R E S E A R C H Open Access
On some A I -convergent difference sequence spaces of fuzzy numbers defined by the
sequence of Orlicz functions
Ekrem Savas*
*Correspondence:
ekremsavas@yahoo.com Department of Mathematics, Istanbul Commerce University, Uskudar, Istanbul, Turkey
Abstract
In this paper, using the difference operator of order m, the sequences of Orlicz functions, and an infinite matrix, we introduce and examine some classes of sequences of fuzzy numbers defined by I-convergence. We study some basic topological and algebraic properties of these spaces. In addition, we shall establish inclusion theorems between these sequence spaces.
MSC: 40A05; 40G15; 46A45
Keywords: ideal; I-convergent; infinite matrix; Orlicz function; fuzzy number;
difference space
1 Introduction
The notion of ideal convergence was introduced first by Kostyrko et al. [] as a generaliza- tion of statistical convergence [, ], which was further studied in topological spaces [].
More applications of ideals can be seen in [–].
The concepts of fuzzy sets and fuzzy set operations were first introduced by Zadeh [].
Subsequently, several authors have discussed various aspects of the theory and applica- tions of fuzzy sets such as fuzzy topological spaces, similarity relations and fuzzy ordering, fuzzy measures of fuzzy events, and fuzzy mathematical programming. In particular, the concept of fuzzy topology has very important applications in quantum particle physics, especially in connection with both string andε∞theory, which were given and studied by El Naschie []. The theory of sequences of fuzzy numbers was first introduced by Matloka []. Matloka introduced bounded and convergent sequences of fuzzy numbers, studied some of their properties, and showed that every convergent sequence of fuzzy numbers is bounded. In [], Nanda studied sequences of fuzzy numbers and showed that the set of all convergent sequences of fuzzy numbers forms a complete metric space. Different classes of sequences of fuzzy real numbers have been discussed by Nuray and Savas [], Altinok, Colak, and Et [], Savas [–], Savas and Mursaleen [], and many others.
The study of Orlicz sequence spaces was initiated with a certain specific purpose in Banach space theory. Lindenstrauss and Tzafriri [] investigated Orlicz sequence spaces in more detail, and they proved that every Orlicz sequence space lMcontains a subspace isomorphic to lp(≤ p < ∞). The Orlicz sequence spaces are the special cases of Orlicz spaces studied in []. Orlicz spaces find a number of useful applications in the theory of nonlinear integral equations. Although the Orlicz sequence spaces are the generalization
© 2012 Savas; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu- tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
of lpspaces, the lp-spaces find themselves enveloped in Orlicz spaces []. Recently, Savas [] generalized c() and l∞() for a single sequence of fuzzy numbers by using the Orlicz function and also established some inclusion theorems.
In the later stage, different classes of Orlicz sequence spaces were introduced and studied by Parashar and Choudhary [], Savas [–], and many others.
Throughout the article, wF denotes the class of all fuzzy real-valued sequence spaces.
Also,N and R denote the set of positive integers and the set of real numbers, respectively.
The operator n: wF → wF is defined by (X)k = Xk; (X)k=Xk = Xk– Xk+; (nX)k=nXk=n–Xk–n–Xk+(n≥ ) for all n ∈ N. The generalized difference has the following binomial expression for n≥ :
nxk=
n ν=
n ν
(–)νxk+ν. ()
In this paper, we study some new sequence spaces of fuzzy numbers using I-convergence, the sequence of Orlicz functions, an infinite matrix, and the difference operator. We es- tablish the inclusion relation between the sequence spaces wI(F)[A,M, m, p], wI(F)[A,M,
m, p], wF[A,M, m, p]∞, and wI(F)[A,M, m, p]∞, where p = (pk) denotes the sequence of positive real numbers for all k∈ N and M = (Mk) is a sequence of Orlicz functions. In addition, we study some algebraic and topological properties of these new spaces.
