On some -convergent difference sequence spaces of fuzzy numbers defined by the sequence of Orlicz functions
© Savas; licensee Springer 2012
Received: 9 October 2011
Accepted: 9 October 2012
Published: 6 November 2012
In this paper, using the difference operator of order m, the sequences of Orlicz functions, and an infinite matrix, we introduce and examine some classes of sequences of fuzzy numbers defined by I-convergence. We study some basic topological and algebraic properties of these spaces. In addition, we shall establish inclusion theorems between these sequence spaces.
MSC:40A05, 40G15, 46A45.
The notion of ideal convergence was introduced first by Kostyrko et al.  as a generalization of statistical convergence [2, 3], which was further studied in topological spaces . More applications of ideals can be seen in [5–9].
The concepts of fuzzy sets and fuzzy set operations were first introduced by Zadeh . Subsequently, several authors have discussed various aspects of the theory and applications of fuzzy sets such as fuzzy topological spaces, similarity relations and fuzzy ordering, fuzzy measures of fuzzy events, and fuzzy mathematical programming. In particular, the concept of fuzzy topology has very important applications in quantum particle physics, especially in connection with both string and theory, which were given and studied by El Naschie . The theory of sequences of fuzzy numbers was first introduced by Matloka . Matloka introduced bounded and convergent sequences of fuzzy numbers, studied some of their properties, and showed that every convergent sequence of fuzzy numbers is bounded. In , Nanda studied sequences of fuzzy numbers and showed that the set of all convergent sequences of fuzzy numbers forms a complete metric space. Different classes of sequences of fuzzy real numbers have been discussed by Nuray and Savas , Altinok, Colak, and Et , Savas [16–20], Savas and Mursaleen , and many others.
The study of Orlicz sequence spaces was initiated with a certain specific purpose in Banach space theory. Lindenstrauss and Tzafriri  investigated Orlicz sequence spaces in more detail, and they proved that every Orlicz sequence space contains a subspace isomorphic to (). The Orlicz sequence spaces are the special cases of Orlicz spaces studied in . Orlicz spaces find a number of useful applications in the theory of nonlinear integral equations. Although the Orlicz sequence spaces are the generalization of spaces, the -spaces find themselves enveloped in Orlicz spaces . Recently, Savas  generalized and for a single sequence of fuzzy numbers by using the Orlicz function and also established some inclusion theorems.
Throughout the article, denotes the class of all fuzzy real-valued sequence spaces. Also, N and R denote the set of positive integers and the set of real numbers, respectively.
In this paper, we study some new sequence spaces of fuzzy numbers using I-convergence, the sequence of Orlicz functions, an infinite matrix, and the difference operator. We establish the inclusion relation between the sequence spaces , , , and , where denotes the sequence of positive real numbers for all and is a sequence of Orlicz functions. In addition, we study some algebraic and topological properties of these new spaces.
2 Definitions and notations
Before continuing with this paper, we present some definitions and preliminaries which we shall use throughout this paper.
Let X and Y be two nonempty subsets of the space w of complex sequences. Let () be an infinite matrix of complex numbers. We write if converges for each n. (Throughout, denotes summation over k from to ). If , we say that A defines a (matrix) transformation from X to Y and we denote it by .
Let X be a nonempty set, then a family of sets (the class of all subsets of X) is called an ideal if and only if for each , we have , and for each and each , we have . A nonempty family of sets is a filter on X if and only if , for each , we have , and for each and each , we have . An ideal I is called non-trivial ideal if and . Clearly, is a non-trivial ideal if and only if is a filter on X. A non-trivial ideal is called admissible if and only if . A non-trivial ideal I is maximal if there cannot exist any non-trivial ideal containing I as a subset. Further details on ideals of can be found in Kostyrko et al. .
The additive identity and multiplicative identity of are defined by and respectively.
A metric on is said to be a translation invariant if for .
The proof is easy and so it is omitted.
A sequence of fuzzy numbers is said to converge to a fuzzy number if for every , there exists a positive integer such that for all .
The fuzzy number is called I-limit of the sequence of fuzzy numbers, and we write .
Example 2.1 If we take , then is a non-trivial admissible ideal of N, and the corresponding convergence coincides with the usual convergence.
