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The Multi Rough Ideal Convergence of Difference Strongly of χ² In p-Metric Spaces Defined by Orlicz Functions

Ayhan Esi*, Subramanian Nagarajan, Ayten Esi

Department of Mathematics, Adiyaman University, Adiyaman, 02040, Turkey

Abstract: The aim of this paper is to introduce multi rough and study a new concept of the χ² space via ideal convergence of difference operator defined by Orlicz. Some topological properties of the resulting sequence spaces are also discussed.

Keywords: Orlicz function, double sequences, χ² space, p-metric space, multi rough ideal

Citation: Ayhan Esi, Subramanian Nagarajan, Ayten Esi, The Multi Rough Ideal Convergence of Difference Strongly of χ² In p-Metric Spaces Defined by Orlicz Functions. Turkish Journal of Analysis and Number Theory, 5 (3). 93-100. January 2017. doi:10.12691/tjant-5-3-3

1. Introduction

The idea of rough convergence was introduced by Phu [3], who also introduced the concepts of rough limit points and roughness degree. The idea of rough convergence occurs very naturally in numerical analysis and has interesting applications. Aytar [1] extended the idea of rough convergence into rough statistical convergence using the notion of natural density just as usual convergence was extended to statistical convergence. Pal et al. [2] extended the notion of rough convergence using the concept of ideals which automatically extends the earlier notions of rough convergence and rough statistical convergence. Statistical convergence and ideal convergence was introduced by Mursaleen [7] and Tripathy et al. [8,9,10,11,12,13] and many others. In this paper we have introduced some Orlicz sequence spaces of fuzzy number using the notion of rough I- convergence and studied some algabraic and topological properties of these spaces.

Throughout and denote the classes of all, gai and analytic scalar valued single sequences respectively. We write for the set of all complex double sequences where the set of positive integers. Then, is a linear space under the coordinate wise addition and scalar multiplication.

Let be a double sequence of real or complex numbers. Then the series is called a double series. The double series give one space is said to be convergent if and only if the double sequence is convergent, where

A double sequence is said to be double analytic if

The vector space of all double analytic sequences are usually denoted by A sequence is called double entire sequence if

The vector space of all double entire sequences are usually denoted by Let the set of sequences with this property be denoted by and is a metric space with the metric

[picture] | (1.1)

for all and in Let

Consider a double sequence The section of the sequence is defined by for all

with 1 in the position and zero otherwise.

An Orlicz function is a function which is continuous, non-decreasing and convex with for and as If convexity of Orlicz function is replaced by then this function is called modulus function. An Orlicz function is said to satisfy -condition for all values if there exists such that

Remark 1: An Orlicz function satis.es the inequality for all with

1.1. Lemma. Let f be an Orlicz function which satisfies - condition and let Then for each we have for some constant

Let M and be mutually complementary Orlicz functions. Then, we have

(i) For all

[picture] | (1.2)

[4]

(ii) For all

[picture] | (1.3)

(iii) For all and

[picture] | (1.4)

[5] used the idea of Orlicz function to construct Orlicz sequence space

The space with the norm

becomes a Banach space which is called an Orlicz sequence space. For the spaces coincide with the classical sequence space

A sequence of Orlicz function is called a Musielak-Orlicz function. A sequence defined by

is called the complementary function of a Musielak-Orlicz function f. For a given Musielak Orlicz function f, the Musielak-Orlicz sequence space tf is defined by

where is a convex modular defined by

We consider equipped with the Luxemburg metric space.

Let be a family of metric spaces such that each two elements of the family are disjoint. Denote If we define

then the pair is a Luxemburg metric space. The notion of difference sequence spaces (for single sequences) was introduced by Kizmaz [6] as follows

for and where for all

Here and denote the classes of convergent, null and bounded scalar valued single sequences respectively. The difference sequence space of the classical space is introduced and studied in the case by Başar and Altay and in the case The spaces and are Banach spaces normed by

Later on the notion was further investigated by many others. We now introduce the following difference double sequence spaces defined by

where and for all The generalized difference double notion has the following representation: and also this generalized difference double notion has the following binomial representation:

2. Definitions and Preliminaries

Let be a non empty set. A non-void class (power set, of ) is called an ideal if is additive (i.e ) and hereditary (i.e and ). A non-empty family of sets is said to be a filter on if and For each ideal there is a filter given by A non-trivial ideal is called admissible if and only if

A double sequence space is said to be solid or normal if whenever and for all double sequences of scalars with for all

Let and be a real vector space of dimension where A real valued function on satisfying the following four conditions:

(i) if and only if are linearly dependent,

(ii) is invariant under permutation,

(iii)

(iv)

for (or)

(v)

for is called the p-product metric of the Cartesian product of n-metric spaces is the p-norm of the n-vector of the norms of the n-sub spaces.

