Generalized difference strongly summable sequence spaces of fuzzy real numbers defined by ideal...

Kılıçman and Borgohain Advances in Difference Equations 2013, 2013:288http://www.advancesindifferenceequations.com/content/2013/1/288

RESEARCH ART ICLE Open Access

Generalized difference strongly summablesequence spaces of fuzzy real numbersdefined by ideal convergence and OrliczfunctionAdem Kılıçman1 and Stuti Borgohain2*

*Correspondence:[email protected] of Mathematics,Indian Institute of Technology,Bombay Powai: 400076, Mumbai,Maharashtra, IndiaFull list of author information isavailable at the end of the article

AbstractWe study some new generalized difference strongly summable sequence spaces offuzzy real numbers using ideal convergence and an Orlicz function in connectionwith de la Vallèe Poussin mean. We give some relations related to these sequencespaces also.MSC: 40A05; 40A25; 40A30; 40C05

Keywords: Orlicz function; difference operator; ideal convergence; de la VallèePoussin mean

1 IntroductionLet �∞, c and c be the Banach space of bounded, convergent and null sequences x = (xk),respectively, with the usual norm ‖x‖ = supn |xn|.A sequence x ∈ �∞ is said to be almost convergent if all of its Banach limits coincide.Let c denote the space of all almost convergent sequences.Lorentz [] proved that,

c ={x ∈ �∞ : lim

mtm,n(x) exists uniformly in n

},

where

tm,n(x) =xn + xn+ + · · · + xm+n

m + .

The following space of strongly almost convergent sequence was introduced by Mad-dox []:

[c] ={x ∈ �∞ : lim

mtm,n(|x – Le|) exists uniformly in n for some L

},

where, e = (, , . . .).Let σ be a one-to-one mapping from the set of positive integers into itself such that

σm(n) = σm–(σ (n)),m = , , , . . . , where σm(n) denotes themth iterative of the mappingσ in n, see [].

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Schaefer [] proved that

Vσ ={x ∈ �∞ : lim

ktkm(x) = L uniformly inm for some L = σ – limx

},

where,

tkm(x) =x + xσ (m) + · · · + xσ k (m)

k + , t–,m = .

Thus, we say that a bounded sequence x = (xk) is σ -convergent if and only if x ∈ Vσ suchthat σ k(n) �= n for all n ≥ , k ≥ .A sequence x = (xk) is said to be strongly σ -convergent (Mursaleen []) if there exists a

number � such that

k

k∑i=

|xσ i(m) – �| → , as k → ∞ uniformly inm. ()

We write [Vσ ] to denote the set of all strong σ -convergent sequences, and when ()holds, we write [Vσ ] – limx = �.Taking σ (m) =m + , we obtain [Vσ ] = [c]. Then the strong σ -convergence generalizes

the concept of strong almost convergence.We also note that

[Vσ ]⊂ Vσ ⊂ �∞.

The notion of ideal convergence was first introduced by Kostyrko et al. [] as a general-ization of statistical convergence, which was later studied by many other authors.An Orlicz function is a function M : [,∞) → [,∞), which is continuous, non-

decreasing and convex withM() = ,M(x) > , for x > andM(x)→ ∞, as x → ∞.Lindenstrauss andTzafriri [] used the idea of Orlicz function to construct the sequence

space,

�M ={(xk) ∈ w :

∞∑k=

M( |x|

ρ

)<∞, for some ρ >

}.

The space �M with the norm

‖x‖ = inf

{ρ > :

∞∑k=

M( |xk|

ρ

)≤

}

becomes a Banach space, which is called an Orlicz sequence space.Kizmaz [] studied the difference sequence spaces �∞(�), c(�) and c(�) of crisp sets.

The notion is defined as follows:

Z(�) ={x = (xk) : (�xk) ∈ Z

},

for Z = �∞, c and c, where �x = (�xk) = (xk – xk+), for all k ∈ N .



The spaces above are Banach spaces, normed by

‖x‖� = |x| + supk

|�xk|.

