Lacunary Statistical Convergence of Multiple Sequences in Intuitionistic Fuzzy Normed Spaces

In this paper we introduce the notion ideal statistical convergence and ideal lacunary statistical convergence of multiple sequences with respect to the intuitionistic fuzzy norm   , v  , investigate their relationships, and make some observations about these classes. The notion of statistical convergence is a very useful functional tool for studying the convergence problems of numerical sequences/matrices (double sequences) through the concept of density. It was first introduced by Fast [1], and Schoenberg [2], independently for the real sequences. Later on it was further investigated from sequence point of view and linked with the summability theory by Fridy [3], Salat [4] and many others. The idea is based on the notion of natural density of subsets of  , the set of positive integers, which is defined as follows: The natural density of a subset K of  is denoted by   K  and is defined by     1 lim : , n K k K k n n      (1) where the vertical bar denotes the cardinality of the respective set. (see, [5]). In another direction, a new type of convergence, called lacunary statistical convergence, was introduced in [6] as follows. A lacunary sequence is an increasing integer sequence

The notion of statistical convergence is a very useful functional tool for studying the convergence problems of numerical sequences/matrices (double sequences) through the concept of density.It was first introduced by Fast [1], and Schoenberg [2], independently for the real sequences.Later on it was further investigated from sequence point of view and linked with the summability theory by Fridy [3], Salat [4] and many others.The idea is based on the notion of natural density of subsets of  , the set of positive integers, which is defined as follows: The natural density of a subset K of  is denoted by   K  and is defined by where the vertical bar denotes the cardinality of the respective set.( see, [5]).
In another direction, a new type of convergence, called lacunary statistical convergence, was introduced in [6] as follows.A lacunary sequence is an increasing integer sequence     A sequence   k x of real numbers is said to be lacunary statistically convergent to L (or S  -convergent to L ) if, for any 0 where |A| denotes the cardinality of A ⊂  .In [6], the relation between lacunary statistical convergence and statistical convergence was established, among other things.
Recently Savas and Patterson [7] introduced and studied lacunary statistical convergence for double sequences and also some inclusion theorems are presented for real sequences.Recently, Mohiuddine and Aiyub [8] studied lacunary statistical convergence as generalization of the statistical convergence introduced the concept θstatistical convergence in random 2-normed space.In [9], Mursaleen and Mohiuddine extended the idea of lacunary statistical convergence with respect to the intuitionistic fuzzy normed space.Also lacunary statistically convergent double sequences in probabilistic normed space was studied by Mohiuddine and Savaş in [10].Maio and Kocinac [11] introduced statistical convergence in topology.In [12], some new double sequence spaces of fuzzy real numbers by combining I-convergence, Orlicz function and four dimensional matrix are introduced.More results on this convergence can be found in [13,14,15,16,17,18].The concepts of fuzzy sets and fuzzy set operations were first introduced by Zadeh [19].Subsequently several authors have discussed various aspects of the theory and applications of fuzzy sets.The theory of intuitionistic fuzzy sets was introduced by Atanassov [20]; it has been extensively used in decision-making problems [21].The concept of an intuitionistic fuzzy metric space was introduced by Park [22].Further, Saadati and Park [23] gave the notion of an intuitionistic fuzzy normed space.Some works related to the convergence of sequences in several normed linear spaces in a fuzzy setting can be found in [24,25,26,27,28,29].
In this paper, as a variant of double statistical convergence, the notions of ideal double statistical convergence and ideal lacunary double statistical convergence are introduced in an intuitionistic fuzzy normed linear space, which naturally extend the notions of double statistical convergence and double lacunary statistical convergence and some important results are established.Furthermore, we try to establish the relations between these two summability notions.
Throughout the paper,  will denote the set of all natural numbers.First we shall give some basic definitions used in this paper.

Definition 1.1. ([30]
) A triangular norm (t-norm) is a continuous mapping  is an abelian monoid with unit one and ] is said to be a continuous t-conorm if it satisfies the following conditions: (i)  is associate and commutative, (ii)  is continuous, (iii) 0 aa  for all

 
, 0,1 .ab  Using the continuous t-norm and t-conorm, Saadati and Park [23] has recently introduced the concept of intuitionistic fuzzy normed space as follows.

Definition1.3. ([23]) The 5-tuple  
, , , , Xv   is said to be an intuitionistic fuzzy normed space (for short, IFNS) if X is a vector space,  is a continuous t-norm, ♢ is a continuous t-conorm, and , v  are fuzzy sets on   0, X  satisfying the following conditions for every , x y X  , and , 0 : Then observe that   , , , , Xv   is an intuitionistic fuzzy normed space.
We also recall that the concept of double convergence in an intuitionistic fuzzy normed space is studied in [31].

I-DOUBLE STATISTICAL AND I-DOUBLE LACUNARY STATISTICAL CONVERGENCE ON IFNS
In this section we deal with the relation between these two new methods as also the relation between I  -double statistical convergence and I -double statistical convergence in an intuitionistic fuzzy normed space   ,. v  Before proceeding further, we should recall some notation on the I -double statistical convergence and ideal convergence.
Statistical convergence of double sequences has been defined and studied by Mursaleen and Edely [32]; and for fuzzy numbers by Savas and Mursaleen [33].
Let K   be a two-dimensional set of positive integers and let   Then the two-dimensional analogue of natural density can be defined as follows [32].
The lower asymptotic density of the set K   is defined as In case the sequence , / K m n mn has a limit then we say that K has a double natural density and is defined as has double natural density zero.In this case we write 2, lim .kl kl st x L  P. Kostyrko et al [33] and [34] introduced the concept of I-convergence of sequences in a metric space and studied some properties of such convergence.Note that Iconvergence is an interesting generalization of statistical convergence.More investigations in this direction and more applications of ideals can be found in [36,37,38,39,40,41 ].Before proceeding further, we should recall some notation on the ideal.A family 2 Y I  of subsets of a nonempty set Y is said to be an ideal in Y if i) ), then the family of sets or equivalently if for each 0 where : and In this case we write .


In this case we write , , , r r r s s s q k k q l l   and , r s r s q q q  .We will denote the set of all double lacunary sequences by .
In this case, we write The class of all I-lacunary statistically convergent sequences will be denoted by     As the proofs for both the assertions are similar, we present the proof for , where stands for the space of all double bounded sequences of intuitionistic fuzzy norm   We need to show that ( , ) {( , ) : .
. 44 , and this completes the proof of the theorem.
, and the class of such sequences will be denoted simply by ( , )   2 , . , . .
This proves the result.
() b In order to establish that the inclusion ( is proper, let  be given and let x defined as follows: Then, for any 0 and for any 0 . , This proves the result.iii) Follows from (i) and (ii).Theorem 2.3.For any double lacunary sequence θ, I-double statistical convergence with respect to the intuitionistic fuzzy norm (μ, v) implies I-double lacunary statistical convergence with respect to the intuitionistic fuzzy norm (μ, v) if and only if liminf 1 rr q  and inf 1. ss q  If liminf 1 rr q  , and liminf 1 ss q  , then there exists a bounded double sequence   kl xx  which is I-double statistically convergent but not I-double lacunary statistically convergent.
Proof.Suppose first that liminf 1 rr q  and liminf 1. ss q  Then there exists 0   such that 1 r q   for sufficiently large r ve 1 s q   for sufficiently large s which implies that .
for every 0, 0, t   and for sufficiently large r,s, we have Then, for any 0, This proves the sufficiency.Conversely, suppose that lim inf 1 rr q  and lim inf 1 ss q  .We can select a subsequence    x is I -double statistically convergent with respect to the intuitionistic fuzzy norm   , v  for any admissible ideal I .

CONCLUDING REMARKS
Among various developments of the theory of fuzzy sets [19] a progressive development has been made to find the fuzzy analogues of the classical set theory.In fact the fuzzy theory has become an area of active research for the last 40 years.It has a wide range applications in the field of science and and engineering ,e.g., population dynamics, chaos control, computer programming, nonlinear dynamical systems, fuzzy physics, fuzzy topology, etc.
Recently fuzzy topology proves to be a very useful tool to deal with such situation where the use of classical theories breaks down.
In this paper, the notions of ideal double statistical convergence and ideal lacunary double statistical convergence are introduced in an intuitionistic fuzzy normed linear space, which naturally extend the notions of double statistical convergence and double lacunary statistical convergence and some important results are established.
.The class of all I-double statistically convergent sequences will be denoted by simply   with double statistical convergence,[32].Definition 2.3.[25].Let   , , , , Xv   be an IFNS.Then, a sequence   , kl xx  is said to be I -double statistically convergent sequences to L with respect to the intuitionistic fuzzy normed space  

Definition 2 . 4 .
The class of all I -double statistically convergent sequences to L with respect to the intuitionistic fuzzy normed space   , v  will be denoted simply by     with double statistical convergence with respect to the intuitionistic fuzzy normed space   , v  ,[31].The double sequence


It can be checked as in the case of I-double statistically and I-double lacunary statistically convergent sequences with respect to the intuitionistic fuzzy normed space of the space of real sequences.
on the right-hand side is a finite set and so belongs to I, it follows that is called the filter associated with the ideal.Throughout, I will stand for a proper admissible ideal of  .
We now ready to obtain our main definitions and results.