈)),𝜔𝜆(𝑥,𝑇(𝜈)).(2.14) Suppose that, for each 𝑛1, 𝜔𝜆(T𝑛𝑘(𝑥),𝑇(𝜈))max𝑛𝛿𝜔(𝑥),𝑘𝜔𝜆(𝜈,𝑇(𝜈)),𝑘𝑛𝜔𝜆(𝑥,𝑇(𝜈)).(2.15) Then we have 𝜔𝜆𝑇𝑛+1𝜔(𝑥),𝑇(𝜈)𝑘max𝜆(𝑇𝑛(𝑥),𝜈),𝜔𝜆𝑇𝑛(𝑥),𝑇𝑛+1(𝑥),𝜔𝜆(𝜈,𝑇(𝜈)),𝜔𝜆(𝑇𝑛(𝑥),𝑇(𝜈)),𝜔𝜆𝑇𝑛+1.(𝑥),𝜈(2.16) Hence we have 𝜔𝜆𝑇𝑛+1𝑘(𝑥),𝑇(𝜈)𝑘max𝑛𝛿𝜔(𝑥),𝑘𝜔𝜆(𝜈,𝑇(𝜈)),𝜔𝜆(𝑇𝑛.(𝑥),𝑇(𝜈))(2.17) Using our previous assumption, we get 𝜔𝜆𝑇𝑛+1𝑘(𝑥),𝑇(𝜈)max𝑛+1𝛿𝜔(𝑥),𝑘𝜔𝜆(𝜈,𝑇(𝜈)),𝑘𝑛+1𝜔𝜆.(𝑥,𝑇(𝜈))(2.18) Thus, by induction, we have 𝜔𝜆(𝑇𝑛𝑘(𝑥),𝑇(𝜈))max𝑛𝛿𝜔(𝑥),𝑘𝜔𝜆(𝜈,𝑇(𝜈)),𝑘𝑛𝜔𝜆(𝑥,𝑇(𝜈))(2.19) for any 𝑛1 and 𝜆(0,). Therefore, we have limsup𝑛𝜔𝜆(𝑇𝑛(𝑥),𝑇(𝑥))𝜔(𝜈,𝑇(𝜈))(2.20) for all 𝜆(0,). Using the Fatou property for the metric modular 𝜔, we get 𝜔𝜆(𝜈,𝑇(𝜈))liminf𝑛𝜔𝜆(𝑇𝑛(𝑥),𝑇(𝜈))𝑘𝜔(𝜈,𝑇(𝜈))(2.21) for all 𝜆(0,). Since 𝑘<1, we get 𝜔𝜆(𝜈,𝑇(𝜈))=0 for all 𝜆(0,), and so 𝑇(𝜈)=𝜈.
Let 𝜈 be another fixed point of 𝑇 such that 𝜔𝜆(𝜈,𝜈)< for all 𝜆(0,). Then we have 𝜔𝜆𝜈,𝜈=𝜔𝜆𝑇𝜈(𝜈),𝑇𝑘𝜔𝜆𝜈,𝜈,(2.22) which implies that 𝜔𝜆𝜈,𝜈=0(2.23) for all 𝜆(0,). Hence 𝜈=𝜈. This complete the proof.

Acknowledgments

The authors are thankful to the anonymous referees and the area editor Professor Rudong Chen for their critical remarks which helped greatly to improve the presentation of this paper. The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2011-0021821).

References

  1. V. V. Chistyakov, “Modular metric spaces I: basic concepts,” Nonlinear Analysis, vol. 74, pp. 1–14, 2010. View at Publisher · View at Google Scholar