2 Definitions and notations
Before continuing with this paper, we present some definitions and preliminaries which we shall use throughout this paper.
Let X and Y be two nonempty subsets of the space w of complex sequences. Let A = (ank) (n, k = , , . . .) be an infinite matrix of complex numbers. We write Ax = (An(x)) if An(x) =
kankxkconverges for each n. (Throughout,
kdenotes summation over k from k = to k =∞). If x = (xk)∈ X ⇒ Ax = (An(x))∈ Y , we say that A defines a (matrix) transformation from X to Y and we denote it by A : X→ Y .
Let X be a nonempty set, then a family of sets I⊂ X (the class of all subsets of X) is called an ideal if and only if for each A, B∈ I, we have A ∪ B ∈ I, and for each A ∈ I and each B⊂ A, we have B ∈ I. A nonempty family of sets F ⊂ Xis a filter on X if and only if
/∈ F, for each A, B ∈ F, we have A ∩ B ∈ F, and for each A ∈ F and each A ⊂ B, we have B∈ F. An ideal I is called non-trivial ideal if I = and X /∈ I. Clearly, I ⊂ Xis a non-trivial ideal if and only if F = F(I) ={X – A : A ∈ I} is a filter on X. A non-trivial ideal I ⊂ Xis called admissible if and only if{{x} : x ∈ X} ⊂ I. A non-trivial ideal I is maximal if there cannot exist any non-trivial ideal J = I containing I as a subset. Further details on ideals of Xcan be found in Kostyrko et al. [].
Let D denote the set of all closed and bounded intervals X = [x, x] on the real lineR.
For X, Y∈ D, we define X ≤ Y if and only if x≤ yand x≤ y,
d(X, Y ) = max
|x– y|, |x– y|
, where X = [x, x] and Y = [y, y].
Then it can be easily seen that d defines a metric on D and (D, d) is a complete metric space (see []). Also, the relation ‘≤’ is a partial order on D. A fuzzy number X is a fuzzy subset of the real lineR, i.e., a mapping X : R → J (= [, ]) associating each real number
t with its grade of membership X(t). A fuzzy number X is convex if X(t)≥ X(s) ∧ X(r) = min{X(s), X(r)}, where s < t < r. If there exists t∈ R such that X(t) = , then the fuzzy number X is called normal. A fuzzy number X is said to be upper semicontinuous if for each > , X–([, a +)) for all a ∈ [, ] is open in the usual topology in R. Let R(J) denote the set of all fuzzy numbers which are upper semicontinuous and have compact support, i.e., if X∈ R(J), then for any α ∈ [, ], [X]αis compact, where
[X]α=
t∈ R : X(t) ≥ α, if α ∈ [, ] , [X]= closure of
t∈ R : X(t) > α, if α = .
The setR of real numbers can be embedded in R(J) if we define r ∈ R(J) by
r(t) =
⎧⎨
⎩
, if t = r;
, if t = r.
The additive identity and multiplicative identity ofR(J) are defined by and respec- tively.
The arithmetic operations onR(J) are defined as follows:
(X⊕ Y)(t) = sup
X(s)∧ Y(t – s)
, t∈ R, (X Y)(t) = sup
X(s)∧ Y(s – t)
, t∈ R, (X⊗ Y)(t) = sup
X(s)∧ Y
t s
, t∈ R,
X Y
(t) = sup
X(st)∧ Y(s)
, t∈ R.
Let X, Y ∈ R(J) and the α-level sets be [X]α= [xα, xα], [Y ]α= [yα, yα],α ∈ [, ]. Then the above operations can be defined in terms ofα-level sets as follows:
[X⊕ Y]α=
xα + yα, xα+ yα , [X Y]α=
xα – yα, xα– yα , [X⊗ Y]α=
i∈{,}minxαiyαi, max
i∈{,}xαiyαi ,
X–α
=
xα –
, xα –
, xαi > , for each <α ≤ .
