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The Scientific World Journal
Volume 2013 (2013), Article ID 205275, 4 pages
Some Analogies of the Banach Contraction Principle in Fuzzy Modular Spaces
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand
Received 5 April 2013; Accepted 29 April 2013
Academic Editors: J. Fernández, J. Liu, K. Sadarangani, and H. Xu
Copyright © 2013 Kittipong Wongkum 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 established some theorems under the aim of deriving variants of the Banach contraction principle, using the classes of inner contractions and outer contractions, on the structure of fuzzy modular spaces.
1. Introduction and Preliminaries
The concept of a modular space was introduced by Nakano . Soon after, Musielak and Orlicz  redefined and generalized the notation of modular space. After that, Kumam et al. [3–7] studied fixed points and some properties in modular spaces. In 2007, Nourouzi  proposed probabilistic modular spaces based on the theory of modular spaces and some researches on the Menger's probabilistic metric spaces. A pair is called a probabilistic modular space if is a real vector space and is a mapping from into the set of all distribution functions (for , the distribution function is denoted by , and is the value at ) satisfying the following conditions: (PM1) ; (PM2) for all if and only if ; (PM3) ; (PM4) for all and , .For every , and , if where , then we say that is -homogeneous.
Recently, further studies have been made on the probabilistic modular spaces. Nourouzi  extended the well-known Baire's Theorem to probabilistic modular spaces by using a special condition. Nourouzi  investigated the continuity and boundedness of linear operators defined between probabilistic modular spaces in the probabilistic sense. After that, Shen and Chen  following the idea of probabilistic modular space and the definition of fuzzy metric space in the sense of George and Veeramani , applied fuzzy concept to the classical notions of modular and modular spaces, and in 2013, Shen and Chen  introduced the concept of a fuzzy modular space.
Definition 1 (Vasuki ). A fuzzy set in is a function with domain and value in .
Definition 2 (Arul Selvaraj and Sivakumar ). A triangular norm is a function satisfying, for each , the following conditions: (1); (2) whenever , ; (3) and .
Definition 3 (Shen and Chen ). Let be a real or complex vector space with a zero , a continuous triangular norm, and a fuzzy set on the product . Suppose that the following properties hold for and : (FM1) ; (FM2) for all if and only if ; (FM3) ; (FM4) whenever is the convex combination between and ; (FM5) the mapping is continuous at each fixed . Then, we write to represent the space with the pre-defined properties. In particular, we call a fuzzy modular and the triple a fuzzy modular space.
It is worth noting that every fuzzy modular is non-decreasing with respect to .
Example 4. Let be a real or complex vector space and be a modular on . Take the -norm = . For every , define for all . Then is a -modular space.
Remark 5. Note that the above conclusion still holds even if the -norm is replaced by and , respectively.
Shen and Chen  also studied the topological properties of a fuzzy modular space with a special property that for every and a non-zero real , the equality holds for some fixed . If the fuzzy modular has this property, we shall say that it is -homogeneous.
The -ball in is the set of the form where and . Now, suppose that is -homogeneous for some . According to Shen and Chen , the family of all -balls forms a base for a first-countable Hausdorff topology, written as . With the notion of the -balls, it is easy to see that a sequence in -converges (i.e., it converges in the topology ) to its -limit if and only if as for all . Note here that the -limit is unique if it does exists after all. It is then natural to say that is -Cauchy if for any given and , there exists with whenever .
At this point, let us turn to a typical example of a triangular norm which is defined by . This triangular norm has a very special property that if is an arbitrary triangular norm, then for all . With this property, it is suitable to call this a strongest triangular norm. As is claimed by Shen and Chen , if is a real vector space equipped with a -homogeneous fuzzy modular and a strongest triangular norm , then a -convergent sequence is -Cauchy. The authors also mentioned that if is not the strongest one, such implementation is not always true.
The space is said to be -complete if -Cauchy sequences actually -converges. We note that it makes more sense if we require a complete fuzzy modular space to be equipped with the strongest triangular norm.
A mapping from into itself is said to be an inner contraction if there exists a positive constant such that for all and .
On the other hand, is said to be an outer contraction if there exists a positive constant such that for all and .
In this paper, we shall be working under the aim of deriving some variants of the Banach contraction principle, by using the classes of mappings defined above, on the structure of fuzzy modular spaces.
2. Main Results
We divide this section into two parts, discussing independently about the two main categories of our contractions predefined in the previous section. Notice that it is clear from the definition that every fuzzy metric space is in turn a fuzzy modular space. Hence, our results are also supplied with corollaries in fuzzy metric spaces. We shall, however, omit such consequences since they are obvious.
2.1. Fixed-Point Theorem for an Inner Contraction
Theorem 6. Let be a real vector space equipped with a -homogeneous fuzzy modular and the strongest triangular norm such that is -complete. Suppose that at each , as . If : is an inner contraction with constant , then has a unique fixed point.
Proof. Given a point , we suppose that for all . Let , observe that
As , we have for every . That is, for any given and , there exists such that
We now claim to show by induction that for all . Let us assume first that holds at some . Observe that We have thus proved our claim.
Next, we shall show that is Cauchy. Let and be arbitrary, and we choose according to the claim given above. For , we may write and , for some . Now, consider that Thus, is Cauchy, and so the -completeness yields that for some . It follows that This means , since is Hausdorff. To show that the fixed point of is unique, assume that is a fixed point of as well. Finally, we obtain that Therefore, it must be the case that , and so the conclusion is fulfilled.
2.2. Fixed-Point Theorem for an Outer Contraction
For this part, we consider a weaker form of a -Cauchy sequence, namely, a -G-Cauchy sequence. This concept has been used all along together with the notion of fuzzy spaces. For a fuzzy modular space , is called a --Cauchy sequence if for each fixed and , we have . If every --Cauchy sequence -converges, is said to be -G-complete. It is to be noted that the notion of -G-completeness is slightly stronger than the ordinary completeness. It is enough to see that every -Cauchy sequence is also a -G-Cauchy sequence. For our result, it is still a question whether or not the -G-completeness assumption can be weakened.
Theorem 7. Let be a real vector space equipped with a -homogeneous fuzzy modular and the strongest triangular norm such that is --complete. If is an outer contraction with constant , then has a unique fixed point.
Proof. Given a point , we suppose that for all . By the definition of an outer contraction, we can rewrite this notion in the following:
for all and .
Let , observe that As , we have Next, we shall show that is Cauchy. Let and be arbitrary, and we choose . For , and for each . Now, consider that Thus, the sequence is Cauchy, and so the -completeness yields that for some . It follows that Taking , we have This means , since is Hausdorff. To show that the fixed point of is unique, assume that is a fixed point of as well. Finally, we obtain that Hence, we have which implies that . Therefore, it must be the case that , and so the conclusion is fulfilled.
The authors would like to thank the Higher Education Research Promotion and National Research University Project of Thailand's Office of the Higher Education Commission for financial support (under NRU-CSEC Project no. 56000508).
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