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Double almost lacunary statistical convergence of order α
Advances in Difference Equations volume 2013, Article number: 254 (2013)
In this paper, we define and study lacunary double almost statistical convergence of order α. Further, some inclusion relations have been examined. We also introduce a new sequence space by combining lacunary double almost statistical convergence and Orlicz function.
The notion of convergence of a real sequence was extended to a statistical convergence by Fast  (see also Schoenberg ) as follows. If ℕ denotes the set of natural numbers and , then denotes the cardinality of the set . The upper and lower natural density of the subset K is defined by
If , then we say that the natural density of K exists, and it is denoted simply by . Clearly .
A sequence of real numbers is said to be statistically convergent to L if for arbitrary , the set has a natural density zero.
Statistical convergence turned out to be one of the most active areas of research in summability theory after the works of Fridy  and Šalát . For some very interesting investigations concerning statistical convergence, one may consult the papers of Cakalli , Miller , Maddox  and many others, where more references on this important summability method can be found.
On the other hand, in [8, 9], a different direction was given to the study of statistical convergence, where the notion of statistical convergence of order α, was introduced by replacing n by in the denominator in the definition of statistical convergence. It was observed in  that the behaviour of this new convergence was not exactly parallel to that of statistical convergence, and some basic properties were obtained. One can also see  for related works.
In this paper, we define and study lacunary double almost statistical convergence of order α. Also some inclusion relations have been examined.
Let be the set of all real or complex double sequences. By the convergence of a double sequence, we mean the convergence on the Pringsheim sense, that is, double sequence has a Pringsheim limit L, denoted by , provided that given , and there exists such that whenever . We shall describe such an x more briefly as ‘P-convergent’ (see, ). We denote by the space of P-convergent sequences. A double sequence is bounded if . Let and be the set of all real or complex bounded double sequences and the set bounded and convergent double sequences, respectively. Moricz and Rhoades  defined the almost convergence of double sequence as follows: is said to be almost convergent to a number L if
that is, the average value of taken over any rectangle
tends to L as both p and q tend to ∞, and this convergence is uniform in m and n. We denote the space of almost convergent double sequence by , as
The notion of almost convergence for single sequences was introduced by Lorentz  and some others.
A double sequence x is called strongly double almost convergent to a number L if
By , we denote the space of strongly almost convergent double sequences.
The notion of strong almost convergence for single sequences has been introduced by Maddox .
The idea of statistical convergence was extended to double sequences by Mursaleen and Edely . More recent developments on double sequences can be found in [8, 15–18]. For the single sequences; statistical convergence of order α and strongly p-Cesàro summability of order α introduced by Çolak . Quite recently, in , Çolak and Bektaş generalized this notion by using de la Valée-Poussin mean.
Let be a two-dimensional set of positive integers, and let be the numbers of in K such that and .
Then the lower asymptotic density of K is defined as
In the case when the sequence has a limit, we say that K has a natural density and is defined as
For example, let , where ℕ is the set of natural numbers. Then
(i.e., the set K has a double natural density zero).
Mursaleen and Edely  presented the notion of a statistical convergence for the double sequence as follows: A real double sequence is said to be statistically convergent to L, provided that for each
We now write the following definition.
The double statistical convergence of order α is defined as follows. Let be given. The sequence is said to be statistically convergent of order α if there is a real number L such that
for every , in this, case we say that x is double statistically convergent of order α to L. In this case, we write . The set of all double statistically convergent sequences of order α will be denoted by . If we take in this definition , we can have the previous definition.
By a lacunary ; , where , we shall mean an increasing sequence of nonnegative integers with as . The intervals determined by θ will be denoted by and . The ratio will be denoted by .
Fridy and Orhan  introduced the idea of lacunary statistical convergence for single sequence as follows.
The number sequence is said to be lacunary statistically convergent to the number ℓ if for each ,
In this case, we write , and we denote the set of all lacunary statistically convergent sequences by .
Definition 1.1 By a double lacunary , , where and , we shall mean two increasing sequences of nonnegative integers with
Let us denote , and the intervals determined by will be denoted by , , , and . We will denote the set of all double lacunary sequences by .
Let have double lacunary density if
Example 1 Let and . Then . But it is obvious that .
In 2005, Patterson and Savaş  studied double lacunary statistical convergence by giving the definition for complex sequences as follows.
Definition 1.2 Let be a double lacunary sequence; the double number sequence x is -convergent to L, provided that for every ,
In this case, write or .
