On the Occasion of his 80th Birthday Applications of Lacunary Sequences to develop Fuzzy Sequence Spaces for Ideal Convergence and Orlicz Function

In the present paper, we introduce and study ideal convergence of some fuzzy sequence spaces via lacunary sequence, infinite matrix and Orlicz function. We study some topological and algebraic properties of these spaces. We also make an effort to show that these spaces are normal as well as monotone. Further, it is very interesting to show that if I is not maximal ideal then these spaces are not symmetric. 2020 Mathematics Subject Classifications: 46A45, 40A05, 03E72


Introduction and preliminaries
The concept of ordinary convergence of a sequence of fuzzy numbers was introduced by Matloka [18] and proved some basic theorems for sequences of fuzzy numbers. Later on Nanda [28] introduced sequences of fuzzy numbers and studied that the set of all convergent sequences of fuzzy numbers forms a complete metric space. Recently, Nuray and Savaş [30] studied statistical convergence and statistically Cauchy for sequence of fuzzy numbers. They proved that a sequence of fuzzy numbers is statistically convergent if and only if it is statistically Cauchy. Initially the idea of I-convergence was introduced by Kostyrko et al. [15]. A lot of developments have been made in this area, one may refer to the articles (see [1-4, 10, 11, 16, 19, 23, 37]).
Let X be a non empty set. Then a family of sets I ⊆ 2 X (power set of X) is said to be an ideal if I is additive i.e U 1 , U 2 ∈ I ⇒ U 1 ∪ U 2 ∈ I and U 1 ∈ I, U 2 ⊆ U 1 ⇒ U 2 ∈ I. A non empty family of sets G ⊆ 2 X is said to be filter on X if and only if Φ / ∈ G, for U 1 , U 2 ∈ G we have U 1 ∩ U 2 ∈ G and for each U 1 ∈ G and U 1 ⊆ U 2 implies U 2 ∈ G. An ideal I ⊆ 2 X is called non trivial if I = 2 X . A non-trivial ideal I ⊆ 2 X is called admissible if {{x} : x ∈ X} ⊆ I. A non-trivial ideal is maximal if there cannot exist any non-trivial ideal J = I containing I as a subset.
Let L(R) denotes the set of all fuzzy numbers. The α-level set of a fuzzy real number u, for 0 < α ≤ 1 denoted by u α is defined as [u] α = {x ∈ R : u(x) ≥ α}, for α = 0 it is the closure of the strong 0 cut (i.e. closure of the set {t ∈ R : u(t) > 0}). For each r ∈ R, r ∈ L(R) is defined bȳ . In this case, (L(R), d) is a complete metric space. The additive identity and multiplicative identity in L(R) are denoted by 0 and 1, respectively.
An Orlicz function M : [0, ∞) → [0, ∞) is convex, continuous and non-decreasing function which also satisfy M (0) = 0, M (x) > 0 for x > 0 and M (x) → ∞ as x → ∞. If convexity of Orlicz function is replaced by M (x + y) ≤ M (x) + M (y), then this function is called the modulus function and characterized by Nakano [27] and followed by Ruckle [33] and others. An Orlicz function M is said to satisfy ∆ 2 -condition for all values of u, if there exists R > 0 such that M (2u) ≤ RM (u), u ≥ 0. Lindenstrauss and Tzafriri [17] used the idea of Orlicz function to define the following sequence space which is called as an Orlicz sequence space. The space M is a Banach space with the norm An increasing non-negative integer sequence θ = (i r ) with i 0 = 0 and h r = (i r −i r−1 ) → ∞ as r → ∞ is known as lacunary sequence. The intervals determined by θ are denoted by I r = (i r−1 , i r ] and the ratio i r /i r−1 will be denoted by q r . Freedman et al. [7] defined the space N θ in the following way: Fridy and Orhan [8] defined and studied the idea of lacunary statistical for sequence of real number. Nuray [29] and Mursaleen and Mohiuddine [25] defined this notion, respectively, for sequences of fuzzy numbers and in the setting of intuitionistic fuzzy normed space. Most recently, Mohiuddine and Alamri [20] defined the notion of weighted lacunary equistatistical convergence and, as an application, proved some approximation theorems. In [14] Kızmaz introduced the notion of difference sequence spaces and studied ∞ (∆), c(∆) and c 0 (∆) which has been recently used to define statistical convergence [12,21]. Further this notion was generalized by Et and Ç olak [6] by introducing the spaces ∞ (∆ m ), c(∆ m ) and c 0 (∆ m ). Later on, another type of generalization of the difference sequence spaces is due to Tripathy and Esi [39] who studied the spaces ∞ (∆ ν ), c(∆ ν ) and c 0 (∆ ν ). Recently, Esi et al. [5] and Tripathy et al. [40] have introduced a new type of generalized difference operators and unified those as follows: Let ν, m be non-negative integers, then for Z a given sequence space, we have x k+1 ) and ∆ 0 ν x k = x k for all k ∈ N, which is equivalent to the following binomial representation Taking ν = 1, we get the spaces ∞ (∆ m ), c(∆ m ) and c 0 (∆ m ) studied by Et and Ç olak [6]. Taking m = ν = 1, we get the spaces ∞ (∆), c(∆) and c 0 (∆) introduced and studied by Kızmaz [14]. For more details about sequence spaces (see [9,31,32,34,36,38,41]) and references therein.
Let λ and η be two sequence spaces and A = (a nk ) be an infinite matrix of real or complex numbers a nk , where n, k ∈ N. Then we say that A defines a matrix mapping from λ into η if for every sequence By (λ, η), we denote the class of all matrices A such that A : λ → η. Thus, A ∈ (λ, η) if and only if the series on the right-hand side of (1.1) converges for each n ∈ N and every x ∈ λ.
The matrix domain λ A of an infinite matrix A in a sequence space λ is defined by The approach constructing a new sequence space by means of the matrix domain of a particular limitation method has recently been employed by several authors (see [35]). Kumar and Kumar [16] defined the notion of ideal (or, I-) convergence for sequence of fuzzy numbers and recently studied by Mursaleen and Mohiuddine [26] in probabilistic normed spaces (see also [22]).