Example 2.2 If we take , where denotes the asymptotic density of the set A, then is a non-trivial admissible ideal of N, and the corresponding convergence coincides with the statistical convergence.
Lemma 2.1 (Kostyrko, Salat, and Wilczynski , Lemma 5.1)
If is a maximal ideal, then for each , we have either or .
Recall in  that the Orlicz function is a continuous, convex, non-decreasing function such that and for , and as . If convexity of the Orlicz function is replaced by , then this function is called the modulus function and characterized by Ruckle . An Orlicz function M is said to satisfy the -condition for all values of u, if there exists such that , .
for all and . Also, for all .
3 Some new sequence spaces of fuzzy numbers
If , then the above classes of sequences are denoted by , , , and , respectively.
If for all , then the above classes of sequences are denoted by , , , and , respectively.
If , then we denote the above spaces by , , , and .
If we take , i.e., the Cesàro matrix, then the above classes of sequences are denoted by , , , and , respectively.
- (v)If we take is a de la Valée Poussin mean, i.e.,
- (vi)By a lacunary , where , we shall mean an increasing sequence of non-negative integers with as . The intervals determined by θ will be denoted by and . As a final illustration, let
4 Main results
In this section, we examine the basic topological and algebraic properties of the new sequence spaces and obtain the inclusion relation related to these spaces.
Theorem 4.1 Let be a bounded sequence. Then the sequence spaces , , and are linear spaces.
Proof We will prove the result for the space only and others can be proved in a similar way.
This completes the proof. □
Let , where , , and let as . To prove that as , let , where and as .
Note that for all .
Hence, by our assumption, the right-hand side tends to 0 as . This completes the proof of the theorem.
Theorem 4.3 Let I be an admissible ideal and be a sequence of Orlicz functions. Then the following hold:
; ; for and the inclusions are strict.
In general, for all , the following hold:
; ; and the inclusions are strict.
The inclusion is strict, it follows from the following example.
respectively. It is easy to see that the sequence is not I-bounded although is I-bounded.
Theorem 4.4 (a) Let . Then
(b) Let . Then
- (b)Let be an element in . Since , then for each , there exists a positive integer such that
The other part can be proved in a similar way.
The following corollary follows immediately from the above theorem.
Corollary 4.5 Let , i.e., the Cesàro matrix, and be a sequence of Orlicz functions.
(a) Let . Then
(b) Let . Then
- Kostyrko P, Salat T, Wilczyński W: I -Convergence. Real Anal. Exch. 2000, 26(2):669–686.Google Scholar
- Fast H: Sur la convergence statistique. Colloq. Math. 1951, 2: 241–244.MathSciNetGoogle Scholar
- Steinhaus H: Sur la convergence ordinarie et la convergence asymptotique. Colloq. Math. 1951, 2: 73–74.MathSciNetGoogle Scholar
- Pratulananda D, Lahiri BK: I and convergence in topological spaces. Math. Bohem. 2005, 130(2):153–160.MathSciNetGoogle Scholar
- Pratulananda D, Kostyrko P, Wilczyński W, Malik P: I and -convergence of double sequences. Math. Slovaca 2008, 58: 605–620. 10.2478/s12175-008-0096-xMathSciNetGoogle Scholar
- Savas E, Pratulananda D: A generalized statistical convergence via ideals. Appl. Math. Lett. 2011. doi:10.1016/j.aml.2010.12.022Google Scholar
- Savas E:-strongly summable sequences spaces in 2-normed spaces defined by ideal convergence and an Orlicz function. Appl. Math. Comput. 2010, 217: 271–276. 10.1016/j.amc.2010.05.057MathSciNetView ArticleGoogle Scholar