The generalized difference is defined as follows:Form ≥ and n≥ ,

Z(�n

m)=

{x = (xk) :

(�n

mxk) ∈ Z

}for Z = �∞, c and c.This generalized difference has the following binomial representation:

�nmxk =

n∑r=

(–)r(nr

)xk+rm.

The concept of fuzzy set theory was introduced by Zadeh in the year . It has beenapplied for the studies in almost all the branches of science, where mathematics is used.Workers on sequence spaces have also applied the notion and introduced sequences offuzzy real numbers and studied their different properties.

2 Definitions and preliminariesA fuzzy real number X is a fuzzy set on R, i.e., a mapping X : R → I (= [, ]) associatingeach 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 ∈ R such that X(t) = , then the fuzzy real number X is called normal.A fuzzy real numberX is said to be upper semicontinuous if for each ε > ,X–([,a+ε)),

for all a ∈ I , it is open in the usual topology of R.The class of all upper semicontinuous, normal, convex fuzzy real numbers is denoted

by R(I).Define d : R(I)×R(I)→ R by d(X,Y ) = sup<α≤ d(Xα ,Y α), forX,Y ∈ R(I). Then it is well

known that (R(I),d) is a complete metric space.A sequence X = (Xn) of fuzzy real numbers is said to converge to the fuzzy number X,

if for every ε > , there exists n ∈N such that d(Xn,X) < ε for all n ≥ n.Let X be a nonempty set. Then a family of sets I ⊆ X (power sets of X) is said to be an

ideal if I is additive, i.e., A,B ∈ I ⇒ A∪ B ∈ I and hereditary, i.e., A ∈ I , B ⊆ A⇒ B ∈ I .A sequence (Xk) of fuzzy real numbers is said to be I-convergent to a fuzzy real number

X ∈ X if for each ε > , the set

E(ε) ={k ∈N : d(Xk ,X)≥ ε

}belongs to I.

The fuzzy number X is called the I-limit of the sequence (Xk) of fuzzy numbers, andwe write I – limXk = X.The generalized de la Vallé-Poussin mean is defined by

tn(x) =λn

∑k∈In

xk ,

where In = [n – λn + ,n] for n = , , . . . .



Then a sequence x = (xk) is said to be (V ,λ)-summable to a number L [] if tn(x)→ L asn→ ∞, and we write

[V ,λ] ={x : lim

n

λn

∑k∈In

|xk| = },

[V ,λ] ={x : x – �e ∈ [V ,λ] for some � ∈ C

},

[V ,λ]∞ ={x : sup

n

λn

∑k∈In

|xk| <∞}

for the sets of sequences that are, respectively, strongly summable to zero, stronglysummable, and strongly bounded by de la Vallé-Poussin method.We also note that Nuray and Savas [] defined the sets of sequence spaces such as

strongly σ -summable to zero, strongly σ -summable and strongly σ -bounded with respectto the modulus function, see [].In this article, we define somenew sequence spaces of fuzzy real numbers by usingOrlicz

function with the notion of generalized de la Vallèe Poussin mean, generalized differencesequences and ideals. We will also introduce and examine certain new sequence spacesusing the tools above.

3 Main resultsLet I be an admissible ideal of N , letM be an Orlicz function. Let r = (rk) be a sequence ofreal numbers such that rk > for all k, and supk rk <∞. This assumption is made through-out the paper.In this article, we have introduced the following sequence spaces,

[Vσ ,λ,�q

p,M, r]I(F)

={(Xk) ∈ wF : ∀ε >

{n ∈N : lim

n

λn

∑k∈In

{M

(d(�qpXσ k (m), )

ρ

)}rk≥ ε,

uniformly inm}

∈ I}

for some ρ > ,

[Vσ ,λ,�q

p,M, r]I(F)

={(Xk) ∈ wF : ∀ε >

{n ∈N : lim

n

λn

∑k∈In

{M

(d(�qpXσ k (m),X)

ρ

)}rk≥ ε

}∈ I

}

for some ρ > ,X ∈ R(I),[Vσ ,λ,�q

p,M, r]I(F)∞

={(Xk) ∈ wF : ∃K > , s.t.

{supn,m

λn

∑k∈In

{M

(d(�qpXσ k (m),

ρ

)}rk≥ K

}∈ I

}

for some ρ > .



