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On generalized sequence spaces via modulus function
Journal of Inequalities and Applications volume 2014, Article number: 101 (2014)
In this paper, we introduce and study the concept of lacunary strongly -convergence with respect to a modulus function and lacunary -statistical convergence and examine some properties of these sequence spaces. We establish some connections between lacunary strongly -convergence and lacunary -statistical convergence.
Let s denote the set of all real and complex sequences . By and c, we denote the Banach spaces of bounded and convergent sequences normed by , respectively. A linear functional L on is said to be a Banach limit  if it has the following properties:
if (i.e. for all n),
, where ,
, where the shift operator D is defined by .
Let B be the set of all Banach limits on . A sequence is said to be almost convergent if all Banach limits of x coincide. Let denote the space of almost convergent sequences. Lorentz  has shown that
By a lacunary ; , where , we shall mean an increasing sequence of non-negative integers with as . The intervals determined by θ will be denoted by and . The ratio will be denoted by . The space of lacunary strongly convergent sequences was defined by Freedman et al.  as follows:
In the special case where (see ) we have , which is defined by
Das and Mishra  have introduced the space of lacunary almost convergent sequences and the space of lacunary strongly almost convergent sequences as follows:
Ruckle used the idea of a modulus function f to construct a class of FK spaces,
The space is closely related to the space , which is an space with for all real .
In 1999, Savaş  generalized the concept of strong almost convergence by using a modulus f and is a sequence of strictly positive real numbers as follows:
Following Ruckle , a modulus function f is a function from to such that
if and only if ,
for all ,
f is continuous from the right at zero.
By a φ-function we understand a continuous non-decreasing function defined for and such that , , for and as .
A φ-function φ is called non-weaker than a φ-function ψ if there are constants such that (for all large u) and we write .
A φ-function φ and ψ are called equivalent if there are positive constants , , c, , , l such that (for all large u) and we write .
In this paper, we introduce and study some properties of the following sequence space which is generalization of Savaş .
2 Main results
Let φ and f be a given φ-function and modulus function, respectively, and let be a sequence of positive real numbers. Moreover, let be the generalized three parametric real matrix with and a lacunary sequence θ be given. Then we define the following sequence spaces:
If , the sequence x is said to be lacunary strong -convergent to zero with respect to a modulus f. When for all x, we obtain
If we take , we write
If we take , for all k, we have
If we take and , respectively, then we have
If we define the matrix as follows:
then we have
then we have
If , the sequence x is said to be almost lacunary strong φ-convergent to zero with respect to a modulus f. In the next theorem we establish inclusion relations between and . We now have the following.
Theorem 2.1 Let f be any modulus function and let there be a φ-function φ and a generalized three parametric real matrix A; let be a sequence of positive real numbers and the sequence θ be given. If
then the following relations are true:
If then we have .
If , then we have .
, then we have .
Proof (a) Let us suppose that . There exists such that for all and we have for sufficiently large r. Then, for all i,
(b) If then there exists such that for all . Let and ε is an arbitrary positive number, then there exists an index such that for every and all i,
Thus, we can also find such that for all . Now let m be any integer with , then we obtain, for all i,
It is easy to see that
Moreover, we have for all i
Thus . Finally, .
The proof of (c) follows from (a) and (b). This completes the proof. □
Theorem 2.2 Let f, , be modulus functions. Then we have
Proof This can be proved by using techniques similar to those used in the theorem of Savaş . □
Recently Savaş  defined -statistical convergence as follows.
Let θ be a lacunary sequence, and let be the generalized three parametric real matrix, the sequence , the φ-function and a positive number be given. We write, for all i,
The sequence x is said to be -statistically convergent to a number zero if for every
where denotes the number of elements belonging to . We denote by , the set of sequences which are lacunary -statistical convergent to zero and we write
We now establish inclusion relations between and .
In the following theorem we assume that .
Theorem 2.3 (a) If the matrix A and the sequence θ and functions f and φ are given, then
(b) If the φ-function and the matrix A are given, and if the modulus function f is bounded, then
Proof (a) Let f be a modulus function and let ε be a positive numbers. We write the following inequalities, for all i,
Finally, if then .
Let us suppose that . If the modulus function f is a bounded function, then there exists an integer K such that for . Let us take
Thus we have, for all i,
Taking the limit as , we observe that .
This completes the proof. □
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I would like to express my gratitude to the referee of the paper for his useful comments and suggestions towards the quality improvement of the paper. This paper was presented during the ‘2nd International Eurasian Conference on Mathematical Sciences and Applications’ held on Sarajevo, Bosnia and Herzegovina on 26th to 29th August 2013, and it was submitted for the conference proceedings.
The author declares that they have no competing interests.
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Cite this article
Savaş, E. On generalized sequence spaces via modulus function. J Inequal Appl 2014, 101 (2014) doi:10.1186/1029-242X-2014-101
- modulus function
- almost convergence
- lacunary sequence
- statistical convergence