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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 907951, 5 pages
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.
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 is a nonempty set and . The notion of a metric modular was introduced by Chistyakov  as follows.
Definition 1.1. A function is said to be a metric modular on (or, simply, a modular if no ambiguity arises) if it satisfies three axioms:(i)for any , for all if and only if ;(ii) for all , and ;(iii) for all and .
Definition 1.2. Let be a metric modular space.(1)A sequence in is said to be -convergent to a point if, for all , as .(2)A subset of is said to be -closed if the -limit of a -convergent sequence of always belongs to .(3)A subset of is said to be -complete if every -Cauchy sequence in is -convergent and its -limit is in .
Definition 1.3. The metric modular is said to have the Fatou property if for all and , where -converges to .
2. Main Results
Definition 2.1. Let be a metric modular space, and let be a nonempty subset of . The self-mapping is said to be quasi-contraction if there exists such that for any and .
Let be a mapping, and let be a nonempty subset of . For any , define the orbit and its -diameter by
Lemma 2.2. Let be a metric modular space, and let be a nonempty subset of . Let be a quasi-contractive mapping, and let be such that . Then, for any , one has where is the constant associated with the mapping of . Moreover, one has for any and .
Proof. For each , we have for any and . This obviously implies that for any . Hence, for any , we have Moreover, for any , we have This completes the proof.
The next lemma is helpful to prove the main result in this paper.
Lemma 2.3. Let be a modular metric space, and let be a -complete nonempty subset of . Let be quasi-contractive mapping, and let be such that . Then -converges to a point . Moreover, one has for all and .
Proof. From Lemma 2.2, we know that is a -Cauchy sequence in . Since is -complete, then there exists such that -converges to . Since for any and satisfies the Fatou property, and letting , we have This completes the proof.
Next, we prove that is, in fact, a fixed point of and it is unique provided some extra assumptions.
Theorem 2.4. Let , and be as in Lemma 2.3. Suppose that and for all . Then the -limit of is a fixed point of , that is, . Moreover, if is any fixed point of in such that for all , then one has .
Proof. We have
From Lemma 2.3, it follows that
Suppose that, for each ,
Then we have
Hence we have
Using our previous assumption, we get
Thus, by induction, we have
for any and . Therefore, we have
for all . Using the Fatou property for the metric modular , we get
for all . Since , we get for all , and so .
Let be another fixed point of such that for all . Then we have which implies that for all . Hence . This complete the proof.
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).
- V. V. Chistyakov, “Modular metric spaces I: basic concepts,” Nonlinear Analysis, vol. 74, pp. 1–14, 2010.