Some Generalized Lacunary statistically difference double semi-normed sequence spaces defined by Orlicz function

In this article, we have introduced the idea of statistically convergent generalized difference lacunary double sequence spaces 2 [ ( , , , )] n w M p q θ Δ , 2 0 [ ( , , , )] n w M p q θ Δ and defined over a semi norm space (X, q). Also we have study some basic properties and obtained some inclusion relations between them.


Introduction
The concept involving statistical convergence plays a vital role not only in pure mathematics but also in other branches of mathematics especially in information theory, computer science and biological science.
Let ∞  ,c and 0 c be the Banach spaces of bounded, convergent and null sequences ( ) . In order to extend the notion of convergence of sequences, statistical convergence was introduced by Fast (1951) and Schoenberg (1959) independently.Later on it was further investigated by Fridy (1985), Mursaleen and Mohiuddine (2009a and b), Mohiuddine and Danish Lohani (2009), Mohiuddine et al. (2010), Šalát (1980Šalát ( ), Tripathy (2003)), Tripathy and Sen (2001) and many others.The idea depends on the notion of density (natural or asymptotic) of subsets of N. A subset E of N is said to have natural density ( ) has natural density zero.Kizmaz (1981) introduced the notion of difference sequence spaces as follows: for X= ∞  ,c and c o .Later on, the notion was generalized by Et and Çolak (1995) as follows: for X= ∞  ,c and c o , where and also this generalized difference notion has the following binomial representation: Subsequently, difference sequence spaces were studied by Esi (2009a and b), Esi andTripathy (2008), Tripathy et al. (2005) and many others.
Lindenstrauss and Tzafriri (1971) used the idea of Orlicz function to construct the sequence space The space M l is closely related to the space p l , which is an Orlicz sequence space with In a later stage different Orlicz sequence spaces were introduced and studied by Tripathy and Mahanta (2004), Esi (1999Esi ( , 2009aEsi ( and b, 2010)), Esi and Et (2000), Parashar and Choudhary (1994) and many others.
The following well-known inequality will be used throughout the article.Let p = (p k ) be any sequence of positive real numbers with for all a ∈ .
Let w 2 denote the set of all double sequences of complex numbers.By the convergence of a double sequence we mean the convergence in the Pringsheim sense that is, a double sequence x = (x k,l ) has Pringsheim limit L (denoted by lim P x L − = ) provided that given ε > 0 there exists , 1900).And we called it as "Pconvergent".We shall denote the space of all Pconvergent sequences by c 2 .The double sequence x = (x k,l ) is bounded if and only if there exists a positive number M such that for all k and l.We shall denote the space of all bounded double sequences by 2 ∞ l .The zero single sequence will be denoted by θ = (θ, θ, θ,…) and the zero double sequence will be denoted by θ 2 = (θ).
The notion of asymtotic density for subsets of The notion of statistically convergent double sequences was introduced by Mursaleen and Edely (2003) andTripathy (2003) independently.
A double sequence (x k,l ) is said to be statistically convergent to  in Pringsheim's sense if for every ε > 0, The double sequence , {( , )}

Notations:
, , , The set of all double lacunary sequences is denoted by In this presentation our goal is to extend a few results known in the literature from ordinary (single) difference sequences to difference double sequences.Some studies on double sequence spaces can be found in Gökhan and Çolak (2004Gökhan and Çolak ( , 2005Gökhan and Çolak ( 2006)).
Let M be an Orlicz function and , ( ) factorable double sequence of strictly positive real numbers and , r s θ be a lacunary sequence.Let X be a seminormed space over the complex field  with the seminorm q.We now define the following new statistically convergent generalized difference lacunary double sequence spaces: and where: ) and also this generalized difference double notion has the following binomial representation: ( 1)


Some double sequence spaces are obtained by specializing , r s θ M, p, q and n.Here are some examples: then we obtain the double sequence spaces then we obtain the double sequence then we obtain the double sequence spaces and where ( ) ( ) ) (3.4) x Δ is a Cauchy sequence in (X, q).Since (X, q) is complete, there exists x k,l ∈ X such that , , lim ( ) is continuous, so for 0 i m ≥ , on taking limit as j → ∞ we have from (3.4), On taking the infimum of such ρ′ s, we have . By linearity of the space 2 0 [ ( , , , )] Proof.The first part of the result follows from the inequality and the second part of the result follows from the inequality Proof.We prove it for ( ) [ ( , , , )] Thus from the second term in (3.6) we have ( ) ( ) [ ( , , , )] M and 2 M be Orlicz functions, q, 1 q and 2 q be seminorms.Then The proofs of (ii) and (iii) follow obviously.The proof of the following result is routine work.
Proposition 3.7.For any Orlicz function M, if and 2 w ∞ . Let ( ( )) ( )  which leads us to the desired results.

Conclusion
In this article we defined some new sequence spaces by double lacunary summability method by combining the concept of Orlicz function and statistical convergence.Further, we proved some topological and algebraic properties of the resulting spaces.