Generalized lacunary Δm statistically convergent sequences of fuzzy numbers using a modulus function

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R E S E A R C H

Open Access

Generalized lacunary

m

-statistically

convergent sequences of fuzzy numbers

using a modulus function

Parmeshwary Dayal Srivastava

*

_{and Sushomita Mohanta}

*_{Correspondence:} [email protected] Department of Mathematics, Indian Institute of Technology, Kharagpur, 721302, India

Abstract

In this paper, we introduce the space of lacunary strongly

m

(p)-summable sequences

of fuzzy numbers and discuss relations between

m_{-statistically convergent}

sequences and lacunary

m_{-statistically convergent sequences of fuzzy numbers. We}

also study inclusion relations using diﬀerent arbitrary lacunary sequences.

MSC: 40A05; 40C05; 46A45

Keywords: fuzzy number; modulus function; lacunary

m_{-statistical convergence;}

lacunary reﬁnement

1 Introduction

The concepts of fuzzy sets and fuzzy set operations were ﬁrst introduced by Zadeh [], and subsequently several authors have discussed various aspects of the theory and appli-cations of fuzzy sets such as topological spaces, similarity relations and fuzzy orderings, fuzzy mathematical programmingetc.Later on, various types of sequence spaces of fuzzy numbers have been constructed by several authors such as Matloka [], Nanda [], Nu-ray and Savaş [], Mursaleen and Basarır [], Malkowskyet al.[], Tripathy and Chandra [], Tripathy and Borgogain [] and so on. Later on, the fuzzy sequence space got mo-mentum after the introduction of new convergence methods and theories in the process as well as the requirement. Some of them are statistical convergence, lacunary statistical convergenceetc.

Nuray and Savaş [] introduced the idea of statistical convergence of fuzzy numbers and Nuray [] introduced the related concept of convergence with the help of a lacunary sequence. Using their ideas, many authors such as Kwon and Shim [], Bilgin [], Altin

et al.[], Esi [], Tripathy and Baruah [, ], Tripathy and Dutta [, ] and others constructed diﬀerent types of sequence spaces.

2 Deﬁnitions and preliminaries

Deﬁnition . A fuzzy numberXis a mappingX:R→[, ] associating each real num-bertwith its grade of membershipX(t).

Deﬁnition . If there existst∈Rsuch thatX(t) = , then the fuzzy numberXis called normal.

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Deﬁnition . A fuzzy numberXis said to be convex ifX(t)≥X(s)∧X(r) =min{X(s),

X(r)}, wheres<t<r.

Deﬁnition . A fuzzy numberXis said to be upper semi-continuous if for eachε> ,

X–([,a+ε)) for alla∈[, ] is open in the usual topology ofR.

L(R) denotes the set of all upper semi-continuous, normal, convex fuzzy numbers such that [X]α₌_{_t_∈_R_:_X₍_t_{) > }_}_{is compact, where}_{_t_∈_R_:_X₍_t_{) > }_}_{denotes the closure of the}

set{t∈R:X(t) > }in the usual topology ofR.

Deﬁnition . The setL(R) forms a linear space under addition and scalar multiplication in terms ofα-level sets as deﬁned below:

[X+Y]α= [X]α+ [Y]α and [λX]α=λ[X]α for each ≤α≤,

whereXα_{is given as}

Xα=

t:X(t)≥α ifα∈(, ],

t:X(t) >  ifα= .

For eachα∈[, ], the setXα_{is a closed, bounded and nonempty interval of}_R_.

LetDdenote the set of all closed and bounded intervalsA= [a,a] on the real lineR. ForA,B∈D, (D,d) is a complete metric space where the metricdis deﬁned as

d(A,B) =max|a–b|,|a–b|

for anyA= [a,a] andB= [b,b].

It is easy to verify thatd:L(R)×L(R)→R, deﬁned by

d(X,Y) = sup

≤α≤

dXα,Yα,

is a metric onL(R).

Deﬁnition . LetX= (Xk) be a sequence of fuzzy numbers. Then the sequenceX= (Xk)

is said to bem-convergent to the fuzzy numberX, denoted aslimk→∞mXk=X, if for

everyε> , there exists a positive integerksuch thatd(mXk,X) <εfor allk>k.

Deﬁnition . The sequenceX= ((Xk)∞k=) of fuzzy numbers is said to bem-statistically convergent to the fuzzy numberξif for everyε> ,

lim

n→∞



nk≤n:d

mXk,ξ

≥ε= .

In this case, we writeXk→ξ(SF(m)) andSF(m) denotes the set of allm-statistically

convergent sequences of fuzzy numbers.

By a lacunary sequence we mean an increasing sequence of integersθ = (kr) such that

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Deﬁnition . Letθ be a lacunary sequence. A lacunary reﬁnement ofθ= (kr) is a

lacu-nary sequenceθ= (k_r) satisfying{kr} ⊆ {kr}.

Deﬁnition . The sequenceX= (Xk) of fuzzy numbers is said to be lacunary m

-statistically convergent to the fuzzy numberξif for everyε> ,

lim

r→∞



hr

k∈Ir:d

mXk,ξ

≥ε= .

In this case, we write Xk →ξ(SFθ(m)), andSFθ(m) denotes the set of all lacunary

m_{-statistically convergent sequences of fuzzy numbers.}

Deﬁnition . A metricdonL(R) is said to be translation invariant ifd(X+Z,Y+Z) =

d(X,Y) for allX,Y,Z∈L(R).

Lemma .(Mursaleen and Basarır []) If d is a translation invariant metric on L(R),

then

(i) d(X+Y, )≤d(X, ) +d(Y, ), (ii) d(λX, )≤ |λ|d(X, ),|λ|> .

