On Some I-convergence of Difference Double Sequence Classes of Fuzzy Real Numbers Defined by Modulus Function

Manmohan Das* and Sanjay Kumar Das

*Department of Mathematics, Bajali College, Assam, INDIA.

Department of Mathematics, Handique Girls' College, Assam, INDIA.
email: mdas.bajali@gmail.com, sanjay_nlb@rediffmail.com

(Received on: December 3, 2017)

ABSTRACT

In this article our aim to introduce some new I-convergence of difference double sequence spaces of fuzzy real numbers defined by modulus function and studies their some topological and algebraic properties. Also we establish some inclusion relations.

AMS Classification No.: 40A05, 40D25, 46A45, 46E30.

Keywords:Fuzzy real number, I- convergence, double sequence, modulus function.

1.INTRODUCTION

INTRODUCTION The notion of fuzzy sets was introduced by Zadeh32. After that many authors have studied and generalized this notion in many ways, due to the potential of the introduced notion. Also it has wide range of applications in almost all the branches of science where mathematics has been used. It attracted many workers to introduce different types of fuzzy sequence spaces. Bounded and convergent sequences of fuzzy numbers were studied by Matloka8. Later on sequences of fuzzy numbers have been studied by Kaleva and Seikkala2, Tripathy and Sarma13,14 and many others.
I-convergence of real valued sequence was studied at the initial stage by Kostyrko, salat and Wilczynski4 which generalizes and unifies different notions of convergence of sequences. The notion was further studied by salat, Tripathy and Ziman9.
Let X be a non-empty set, then a non-void class I ⊆ 2X (power set of X ) is called an ideal if I is additive (i.e. A, B ∈ I ∈ A=>B=>I) and hereditary (i.e. A∈I and B=> A=> B∈I). An ideal I ⊆ 2X is said to be non-trivial if I ± 2X . A non-trivial ideal I is said to be admissible if I contains every finite subset of N. A non-trivial ideal I is said to be maximal if there does not exist any non-trivial ideal J ± I containing I as a subset.

A sequence X = (Xk) of fuzzy numbers is a function X from the set N of all positive integers into L(R). The fuzzy number Xk denotes the value of the function at k∈N and is called the k-th term or general term of the sequence. The set of all sequences of fuzzy numbers is denoted by wF . A sequence (Xk) of fuzzy real numbers is said to be convergent to the fuzzy real number X 0 , if for every ∈ >0, there exists k0 ∈ N such that d ( X k , X 0 )< ∈ for all k ≥ k0 . A sequence X = (Xk) of fuzzy numbers is said to be I- convergent if there exists a fuzzy number X 0 such that for all ∈ > 0, the set {k∈N: d (Xk, X 0 ) ≥ ∈ }∈I. We write I-lim X k = X 0 . A sequence (Xk) of fuzzy numbers is said to be I- bounded if there exists a real number μsuch that the set {k∈N : d (Xk, 0 ) > μ}∈I. We follow Tripathy and Hazarika18,19,23,25, Tripathy and Sen31, Tripathy and Mahanta22, Tripathy and Tripathy12 for the ideals considered throughout the article. If I = I f , then I f convergence coincides with the usual convergence of fuzzy sequences. If I = I d ( Iδ ), then I d ( Iδ ) convergence coincides with statistical convergence (logarithmic convergence) of fuzzy sequences. If I = Iu , Iu convergence is said to be uniform convergence of fuzzy sequences.







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