Definition 1.
A sequence X = (X k ) of fuzzy numbers is said to be I-convergent to a fuzzy number X 0 , if for every ε > 0 such that The fuzzy number X 0 is called I-limit of the sequence (X k ) of fuzzy numbers and we write I-lim X k = X 0 .
Definition 3. Let θ = (k r ) be lacunary sequence. Then a sequence (X k ) of fuzzy numbers is said to be lacunary I-convergent if for every ε > 0 such that r ∈ N : 1 hr k∈Ir Then I F is a nontrival admissible ideal of N and the corresponding convergence coincide with the usual convergence.
(ii) If we take where δ(A) denote the asymptotic density of the set A. Then I δ is a non-trival admissible ideal of N and the corresponding convergence coincide with the statistical convergence.

Lemma 1. [24] If d is a translation invariant metric. Then
(For the crisp set case, one may refer to Kamthan and Gupta [13]).

Some fuzzy sequence spaces
Throughout the paper w F denote the class of all fuzzy real-valued sequences. By N and R we denote the set of natural and real numbers respectively. Let I be an admissible ideal of N and θ = (i r ) be lacunary sequence. Suppose p = (p k ) is a bounded sequence of positive real numbers, u = (u k ) be a sequence of nonzero, nonnegative real numbers, A = (a nk ) an infinite matrix and M = (M k ) be a sequence of Orlicz functions. In this paper, we define the following sequence spaces as follows: for some ρ > 0, s ≥ 0 and X 0 ∈ L(R) , for some ρ > 0 and s ≥ 0 , For instance take m = v = 1, then the α−level sets of (X k ) and (∆X k ) are and Let A = (C, 1), the Cesàro matrix, M(x) = x, s = 0, u = (u k ) = 1, p = (p k ) = 1, for all k ∈ N, ρ = 1 and θ = 2 r , we have sup n k∈Ir is not an Ideal convergent.
Let us consider a few special cases of the above sequence spaces: (ii) If p = (p k ) = 1, for all k, then we have w (iii) If we take A = (C, 1), i.e., the Cesàro matrix, then the above classes of sequences are denoted by w (iv) If we take A = (a nk ) a de la Vallée-Poussin mean, i.e., where (λ n ) is a non-decreasing sequence of positive numbers tending to ∞ and λ n+1 ≤ λ n + 1, λ 1 = 1, then the above classes of sequences are denoted by w = (x k ) ∈ w F : ∀ε > 0, n, r ∈ N : for some ρ > 0 and s ≥ 0 .
The following inequality will be used throughout the paper. Let p = (p k ) be a sequence of positive real numbers with 0 < p k ≤ sup k p k = H and let D = max 1, 2 H−1 . Then, for the factorable sequences (a k ) and (b k ) in the complex plane, we have Also |a k | p k ≤ max 1, |a| H for all a ∈ C.
The main purpose of this paper is to introduced and study some lacunary I-convergent sequence spaces of fuzzy numbers by using an infinite matrix and a sequence of Orlicz functions in more general setting. We also make an effort to study some properties like linearity, paranorm, solidity and some interesting inclusion relations between the spaces w

Main Results
In the current section we study some topological properties and some inclusion relations between the sequence spaces which we have defined above.
Let α and β be two scalars. Then by using the inequality (3) and continuity of the function From the above relation we obtain the following: This completes the proof.
Theorem 2. Let M = (M k ) be a sequence of Orlicz functions, p = (p k ) be a bounded sequence of positive real numbers and u = (u k ) be a sequence of strictly positive real numbers. Then the spaces w Proof. Clearly, g ∆ (−X) = g ∆ (X) and g ∆ (θ) = 0. Let X = (X k ) and Y = (Y k ) be two elements in w Then for every ρ > 0 we write Let ρ 1 ∈ A 1 and ρ 2 ∈ A 2 . If ρ = ρ 1 + ρ 2 , then we get the following 1 hr Thus, we have If ρ k ∈ A 3 and ρ k ∈ A 4 then by inequality (3) and continuity of the function M = (M k ), we have that From the above inequality it follows that The other part can be proved in the same way.
(ii) Let X = (X k ) be an element in w Then for each 0 < ε < 1 there exists a positive integer n 0 such that This implies that Therefore, we have n, r ∈ N : 1 hr k∈Ir a nk k −s M k The other part can be proved in the similar way. This completes the proof.
Proof. The inclusion w Then there is some fuzzy number X 0 , such that Now, by inequality (3), we have This implies that X = (X k ) ∈ w F θ [A, M, p, u, ∆ m v ] ∞ . This completes the proof.
Theorem 5. Let M = (M k ) and S = (S k ) be a sequence of Orlicz functions. Then Thus, X = (X k ) ∈ w again the set Hence, B 1 ∈ I and so Y = (Y k ) ∈ w   If m = 0, then ∆ m v X k = 1. Let A = (C, 1), the Cesàro matrix, M(x) = x, u = (u k ) = 1, s = 0, p = (p k ) = 1, for all k ∈ N, ρ = 1 and θ = 2 r then we have (X k ) ∈ w Since I is not maximal by Lemma 3, their exist a subset K of N such that K / ∈ I and N − K / ∈ I. Let us define sequence Y = (Y k ) by otherwise.