IDEAL CONVERGENT SEQUENCE SPACES WITH RESPECT TO INVARIANT MEAN AND A MUSIELAK-ORLICZ FUNCTION OVER n-NORMED SPACES

In the present paper we defined I-convergent sequence spaces with respect to invariant mean and a Musielak-Orlicz function M = (Mk) over n-normed spaces. We also make an effort to study some topological properties and prove some inclusion relation between these spaces.

V σ denotes the set of bounded sequences all of whose invariant means are equal which is also called as the space of σ-convergent sequences.In [26], it is defined by . σ-mean is called a Banach limit if σ is the translation mapping n → n + 1.In this case, V σ becomes the set of almost convergent sequences which is denoted by ĉ and defined in [11] as ĉ = x ∈ ∞ : lim k d kn (x) exists uniformly in n , The space of strongly almost converegnt sequences was introduced by Maddox [12] as follow: ĉ = x ∈ ∞ : lim k d kn (|x − e|) exists uniformly in n for some .
The notion of ideal convergence was first introduced by P. Kostyrko [8] as a generalization of statistical convergence which was further studied in topological spaces by Das, Kostyrko, Wilczynski and Malik see [1].
More applications of ideals can be seen in ( [1], [2]).Mursaleen and Sharma [19] continue in this direction and introduced I-convergence of generalized sequences with respect to Musielak-Orlicz function.
A family I ⊂ 2 X of subsets of a non empty set X is said to be an ideal in X if while an admissible ideal I of X further satisfies {x} ∈ I for each x ∈ X see [8].
Let A = A ij be an infinite matrix of complex numbers a ij , where i, j, ∈ N. We write Ax = (A i (x)) if a ij x j converges for each i ∈ N. Throughout the paper, by t kn (Ax), we mean Let X be a sequence space and A canonical preimage of a sequence (x kn ) ∈ Z X K is a sequence (y n ) ∈ w defined by A sequence space X is monotone if it contains the canonical preimages of all its step spaces.
Lindenstrauss and Tzafriri [10] used the idea of Orlicz function to define the following sequence space.Let w be the space of all real or complex sequences x = (x k ), then which is called as an Orlicz sequence space.The space M is a Banach space with the norm It is shown in [10] that every Orlicz sequence space M contains a subspace isomorphic to p (p ≥ 1).The ∆ 2 −condition is equivalent to M (Lx) ≤ kLM (x) for all values of x ≥ 0, and for L > 1.
A sequence M = (M k ) of Orlicz function is called a Musielak-Orlicz function see ( [13], [20]).A sequence is called the complementary function of a Musielak-Orlicz function M. For a given Musielak-Orlicz function M, the Musielak-Orlicz sequence space t M and its subspace h M are defined as follows where I M is a convex modular defined by We consider t M equipped with the Luxemburg norm or equipped with the Orlicz norm For more details about sequence spaces defined by Orlicz function see ( [22], [23], [24]) and reference therein.
The concept of 2-normed spaces was initially developed by Gähler [3] in the mid of 1960's, while that of n-normed spaces one can see in Misiak [14].Since then, many others have studied this concept and obtained various results, see Gunawan ([4], [5]) and Gunawan and Mashadi [6].Let n ∈ N and X be a linear space over the field K, where K is field of real or complex numbers of dimension d, where •|| on X n satisfying the following four conditions: (1) For example, we may take X = R n being equipped with the Euclidean n-norm ume of the n-dimensional parallelopiped spanned by the vectors x 1 , x 2 , • • • , x n which may be given explicitly by the formula where If every cauchy sequence in X converges to some L ∈ X, then X is said to be complete with respect to the n-norm.Any complete n-normed space is said to be n-Banach space.
In the present paper, we define some new sequence spaces by using the concept of ideal convergence, invariant mean, Musielak-Orlicz function, n-normed and A transform as follows: If we take p = (p k ) = 1, we get the spaces The following inequality will be used throughout the paper. then The main goal of this paper is to introduce the sequence spaces defined by a Musielak-Orlicz function M = (M k ) over n-normed spaces.We also make an effort to study some topological properties and prove some inclusion relation between these spaces.

Main Results
Theorem 2.1 Let M = (M k ) be a Musielak-Orlicz function, p = (p k ) be a bounded sequence of positive real numbers.Then the spaces and let α, β be scalars.Then there exist positive numbers ρ 1 and ρ 2 such that for every > 0 Let ρ 3 = max 2|α|ρ 1 , 2|β|ρ 2 .Since M = (M k ) is non-decreasing, convex function and so by using inequality (1.1), we have Now by (2.1) and (2.2), we have Therefore αx Proof.The first inclusion is obvious.For second inclusion, let x ∈ I − c σ (A, M, p, ||•, • • • , •||).Then there exists ρ 1 > 0 such that for every > 0 Let us define ρ = 2ρ 1 .Since M = (M k ) is non-decreasing and convex, we have Hence by above inequality and (1.1), we have This completes the proof of the theorem.
Proof.It is clear that g(x) = g(−x).Since M k (0) = 0, we get g(0) = 0. Let us take x, y ∈ I − Let ρ 1 ∈ B(x) and ρ 2 ∈ B(y).If ρ = ρ 1 + ρ 2 , then we have Let η s → η where η, η s ∈ C and let g(x s − x) → 0 as s → ∞.We have to show that g(η s x s − ηx) → 0 as If ρ s ∈ B(x s ) and ρ s ∈ B(x s − x) then we observe that From the above inequality, it follows that and consequently, This completes the proof of the theorem.
Proof.(i) We prove the theorem in two parts.Firstly, let nondecreasing, convex and satisfies ∆ 2 -condition, we have where K ≥ 1 and δ < 1.From the last inequality, the inclusion , then the set in the right side of the above inclusion belongs to the ideal and so This completes the proof of (i) part.Similarly, we can prove other parts.
This completes the proof of (i) part of the theorem.Similarly, we can prove (ii) and (iii) part.Write α k = p k q k .By hypothesis, we have 0 holds.This implies the inclusion