Research Article | Open Access

# Some New Lacunary Strong Convergent Vector-Valued Sequence Spaces

**Academic Editor:**Hassan Eltayeb

#### Abstract

We introduce some vector-valued sequence spaces defined by a Musielak-Orlicz function and the concepts of lacunary convergence and strong ()-convergence, where is an infinite matrix of complex numbers. We also make an effort to study some topological properties and some inclusion relations between these spaces.

#### 1. Introduction and Preliminaries

An Orlicz function is convex and continuous such that , for . Lindenstrauss and Tzafriri [1] used the idea of Orlicz function to define the following sequence space: which is called as an Orlicz sequence space. The space is a Banach space with the norm It is shown in [1] that every Orlicz sequence space contains a subspace isomorphic to . An Orlicz function satisfies -condition if and only if, for any constant , there exists a constant such that for all values of . An Orlicz function can always be represented in the following integral form: where is known as the kernel of and is right differentiable for , , ; is nondecreasing and as .

A sequence of Orlicz functions is called a Musielak-Orlicz function; see ([2, 3]). A sequence defined by is called the complementary function of a Musielak-Orlicz function . For a given Musielak-Orlicz function , the Musielak-Orlicz sequence space and its subspace are defined as follows: where is a convex modular defined by We consider equipped with the Luxemburg norm or equipped with the Orlicz norm A Musielak-Orlicz function is said to satisfy -condition if there exist constants , and a sequence (the positive cone of ) such that the inequality holds for all and , whenever .

Let be a linear metric space. A function : is called paranorm, if(1), for all ;(2), for all ;(3), for all , ;(4)if is a sequence of scalars with as and is a sequence of vectors with , then as .

A paranorm for which implies is called total paranorm and the pair is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [4], Theorem , P-183). For more details about sequence spaces, see [5–11] and references therein.

The space of lacunary strong convergence has been introduced by Freedman et al. [12]. A sequence of positive integers is called “lacunary” if , and , as . The intervals determined by are denoted by and the ratio will be denoted by . The space of lacunary strongly convergent sequences is defined by Freedman et al. [12] as follows: The space of strongly Cesàro summable sequences is In case, when , . Recently, Bilgin [13] in his paper generalized the concept of lacunary convergence and introduced the space , as where is a modulus function and ; converges for each . Later Bilgin [14] generalized lacunary strongly -convergent sequences with respect to a sequence of modulus function as follows: We write for the zero sequences.

Mursaleen and Noman [15] introduced the notion of -convergent and -bounded sequences as follows.

Let be a strictly increasing sequence of positive real numbers tending to infinity, that is, and said that a sequence is -convergent to the number , called the -limit of if as , where The sequence is -bounded if . It is well known [15] that if in the ordinary sense of convergence, then This implies that which yields that and hence is -convergent to .

We now introduce the concept of lacunary strongly -convergence for sequences with the elements chosen from a Banach space over the complex field , with respect to Musielak-Orlicz functions .

Let be an infinite matrix of complex numbers and be a Musielak-Orlicz function. In the present paper we define the following sequence spaces:

If we take , for all , we have where A sequence is said to be -lacunary strong -convergent with respect to if there is a number , such that .

We have generalized the strongly Cesàro-summable sequence space into -strongly Cesàro-summable vector-valued sequence space as where is a Cesàro matrix, that is, Then it can be shown that is a paranormed space with respect to the paranorm

#### 2. Topological Properties of the Spaces and

Theorem 1. *Let be an infinite matrix of complex numbers and let be a Musielak-Orlicz function. Then and are linear spaces over the field of complex number .*

*Proof. *It is easy to prove.

Theorem 2. *Let be an infinite matrix of complex numbers and let be a Musielak-Orlicz function. Then is normal spaces, when is normal.*

*Proof. *Let . Let . Then
Since is increasing,
Consequently, . This completes the proof of the theorem.

Theorem 3. *The spaces and are paranormed spaces, with respect to the paranorm
*

*Proof. *It is easy to prove, so we omit the details.

#### 3. Relation between the Spaces and

The main purpose of this section is to study relation between and .

Theorem 4. *Let be an infinite matrix of complex numbers and let be a Musielak-Orlicz function satisfying -condition. If is -lacunary strong -convergent to , with respect to and is a normal Banach space, then .*

*Proof . *Let and , where . Then
We define two sequences and such that
Hence,
Now,
Since is normal, . Let . Then
Hence . This completes the proof of the theorem.

Theorem 5. *Let be an infinite matrix of complex numbers and let be a Musielak-Orlicz function satisfying -condition. If
**
then .*

*Proof . *If , then there exists a number such that
Let and , where , . Then clearly
Hence . This completes the proof.

#### 4. Relation between the spaces and

In this section of the paper we study relation between the spaces and .

Lemma 6. * if and only if .*

*Proof . *First suppose that . Then there exist such that for all . Let . Then
Now, . So we have
Also
Since , then
and hence
that is, . By linearity, it follows that .

