International Journal of Analysis

Volume 2014 (2014), Article ID 631301, 7 pages

http://dx.doi.org/10.1155/2014/631301

## Some New Difference Sequence Spaces of Invariant Means Defined by Ideal and Modulus Function

^{1}School of Mathematics and Computer Applications, Thapar University, Patiala, Punjab 147004, India^{2}Department of Mathematics, Haryana College of Technology and Management, Kaithal, Haryana 136027, India

Received 15 February 2014; Accepted 9 May 2014; Published 28 May 2014

Academic Editor: Shamsul Qamar

Copyright © 2014 Sudhir Kumar 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

The main objective of this paper is to introduce a new kind of sequence spaces by combining the concepts of modulus function, invariant means, difference sequences, and ideal convergence. We also examine some topological properties of the resulting sequence spaces. Further, we introduce a new concept of -convergence and obtain a condition under which this convergence coincides with above-mentioned sequence spaces.

#### 1. Introduction and Background

Let and be the Banach spaces of real bounded and convergent sequences with the usual supremum norm.

Let be the mapping of the set of all positive integers into itself. A continuous linear functional on is said to be an invariant mean or -mean if and only if(i), when the sequence has , for all ;(ii), where ;(iii), for all .

If , we write . It can be shown by Schaefer [1] that where .

In case is the translation mapping , -mean is often called a Banach limit and , the set of bounded sequences all whose invariant means are equal, is the set of almost convergent sequences (see Lorentz [2]). Using the concept of invariant means Mursaleen et al. [3] introduced the following sequence spaces as a generalization of Das and Sahoo [4]: and investigated some of its properties.

The notion of statistical convergence for number sequences was studied at the initial stage by Fast [5] and later investigated by Connor [6], Fridy [7], Maddox [8], Šalát [9], and many others.

*Definition 1 (see [5]). *A number sequence is said to be statistically convergent to a number (denoted by ) provided that, for every ,
where the vertical bars denote the cardinality of the enclosed set.

By a lacunary sequence, we mean an increasing sequence of positive integers such that and as . The intervals determined by will be denoted by , where the ratio is denoted by . The space of lacunary strongly convergent sequence was defined by Freedman et al. [10] as follows:

Fridy and Orhan [11] generalized the concept of statistical convergence by using lacunary sequence which is called lacunary statistical convergence. Further, lacunary sequences have been studied by Fridy and Orhan [12], Pehlivan and Fisher [13], Et and Gökhan [14], and Tripathy and Dutta [15]. Quite recently, Karakaya [16] combined the approach of lacunary sequence with invariant means and introduced the notion of strong -lacunary statistically convergence as follows.

*Definition 2 (see [16]). *Let be a lacunary sequence. A sequence is said to be lacunary strong -lacunary statistically convergent if, for every ,
where denotes the set of all lacunary strong -lacunary statistically convergent sequences.

Another interesting generalization of statistical convergence was introduced in [17] with the help of ideals of subsets of . Let denote the power set of . A family of sets is called an ideal in if and only if (i) ; (ii) imply ; (iii) and imply . A nonempty family of sets is called a filter on if and only if (i) ; (ii) imply ; (iii) and imply . An ideal is called nontrivial if and . It immediately implies that is a nontrivial ideal if and only if the class is a filter on . The filter is called the filter associated with the ideal . A nontrivial ideal is called an admissible ideal in if and only if it contains all singletons, that is, if it contains . Throughout the present work, denotes a nontrivial admissible ideal.

*Definition 3 (see [17]). *Let be a nontrivial ideal in and let be a metric space. A sequence in is said to be -convergent to if, for each , the set . In this case, we write .

Recently, Das et al. [18] unified the idea of lacunary statistical convergence with ideal convergence and presented the following interesting generalization of statistical convergence.

*Definition 4 (see [18]). *Let be a lacunary sequence. A sequence of numbers is said to be -lacunay statistical convergent or -convergent to , if, for every and ,
In this case, we write or . The set of all -lacunary statistically convergent sequences will be denoted by .

*Definition 5 (see [18]). *Let be a lacunary sequence. A sequence of numbers is said to be -convergent to if, for every , we have
It is denoted by .

In 1981, Kızmaz [19] introduced the notion of difference sequence space as follows:
for and , where , for all .

Continuing on this way, the notion was further generalized by Et and Çolak [20] by introducing the sequence spaces as follows:
for , and , where and , so that . For extensive view in this area, we refer to the series of papers ([21–27]).

The notion of modulus function was introduced by Nakano [28] as follows: by a modulus function, we mean a function from to such that (i) if and only if , (ii) , for all , (iii) is increasing, and (iv) is continuous from right at 0. It follows that must be continuous everywhere on . A modulus function may be bounded or unbounded. In the recent past the notion of modulus function was investigated from different aspects and sequence spaces have been studied by Ruckle [29], Maddox [30], Et [31], Pehlivan and Fisher [13], Savas [32], Et and Gökhan [14], Kumar et al. [33], and many others.

The following well-known lemma is required for establishing a very important result in our paper.

Lemma 6. *Let be a modulus function and let . Then, for each , we have .*

The following inequality will be used throughout the paper. Let be a positive sequence of real numbers with and . Then, for all , , for all , we have

Inspired by the above works, we presently introduce some new kind of sequence spaces by using ideal convergence, modulus function, and invariant mean. Further, we also obtain some relevant connections of these spaces with -convergence.

#### 2. Main Results

Throughout the paper, is considered a nontrivial admissible ideal and denotes the space of all sequences .

*Definition 7. *Let be an admissible ideal, let be a modulus function, and let be any sequence of strictly positive real numbers. Then, for each , we define the following sequence spaces:
uniformly in .

*Remark 8. *By taking some particular cases, we obtain the following. (i)If we take , for all , then the above spaces reduce to , , and , respectively.(ii)If we choose , for all and , then we obtain , , and instead of , , and .(iii)By taking , , , , and , for all , then we obtain defined by Mursaleen et al*. *[3] instead of and reduces to defined in Das and Sahoo [4].

Theorem 9. *Let be a modulus function and is a bounded sequence of strictly positive real numbers; then, , , and are linear spaces over .*

*Proof. *We will prove the assertion only for and others can be treated similarly. Suppose that , . Then, for every and uniformly in , the sets
belong to .

Let , and is linear; then,
where , are two positive numbers such that and .

Then, for given , we have the following containment:
uniformly in .

Since , it follows that the later sets belong to . By using the property of an ideal the set on the left hand side in the above expression also belongs to . This completes the proof.

Theorem 10. *For , then the inclusion
**
is strict.*

*Proof. *We will prove the result for only. Suppose that ; by definition, for each ,
uniformly in .

Since is a modulus function, therefore we have the following inequality:
uniformly in .

Now, for given , we have the following containment:
uniformly in .

Both the sets on the right hand side in the above containment belong to by (16). It follows that .

Since is an admissible ideal and the inclusion is strict as the sequence belongs to , it does not belong to , for , , , and , for all .

Theorem 11. *Let be a bounded sequence of strictly positive real numbers and , are modulus functions. If
**
then .*

*Proof. *Assume that ; there exists a positive number such that , for all , which implies that
uniformly in .

Thus, for any ,
uniformly in . Therefore, the above containment gives the result.

Theorem 12. *If , and are modulus functions, then *(i)*,
*(ii)*.*

*Proof. *(i) Let and let be given. Choose such that , for all , since such that
uniformly in .

On the other hand, we have
uniformly in .

Then, for any ,
where .

Since , so by (22) the latter set belongs to and therefore the theorem is proved.

(ii) This result can be proved by the following inequality:
uniformly in , where and .

Theorem 13. *If is a modulus function and is a sequence of positive real numbers, then .*

*Proof. *This can be proved similarly as in Theorem 12(i).

Theorem 14. *Let be a modulus function. If , then
*

*Proof. *Suppose that . It is given that ; there exists a constant such that , for all ,

which implies that
uniformly in .

Then, for each ,
uniformly in .

Since , therefore the latter set belongs to . It follows that .

Theorem 15. *If and are bounded, then .*

*Proof. *The proof of this theorem is easy and so it is omitted.

#### 3. -Convergence

In this section, we define the notion of -convergence with the help of ideal and invariant means and difference sequences. Further, we also establish some relations between -convergence and .

*Definition 16. *Let be a nontrivial ideal. A sequence is said to be -strong lacunary -statistically convergent or -convergent to a number , provided that, for every and ,
In this case, we write or . Let denote the set of all -convergent sequences.

Theorem 17. *Let be a modulus function and let be a sequence of strictly positive real numbers. If , then .*

*Proof. *Assume that and is given. Then, we have
uniformly in .

Then, for every , we have the following containment:
uniformly in , where , since , which implies that .

Theorem 18. *Let be a bounded modulus function and ; then, .*

*Proof. *Using the same technique of [26, Theorem 3.3], it is easy to prove.

Theorem 19. *If , then if and only if is bounded.*

*Proof. *This part is the direct consequence of Theorems 17 and 18.

*Conversely*. Suppose that is unbounded defined by , for all , and let be a lacunary sequence. We take a fixed set and define as follows:
where and .

For given , we have for all , and uniformly in .

Hence, for , there exists a positive integer such that Now, we have Since is an admissible ideal, it follows that .

If we take , for all , then . This contradicts the fact that .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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