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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.36 no.3 Antofagasta Sept. 2017 


On fuzzy normed linear space valued statistically convergent sequences*

Paritosh Chandra Das1 

1Rangia College, Department of Mathematics, India, email:


In this article we define the notion of statistically convergent and statistically null sequences with the concept of fuzzy norm and discuss some of their properties such as completeness, monotone, solidness, symmetricity sequence algebra and convergence free.

Keywords: Fuzzy real number; statistical convergence; fuzzy normed linear space; monotone; solid space; symmetricity; sequence algebra and convergence free.


The concept of fuzzy set, a set whose boundary is not sharp or precise has been introduced by L. A. Zadeh in 1965. It is the origin of new theory of uncertainty, distinct from the notion of probability. After the introduction of fuzzy sets, the scope for studies in different branches of pure and applied mathematics increased widely. The notion of fuzzy sets has successfully been applied in studying sequence spaces with classical metric by Nanda (7), Savas (10),Tripathy and Baruah (15), Tripathy and Dutta (16) and many others. Using the concept of fuzzy metric, different authors namely Kelava and Seikkala (6), Syau (13) and many others have worked in different fields but, only a few works have been done on fuzzy norm by Das, Das and Choudhury (1), Esi (2), Felbin (4) and Tripathy and Nanda (18) and some others.

The notion of statistical convergence was introduced by Fast (3) and Schoenberg (11) independently. The potential of the introduced notion was realized in eighties by the workers on sequence spaces. Since than, a lot of work has been done on classical statistically convergent sequences. It is evidenced by the works of Fridy (5), Nuary and Savas (8), Salát (9), Tripathy (14), Tripathy and Sen (17) and many others. There are many works done on statistically convergent sequences of fuzzy real numbers under classical metric, but we dont find work on statistically convergent sequences with the concept of fuzzy norm. A sizable work can be investigated in the direction of statistically convergent sequences with fuzzy norm.

2.Definitions and preliminaries

A fuzzy real number X is a fuzzy set on R, i.e. a mapping X: RI(= 0,1 ) associating each real number t with its grade of membership X(t).

A fuzzy real number X is called convex if X, (t) ≥ X (s) ˄ X (r) = min (X(s), X(r)) where s < t < r

If there exists t 0R such that X(t 0) = 1, 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 ɛ > 0, X¯1( 0,ɑ + ɛ )), for all ɑI is open in the usual topology of R.

The set of all upper-semi continuous, normal, convex fuzzy real numbers is denoted by R(I). Throughout the article, by a fuzzy real number we mean that the number belongs to R(I).

The α-level set X of the fuzzy real number X, for 0 < ⍺ ≤ 1, defined as X = {tR: X (t) ≥ ⍺}. If ⍺ = 0, then it is the closure of the strong 0-cut. (The strong ⍺-cut of the fuzzy real number X, for 0 ≤ ⍺ ≤ 1 is the set {tR: X(t) > ⍺}).

The absolute value, |X| of XR(I) is defined by (see for instance Kaleva and Seikkala (6))

A fuzzy real number X is called non-negative if X(t)=0, for all t < 0. The set of all non-negative fuzzy real numbers is denoted by R (I).

Statistically convergent sequence

A subset E of N is said to have natural density δ(E) if

where χE is the characteristic function of E. Clearly all finite subsets of Ν have zero natural density and δ(Ec) = δ(N ̶ E) = 1 ̶ δ(E).

A given complex sequence x = ( xk ) is said to be statistically convergent to L if for every

Let ( xk ) and (yk) be two sequences, then we say that xk = yk for almost all k (in short ɑ. ɑ. k.) if δ (kN: xk ≠ yk) = 0.

Fuzzy Normed Linear Space

Let X be a linear space over R and the mapping || · ||: XR (I) and the mappings L, M: 0,1 0,1 0,1 be symmetric, non-decreasing in both arguments and satisfy L (0,0) = 0 and M(1,1) = 1. Write for xX, 0 < α ≤ 1 and suppose for all xX, x ≠ 0 there exists α0 ∈ (0,1 independent of X such that for all α ≤ α0,

In the sequel we take L(x, y) = min(x, y) and M(x, y) = max(x, y) for x, y, ∈ 0,1 and we denote (X, ‖ · ‖, L, M) by (X, ‖ · ‖) or simply by X in this case.

With these L(x, y) = min(x, y) and M(x, y) = max(x, y) for x, y, ∈ 0,1 we have (refer to Felbin (4)) in a fuzzy normed linear space (X, ‖ · ‖), the triangle inequality (iii) of the definition of fuzzy normed linear space is equivalent to

x + y‖ ≤ ‖x‖ ⊕ ‖y

The set ⍵(X) of all sequences in a vector space X is a vector space with respect to pointwise addition and scalar multiplication. Any subspace λ(X) of ⍵(X) is called vector valued sequence space. When (X, ‖ · ‖) is a fuzzy normed linear space, then λ(X) is called a fuzzy normed linear space-valued sequence space.

Throughout fnls denotes fuzzy normed linear space.

A fnls-valued sequence space EF (X) is said to be normal (or solid) if ( yk ) ∈ EF (X), whenever ‖ yk ‖ ≤ ‖ xk ‖, for all kN and ( xk ) ∈ EF (X).

A fnls-valued sequence space EF (X) is said to be monotone if EF (X) contains the canonical pre-images of all its step spaces.

From the above definitions we have following remark.

Remark 2.1. A fnls-valued sequence space EF (X) is solid ⇒ EF (X) is monotone.

A fnls-valued sequence space EF (X) is said to be symmetric if ( ( n )) ∈ EF (X), whenever ( xk ) ∈ EF (X), where π is a permutation of N.

A fnls-valued sequence space EF (X) is said to be convergence free if ( yk ) ∈ EF (X) whenever ( xk ) ∈ EF (X) and xk = 0 implies yk = 0.

3.Main results

We show that x mF (X).

Now for each m, n > n 0N and from the inequalities (3.1), (3.2) and (3.3), we get

Thus ( ɑn ) is a Cauchy sequence in X. Since X complete, so there exists such a number a ϵ X such that


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*The work of the author is supported by University Grants Commission of India videproject No. F. 42-28/2013 (SR), dated 12 March, 2013, Rangia College, India

Received: January 2017; Accepted: April 2017

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