Generalized difference strongly summable sequence spaces of fuzzy real numbers defined by ideal convergence and Orlicz function

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R E S E A R C H A R T I C L E

Open Access

Generalized difference strongly summable

sequence spaces of fuzzy real numbers

deﬁned by ideal convergence and Orlicz

function

Adem Kılıçman

1

_{and Stuti Borgohain}

2*

*_{Correspondence:} stutiborgohain@yahoo.com 2_{Department of Mathematics,} Indian Institute of Technology, Bombay Powai: 400076, Mumbai, Maharashtra, India

Full list of author information is available at the end of the article

Abstract

We study some new generalized diﬀerence strongly summable sequence spaces of fuzzy real numbers using ideal convergence and an Orlicz function in connection with de la Vallèe Poussin mean. We give some relations related to these sequence spaces also.

MSC: 40A05; 40A25; 40A30; 40C05

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

1 Introduction

Let∞,candcbe the Banach space of bounded, convergent and null sequencesx= (xk),

respectively, with the usual normx=sup_n|xn|.

A sequencex∈_∞is said to be almost convergent if all of its Banach limits coincide. Letcˆdenote the space of all almost convergent sequences.

Lorentz [] proved that,

ˆ

c=

x∈∞:lim

m tm,n(x) exists uniformly inn

,

where

tm,n(x) =

xn+xn++· · ·+xm+n

m+  .

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

[cˆ] =

x∈_∞:lim

m tm,n(|x–Le|) exists uniformly innfor someL

,

where,e= (, , . . .).

Letσ 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 the}_m_{th iterative of the mapping}

σinn, see [].

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Schaefer [] proved that

Vσ =

x∈∞:lim

k tkm(x) =Luniformly inmfor someL=σ–limx

,

where,

tkm(x) =

x+xσ(m)+· · ·+xσk₍_m₎

k+  , t–,m= .

Thus, we say that a bounded sequencex= (xk) isσ-convergent if and only ifx∈Vσ such thatσk(n)=nfor alln≥,k≥.

A sequencex= (xk) is said to be stronglyσ-convergent (Mursaleen []) if there exists a

numbersuch that



k

i=

|xσi₍_m₎–| →, ask→ ∞uniformly inm. ()

We write [Vσ] to denote the set of all strongσ-convergent sequences, and when () holds, we write [Vσ] –limx=.

Takingσ(m) =m+ , we obtain [Vσ] = [cˆ]. Then the strongσ-convergence generalizes the concept of strong almost convergence.

We also note that

[Vσ]⊂Vσ ⊂∞.

The notion of ideal convergence was ﬁrst introduced by Kostyrkoet 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) > , forx>  andM(x)→ ∞, asx→ ∞.

Lindenstrauss and Tzafriri [] used the idea of Orlicz function to construct the sequence space,

M=

(xk)∈w: ∞

k=

M

_|

x|

ρ

<∞, for someρ> 

.

The spaceMwith 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() andc() of crisp sets.

The notion is deﬁned as follows:

Z() = x= (xk) : (xk)∈Z

,

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The spaces above are Banach spaces, normed by

x=|x|+sup

k |xk|.

The generalized diﬀerence is deﬁned as follows: Form≥ andn≥,

Zn_m= x= (xk) :

n_mxk

∈Z

forZ=_∞,candc.

This generalized diﬀerence has the following binomial representation:

n_mxk= n

r=

(–)r

n r

xk+rm.

The concept of fuzzy set theory was introduced by Zadeh in the year . It has been applied 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 of fuzzy real numbers and studied their diﬀerent properties.

2 Deﬁnitions and preliminaries

A fuzzy real numberXis a fuzzy set onR,i.e., a mappingX:R→I(= [, ]) associating each real numbertwith its grade of membershipX(t).

A fuzzy real numberXis calledconvexifX(t)≥X(s)∧X(r) =min(X(s),X(r)), where

s<t<r.

If there existst∈Rsuch thatX(t) = , then the fuzzy real numberXis callednormal.

A fuzzy real numberXis said to beupper semicontinuousif for eachε> ,X–_([,_a₊_ε_)),

for alla∈I, it is open in the usual topology ofR.

The class of all upper semicontinuous, normal, convexfuzzy real numbers is denoted byR(I).