In particular, if we take rk = for all k, we have

[Vσ ,λ,�q

p,M]I(F)

={(Xk) ∈ wF : ∀ε >

{n ∈N : lim

n

λn

∑k∈In

{M

(d(�qpXσ k (m))

ρ

)}≥ ε,

uniformly inm}

∈ I}

for some ρ > ,

[Vσ ,λ,�q

p,M]I(F)

={(XK ) ∈ wF : ∀ε >

{n ∈N : lim

n

λn

∑k∈In

{M


ρ

)}≥ ε

}∈ I

}

for some ρ > ,X ∈ R(I),[Vσ ,λ,�q

p,M]I(F)∞

={(Xk) ∈ wF : ∃K >

{supn,m

λn

∑k∈In

{M


ρ

)}≥ K

}∈ I

}.

Similarly, when σ (m) =m+ , then [V ,λ,�qp,M, r]I(F) , [Vσ ,λ,�q

p,M, r]I(F) and [Vσ ,λ,�qp,

M, r]I(F)∞ are reduced to

[V ,λ,�q

p,M, r]I(F)

={(Xk) ∈ wF : ∀ε >

{n ∈N : lim

n

λn

∑k∈In

{M

(d(�qpXk+m, )

ρ

)}rk≥ ε

}∈ I

}

uniformly inm for some ρ > .[V ,λ,�q

p,M, r]I(F)

={(Xk) ∈ wF : ∀ε >

{n ∈N : lim

n

λn

∑k∈In

{M

(d(�qpXk+m,X)

ρ

)}rk≥ ε

}∈ I

}

for some ρ > ,X ∈ R(I).

[V ,λ,�q

p,M, r]I(F)∞ =

{x : ∃K > , s.t.

{supn,m

λn

∑k∈In

{M

(d(�qpXk+m, ρ

)}rk≥ k

}∈ I

}

for some ρ > .

In particular, if we put rk = r, for all k, then we have the spaces

[V ,λ,�q

p,M, r]I(F) =

[V ,λ,�q

p,M]I(F) ,[

V ,λ,�qp,M, r

]I(F) = [V ,λ,�q

p,M]I(F),[

V ,λ,�qp,M, r

]I(F)∞ =

[V ,λ,�q

p,M]I(F)∞ .

Further, when λn = n, for n = , , . . . , the sets [V ,λ,�qp,M]I(F) and [V ,λ,�q

p,M]I(F) arereduced to [c(M,�q

p)]I(F) and [c(M,�qp)]I(F), respectively.



Now, if we considerM(x) = x, then we can easily obtain

[Vσ ,λ,�q

p, r]I(F)

={(Xk) ∈ wF : ∀ε >

{n ∈N : lim

n

λn

∑k∈In

(d(�q

pXσ k (m), ))rk ≥ ε,

uniformly inm}

∈ I},

[Vσ ,λ,�q

p, r]I(F)

={(Xk) ∈ wF : ∀ε >

{n ∈N : lim

n

λn

∑k∈In

(d(�q

pXσ k (m),X))rk ≥ ε

}∈ I

}

for X ∈ R(I).

[Vσ ,λ,�q

p, r]I(F)∞ =

{(Xk) ∈ wF : ∃K > s.t.

{supn,m

λn

∑k∈In

(d(�q

pXσ k (m), ))rk ≥ K

}∈ I

}.

If X ∈ [Vσ ,λ,�qp,M, r]I(F) with {

λn

∑k∈In{M(

d(�qpXσk (m),X)

ρ)}rk ≥ ε} ∈ I as n → ∞ uni-

formly in m, then we write Xk → X ∈ [Vσ ,λ,�qp,M, r]I(F).

The following well-known inequality will be used later.If ≤ rk ≤ sup rk =H and C = max(, H–), then

|ak + bk|rk ≤ C{|ak|rk + |bk|rk

}()

for all k and ak ,bk ∈ C.

Lemma . (See []) Let rk > , sk > . Then c(s) ⊂ c(r) if and only if limk→∞ inf rksk> ,

where c(r) = {x : |xk|rk → as k → ∞}.

