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 [511] 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.