For r∈ R and X ∈ R(J), the product rX is defined as follows:
rX(t) =
⎧⎨
⎩
X(r–t), if r = ;
, if r = .
The absolute value|X| of X ∈ R(J) is defined by
|X|(t) =
⎧⎨
⎩
max{X(t), X(–t)}, if t ≥ ;
, if t < .
Define a mapping ¯d :R(J) × R(J) → R+∪ {} by
¯d(X,Y) = sup
≤α≤d
[X]α, [Y ]α .
A metric ¯d on R(J) is said to be a translation invariant if ¯d(X + Z, Y + Z) = ¯d(X, Y) for X, Y , Z∈ R(J).
Proposition . If ¯d is a translation invariant metric on R(J), then (i) ¯d(X + Z, )≤ ¯d(X, ) + ¯d(Y, ),
(ii) ¯d(λX, ) ≤ |λ|¯d(X, ), |λ| > .
The proof is easy and so it is omitted.
A sequence X = (Xk) of fuzzy numbers is said to converge to a fuzzy number Xif for every > , there exists a positive integer nsuch that ¯d(Xk, X) < for all n ≥ n.
A sequence X = (Xk) of fuzzy numbers is said to be bounded if the set {Xk: k∈ N}
of fuzzy numbers is bounded. A sequence X = (Xk) of fuzzy numbers is said to be I- convergent to a fuzzy number Xif for each > such that
A =
k∈ N : ¯d(Xk, X)≥
∈ I.
The fuzzy number Xis called I-limit of the sequence (Xk) of fuzzy numbers, and we write I- lim Xk= X.
A sequence X = (Xk) of fuzzy numbers is said to be I-bounded if there exists M > such that
k∈ N : ¯d(Xk, ¯) > M
∈ I.
Example . If we take I = If={A ⊆ N : A is a finite subset}, then Ifis a non-trivial admis- sible ideal ofN, and the corresponding convergence coincides with the usual convergence.
Example . If we take I = Iδ={A ⊆ N : δ(A) = }, where δ(A) denotes the asymptotic density of the set A, then Iδ is a non-trivial admissible ideal ofN, and the corresponding convergence coincides with the statistical convergence.
Lemma . (Kostyrko, Salat, and Wilczynski [], Lemma .) If I ⊂ Nis a maximal ideal, then for each A⊂ N, we have either A ∈ I or N – A ∈ I.
Recall in [] that the Orlicz function M : [,∞) → [, ∞) is a continuous, convex, non- decreasing function such that M() = and M(x) > for x > , and M(x)→ ∞ as x → ∞.
If convexity of the Orlicz function is replaced by M(x + y)≤ M(x)+M(y), then this function is called the modulus function and characterized by Ruckle []. An Orlicz function M is said to satisfy the-condition for all values of u, if there exists K > such that M(u)≤ KM(u), u≥ .
The following well-known inequality will be used throughout the article. Let p = (pk) be any sequence of positive real numbers with ≤ pk≤ supkpk= G, H = max{, G–}, then
|ak+ bk|pk≤ H
|ak|pk+|bk|pk
for all k∈ N and ak, bk∈ C. Also, |ak|pk≤ max{, |a|G} for all a ∈ C.
3 Some new sequence spaces of fuzzy numbers
In this section, using the sequence of Orlicz functions, an infinite matrix, the difference operatorm, and I-convergence, we introduce the following new sequence spaces and examine some properties of the resulting sequence spaces. Let I be an admissible ideal ofN, and let p = (pk) be a sequence of positive real numbers for all k∈ N and A = (ank) be an infinite matrix. LetM = (Mk) be a sequence of Orlicz functions and X = (Xk) be a sequence of fuzzy numbers. We define the following new sequence spaces:
wI(F)
A,M, m, p
=
(Xk)∈ wF:∀ε > ,
n∈ N :
∞ k=
ank
Mk
¯d(mXk, X) ρ
pk
≥ ε
∈ I,
for someρ > and X∈ R(J)
,
wI(F)
A,M, m, p
=
(Xk)∈ wF:∀ε > ,
n∈ N :
∞ k=
ank
Mk
¯d(mXk, ¯) ρ
pk
≥ ε
∈ I,
for someρ >
,
wF
A,M, m, p
∞
=
(Xk)∈ wF: sup
n
∞ k=
ank
Mk
¯d(mXk, ¯) ρ
pk
<∞, for some ρ >
,
and
wI(F)
A,M, m, p
∞
=
(Xk)∈ wF:∃K > s.t.