Next, suppose that . Since is lacunary we can select a subsequence of such that
where . Define by
where and let , and then for any , ,
So, . But is strongly Cesàro-summable, since if is sufficiently large integer we can find the unique for which and hence
and it follows that also . Hence .

Lemma 7. * if and only if .*

*Proof. *First suppose that if , there exists such that for all . Let and . Then
Then we can find and such that
Then if is any integer with
then
Since as , it follows that
and hence .

Next, suppose that . We construct a sequence in that is not Cesaro -summable. By the idea of Freedman et al. [12] we can construct a subsequence of the lacunary sequence such that , and then define a bounded difference sequence by
where . Let and . Then,
and if ,
Thus . For the above sequence and for
this converges to , but for ,
It proves that , since any sequence in consisting of ’s and ’s has a limit only or .

Theorem 8. *Let be a lacunary sequence. Then if and only if .*

*Proof . *The proof of the theorem follows from Lemmas 6 and 7.

#### 5. Statistical Convergence

The notion of statistical convergence was introduced by Fast [16] and Schoenberg [17] independently. Over the years and under different names, statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory, and number theory. Later on, it was further investigated from the sequence space point of view and linked with summability theory by Fridy [18], Connor [19], Šalát [20], Mursaleen and Edely [21], Isk [22], Mohiuddine and Alghamdi [23], Hazarika et al. [24], Kolk [25], Maddox [26], Alotaibi and Mursaleen [27], Mohiuddine et al. [28], Mohiuddine and Aiyub [29], and many others. In recent years, generalizations of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces. Statistical convergence and its generalizations are also connected with subsets of the Stone-ech compactification of natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability. The notion depends on the density of subsets of the set of natural numbers.

A subset of is said to have the natural density if the following limit exists: where is the characteristic function of . It is clear that any finite subset of has zero natural density and .

A sequence is said to be statistically convergent to the number if for every Bilgin [14] also introduced the concept of statistical convergence in and proved some inclusion relation.

Let be a lacunary sequence and let be an infinite matrix of complex numbers. Then a sequence is said to be -lacunary -statistically convergent to a number , if for any , where We denote it as . The vertical bar denotes the cardinality of the set. The set of all -lacunary -statistical convergent sequences is denoted by .

In this section we study some relation between the spaces and .

Theorem 9. *Let be a Musielak-Orlicz function and let be pointwise convergent. Then if and only if for some .*

*Proof . *Let and . Let , where , . Since , there exists a number such that
Let
Then
Hence it follows that .

Conversely, let us assume that the condition does not hold good. Then there is a number such that for some . Now, we select a lacunary sequence such that for any .

Let , and define the sequence by putting
Therefore,
Thus, we have . But
So .

Theorem 10. *Let be a Musielak-Orlicz function. Then if and only if .*

*Proof . *Let and . Suppose and . Let
Now, for all , . So
Hence, as , it follows that .

Conversely, suppose that
Then we have
so that for . Let . We set a sequence by
Then
Hence and hence .

But
So, .

