On generalized sequence spaces via modulus function
© Savaş; licensee Springer. 2014
Received: 25 September 2013
Accepted: 4 February 2014
Published: 4 March 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.
Keywordsmodulus function almost convergence lacunary sequence φ-function statistical convergence
if (i.e. for all n),
, where ,
, where the shift operator D is defined by .
The space is closely related to the space , which is an space with for all real .
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
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.
If then we have .
If , then we have .
, then we have .
Thus . Finally, .
The proof of (c) follows from (a) and (b). This completes the proof. □
Proof This can be proved by using techniques similar to those used in the theorem of Savaş . □
Recently Savaş  defined -statistical convergence as follows.
We now establish inclusion relations between and .
In the following theorem we assume that .
- (b)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
Taking the limit as , we observe that .
This completes the proof. □
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.
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