2 Main results
In this section, we define lacunary double almost statistically convergent sequences of order α. Also we shall prove some inclusion theorems.
We now have the following.
Definition 2.1 Let be given. The sequence is said to be -statistical convergence of order α if there is a real number L such that
where denote the α th power of . In case is -statistically convergent of order α to L, we write . We denote the set of all -statistically convergent sequences of order α by .
We know that the -statistical convergence of order α is well defined for , but it is not well defined for in general. It is easy to see by taking as fixed.
Definition 2.2 Let be any real number, and let t be a positive real number. A sequence x is said to be strongly -summable of order α, if there is a real number L such that
If we take , the strong -summability of order α reduces to the strong -summability.
We denote the set of all strongly -summable sequence of order α by .
We now state the following theorem.
Theorem 2.1If, then.
Proof Let . Then
for every , and finally, we have that . This proves the result. □
Theorem 2.2For any lacunary sequencesθ, , ifand.
Proof Suppose that and , and , say. Write and . Then there exist a positive integer and such that for and for . Hence for , and ,
Take any , and , say. We prove that . Then for and , we have
Therefore, . □
Remark 2.1 The converse of this result is true for . However, for it is not clear, and we leave it as an open problem.
Theorem 2.3For any double lacunary sequence, ifand.
Proof Suppose that and . Then there exists such that and for all r and s. Suppose that and
by the definition of given , there exists such that for all and . Let
Let n and m be such that and . Therefore, we obtain the following:
This completes the proof of the theorem. □
Theorem 2.4Letandtbe a positive real number, then.
Proof Let . Then given α and β such that and a positive real number t we write
and we get that . □
As a consequence of Theorem 2.4, we have the following.
Corollary 2.1Letandtbe a positive real number. Then:
Theorem 2.5Letαandβbe fixed real numbers such thatand. If a sequence is a strongly-summable sequence of orderα, toL, then it is-statistically convergent of orderβ, toL, i.e., .
Proof For any sequence and , we write
and so that
this shows that if is strongly -summable sequence of order α to L, then it is -statistically convergent of order β to L. This completes the proof. □
We have the following.
Corollary 2.2Letαbe fixed real numbers such thatand.
If a sequence is strongly-summable sequence of orderαtoL, then it is-statistically convergent of orderαtoL, i.e., .
3 New sequence space
In this section, we study the inclusion relations between the set of -statistical convergent sequences of order α and strongly -summable sequences of order α with respect to an Orlicz function M.
The study of Orlicz sequence spaces was initiated with a certain specific purpose in Banach space theory. Lindenstrauss and Tzafriri  investigated Orlicz sequence spaces in more detail, and they proved that every Orlicz sequence space contains a subspace isomorphic to (). The Orlicz sequence spaces are the special cases of Orlicz spaces studied in . Orlicz spaces find a number of useful applications in the theory of nonlinear integral equations. Whereas the Orlicz sequence spaces are the generalization of spaces, the -spaces find themselves enveloped in Orlicz spaces .
Recall in  that an Orlicz function is continuous, convex, non-decreasing function such that and for , and as .
An Orlicz function M is said to satisfy -condition for all values of u, if there exists such that , .
Definition 3.1 Let M be an Orlicz function, be a sequence of strictly positive real numbers, and let be any real number. Now, we write
If , then we say that x is strongly double almost lacunary summable of order α with respect to the Orlicz function M.
If we consider various assignments of M, and t in the sequence spaces above, we are granted the following:
If , , and for all then .
If for all , then .
If for all and , then .
If , then .
In the followings theorems, we shall assume that is bounded and .
Theorem 3.1Letbe real numbers such that, and letMbe an Orlicz function, then.
Proof Let , be given and ∑1 and ∑2 denote the sums over , and , , respectively. Since for each r, s we write
Since , the left hand side of the inequality above tends to zero as uniformly in p, q. Hence the right hand side tends to zero as uniformly in p, q, and, therefore, . This proves the result. □
Corollary 3.1LetandMbe an Orlicz function, then.
We finally prove the following theorem.
Theorem 3.2LetMbe an Orlicz function, and letbe a bounded sequence, then.
Proof Suppose that and . Since , then there is a constant such that . Given , we write for all p, q
Therefore, . This proves the result. □
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The author would like to thank the referees for their help and useful discussions.
The author declares that they have no competing interests.
About this article
- statistical convergence
- Orlicz function
- double statistical convergence of order α
- lacunary statistical convergence
- double almost statistical convergence