Deﬁned:R(I)×R(I)→Rbyd(X,Y) =sup_<_α_≤d(Xα_,_Yα_{), for}_X_,_Y_∈_R₍_I_{). Then it is well} known that (R(I),d) is a complete metric space.

A sequenceX= (Xn) of fuzzy real numbers is said to converge to the fuzzy numberX,

if for everyε> , there existsn∈Nsuch thatd(Xn,X) <εfor alln≥n.

LetXbe a nonempty set. Then a family of setsI⊆X_{(power sets of}_X_{) is said to be an}

idealifIis additive,i.e.,A,B∈I⇒A∪B∈Iand hereditary,i.e.,A∈I,B⊆A⇒B∈I. A sequence (Xk) of fuzzy real numbers is said to beI-convergentto a fuzzy real number

X∈Xif for eachε> , the set

E(ε) = k∈N:d(Xk,X)≥ε

belongs toI.

The fuzzy numberXis called theI-limit of the sequence (Xk) of fuzzy numbers, and

we writeI–limXk=X.

The generalized de la Vallé-Poussin mean is deﬁned by

tn(x) =



λn

k∈In

xk,

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Then a sequencex= (xk) is said to be (V,λ)-summable to a numberL[] iftn(x)→Las

n→ ∞, 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, strongly summable, and strongly bounded by de la Vallé-Poussin method.

We also note that Nuray and Savas [] deﬁned the sets of sequence spaces such as stronglyσ-summable to zero, stronglyσ-summable and stronglyσ-bounded with respect to the modulus function, see [].

In this article, we deﬁne some new sequence spaces of fuzzy real numbers by using Orlicz function with the notion of generalized de la Vallèe Poussin mean, generalized diﬀerence sequences and ideals. We will also introduce and examine certain new sequence spaces using the tools above.

3 Main results

LetIbe an admissible ideal ofN, letMbe an Orlicz function. Letr= (rk) be a sequence of

real numbers such thatrk>  for allk, andsupkrk<∞. This assumption is made

through-out the paper.

In this article, we have introduced the following sequence spaces,

Vσ,λ,qp,M,r

I(F) 

=

(Xk)∈wF:∀ε> 

n∈N:lim

n



λn

k∈In

M

_d₍q

pXσk₍_m₎, ) ρ

rk ≥ε,

uniformly inm

∈I

for someρ> ,

Vσ,λ,q_p,M,r I(F)

=

(Xk)∈wF:∀ε> 

n∈N:lim

n



λn

k∈In

M

d(qpXσk(m),X) ρ

rk ≥ε

∈I

for someρ> ,X∈R(I),

Vσ,λ,qp,M,r

I(F) ∞

=

(Xk)∈wF:∃K> , s.t.

sup

n,m



λn

k∈In

M

_d₍q

pXσk₍_m₎,  ρ

rk ≥K

∈I

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In particular, if we takerk=  for allk, we have

uniformly inm

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

uniformly inm

The following well-known inequality will be used later. If ≤rk≤suprk=HandC=max(, H–), then

By the deﬁnition of Orlicz function, we have for allε> ,

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Letlimk→∞rk=r> . Suppose thatXk→Y∈[Vσ,λ,qp,M,r]I(F),Xk→Y∈[Vσ,λ,qp,

M,r]I(F)_{and (}_d₍_Y

,Y))rk=a> .

Now, from () and the deﬁnition of Orlicz function, we have

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So,

exists a positive integerNsuch that 

This completes the proof.

Theorem . Let XF(Vσ,λ,qp–)stand for[Vσ,λ,qp–,M,r]I(F), [Vσ,λ,qp–,M,r]I(F)or

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≤D

Hence, we have

Since the set on the right-hand side belongs toI, so does the left-hand side. The inclusion is strict as the sequenceX= (kr_{), for example, belongs to [}_V

the metric deﬁned by

dσ(X,Y) =

The authors declare that they have no competing interests.

Authors’ contributions

Both of the authors contributed equally. The authors also read the galley proof and approved the ﬁnal copy of the manuscript.

Author details

1_{Department of Mathematics and Institute for Mathematical Research, University of Putra Malaysia, Serdang, Selangor} 43400, Malaysia.2_{Department of Mathematics, Indian Institute of Technology, Bombay Powai: 400076, Mumbai,} Maharashtra, India.

Acknowledgements

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

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