n∈ N :
∞ k=
ank
Mk
¯d(mXk, ¯) ρ
pk
≥ K
∈ I,
for someρ >
.
Let us consider a few special cases of the above sets.
(i) If m = , then the above classes of sequences are denoted by wI(F)[A,M, p], wI(F)[A,M, p], wF[A,M, p]∞, and wI(F)[A,M, p]∞, respectively.
(ii) If Mk(x) = x for all k∈ N, then the above classes of sequences are denoted by wI(F)[A,m, p], wI(F)[A,m, p], wF[A,m, p]∞, and wI(F)[A,m, p]∞, respectively.
(iii) If p = (pk) = (, , , . . .), then we denote the above spaces by wI(F)[A,M, m], wI(F)[A,M, m], wF[A,M, m]∞, and wI(F)[A,M, m]∞.
(iv) If we take A = (C, ), i.e., the Cesàro matrix, then the above classes of sequences are denoted by wI(F)[M, m, p], wI(F)[M, m, p], wF[M, m, p]∞, and
wI(F)[M, m, p]∞, respectively.
(v) If we take A = (ank) is a de la Valée Poussin mean, i.e.,
ank=
⎧⎨
⎩
λn, if k∈ In= [n –λn+ , n],
, otherwise,
where (λn) is a non-decreasing sequence of positive numbers tending to∞ and λn+≤ λn+ ,λ= , then the above classes of sequences are denoted by wI(F)λ [A,M, m, p], wI(F)λ [M, m, p], wFλ[M, m, p]∞, and wI(F)λ [M, m, p]∞, respectively (see []).
(vi) By a lacunaryθ = (kr), r = , , , . . . where k= , we shall mean an increasing sequence of non-negative integers with kr– kr–as r→ ∞. The intervals determined byθ will be denoted by Ir= (kr–, kr] and hr= kr– kr–. As a final illustration, let
ank=
⎧⎨
⎩
hr, if kr–< k≤ kr,
, otherwise.
Then we denote the above classes of sequences by wI(F)θ [M, m, p], wI(F)θ [M, m, p], wFθ[M, m, p]∞, and wI(F)θ [M, m, p]∞, respectively.
(vii) If I = If, then we obtain
wF
A,M, m, p
=
(Xk)∈ wF: lim
n→∞
∞ k=
ank
Mk ¯d(mXk, X) ρ
pk
= ,
for someρ > and X∈ R(J)
,
wF
A,M, m, p
=
(Xk)∈ wF: lim
n→∞
∞ k=
ank
Mk ¯d(mXk, ¯) ρ
pk
= , for someρ >
,
wF
A,M, m, p
∞
=
(Xk)∈ wF: sup
n
∞ k=
ank
Mk
¯d(mXk, ¯) ρ
pk
<∞, for some ρ >
.
If X = (Xk)∈ wF[A,M, m, p], then we say that X = (Xk) is strongly A-convergent with respect to the sequence of Orlicz functionsM.