A trivial example of p-product metric of n-metric space is the p-norm space is X = equipped with the following Euclidean metric in the product space is the p-norm:

where for each

If every Cauchy sequence in X converges to some then X is said to be complete with respect to the p-metric. Any complete p-metric space is said to be p-Banach metric space.

2.1. Definition. A sequence is said to be rough convergent (r-convergent) to if for every there exists a positive integer n0 such that for all where is a non negative real number called the convergence degree.

2.2. Definition. A sequence is said to be rough I- convergent to if for each

Here is called the rough I-limit of the sequence and we write

3. Main Results

In this section we introduce the notion of different types of I-convergent double sequences. This generalizes and unifies different notions of convergence for We shall denote the ideal of by

Let be an ideal of f be an Orlicz function. Let and be two non-negative integers and be a sequence of non-zero reals. Then for a sequence be a double analytic sequence of strictly positive real numbers and be an p-product of n metric spaces is the p norm of the n-vector of the norms of the n subspaces. Further denotes -valued sequence space. Now, we define the following sequence spaces:

If for all then we obtain

The following well-known inequality will be used in this study: then

for all and Also for all

3.1. Theorem. The classes of sequences

are linear spaces over the complex field

Proof: Now we establish the result for the case and the others can be proved similarly. Let and Then

Since be an p-product of n metric spaces is the p norm of the n-vector of the norms of the n subspaces and f is an Orlicz function, the following inequality holds:

From the above inequality we get

This completes the proof.

3.2. Theorem. The class of sequence is a paranormed space with respect to the paranorm defined by

Proof: and are easy to prove, so we omit them. Let us take Let

Then we have

Thus

and

Now, where and as We have to prove that as Let

and

We observe that

From this inequality, it follows that

and consequently

Hence by our assumption the right hand side tends to zero as and This completes the proof.

3.3. Theorem. (i) If then

(ii) If then

(iii) If and is double analytic, then

Proof: The proof can be established using standard technique.

The following result is well known.

3.4. Lemma. If a sequence space E is solid, then it is monotone.

3.5. Theorem. The class of sequence is not solid and hence not monotone.

Proof: It is routine verification. Therefore we omit the proof.

3.6. Theorem. Let f, f₁ and f₂ be Orlicz functions. Then we have

(i)

(ii)

Proof: (i) Let For given we first choose such that Now using the continuity of f, choose such that implies Let

We observe that

Thus if then

Hence from above inequality and using continuity of f; we must have

Hence we have

(ii) Let

Then the fact that

This completes the proof.

3.7. Theorem. The class of sequence is a sequence algebra.

Proof: Let

and Then the result follows from the following inclusion relation:

Similarly we can prove the result for other cases.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this research paper.

References

[1]   | S. Aytar 2008. Rough statistical Convergence, Numerical Functional Analysis Optimization, 29(3), 291-303.
[2]   | S.K. Pal, D. Chandra and S. Dutta 2013. Rough ideal Convergence, Hacettepe Journal Mathematics and Statistics, 42(6), 633-640.
[3]   | H.X. Phu 2001. Rough convergence in normed linear spaces, Numerical Functional Analysis Optimization, 22, 201-224.
[4]   | P.K. Kamthan and M. Gupta 1981. Sequence spaces and series, Lecture notes, Pure and Applied Mathematics, 65 Marcel Dekker, In c., New York.
[5]   | J. Lindenstrauss and L. Tzafriri 1971. On Orlicz sequence spaces, Israel Journal of Mathematics, 10, 379-390.
[6]   | H. Kizmaz 1981. On certain sequence spaces, Canadian Mathematical Bulletin, 24(2), 169-176.
[7]   | M. Mursaleen and O.H.H. Edely 2003. Statistical convergence of double sequences, Journal of Mathematical Analysis and Applications, 288(1), 223-231.
[8]   | B.C. Tripathy, M. Sen and S. Nath 2012. I-convergence in probabilistic n-normed space, Soft Computing, 16, 1021-1027.
[9]   | B.C. Tripathy, B. Hazarika and B. Choudhary 2012. Lacunary I-convergent sequences, Kyung-pook Math. Journal, 52(4), 473-482.
[10]   | B.C. Tripathy and B. Sarma 2008. Statistically convergent difference double sequence spaces, Acta Mathematica Sinica, 24(5), 737-742.
[11]   | B.C. Tripathy and B. Sarma 2012. On I-convergent Double sequence spaces of fuzzy numbers, Kyungpook Math. Journal, 52(2), 189-200.
[12]   | B.C. Tripathy 2003. On statistically convergent double sequences, Tamkang Journal of Mathematics, 34(3), 231-237.
[13]   | B.C. Tripathy and B. Hazarika 2008. I-convergent sequence spaces associated with multiplier sequence spaces, Mathematical Inequalities and Applications, 11(3), 543-548.