Note that no other relation between (rk) and (sk) is needed in Lemma ..

Theorem . Let limk→∞ inf rk > . Then Xk → X implies that Xk → X ∈ [Vσ ,λ,�qp,

M, r]I(F). Let limk→∞ rk = r > . If Xk → X ∈ [Vσ ,λ,�qp,M, r]I(F), then X is unique.

Proof Let Xk → X.By the definition of Orlicz function, we have for all ε > ,

{n ∈ N : lim

n

λn

∑k∈In

M(d(�q

pXσ k (m),X)ρ

)≥ ε

}∈ I.

Since limk→∞ inf rk > , it follows that

{n ∈ N : lim

n

λn

∑k∈In

{M


ρ

)}rk≥ ε

}∈ I,

and, consequently, Xk → X ∈ [Vσ ,λ,�qp,M, r]I(F).



Let limk→∞ rk = r > . Suppose thatXk → Y ∈ [Vσ ,λ,�qp,M, r]I(F),Xk → Y ∈ [Vσ ,λ,�q

p,M, r]I(F) and (d(Y,Y))rk = a > .Now, from () and the definition of Orlicz function, we have

λn

∑k∈In

M(d(Y,Y)

ρ

)rk≤ C

λn

∑k∈In

M(d(�q

pXσ k (m),Y)ρ

)rk

+Cλn

∑k∈In

M(d(�q

pXσ k (m),Y)ρ

)rk.

Since

{n ∈ N : lim

n

λn

∑k∈In

M(d(�q

pXσ k (m),Y)ρ

)rk≥ ε

}∈ I,

{n ∈ N : lim

n

λn

∑k∈In

M(d(�q

pXσ k (m),Y)ρ

)rk≥ ε

}∈ I.

Hence,

{n ∈ N : lim

n

λn

∑k∈In

M(d(Y,Y)

ρ

)rk≥ ε

}∈ I. ()

Further,M( d(Y,Y)ρ

)rk →M( aρ)r as k → ∞, and therefore,

limn→∞

λn

∑k∈In

M(d(Y,Y)

ρ

)rk=M

(aρ

)r. ()

From () and (), it follows that M( aρ) = , and by the definition of an Orlicz function,

we have a = .Hence, Y = Y, and this completes the proof. �

Theorem . (i) Let < infk rk ≤ rk ≤ . Then

[Vσ ,λ,�q

p,M, r]I(F) ⊂ [

Vσ ,λ,�qp,M

]I(F).(ii) Let < rk ≤ supk rk <∞. Then

[Vσ ,λ,�q

p,M]I(F) ⊂ [

Vσ ,λ,�qp,M, r

]I(F).Proof (i) Let X ∈ [Vσ ,λ,�q

p,M, r]I(F). Since < infk rk ≤ , we get

λn

∑k∈In

M(d(�q

pXσ k (m),X)ρ

)≤

λn

∑k∈In

M(d(�q

pXσ k (m),X)ρ

)rk.



So,

{n ∈ N : lim

n

λn

∑k∈In

{M


ρ

)}≥ ε, uniformly inm

}

⊆{n ∈N : lim

n

λn

∑k∈In

{M

(d(�qpXσ k (m),X

ρ

)}rk≥ ε, uniformly inm

}∈ I,

and hence, X ∈ [Vσ ,λ,�qp,M]I(F).

(ii) Let r ≥ and supk rk <∞. Let X ∈ [Vσ ,λ,�qp,M]I(F). Then for each k, < ε < , there

exists a positive integer N such that

λn

∑k∈In

M(d(�q

pXσ k (m),X)ρ

)≤ ε <

for allm ≥N . This implies that

λn

∑k∈In

M(d(�q

pXσ k (m),X)ρ

)rk≤

λn

∑k∈In

M(d(�q

pXσ k (m),X)ρ

).

So,{n ∈ N :

λn

∑k∈In

{M


ρ

)}rk≥ ε, uniformly inm

}

⊆{n ∈N :

λn

∑k∈In

{M


ρ

)}≥ ε, uniformly inm

}∈ I.

Therefore, X ∈ [Vσ ,λ,�qp,M, r]I(F).