#### Conflict of Interests

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

#### References

- J. Lindenstrauss and L. Tzafriri, “On Orlicz sequence spaces,”
*Israel Journal of Mathematics*, vol. 10, pp. 379–390, 1971. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. Maligranda,
*Orlicz Spaces and Interpolation*, Seminars in Mathematics 5, Polish Academy of Science, 1989. - J. Musielak,
*Orlicz Spaces and Modular Spaces*, vol. 1034 of*Lecture Notes in Mathematics*, Springer, Berlin, Germany, 1983. View at: MathSciNet - A. Wilansky,
*Summability through Functional Analysis*, vol. 85, North-Holland, Amsterdam, Netherlands, 1984. View at: MathSciNet - M. Mursaleen and A. K. Noman, “On some new sequence spaces of non absolute type related to the spaces l
_{p}and ${l}_{\infty}$ II,”*Mathematical Communications*, vol. 16, pp. 383–398, 2011. View at: Google Scholar - S. D. Parashar and B. Choudhary, “Sequence spaces defined by Orlicz functions,”
*Indian Journal of Pure and Applied Mathematics*, vol. 25, no. 4, pp. 419–428, 1994. View at: Google Scholar | Zentralblatt MATH | MathSciNet - K. Raj and S. K. Sharma, “Some sequence spaces in 2-normed spaces defined by Musielak-Orlicz function,”
*Acta Universitatis Sapientiae: Mathematica*, vol. 3, no. 1, pp. 97–109, 2011. View at: Google Scholar | MathSciNet - K. Raj and S. K. Sharma, “Some generalized difference double sequence spaces defined by a sequence of Orlicz-functions,”
*Cubo*, vol. 14, no. 3, pp. 167–189, 2012. View at: Google Scholar | MathSciNet - K. Raj and S. K. Sharma, “Some multiplier sequence spaces defined by a Musielak-Orlicz function in $n$-normed spaces,”
*New Zealand Journal of Mathematics*, vol. 42, pp. 45–56, 2012. View at: Google Scholar | Zentralblatt MATH | MathSciNet - A. Gökhan, M. Et, and M. Mursaleen, “Almost lacunary statistical and strongly almost lacunary convergence of sequences of fuzzy numbers,”
*Mathematical and Computer Modelling*, vol. 49, no. 3-4, pp. 548–555, 2009. View at: Publisher Site | Google Scholar | MathSciNet - M. Gungor and M. Et, “${\Delta}^{T}$-strongly almost summable sequences defined by Orlicz functions,”
*Indian Journal of Pure and Applied Mathematics*, vol. 34, no. 8, pp. 1141–1151, 2003. View at: Google Scholar | MathSciNet - A. R. Freedman, J. J. Sember, and M. Raphael, “Some Cesàro-type summability spaces,”
*Proceedings of the London Mathematical Society*, vol. 37, no. 3, pp. 508–520, 1978. View at: Publisher Site | Google Scholar | MathSciNet - T. Bilgin, “Lacunary strong $A$-convergence with respect to a modulus,”
*Studia Universitatis Babeş-Bolyai*, vol. 46, no. 4, pp. 39–46, 2001. View at: Google Scholar | MathSciNet - T. Bilgin, “Lacunary strong
*A*-convergence with respect to a sequence of modulus functions,”*Applied Mathematics and Computation*, vol. 151, no. 3, pp. 595–600, 2004. View at: Publisher Site | Google Scholar | MathSciNet - M. Mursaleen and A. K. Noman, “On some new sequence spaces of non-absolute type related to the spaces ${l}_{p}$ and ${l}_{\infty}$ I,”
*Filomat*, vol. 25, no. 2, pp. 33–51, 2011. View at: Publisher Site | Google Scholar | MathSciNet - H. Fast, “Sur la convergence statistique,”
*Colloquium Mathematicae*, vol. 2, pp. 241–244, 1951. View at: Google Scholar | MathSciNet - I. J. Schoenberg, “The integrability of certain functions and related summability methods,”
*The American Mathematical Monthly*, vol. 66, pp. 361–375, 1959. View at: Publisher Site | Google Scholar | MathSciNet - J. A. Fridy, “On statistical convergence,”
*Analysis*, vol. 5, no. 4, pp. 301–313, 1985. View at: Publisher Site | Google Scholar | MathSciNet - J. S. Connor, “A topological and functional analytic approach to statistical convergence,” in
*Applied and Numerical Harmonic Analysis*, vol. 8 of*Analysis of Divergence*, pp. 403–413, 1999. View at: Google Scholar - T. Šalát, “On statistically convergent sequences of real numbers,”
*Mathematica Slovaca*, vol. 30, no. 2, pp. 139–150, 1980. View at: Google Scholar | MathSciNet - M. Mursaleen and O. H. H. Edely, “Generalized statistical convergence,”
*Information Sciences*, vol. 162, no. 3-4, pp. 287–294, 2004. View at: Publisher Site | Google Scholar | MathSciNet - M. Isk, “On statistical convergence of generalized difference sequences,”
*Soochow Journal of Mathematics*, vol. 30, no. 2, pp. 197–205, 2004. View at: Google Scholar | MathSciNet - S. A. Mohiuddine and M. A. Alghamdi, “Statistical summability through a lacunary sequence in locally solid Riesz spaces,”
*Journal of Inequalities and Applications*, vol. 2012, article 225, 2012. View at: Publisher Site | Google Scholar | MathSciNet - B. Hazarika, S. A. Mohiuddine, and M. Mursaleen, “Some inclusion results for lacunary statistical convergence in locally solid Riesz spaces,”
*Iranian Journal of Science Technology*, vol. 38, no. A1, pp. 61–68, 2014. View at: Google Scholar - E. Kolk, “The statistical convergence in Banach spaces,”
*Acta et Commentationes Universitatis Tartuensis*, vol. 928, pp. 41–52, 1991. View at: Google Scholar | MathSciNet - I. J. Maddox, “Statistical convergence in a locally convex space,”
*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 104, no. 1, pp. 141–145, 1988. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Alotaibi and M. Mursaleen, “Statistical convergence in random paranormed space,”
*Journal of Computational Analysis and Applications*, vol. 17, no. 2, pp. 297–304, 2014. View at: Google Scholar | MathSciNet - S. A. Mohiuddine, K. Raj, and A. Alotaibi, “Some paranormed double difference sequence spaces for Orlicz functions and bounded-regular matrices,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 419064, 10 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - S. A. Mohiuddine and M. Aiyub, “Lacunary statistical convergence in random 2-normed spaces,”
*Applied Mathematics & Information Sciences*, vol. 6, no. 3, pp. 581–585, 2012. View at: Google Scholar

#### Copyright

Copyright © 2014 M. Mursaleen 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.