- Savas E: A-sequence spaces in 2-normed space defined by ideal convergence and an Orlicz function. Abstr. Appl. Anal. 2011., 2011: Article ID 741382Google Scholar
- Savas E: On some new sequence spaces in 2-normed spaces using Ideal convergence and an Orlicz function. J. Inequal. Appl. 2010., 2010: Article ID 482392. doi:10.1155/2010/482392Google Scholar
- Zadeh LA: Fuzzy sets. Inf. Control 1965, 8: 338–353. 10.1016/S0019-9958(65)90241-XMathSciNetView ArticleGoogle Scholar
- El Naschie MS:A review of , theory and the mass spectrum of high energy particle physics. Chaos Solitons Fractals 2004, 19(1):209–236. 10.1016/S0960-0779(03)00278-9View ArticleGoogle Scholar
- Matloka M: Sequences of fuzzy numbers. BUSEFAL 1986, 28: 28–37.Google Scholar
- Nanda S: On sequences of fuzzy numbers. Fuzzy Sets Syst. 1989, 33: 123–126. 10.1016/0165-0114(89)90222-4MathSciNetView ArticleGoogle Scholar
- Nuray F, Savas S: Statistical convergence of sequences of fuzzy numbers. Math. Slovaca 1995, 45: 269–273.MathSciNetGoogle Scholar
- Altinok H, Colak R, Et M: λ -Difference sequence spaces of fuzzy numbers. Fuzzy Sets Syst. 2009, 160: 3128–3139. 10.1016/j.fss.2009.06.002MathSciNetView ArticleGoogle Scholar
- Savaş E: A note on sequence of fuzzy numbers. Inf. Sci. 2000, 124: 297–300. 10.1016/S0020-0255(99)00073-0View ArticleGoogle Scholar
- Savaş E: On strongly λ -summable sequences of fuzzy numbers. Inf. Sci. 2000, 125: 181–186. 10.1016/S0020-0255(99)00151-6View ArticleGoogle Scholar
- Savaş E: On statistically convergent sequence of fuzzy numbers. Inf. Sci. 2001, 137: 272–282.Google Scholar
- Savaş E: Difference sequence spaces of fuzzy numbers. J. Fuzzy Math. 2006, 14(4):967–975.MathSciNetGoogle Scholar
- Savaş E: On lacunary statistical convergent double sequences of fuzzy numbers. Appl. Math. Lett. 2008, 21: 134–141. 10.1016/j.aml.2007.01.008MathSciNetView ArticleGoogle Scholar
- Savaş E, Mursaleen M: On statistically convergent double sequence of fuzzy numbers. Inf. Sci. 2004, 162: 183–192. 10.1016/j.ins.2003.09.005View ArticleGoogle Scholar
- Lindenstrauss J, Tzafriri L: On Orlicz sequence spaces. Isr. J. Math. 1971, 101: 379–390.MathSciNetView ArticleGoogle Scholar
- Krasnoselskii MA, Rutitsky YB: Convex Functions and Orlicz Functions. Noordhoff, Groningen; 1961.Google Scholar
- Kamthan PK, Gupta M: Sequence Spaces and Series. Marcel Dekker, New York; 1980.Google Scholar
- Parashar SD, Choudhury B: Sequence space defined by Orlicz functions. Indian J. Pure Appl. Math. 1994, 25(14):419–428.MathSciNetGoogle Scholar
- Savaş E, Patterson RF: An Orlicz extension of some new sequence spaces. Rend. Ist. Mat. Univ. Trieste 2005, 37(1–2):145–154. (2006)Google Scholar
- Savaş E, Savas R: Some sequence spaces defined by Orlicz functions. Arch. Math. 2004, 40(1):33–40.MathSciNetGoogle Scholar
- Savas E, Patterson RF: Some double lacunary sequence spaces defined by Orlicz functions. Southeast Asian Bull. Math. 2011, 35(1):103–110.MathSciNetGoogle Scholar
- Savas E: On some new double lacunary sequences spaces via Orlicz function. J. Comput. Anal. Appl. 2009, 11(3):423–430.MathSciNetGoogle Scholar
- Kaleva O, Seikkala S: On fuzzy metric spaces. Fuzzy Sets Syst. 1984, 12: 215–229. 10.1016/0165-0114(84)90069-1MathSciNetView ArticleGoogle Scholar
- Ruckle WH: FK-spaces in which the sequence of coordinate vectors is bounded. Can. J. Math. 1973, 25: 973–978. 10.4153/CJM-1973-102-9MathSciNetView ArticleGoogle Scholar
- Hazarika B, Savaş E: Some I -convergent lamda-summable difference sequence spaces of fuzzy real numbers defined by a sequence of Orlicz. Math. Comput. Model. 2011, 54: 2986–2998. 10.1016/j.mcm.2011.07.026View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.