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ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 378575, 11 pages
Some Almost Lacunary Double Sequence Spaces Defined by Orlicz Functions in 2-Normed Spaces
Ishik University, Erbil, Iraq
Received 16 November 2011; Accepted 25 December 2011
Academic Editor: O. Miyagaki
Copyright © 2012 Orhan Tuğ. 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.
Das and Patel (1989) introduced two new sequence spaces which are called lacunary almost convergent and lacunary strongly almost convergent sequence spaces. Móricz and Rhoades (1988) defined and studied almost P-convergent double sequences spaces. Savaş and Patterson (2005) introduce the almost lacunary strong P-convergent double sequence spaces by using Orlicz functions and examined some properties of these sequences spaces. In this paper, some almost lacunary double sequences spaces are given by using 2-normed spaces.
By the convergence of double sequence which is known the convergence on the Pringsheim sense, that is, a double sequence has Pringsheim limit denoted by , provided given there exists such that whenever . We write briefly as -convergent [1–3].
Freedman et al.  presented a definition for lacunary refinement as follow: is called a lacunary refinement of the lacunary sequence if and studied many scholar [4–6]. Savaş and Patterson  gave some properties and theorem and also defined -convergence.
By a lacunary ; where , we shall mean an increasing sequence of non-negative integers with as . The intervals determined by denoted by and . The ratio is denoted by .
An Orlicz Function, which was presented by Krasnoselskii and Rutisky , is continuous, convex, non-decreasing 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 non-decreasing and as .
be a normed space and a sequence of elements of is called to be statistically convergent to if the set has natural density zero for each .
Let be a real vector space of dimension , where . A 2-norm on is a function which satisfy the following four conditions;(i) if and only if and are linear dependent.(ii)(iii)(iv)
The sequence in a 2-normed space is said to be convergent to in if for every . In this case, we write .
2. Notations and Known Results
Almost -convergent sequences have been defined by Móricz and Rhoades  as follow:
Definition 2.1. A double sequence of real numbers is called almost -convergent to a limit if
That is, the average take over any rectangle tends to as both and tend to , and this -convergence is uniform in and . The set of sequence which satisfy this property was denoted as by Savaş and Paterson .
We can define the set of almost -convergent double sequence in similar to above definition as follow:
Definition 2.2. Let be an Orlicz function, be any factorable double sequence of strictly positive reel numbers and denote all double sequence in 2-normed space we can define the following double sequence space
If we choose and for all and , then which was defined above.
Definition 2.3. The double sequnce is called double lacunary if there exist two increasing sequences of integers such that
Let , and is defined by
Definition 2.4. Let be an Orlicz function, denote all double sequence in 2-normed space, and be any factorable double sequence of strictly positive reel numbers, now we can define the following sequence spaces in 2-normed space as follows:
When for all and , we shall denote and as and . That is,
If we shall say that is almost lacunary strongly -convergent with respect to the Orlicz function in 2-normed space. In addition if and for all and , then and which are defined above. Also note that if for all and , then and which are defined above.
Let us generalized almost -convergent double sequence to Orlicz function in 2-normed spaces.
3. Main Results
Theorem 3.1. For any Orlicz function and a bounded factorable positive double sequence , and are linear spaces.
Proof. Suppose that , and , . So we have Since is an Orlicz function we have the following inequality where . When we take the limit of each side as So this is the result.
Definition 3.2. An Orlicz function is said to be satisfy -condition for all values of , if there exists a constant such that
Lemma 3.3. Let be an Orlicz function which satisfies -condition and . Then for each and some constant we have
Theorem 3.4. For any Orlicz function which satisfies -condition, we have
Proof. Let . For each and Let and choose with such that for every with . For every , we get From Lemma 3.3 as and goes to infinity in Pringsheim sense, for each and we are granted .
Theorem 3.5. Let be a double lacunary sequence with and then for any Orlicz function , .
Proof. Since and , then there exists such that and . This mean , . Then for , we can write for each and , some and every Since the last two terms tends to 0 uniformly in and in Pringsheim sense. Thus for each and Since we get for each and the following inequalities as follow: Thus the terms are both convergent to in Pringsheim sense for all and , every and some . Therefore is a convergent sequence in Pringsheim sense for each and . So and this is the proof.
Theorem 3.6. Let be a double lacunary sequence with and then for any Orlicz function , .
Proof. Since and there exists such that and for all and . Let and . There exists and such that for every and , and all and , for every Let and let and . Hence we get Since and both tends to infinity as both and tends to infinity, uniformly in and , and for every , Therefore .
Theorem 3.7. Let be a double lacunary sequence with , then for any Orlicz function .
Gähler  defined almost lacunary statistical convergence for single sequence, then Savaş and Patterson  defined almost lacunary statistical convergence for double sequence by combining lacunary sequence and almost convergence. Now we can define this definition in 2-normed space as follow:
Definition 3.8. Let be a double lacunary sequence; the double number sequence is -convergent to provided that for every and So we can write .
Theorem 3.9. Let be a double lacunary sequence then(1) implies (2) is a proper subset of (3)If and then (4)where is the space of all bounded double sequence.
Proof. (1) Since for all and , and every
and for all and
This show that for all and
this completes the proof of (1)(2) Let be defined as follows:
It is obvious that is an unbounded double sequence and for , for all and , and for every Thus . But Therefore which is the proof of (2).(3)Let and . Assuming that for all and , and every for all . And also for given and and large for all and , and every we get the following inequality as follow: Therefore and , this shows that .(4)from (1), (2) and (3), we get .
Theorem 3.10. For any Orlicz function ,
Proof. Let and . Then for all and , and every This shows that .
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