Lacunary Statistical Convergence in Fuzzy Normed Linear Spaces

: In this paper


Introduction
The concept of statistical convergence of real number sequences was introduced Fast [7] and Steinhaus [18] and later reintroduced by Schoenberg [15] independently. Many mathematicians such as Connor [5], Fridy [9], Colak [4], Et and Cinar [3], T. Šalát [19] etc studied the concept of statistical convergence in summability theory. in X is statistically convergent to L X ∈ with respect to the fuzzy norm on X provided that for each 0 This implies that for each 0 ε > , the set has natural density zero; namely, for each 0 < for almost all k [17].
is a lacunary sequence then we will take an increasing integer sequence such that 0 0 k = and  [16]. (4) Fuzzy sets are considered with respect to a nonempty base set X of elements of interest. The essential idea is that each element x X ∈ is assigned a membership grade ( ) u x taking values in [0,1] , with ( ) = 0 u x corresponding to nonmembership, 0 < ( ) < 1 u x to partial membership, and ( ) = 1 u x to full membership. According to Zadeh [21] a fuzzy subset of X is a nonempty subset {( , ( )) : u itself is often used for the fuzzy set. A fuzzy set u on R is called a fuzzy number if it has the following properties: 1. u is normal, that is, there exists an 0 x ∈ R such that 0 ( ) = 1; u x 2. u is fuzzy convex, that is, for , x y ∈ R and 0 1, Some arithmetic operations for α − level sets are defined as follows: , x of fuzzy numbers is said to be convergent to the fuzzy number 0 , 6,11,12] where C x is the ordinary norm of ( ), x θ ≠ 0 < < 1 a and x ɶ Hence ( ) , . X is a fuzzy normed linear space. [8] Sençimen [17] was defined convergence in fuzzy normed spaces by taking advantage of Kaleva [11,8], as follows; with respect to the fuzzy norm on X and we denote by , n N ε ≥ The purpose of this paper is to introduce and study a concept of lacunary convergence sequence with respect to fuzzy norm.

Main Result
In this section, we introduce notions of lacunary statistical convergence and lacunary summable in fuzzy normed linear spaces and we give some results.

Theorem
Let { } r k θ = be a lacunary sequence; then where l ∞ is the set of bounded sequences.

Proof.
(1) a) If 0 If we pay attention here, the example given in (1) we have got an unbounded sequence. Afterwards we added boundedness in (2). In this case the hypothesis given (1) for the bounded sequence already provide.
(3) When the boundedness condition is added as a result of (1) and (2)

Conclusion
In this study, we introduced the concept of lacunary statistical convergence with respect to a fuzzy norm. We also studied the relation between lacunary summabilty and lacunary convergence in fuzzy normed space. As a result, we have seen that results similar to the results in classical normed spaces are also obtained in fuzzy normed spaces.