Triple sets of χ 3-summable sequences of fuzzy numbers defined by an Orlicz function

In this paper we introduce the 3 fuzzy numbers defined by an Orlicz function and study some of their properties and inclusion results. Subjects: Science; Mathematics & Statistics; Foundations & Theorems


ABOUT THE AUTHORS
Vandana is a Research Scholar at School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur-492010, (C.G.) India. Her research interests are in the areas of applied mathematics including Optimization, Mathematical Programming, Inventory control, Supply Chain Management, Operation Research, etc. She is member of several scientific committees, advisory boards as well as member of editorial board of a number of scientific journals.
Deepmala is Visiting Scientist at SQC & OR Unit at Indian Statistical Institute, Kolkata, India. Her research interests are in the areas of Optimization, Mathematical Programming, Fixed Point Theory and Applications, Operator theory, Approximation Theory etc. She is member of several scientific committees and also member of editorial board of a number of scientific journals.
N. Subramanian received PhD degree in Mathematics from Alagappa University at Karaikudi, Tamil Nadu, India and also getting Doctor of Science (D.Sc) degree in Mathematics from Berhampur University, Berhampur, India. His research interests are in the areas of summability through functional analysis of applied mathematics and pure mathematics.

PUBLIC INTEREST STATEMENT
In this paper, we introduced the 3 fuzzy numbers defined by an Orlicz function and study some of their properties with inclusion results. Furthermore we provided an example of triple sequence of gai which is not symmetric, not solid, not monotone and not convergent free. Our result unifies the results of several author's in the case of classical Orlicz spaces. One can extend our results for more general spaces.
1 m+n+k → 0 as m, n, k → ∞. The triple gai sequences will be denoted by 3 . This paper deals with introducing the 3 -fuzzy number defined by an Orlicz function and study some topological properties, inclusion relations and give some examples. Some interesting results may be seen in Alzer et al. (2006), Bor et al. (2012), Choi and Srivastava (1991), Liu and Srivastava (2006).

Definitions and preliminaries
Definition 2.1 An Orlicz function (see Kamthan & Gupta, 1981) is a function M: 0, ∞ → 0, ∞ which is continuous, non-decreasing and convex with M 0 , then this function is called modulus function. Lindenstrauss and Tzafriri (1971) used the idea of Orlicz function to construct Orlicz sequence space.
Throughout a triple sequence is denoted by ⟨ X mnk ⟩ , a triple infinite array of fuzzy real numbers.
Let D denote the set of all closed and bounded intervals X = a 1 , a 2 , a 3 on the real line ℝ. For It is known that D, d is a complete metric space.
A fuzzy real number X is a fuzzy set on ℝ, that is, a mapping X:ℝ × ℝ × ℝ → I × I × I = 0, 1 associating each real number t with its grade of membership X(t).
The -level set X , of the fuzzy real number X, for 0 < ≤ 1; is defined by The 0-level set is th closure of the strong 0-cut that is, cl t ∈ ℝ: If there exists t 0 ∈ ℝ such that X t 0 = 1 then, the fuzzy real number X is called normal.
A fuzzy real number X is said to be upper-semi continuous if, for each < 0, X −1 0, a + is open in the usual topology of ℝ for all a ∈ I.
The set of all upper-semi continuous, normal, convex fuzzy real numbers is denoted by L(ℝ).
Then, d defines a metric on L(ℝ) and it is well-known that L(ℝ),d is a complete metric space.
A triple sequence ⟨ X mnk ⟩ of fuzzy real numbers is said to be gai in Pringsheim's sense to a fuzzy number 0 if lim m,n,k→∞ m + n + k !X mnk 1∕m+n+k = 0.
A triple sequence ⟨ X mnk ⟩ is said to regularly if it converges in the Prinsheim's sense and the following limts zero:

as follows:
A canonical pre-image of a step space E K is a set of canonical pre-images of all elements in E K .
A sequence set E F is said to be monotone if E F contains the canonical pre-images of all its step spaces.
We define the following classes of sequences: Also, we define the classes of sequences 3F R f as follows : f and the following limits hold

Sufficiency:
Let there exists a positive number such that P mnk > for sufficiently large positive integer m, n, k. Hence for any ∈ ℂ, we may write | | P mnk ≤ max | | M , | | for suffciently large positive integers m, n, k ≥ N. Therefore, we obtain Using this, one can prove that X → , whenever X is fixed and → 0 or → 0 and X → , or is fixed and X → .
Because a paranormed space is a vector space. 3F R f p is a set of sequences of fuzzy numbers. But the set w F = ⟨ X mnk ⟩ :X mnk ∈ L(R) of all sequences of fuzzy numbers is not a vector space. That is why, in order to say that 3F R f p is a vector subspace (that is a sequence space) it is not sufficient to show that 3F R f p is closed under addition and scalar multiplication. Consequently since w F is not a vector space, then 3F R f p is not a vector subspace so that it is not a sequence space. Therefore it cannot be a paranormed space.

Proof
. Then for every > 0 0 < < 1 there exists a positive integer s 0 such that for all k, , r, t > s 0 .
By (3.2) there exists a positive integer n 0 such that for all k, , r, t > s 0 and for N > n 0 . Hence we obtain so that for each k, 1 ∈ N × N × N for every > 0 0 < < 1 there exists a positive integer n 1 ∈ N such that By (3.6) and (3.7) we obtain max s 0 , s 1 and m, n, k > max n 0 , n 1 . This implies that X ∈ 3F R f p .