(viii) If I = Iδis an admissible ideal ofN, then we obtain
wI(F)
A,M, m, p
=
(Xk)∈ wF:∀ε > ,
n∈ N :
∞ k=
ank
Mk
¯d(mXk, X) ρ
pk
≥ ε
∈ Iδ,
for someρ > and X∈ R(J)
,
wI(F)
A,M, m, p
=
(Xk)∈ wF:∀ε > ,
n∈ N :
∞ k=
ank
Mk ¯d(mXk, ¯) ρ
pk
≥ ε
∈ Iδ,
for someρ >
,
and
wI(F)
A,M, m, p
∞
=
(Xk)∈ wF:∃K >
s.t.
n∈ N :
∞ k=
ank
Mk ¯d(mXk, ¯) ρ
pk
≥ K
∈ Iδ, for someρ >
.
4 Main results
In this section, we examine the basic topological and algebraic properties of the new se- quence spaces and obtain the inclusion relation related to these spaces.
Theorem . Let (pk) be a bounded sequence. Then the sequence spaces wI(F)[A,M, m, p], wI(F)[A,M, m, p], and wI(F)[A,M, m, p]∞are linear spaces.
Proof We will prove the result for the space wI(F)θ [M, m, p]only and others can be proved in a similar way.
Let X = (Xk) and Y = (Yk) be two elements in wI(F)θ [M, m, p]. Then there existρ> andρ> such that
Aε =
r∈ N :
∞ k=
ank
Mk
¯d(mXk, ¯) ρ
pk
≥ε
∈ I
and
Bε
=
r∈ N :
∞ k=
ank
Mk
¯d(mYk, ¯) ρ
pk
≥ε
∈ I.
Letα, β be two scalars. By the continuity of the function M = (Mk), the following in- equality holds:
∞ k=
ank
Mk ¯d(m(αXk+βYk, ¯))
|α|ρ+|β|ρ
pk
≤ D
∞ k=
ank
|α|
|α|ρ+|β|ρ
Mk ¯d(mXk, ¯) ρ
pk
+ D
∞ k=
ank
|β|
|α|ρ+|β|ρ
Mk ¯d(mYk, ¯) ρ
pk
≤ DK
∞ k=
ank
Mk
¯d(mXk, ¯) ρ
pk
+ DK
∞ k=
ank
Mk
¯d(mYk, ¯) ρ
pk
,
where K = max{, (|α|ρ|α|+|β|ρ)G, (|β|ρ|α|
+|β|ρ)G}.
From the above relation, we obtain the following:
n∈ N :
∞ k=
ank
Mk
¯d(m(αXk+βYk, ¯))
|α|ρ+|β|ρ
pk
≥ ε
⊆
n∈ N : DK
∞ k=
ank
Mk
¯d(mXk, ¯) ρ
pk
≥ ε
∪
n∈ N : DK
∞ k=
ank
Mk ¯d(mYk, ¯) ρ
pk
≥ε
∈ I.
This completes the proof.
Theorem . wI(F)[A,M, m, p], wI(F)[A,M, m, p], and wI(F)[A,M, m, p]∞ are linear topological spaces with the paranorm gdefined by
g(X) = inf
ρpnH :
∞
k=
ank
Mk
¯d(mXk, ¯) ρ
pk/H
≤ ,
for someρ > , n = , , , . . .
,
where H = max{, supkpk}.
Proof Clearly, g(–X) = g(X) and g(θ) = . Let X = (Xk) and Y = (Yk) be two elements in wI(F)[A,M, m, p]∞. Then for everyρ > , we write
A=
ρ > :
∞
k=
ank
Mk
¯d(mXk, ¯) ρ
pkH
≤
and
A=
ρ > :
∞
k=
ank
Mk ¯d(mXk, ¯) ρ
pkH
≤
.
Letρ∈ Aandρ∈ A. Ifρ = ρ+ρ, then we get the following:
∞
k=
ank
Mk
¯d(mXk, ¯) ρ
≤ ρ
ρ+ρ
∞
k=
ank
Mk
¯d(mXk, ¯) ρ
+ ρ
ρ+ρ
∞
k=
ank
Mk ¯d(mXk, ¯) ρ
.
Hence, we obtain
∞ k=
ank
Mk
¯d(mXk, ¯) ρ
pk
≤
and
g(x + y) = inf
(ρ+ρ)pnH :ρ∈ A,ρ∈ A
≤ inf
(ρ)pnH :ρ∈ A
+ inf
(ρ)pnH :ρ∈ A
= g(x) + g(y).
Let umk → t, where umk, u∈ C, and let g(Xkm– Xk)→ as m → ∞. To prove that g(umkXmk – uXk)→ as m → ∞, let uk→ u, where uk, u∈ C and g(Xkm– Xk)→ as m → ∞.
We have
A=
ρk> :
∞ k=
ank
Mk
¯d(mXk, ¯) ρk
pk
≤
and
A=
ρk> :
∞ k=
ank
Mk
¯d(mXk, ¯) ρk
pk
≤
.
Ifρk∈ Aandρk∈ Aand by continuity of the functionM = Mk, we have that
Mk
¯d(m(umXkm– uX), ¯)
|um– u|ρk+|u|ρk
≤ Mk
¯d(m(umXkm– uXk), ¯)
|um– u|ρk+|u|ρk
+ Mk
¯d(m(uXk– uX), ¯)
|um– u|ρk+|u|ρk
≤ |um– u|ρk
|um– u|ρk+|u|ρkMk ¯d(mXkm, ¯) ρk
+ |u|ρk
|um– u|ρk+|u|ρkMk
¯d(m(Xmk – Xk), ¯) ρk
.
From the above inequality, it follows that
∞ k=
ank
Mk
¯d(m(umXkm– uX), ¯)
|um– u|ρk+|u|ρk
pk
≤ ,
and consequently
g
umkxk– uxk
= infumk – uρk+|u|ρk
pnG
:ρk∈ A,ρk∈ A
≤umk – upnG inf
(ρk)pnG :ρk∈ A
+|u|pnG inf
ρk pnG
:ρk∈ A
≤ max
|u|, |u|pnG g
xmk – xk
.
Note that g(xm)≤ g(x) + g(xm– x) for all m∈ N.
Hence, by our assumption, the right-hand side tends to as m→ ∞. This completes the proof of the theorem.
Theorem . Let I be an admissible ideal and M = (Mk) be a sequence of Orlicz functions.
Then the following hold:
wI(F)[A,M, m–, p]⊂ wI(F)[A,M, m, p]; wI(F)[A,M, m–, p]⊂ wI(F)[A,M, m, p];
wI(F)[A,M, m–, p]∞⊂ wI(F)[A,M, m, p]∞for m ≥ and the inclusions are strict.
In general, for all i = , , , . . . , m – , the following hold:
wI(F)[A,M, i, p]⊂ wI(F)[A,M, m, p]; wI(F)[A,M, i, p]⊂ wI(F)[A,M, m, p];
wI(F)[A,M, i, p]∞⊂ wI(F)[A,M, m, p]∞and the inclusions are strict.
Proof Let X = (Xk) be an element in wI(F)[A,M, m–, p]∞. Then there exists K > , and for givenε > , ρ > , we have
n∈ N :
∞ k=
ank
Mk
¯d(m–Xk, ¯) ρ
pk
≥ K
∈ I.
SinceM = (Mk) is non-decreasing and convex, it follows that
∞ k=
ank
Mk
¯d(mXk, ¯)
ρ
pk
≤
∞ k=
ank
Mk
¯d(m–Xk+–m–Xk, ¯)
ρ
pk
≤ D
∞ k=
ank
Mk
¯d(m–Xk+, ¯) ρ
pk
+ D
∞ k=
ank
Mk
¯d(m–Xk, ¯) ρ
pk
≤ D
∞ k=
ank
Mk
¯d(m–Xk+, ¯) ρ
pk
+ D
∞ k=
ank
Mk
¯d(m–Xk, ¯) ρ
pk
.
