Journal of Function Spaces and Applications

Volume 2013, Article ID 509613, 8 pages

http://dx.doi.org/10.1155/2013/509613

## On a New Space of Double Sequences

Department of Mathematics, Art and Science Faculty, Ondokuz Mayıs University, Kurupelit Campus, Samsun, Turkey

Received 3 May 2013; Accepted 17 June 2013

Academic Editor: Józef Banaś

Copyright © 2013 Cenap Duyar and Oğuz Oğur. 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 introduce a new space of double sequences related to -absolute convergent double sequence space, combining an Orlicz function and an infinity double matrix. We study some properties of and obtain some inclusion relations involving .

#### 1. Introduction

Throughout this work, and denote the set of positive integers and complex numbers, respectively. A complex double sequence is a function from into and briefly denoted by . Throughout this work, and denote the spaces of single complex sequences and double complex sequences, respectively. If, for all , there is such that where and , then a double sequence is said to be converging (in terms of Pringsheim) to . A real double sequence is nondecreasing, if for . A double series is infinity sum , and its convergence implies the convergence by of partial sums sequence , where (see [1–4]).

A double sequence space is said to be solid if for all double sequences of scalars such that for all whenever .

Now let be a family of subsets having most elements in . Also denotes the class of subsets in such that the element numbers of and are most and , respectively. Besides is taken as a nondecreasing double sequence of the positive real numbers such that An Orlicz function is a function which is continuous, nondecreasing, and convex with , for , and as .

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

An Orlicz function is said to be satisfied -condition for all values of , if there exists a constant such that for all .

Lindenstrauss and Tzafriri [5] used the idea of Orlicz functions to construct Orlicz sequence space The sequence space is a Banach space according to the norm defined by This space is called an Orlicz sequence space. The space is closely related to the space , which is an Orlicz sequence space with for .

The double sequence spaces in the various forms defined by Orlicz functions were introduced and studied by Khan and Tabassum in [6–12] and by Khan et al. in [13].

The space , introduced by Sargent in [14], is in the form Sargent studied some properties of this space and examined relationship between this space and -space. Similar sequence classes were studied by many mathematicians using Orlicz functions (see [15–17]).

Later on, this space was investigated from sequence space point of view by Rath [18], Rath and Tripathy [19], Tripathy and Sen [20], Tripathy and Mahanta [17], and others. Recently Altun and Bilgin [15] introduced and studied the following sequence space .

Let be an infinite matrix of complex numbers, an Orlicz space, and a bounded sequence of positive real numbers such that . Then the space is defined by where if converges for each .

Let be a double sequence. A set is defined by

If for all , then is said to be symmetric.

In this work, we introduce the following sequence space.

Let be an infinite double matrix of complex numbers, an Orlicz function and bounded double sequence of positive real numbers such that . Then the space is defined by where if converges for each .

Also, we introduce and investigate the following space:

In this work, we also use the following sequence spaces:

The following inequality will be used throughout this paper: where and.

#### 2. Main Results

*Definition 1. *Let be a set of increasing positive integer binaries, namely, if and only if and , and be a double sequence space. A -set space is a double sequence space, defined by
The canonical preimage of a double sequence is a double sequence with
The canonical preimage of a set space is a set of canonical preimages of all elements in .

*Definition 2. *If a double sequence space contains the canonical preimages of all set spaces, then is said to be monotone.

The following lemma is an easy result of the definitions.

Lemma 3. *If a double sequence space is solid, then is monotone. *

Proposition 4. *The space is a -linear space. *

*Proof. *Let , be in and , in . Then there exist positive numbers and such that
Let . Using that is nondecreasing convex function, we have
Thus we can write
This shows that . Hence is a linear space.

Proposition 5. *The space is a paranormed space with the paranorm
*

*Proof. *It is clear that and if . If there are and such that
then
where . This shows that . Using this triangle inequality we can write
Separately, we obtain
Hence where and as . Consequently is a paranom on .

Proposition 6. *The class of double sequences is solid. *

*Proof. *Let be a double sequence of scalars such that and . Then we can write

This implies that , and hence the class is solid.

Corollary 7. *The space is monotone. *

Theorem 8. *Let be another double sequence like . Then if and only if . *

*Proof. *Let . Then for all . If , then
for some . Thus
for some , and hence . This shows that .

Conversely, let . We say for all and suppose . Then there exists a subsequence of such that . If , then we have

This is a contradiction as . This completes the proof.

Corollary 9. * if and only if and . *

Theorem 10. *Let , , and be Orlicz functions satisfying -condition. Then *(a)*,*(b)*.*

*Proof. *(a) Let . Then there exists such that
By the continuity of , we select a number with such that , whenever , for arbitrary . Now let . We can write
By the properties of , we have

Again we can write
for . If we use that satisfies -condition, then we find
and so
Hence we get

Finally, we have , and hence .(b) Let . Then there exists such that
By the inequality
we have .

Theorem 11. (a)*. *(b)* if and only if .*(c)* if and only if .*

*Proof. *(a) Let , and let a set be defined as follows:

Since is a nondecreasing double sequence, is a nonincreasing double sequence. So we obtain
for all and hence
Thus we have .