Lemma .(Maddox []) Let ak,bkfor all k be sequences of complex numbers,and let

(pk)be a bounded sequence of positive real numbers,then

|ak+bk|pk≤C

|ak|pk+|bk|pk

and

|λ|pk≤_max_,|_λ|H_,

where C=max(, H–_),_H₌_sup_p

kandλis any complex number.

Lemma .(Maddox []) Let ak≥,bk≥for all k be sequences of complex numbers

and≤pk≤suppk<∞,then

k

|ak+bk|pk



M

≤

k

|ak|pk



M +

k

|bk|pk



M ,

where M=max(,H),H=suppk.

3 Main results

Now, we introduce the spaceNθF(m(p),f) as follows:

NθF

m₍_p₎,f=

X= (Xk) :Xk∈L(R) such that lim r→∞



hr_k_∈_I

r

fdmXk,ξ

pk = 

,

whereθ = (kr) is a lacunary sequence,f is any modulus function andp= (pk) is any

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Now we deﬁne a lacunary stronglym

p-summable sequence of fuzzy numbers as follows.

The sequenceX= (Xk) is said to be lacunary stronglym₍_p₎-summable to the fuzzy

num-berξ∈L(R) if for everyε> ,

lim

r→∞



hr_k_∈_I

r

fdmXk,ξ

pk = .

In this case, we writeXk→ξ(NθF(m(p),f)). Thus the classNθF(m(p),f) denotes the set of all lacunary stronglym

(p)-summable sequences of fuzzy numbers.

For suitable choices ofm,f andpk, some of the known sequence spaces, which are

ob-tained from the spaceNF

θ(m(p),f), are as follows.

(i) Letf(x) =x,m= andpk≡pfor allk∈N, the sequence spaceNθF(m(p),f)

represents the spaceNθstudied by Kwon and Shim [].

(ii) Letf(x) =x,m= andpk≡for allk∈N, the sequence spaceNθF(m(p),f)denotes

the spaceNθ()investigated by Bilgin []. (iii) Letf(x) =x, the sequence spaceNF

θ(m(p),f)denotes the spaceNθ(mp)studied by

Altinet al.[].

(iv) Letf(x) =xandpk≡for allk∈N, the sequence spaceNθF(m(p),f)reduces to the

spaceNθ(m,F)investigated by Esi [].

Thus, the study of the sequence space NθF(m(p),f)gives a uniﬁed approach to many of the

earlier known spaces.

Theorem . Let(pk)be a bounded sequence of positive real numbers. Then the class

NθF(m(p),f)is a linear space overR.

Proof Using Lemma ., Lemma ., the subadditivity property of a modulus functionf

and the resultf(λx)≤( + [|λ|])f(x), it is easy to show thatNF

θ(m(p),f) is a linear space over

the real ﬁeldR.

Theorem . Let(pk)be a bounded sequence of positive real numbers such thatinfpk> .

Then the sequence space NF

θ(m(p),f)is a complete metric space with respect to the metric

h(X,Y) =

m

i=

fd(Xi,Yi)

+sup

r



hr_k_∈_I

r

fdmXk,mYk

pk



M ,

where M=max{,suppk}.

Proof It is easy to verify thathgives a metric onNF

θ(m(p),f). To show completeness, let us assume that (Xu_{) is a Cauchy sequence in}_NF

θ(m(p),f), whereXu= (Xku)∞k=∈NθF(m(p),f) for eachu∈N. So, for givenε> , there existsu∈Nsuch that

gXu_,_Xv_<_ε _{for all}_u_,_v_≥_u

,

i.e.,

m

i=

fdX_iu,X_iv+sup

r



hr_k_∈_I

r

fdmXu_k,mX_kvpk



M

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This means that

m

i=

fdX_iu,X_iv<ε for allu,v≥u (.)

as well as



hr_k_∈_I

r

fdmXu_k,mX_kvpk

<ε for allu,v≥uand for allr. (.)

Sincef is a modulus function, so equation (.) givesd(Xu_i,X_iv) <εfor allu,v≥u, for someεsuch thatε>ε>  and for eachi= , , . . . ,m,i.e.,

X_iuis a Cauchy sequence inL(R) for eachi= , , . . . ,m. (.)

Similarly, asf is a modulus function, so by choosing suitableε> , equation (.) gives

d(m_Xu

k,mXkv) <εfor allu,v≥uand for eachk,i.e.,

mX_kuis a Cauchy sequence inL(R) for eachk∈N. (.)

Using equation (.) and equation (.), it can be easily shown that (X_ku) is a Cauchy sequence inL(R). ButL(R) is complete, so the sequence (X_ku) converges to X= (Xk) in

L(R) asu→ ∞.

Keepinguﬁxed and lettingv→ ∞in equation (.) and equation (.), we get

m

i=

fdX_iu,Xi

<ε for allu≥u

and



hr_k_∈_I

r

fdmXu_k,mXk

pk

<ε for allu≥uand for allr, (.)

i.e.,

gXu,X<ε for allu≥u.

Next we show that the limit pointX∈NF

θ(m(p),f), for which the proof is as follows. SinceXu= (X_ku)∈NθF(m(p),f) foru∈N, so for eachu, there existLu∈L(R) andru∈N

such that



hr_k_∈_I

r

fdmXu_k,Lupk

<ε for allr≥ru. (.)

Similarly, for eachv, there existLv_∈_L₍_R_{) and}_r

v∈Nsuch that



hr_k_∈_I

r

fdmXv_k,Lvpk

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Now letu,v≥uandr=max(ru,rv). