On Some Double Lacunary Sequence Spaces of Fuzzy Numbers

In this paper we introduce a new concept for lacunary strong P-convergent with respect to an Orlicz function and examine some properties of the resulting sequence space of fuzzy numbers. We also show that if a sequence is lacunary strong P-convergence with respect to an Orlicz function then it is , () r s S F θ-convergent.


INTRODUCTION
Among various developments of the theory of fuzzy sets [28] 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 wide range applications in the field of science and engineering, e.g., population dynamics [1], chaos control [8], computer programming [9], nonlinear dynamical systems [10], fuzzy physics [13], fuzzy topology [22], etc.Recently fuzzy topology proves to be a very useful tool to deal with such situation where the use of classical theories breaks down.The most fascinating application of fuzzy topology in quantum particle physics arises in string and ε ∞ -theory of El-Nashie [4,5,6].
In [18], Nanda studied on sequences of fuzzy numbers and showed that the set of all convergent sequences of fuzzy numbers forms a complete metric space.Nuray [19] proved the inclusion relations between the set of statistically convergent and lacunary statistically convergent sequences of fuzzy numbers.Savas [23] introduced and discussed double convergent sequences of fuzzy numbers and showed that the set of all double convergent sequences of fuzzy numbers is complete.
The idea of statistical convergence of a sequence was introduced by Fast [7].Statistical convergence was generalized by Buck [2] and studied by other authors, using a regular nonnegative summability matrix A in place of Cesáro matrix.The existing literature on double statistical convergence appears to have been restricted to real or complex sequences, but at the first time Savaş and Mursaleen [26] extended the idea to apply to double sequences of fuzzy numbers.
The study of Orlicz sequence spaces was initiated with a certain specific purpose in Banach space theory.Lindenstrauss and Tzafriri [14] investigated Orlicz sequence spaces in more detail, and they proved that every Orlicz sequence space M l contains a subspace isomorphic to (1 ) p l p ≤ < ∞ .The Orlicz sequence spaces are the special cases of Orlicz spaces studied in [12].Orlicz spaces find a number of useful applications in the theory of nonlinear integral equations.Whereas the Orlicz sequence spaces are the generalization of p l -spaces, the p l -spaces find themselves enveloped in Orlicz spaces [11].
Subsequently the notion of Orlicz function was used to define sequence spaces by Parashar and Choudhary [20] and other authors.An Orlicz function M can be represented in the following integral form: where p is the known kernel of M, right differential for 0 t ≥ , (0) 0 p = , ( ) 0 p t > for 0 t > , p is non-decreasing and ( ) p t → ∞ as t → ∞ .Recently Savas generalized ( ) c ∆ and ( ) l ∞ ∆ by using Orlicz function and also established some inclusion theorems.In this paper, using an Orlicz function some sequence spaces of fuzzy numbers have been given.Later on we give a brief overview about fuzzy numbers, Orlicz function and also we define the concepts of double lacunary sequence spaces of fuzzy numbers with respect to the Orlicz function M and also we prove some inclusion relations between , ,  and , ( )

INTRODUCTION AND BACKGROUND
We begin this definition presentation with the statement of a few preliminaries which are useful in the sequel of this paper.
Recall in [12] that an Orlicz function where ⋅ denotes the usual Euclidean norm in n R .It is well known that ( ) A fuzzy number is a function X from n R to [0,1] satisfying (1) X is normal, i.e. there exists an 0 (2) X is fuzzy convex, i.e. for any , These properties imply that for each 0 1 the set of all fuzzy numbers.The linear structure of ( ) Moreover q d is a complete, separable and locally compact metric space [3].
Throughout the paper, d will denote q d with 1 q ≤ < ∞ .
Definition 2.1.The double sequence is called double lacunary if there exist two increasing sequences of integers such that We have the following definitions.( ) , , ( ) : lim 0, for some 0 , , ( ) : lim 0, for some 0 , otherwise.We shall denote , , , respectively when , 1  we shall say that X is strongly double lacunary P-convergent with respect to the Orlicz function M. In this case we write , 0 , , for some 0 With these new concepts we can now consider the following theorem.
 the last two terms trends to zero in the Pringsheim sense, thus (1).
we are granted the following:  be a double lacunary sequence with lim sup r r q < ∞ , and lim sup r r q < ∞ then for any Orlicz function M, , Proof.Since lim sup r r q < ∞ , and lim sup r r q < ∞ there exists H > 0 such that r q H < and s q H < for all r and s.Let , , ,

Mk l r s p u p u k l k l r p r s u s Mk l r s p u p u k l k l p r u s r p r s u s h
non decreasing function define for 0 x > such that (0) 0 M = and ( ) 0 M x > for 0 x > , and ( ) M x → ∞ as x → ∞ .The Hausdorff distance between A and B of ( )

4 .
Let M be an Orlicz function and, double sequence of positive real numbers, we define the following sequence spaces: and , 1 k l p = for all k and l, presented by Savas in[27] as follows:

Theorem 2 . 1 .
If , 0 k l p > and X is strongly double lacunary convergent to 0 X , with respect to the Orlicz function M, that is . Then it follows:( l both approaches infinity as both m and n approaches infinity.Thus ( ) . In[27], Savas defined lacunary statistical convergence for double sequence of fuzzy numbers as follows: Definition 2.5.Let , r s θ be a double lacunary sequence; the double number sequence X is , The following Theorem is an immediate consequence of Theorem 2.2 and Theorem 2.3