Conversely if , then it is clear that
for all , and hence
This shows that if , then . Thus we have .(b) It is clear that where for all . Then we can write . By Theorem 8, we have , and according to alternative (a)
(c) Firstly we show that if for all . Let . Then for and , we can find some such that
This gives the inclusion . Conversely let . Then for , we can find some such that
This shows that . By Theorem 8 and alternative (a), we can write if and only if . This completes the proof.

#### References

- V. A. Khan, “On ${\mathrm{\Delta}}_{v}^{m}$-Cesáro summable double sequences,”
*Thai Journal of Mathematics*, vol. 10, no. 3, pp. 535–539, 2012. View at Google Scholar · View at MathSciNet - V. A. Khan and S. Tabassum, “Statistically pre-Cauchy double sequences,”
*Southeast Asian Bulletin of Mathematics*, vol. 36, no. 2, pp. 249–254, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. A. Khan, S. Tabassum, and A. Esi, “Statistically convergent double sequence spaces in $n$-normed spaces,”
*ARPN Journal of Science and Technology*, vol. 2, no. 10, pp. 991–995, 2012. View at Google Scholar - B. V. Limaye and M. Zeltser, “On the Pringsheim convergence of double series,”
*Proceedings of the Estonian Academy of Sciences*, vol. 58, no. 2, pp. 108–121, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Lindenstrauss and L. Tzafriri, “On Orlicz sequence spaces,”
*Israel Journal of Mathematics*, vol. 10, pp. 379–390, 1971. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. A. Khan and S. Tabassum, “Statistically convergent double sequence spaces in 2-normed spaces defined by Orlicz function,”
*Applied Mathematics*, vol. 2, no. 4, pp. 398–402, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - V. A. Khan and S. Tabassum, “On some new quasi almost ${\mathrm{\Delta}}^{m}$-lacunary strongly P-convergent double sequences defined by Orlicz functions,”
*Journal of Mathematics and Applications*, vol. 34, pp. 45–52, 2011. View at Google Scholar · View at MathSciNet - V. A. Khan and S. Tabassum, “On ideal convergent difference double sequence spaces in 2-normed spaces defined by Orlicz functions,”
*JMI International Journal of Mathematics and Applications*, vol. 1, no. 2, pp. 26–34, 2010. View at Google Scholar - V. A. Khan and S. Tabassum, “Some vector valued multiplier difference double sequence spaces in 2-normed spaces defined by Orlicz functions,”
*Journal of Mathematical and Computational Science*, vol. 1, no. 1, pp. 126–139, 2011. View at Google Scholar · View at MathSciNet - V. A. Khan and S. Tabassum, “On some new double sequence spaces of invariant means defined by Orlicz functions,”
*Communications de la Faculté des Sciences de l'Université d'Ankara A1*, vol. 60, no. 2, pp. 11–21, 2011. View at Google Scholar · View at MathSciNet - V. A. Khan and S. Tabassum, “The strongly summable generalized difference double sequence spaces in 2-normed spaces defined by Orlicz functions,”
*Journal of Mathematical Notes*, vol. 7, no. 2, pp. 45–58, 2011. View at Google Scholar · View at MathSciNet - V. A. Khan and S. Tabassum, “On some new almost double Lacunary ${\Delta}^{m}$-squence spaces dened by Orlicz functions,”
*Journal of Mathematical Notes*, vol. 6, no. 2, pp. 80–94, 2011. View at Google Scholar - V. A. Khan, S. Tabassum, and A. Esi, “${A}_{\sigma}$ double sequence spaces and statistical convergence in 2-normed spaces dened by Orlicz functions,”
*Theory and Applications of Mathematics and Computer Science*, vol. 2, no. 1, pp. 61–71, 2012. View at Google Scholar - W. L. C. Sargent, “Some sequence spaces related to the ${l}_{p}$ spaces,”
*Journal of the London Mathematical Society*, vol. 35, pp. 161–171, 1960. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Altun and T. Bilgin, “On a new class of sequences related to the ${l}_{p}$ space defined by Orlicz function,”
*Taiwanese Journal of Mathematics*, vol. 13, no. 4, pp. 1189–1196, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Esi, “On a class of new type difference sequence spaces related to the space ${l}_{p}$,”
*Far East Journal of Mathematical Sciences*, vol. 13, no. 2, pp. 167–172, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. C. Tripathy and S. Mahanta, “On a class of sequences related to the ${l}_{p}$ space defined by Orlicz functions,”
*Soochow Journal of Mathematics*, vol. 29, no. 4, pp. 379–391, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Rath, “Spaces of $r$-convex sequences and matrix transformations,”
*Indian Journal of Mathematics*, vol. 41, no. 2, pp. 265–280, 1999. View at Google Scholar · View at MathSciNet - D. Rath and B. C. Tripathy, “Characterization of certain matrix operators,”
*Orissa Mathematical Society*, vol. 8, pp. 121–134, 1989. View at Google Scholar - B. C. Tripathy and M. Sen, “On a new class of sequences related to the space ${l}_{p}$,”
*Tamkang Journal of Mathematics*, vol. 33, no. 2, pp. 167–171, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet