ISRN Mathematical AnalysisVolume 2012 (2012), Article ID 524962, 11 pagesdoi:10.5402/2012/524962

Research Article

Some New Double-Sequence Spaces in 2-Normed Spaces Defined by Ideal Convergence and an Orlicz Function

Orhan Tuğ, Mutlay Doğan, and Abdullah Kurudirek

Ishik University, Arbil, Iraq

Received 14 December 2011; Accepted 26 January 2012

Academic Editors: S. Anita and S. Liu

Copyright © 2012 Orhan Tuğ 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.

We generalize some sequence spaces from single to double, we study some topological properties of these double sequence spaces by using ideal convergence, difference sequence spaces, and an Orlicz function in 2-normed spaces, and we give some results related to these sequence spaces.

1. Introduction

The concept of -convergent was introduced by Das et al. [1] and developed by many scholars. B. Tripathy and B. C. Tripathy [2] extended the concept of -convergence from single sequence to double sequence. Some difference double-sequence spaces and paranormed double-sequence spaces defined by Orlicz function were introduced by Tripathy and Sarma [3, 4]. And also some new single- and double-sequence spaces defined in 2-normed spaces using ideal convergence and an Orlicz function were introduced by Savaş [5, 6].

Recall that an Orlicz Function, which was presented by Karasnoselskii and Rutishky [7] is continuous, convex, nondecreasing function such that and for and as .

An Orlicz function can be represented in the following integral form: , where is the known kernel of , right differential for for is nondecreasing, and as .

Ruckle [8] and Maddox [9] presented and discussed that if convexity of Orlicz function is replaced by then this function is called Modulus function.

An Orlicz function is said to satisfy -condition for all values of , if there exists constant , such that . The -condition is equivalent to the inequality for all values of and for being satisfied [7].

The notion of difference double-sequence spaces was introduced by Tripathy and Sarma [3]. These notions are further studied by Tripathy and Sarma [4]. Let be a double sequence. Then the operator is defined as , for all .

Let us recall some well-known concepts that is a normed space and a sequence of elements of is called to be statistically convergent to if the set has zero natural density for each [2].

A real double-sequence is said to be statistically bounded above if there exists a real number such that . A real double-sequence is said to be statistically bounded below if there exists a real number such that . If a real double-sequence is statistically bounded both above and below, then we say that is statistically bounded. It is clear that any bounded double-sequence is also statistically bounded [10].

A family of subset of a nonempty set is said to be an ideal in if(i),(ii) imply ,(iii) imply [11, 12].

is called an admissible ideal in if and only if it contains [13].

Let be a nontrivial ideal and in N. The sequence is said to be -convergent to if for each the set [1].

A real double-sequence is said to be -bounded above if there exists a real number such that . A real double-sequence is said to be -bounded below if there exists a real number such that . If a real double-sequence is -bounded both above and below, then we say that is -bounded. One can observe easily that any bounded double sequence is -bounded [14].

Let be a real vector space of dimension , where . A 2-norm on is a function which satisfied the following four conditions:(i) if and only if and are linear dependent,(ii), (iii), (iv), the pair is then called a 2-normed space [15, 16].

The sequence in a 2-normed space is said to be convergent to in if for every . In this instance, we write [17].

Let be a nontrivial ideal in . The sequence in a-normed space is said to be -convergent to , if for every and in the set belongs to [18].

The following inequalities which will be used throughout the paper can be introduced like in Maddox [19]. If then for all and . And also for all .

2. Main Results

Let and be two nondecreasing sequences of positive real numbers both of which tend to as and approach , respectively. Also let and . Let be an admissible ideal of , an Orlicz function, and a 2-normed spaces. In addition let be a bounded sequence of positive real numbers. With , we symbolize the space of all double sequences defined over . Now we define the following double sequence spaces:

where , , and . Throughout this paper we will denote by and by .

Theorem 2.1. are linear spaces.

Proof. Because all the statements can be proved in a similar way, we will prove the assertion for . Suppose that and . So Since we study   2-normed, is linear and is an Orlicz function, we have the following inequality: where .From the above inequality we get the two set on the right side belonging to and this completes the proof.

Theorem 2.2. For any fixed is a paranormed space with the paranorm defined by

Proof. (i) and (ii) are easy to prove. So we leave them out. (iii) Let us take . Let Let and . Then if then we have Thus   (iv) Let , where and as . We have to show that as . Let If and then we observe that From above inequality it obviously follows that and consequently Note that for all . Hence by our assumption the right-hand side tends to 0 as and the results follow. This completes the proof of the theorem.

Theorem 2.3. Let be an Orlicz function which satisfies the -condition. Then and the inclusions are strict.

Proof. Because it can be proven in a similar way, we introduce the proof for only. Let . Then where and we get from -condition so and this completes the proof.

Theorem 2.4. Let stand for or . Then the inclusion is strict.

Proof. Because this can be proven in a similar way, we will give the proof for only. Let . Then given we have Since is nondecreasing and convex it follows that where . Similarly from the convexity and nondecreasing properties of Since the right-hand side belongs to so does the left hand side.

Theorem 2.5. Let be Orlicz function. Then one has (i) provided is such that ,(ii).

Proof. (i) For given , we can choose such that . Because of the continuity of , we can choose such that . Let . We can write from the definition Hence if , then that is, that is, that is, From this inequality using continuity of we must have which consequently implies that that is, This show that and from the definition ideal the left side belongs to . This proves the result. (ii)  Let . Then the fact give us the result. Thus , hence the proof.

Definition 2.6. Let be sequence space. Then is called solid if whenever for all sequence of scalar with for all .

Theorem 2.7. The sequence spaces are solid.

Proof. We give the proof for only. Let and be a sequence of scalars such that for all . Then we have where . Hence for all sequences of scalars with for all whenever .

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