Hence, we have
n∈ N :
∞ k=
ank
Mk
¯d(mXk, ¯)
ρ
pk
≥ K
⊆
n∈ N : D
∞ k=
ank
Mk
¯d(m–Xk+, ¯) ρ
pk
≥K
∪
n∈ N : D
∞ k=
ank
Mk ¯d(m–Xk, ¯) ρ
pk
≥K
∈ I.
The inclusion is strict, it follows from the following example.
Example . Let Mk(x) = x, pk= for all k∈ N and A = (C, ), i.e., the Cesàro matrix, m = . Consider the sequence X = (Xk) of fuzzy numbers as follows:
Xk(t) =
⎧⎪
⎪⎨
⎪⎪
⎩
–kt–– , for k– ≤ t ≤ ;
–kt+– , for < t≤ k+ ;
, otherwise.
Forα ∈ (, ], α-level sets of Xk,Xk,Xk, andXkare [Xk]α=
( –α) k–
, ( –α) k+
, [Xk]α=
( –α)
–k– k –
, ( –α)
–k– k + ,
Xk
α
=
( –α)(k + ), ( – α)(k + ) ,
Xkα
=
–( –α), ( – α) ,
respectively. It is easy to see that the sequence [Xk]αis not I-bounded although [Xk]α is I-bounded.
Theorem . (a) Let < inf pk≤ pk≤ . Then
wI(F)[A,M, m, p]⊆ wI(F)[A,M, m]; wI(F)[A,M, m, p]⊆ wI(F)[A,M, m]. (b) Let ≤ pk≤ sup pk<∞. Then
wI(F)[A,M, m]⊆ wI(F)[A,M, m, p]; wI(F)[A,M, m]⊆ wI(F)[A,M, m, p].
Proof (a) Let X = (Xk) be an element in wI(F)[A,M, m, p]. Since < inf pk≤ pk≤ , we have
∞ k=
ank
Mk
¯d(mXk, X) ρ
≤
∞ k=
ank
Mk
¯d(mXk, X) ρ
pk
.
Therefore,
n∈ N :
∞ k=
ank
Mk
¯d(mXk, X) ρ
≥ ε
⊆
n∈ N :
∞ k=
ank
Mk ¯d(mXk, X) ρ
pk
≥ ε
∈ I.
The other part can be proved in a similar way.
(b) Let X = (Xk) be an element in wI(F)[A,M, m, p]. Since ≤ pk≤ sup pk<∞, then for each <ε < , there exists a positive integer nsuch that
∞ k=
ank
Mk
¯d(mXk, X) ρ
≤ ε < for all n ≥ n.
This implies that
∞ k=
ank
Mk ¯d(mXk, X) ρ
pk
≤
∞ k=
ank
Mk ¯d(mXk, X) ρ
.
Therefore, we have
n∈ N :
∞ k=
ank
Mk
¯d(mXk, X) ρ
pk
≥ ε
⊆
n∈ N :
∞ k=
ank
Mk
¯d(mXk, X) ρ
≥ ε
∈ I.
The other part can be proved in a similar way.
The following corollary follows immediately from the above theorem.
Corollary . Let A = (C, ), i.e., the Cesàro matrix, and M = (Mk) be a sequence of Orlicz functions.
(a) Let < inf pk≤ pk≤ . Then
wI(F)[M, m, p]⊆ wI(F)[M, m]; wI(F)[M, m, p]⊆ wI(F)[M, m]. (b) Let ≤ pk≤ sup pk<∞. Then
wI(F)[M, m]⊆ wI(F)[M, m, p]; wI(F)[M, m]⊆ wI(F)[M, m, p].
Competing interests
The author declares that they have no competing interests.
Received: 9 October 2011 Accepted: 9 October 2012 Published: 6 November 2012 References
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Cite this article as: Savas: On some AI-convergent difference sequence spaces of fuzzy numbers defined by the sequence of Orlicz functions. Journal of Inequalities and Applications 2012 2012:261.
