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Double almost statistical convergence of order α
Advances in Difference Equations volume 2013, Article number: 62 (2013)
The goal of this paper is to define and study λ-double almost statistical convergence of order α. Further some inclusion relations are examined. We also introduce a new sequence space by combining the double almost statistical convergence and an Orlicz function.
The notion of statistical convergence was introduced by Fast  and Schoenberg  independently. Over the years and under different names, statistical convergence was 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 , Connor , S̆alát , Cakalli , Miller , Maddox  and many others. However, Mursaleen  defined the concept of λ-statistical convergence as a new method and found its relation to statistical convergence, -summability and strong -summability. Recently, for , Çolak and Bektaş  have introduced the λ-statistical convergence of order α and strong -summability of order α for sequences of complex numbers.
In this paper we define and study λ-double almost statistical convergence of order α. Also, some inclusion relations have been examined.
The notion of statistical convergence depends on the density of subsets of N. A subset E of N is said to have density if
Note that if is a finite set, then , and for any set , .
Definition 1.1 A sequence is said to be statistically convergent to ℓ if for every ,
We write in case is st-statistically convergent to L.
Let be the set of all real or complex double sequences. By the convergence of a double sequence we mean the convergence in the Pringsheim sense, that is, the double sequence has a Pringsheim limit L denoted by provided that, given , there exists such that whenever . We will 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 of bounded and convergent double sequences, respectively. Móricz and Rhoades  defined the almost convergence of the 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 sequences 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. It is easy to see that the inclusions strictly hold.
The notion of strong almost convergence for single sequences has been introduced by Maddox .
A linear functional L on is said to be a Banach limit if it has the following properties:
if (i.e., for all i, j),
, where with for all i, j and
, where the shift operators , , are defined by , , .
Let be the set of all Banach limits on . A double sequence is said to be almost convergent to a number L if for all (see ).
The idea of statistical convergence was extended to double sequences by Mursaleen and Edely . More recent developments on double sequences can be found in [16–18], where some more references can be found. For the single sequences, statistical convergence of order α and strong p-Cesàro summability of order α was 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 double natural density zero).
Mursaleen and Edely  presented the notion statistical convergence for a double sequence as follows: A real double sequence is said to be statistically convergent to L provided that for each ,
We now give 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 which 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 previous definition.
Let be a non-decreasing sequence of positive numbers tending to ∞ such that
The generalized de la Valèe-Poussin mean is defined by
where . A sequence is said to be -summable to a number L if as .
In  Mursaleen introduced the idea of λ-statistical convergence for a single sequence as follows:
The number sequence is said to be λ-statistically convergent to the number ℓ if for each ,
In this case, we write and we denote the set of all λ-statistically convergent sequences by .
Definition 1.2 Let and be two non-decreasing sequences of positive real numbers both of which tend to ∞ as m and n approach ∞, respectively. Also, let , and , . We write the generalized double de la Valèe-Poussin mean by
A sequence is said to be -summable to a number L, if as in the Pringsheim sense. Throughout this paper, we denote by and (, ) by .
2 Main results
In this section, we define λ-double almost statistically convergent sequences of order α. Also, we prove some inclusion theorems.
We now have the following.
Definition 2.1 Let be given. The sequence is said to be -statistically convergent of order α if there is a real number L such that
where denotes 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 write if and for .
We know that the -statistical convergence of order α is well defined for , but it is not well defined for in general. For this let be fixed. Then, for an arbitrary number L and , we write
Therefore is not uniquely determined for .
Definition 2.2 Let be any real number and let r 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 sequences of order α by .
We now are ready to state the following theorem.
Theorem 2.1 If , then .
Proof Let . Then
for every , and finally we have that . This proves the result. □
We have the following from the previous theorem.
If a sequence is -statistically convergent of order α to L, then it is -statistically convergent to L, that is, for each ,
Theorem 2.2 if
Proof For given , we write
Using (2.1) and taking the limit as , we have . □
Theorem 2.3 Let and r be a positive real number, then .
Proof Let . Then given α and β such that and a positive real number r, we write
and we get that . □
We have the following corollary which is a consequence of Theorem 2.3.
Corollary 2.2 Let and r be a positive real number. Then
If , then .
for each and .
Theorem 2.4 Let α and β be fixed real numbers such that and . If a sequence is a strongly -summable sequence of order α to L, then it is -statistically convergent of order β to L, i.e., .
Proof For any sequence and , we write
and so that
This shows that if is a 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.
Corollary 2.3 Let α be fixed real numbers such that and .
If a sequence is a strongly -summable sequence of order α to L, then it is -statistically convergent of order α to L, i.e., .
3 Some sequence spaces
In present section, we study the inclusion relations between the set of -statistically convergent sequences of order α and strongly -summable sequences of order α with respect to an Orlicz function M.
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 , .
Lindenstrauss and Tzafriri  used the idea of an Orlicz function to construct the sequence space
The space with the norm
becomes a Banach space called an Orlicz sequence space. The space is closely related to the space which is an Orlicz sequence space with for .
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 almost strongly double λ-summable of order α with respect to the Orlicz function M.
If we consider various assignments of M, and r in the above sequence spaces, we are granted the following:
If , and for all , then .
If for all , then .
If for all and , then .
If , then .
We now have the following theorem.
Theorem 3.1 If and x is almost strongly λ-double convergent to with respect to the Orlicz function M, that is, , then is unique.
The proof of Theorem 3.1 is straightforward. So, we omit it.
In the following theorems, we assume that is bounded and .
Theorem 3.2 Let be real numbers such that and M be an Orlicz function, then .
Proof Let , be given and ∑1 and ∑2 denote the sums over , and , , respectively. Since , for each m, n we write
Since , the left-hand side of the above inequality 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 3.1 Let and M be an Orlicz function, then .
We conclude this paper with the following theorem.
Theorem 3.3 Let M be an Orlicz function and be 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 declares that they have no competing interests.