This completes the proof. �

Theorem . Let XF (Vσ ,λ,�q–p ) stand for [Vσ ,λ,�q–

p ,M, r]I(F) , [Vσ ,λ,�q–p ,M, r]I(F) or

[Vσ ,λ,�q–p ,M, r]I(F)∞ and m ≥ . Then the inclusion XF (Vσ ,λ,�q–

p ) ⊂ XF (Vσ ,λ,�qp) is

strict. In general, XF (Vσ ,λ,�ip) ⊂ X(Vσ ,λ,�q

p) for all i = , , , . . . ,p – and the inclusionis strict.

Proof Let us take [Vσ ,λ,�q–p ,M, r]I(F) .

Let X = (Xk) ∈ [Vσ ,λ,�q–p ,M, r]I(F) . Then for given ε > , we have

{n ∈ N :

λn

∑k∈In

{M

(d(�q–p Xσ k (m), )

ρ

)}rk≥ ε

}∈ I for some ρ > .

SinceM is non-decreasing and convex, it follows that

λn

∑k∈In

{M


ρ

)}rk

=λn

∑k∈In

{M

(d(�q–p Xσ k+(m),�

q–p Xσ k (m))

ρ

)}rk



≤D λn

∑k∈In

([M

(d(�q–p Xσ k+(m), )

ρ

)]rk+

[M

(d(�q–p Xσ k (m), )

ρ

)]rk)

≤D λn

∑k∈In

([M

(d(�q–p Xσ k+(m), )

ρ

)]rk+

[M

(d(�q–p Xσ k (m), )

ρ

)]rk).

Hence, we have

{n ∈ N :

λn

∑k∈In

{M


ρ

)}rk≥ ε

}

⊆{n ∈N :D

λn

∑k∈In

{M

(d(�q–p Xσ k+(m), )

ρ

)}rk≥ ε

}

∪{n ∈ N :D

λn

∑k∈In

{M

(d(�q–p Xσ k (m), )

ρ

)}rk≥ ε

}.

Since the set on the right-hand side belongs to I , so does the left-hand side. The inclusionis strict as the sequence X = (kr), for example, belongs to [Vσ ,λ,�q

p,M]I(F) but does notbelong to [Vσ ,λ,�q–

p ,M]I(F) forM(x) = x and rk = for all k. �

Theorem . [Vσ ,λ,�qp,M, r]I(F) and [Vσ ,λ,�q

p,M, r]I(F) are complete metric spaces,withthe metric defined by

dσ (X,Y ) =pq∑m=

d(Xσ k (m),Yσ k (m))

+ inf

{ρ

rkH : sup

m,n

(λn

∑k∈In

(M

(d(�qpXσ k (m),�

qpYσ k (m))

ρ

))H)≤

for some ρ > },

where H = max(, (supk rk)).

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsBoth of the authors contributed equally. The authors also read the galley proof and approved the final copy of themanuscript.

Author details1Department of Mathematics and Institute for Mathematical Research, University of Putra Malaysia, Serdang, Selangor43400, Malaysia. 2Department of Mathematics, Indian Institute of Technology, Bombay Powai: 400076, Mumbai,Maharashtra, India.

AcknowledgementsThe authors are very grateful to the referees for the very useful comments and for detailed remarks that improved thepresentation and the contents of the manuscript. The work of the authors was carried under the Post Doctoral Fellowunder National Board of Higher Mathematics, DAE, project No. NBHM/PDF.50/2011/64. The first author gratefullyacknowledges that the present work was partially supported under the Post Doctoral Fellow under National Board ofHigher Mathematics, DAE, project No. NBHM/PDF.50/2011/64. The second author also acknowledges that the part of thework was supported by the University Putra Malaysia, Grant ERGS 5527179.

Received: 25 June 2013 Accepted: 10 September 2013 Published: 07 Nov 2013



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10.1186/1687-1847-2013-288Cite this article as: Kılıçman and Borgohain: Generalized difference strongly summable sequence spaces of fuzzy realnumbers defined by ideal convergence and Orlicz function. Advances in Difference Equations 2013, 2013:288


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