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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 642535, 5 pages
http://dx.doi.org/10.1155/2013/642535
Research Article

Strongly Almost Lacunary -Convergent Sequences

1Department of Mathematics and Institute for Mathematical Research, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia
2Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai, Maharashtra 400076, India

Received 16 July 2013; Revised 12 September 2013; Accepted 2 October 2013

Academic Editor: S. A. Mohiuddine

Copyright © 2013 Adem Kılıçman and Stuti Borgohain. 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 study some new strongly almost lacunary -convergent generalized difference sequence spaces defined by an Orlicz function. We give also some inclusion relations related to these sequence spaces.

1. Introduction

The notion of ideal convergence was first introduced by Kostyrko et al. [1] as a generalization of statistical convergence which was later studied by many other authors.

By a lacunary sequence, we mean an increasing integer sequence such that and = as .

Throughout this paper, the intervals determined by will be denoted by , and the ratio will be defined by .

An Orlicz function is a function , which is continuous, nondecreasing, and convex with , , for and , as .

Let , , and be the Banach space of bounded, convergent, and null sequences , respectively, with the usual norm .

A sequence is said to be almost convergent if all of its Banach limits coincide. Let denote the space of all almost convergent sequences.

Lorentz [2] introduced the following sequence space where = .

The following space of strongly almost convergent sequence was introduced by Maddox [3]: where .

Kızmaz [4] studied the difference sequence spaces , , and of crisp sets. The notion is defined as follows: for , and , where = , for all .

The above spaces are Banach spaces, normed by

Tripathy et al. [5] introduced the generalized difference sequence spaces which are defined as, for and ,

This generalized difference has the following binomial representation:

2. Definitions and Preliminaries

Kostyrko et al. [1] introduced the following three definitions.

Let be a nonempty set. Then a family of sets (power sets of ) is said to be if is additive, that is, , and hereditary, that is, .

A sequence in a normed space is said to be I-convergent to if for each , the set

A sequence in a normed space is said to be I-bounded if there exists such that the set belongs to .

Freedman et al. [6] defined the space . For any lacunary sequence ,

The space is a space with the norm

The notion of lacunary ideal convergence of real sequences introduced by Tripathy et al. in [7, 8] and Hazarika [9, 10] introduced the lacunary ideal convergent sequences of fuzzy real numbers and studied some properties. In [5, 7], the lacunary ideal convergence is defined as follows.

Let be a lacunary sequence. Then a sequence is said to be lacunary -convergent if for every , such that

we write .

Lindenstrauss and Tzafriri [11] used the idea of Orlicz function to construct the sequence space:

The space with the norm becomes a Banach space which is called an Orlicz sequence space.

In this paper, we defined some new generalized difference lacunary -convergent sequence spaces defined by Orlicz function. We also introduce and examine some new sequence spaces and study their different properties.

3. Main Results

Esi [12] introduced the strongly almost ideal convergent sequence spaces in 2-normed spaces. In this paper we introduced the strongly almost lacunary ideal convergent sequence spaces using generalized difference operator and Orlicz function.

Let be an admissible ideal of ,   an Orlicz function, and a lacunary sequence. Further, let be a bounded sequence of positive real numbers and a generalized difference operator.

For every and for some , we have introduced the following sequence spaces:

Particular Cases. Consider the following.(1)If , we have , , and .(2)If , then , , and .(3)If for all , , and , then , , and .

Theorem 1. Let the sequence be bounded; then .

Proof. Let . Then, for some , we have where and .
Hence, .
The inclusion is obvious.

Theorem 2. Let the sequence be bounded; then , , and are closed under the operations of addition and scalar multiplication.

Theorem 3. Let be Orlicz functions; then we have(1),(2), (3).

Theorem 4. Let for all , and let be bounded; then we have .

Theorem 5. Let be a lacunary sequence with . Then, for any Orlicz function , .

Proof. Suppose then there exists such that for all .
Then, for , we have
Let
Since , we have and .
So, for and for some ,
Hence, .
Next, suppose that . Then, there exists , such that, for all .
Let and . There exists such that for every ,
Let such that for all . Now let be any integer with , where . Then,
Since as , it follows that
Hence, .

Theorem 6. If and is strongly almost lacunary convergent to , with respect to the Orlicz function , that is, , then is unique.

Proof. Let and suppose that , .
Then there exist and such that
Let . Then we have where and .
Thus, from (21), we get
Further, as , and, therefore,
Hence, .

4. Conclusion

The concept of lacunary -convergence has been studied by various mathematicians. In this paper, we have introduced some fairly wide classes of strongly almost lacunary -convergent sequences of real numbers using Orlicz function with the generalized difference operator. Giving particular values to the sequence and , we obtain some new sequence spaces which are the special cases of the sequence spaces we have defined. There are lots more to be investigated in the future.

Acknowledgments

First of all, the authors sincerely thank the referees for the valuable comments. The first author gratefully acknowledges that part of this research was partially supported by the University Putra Malaysia under the ERGS Grant Scheme having Project no. 5527068. The work of the second author was carried under the Postdoctoral Fellow under National Board of Higher Mathematics, DAE (Government of India), Project no. NBHM/PDF.50/2011/64.

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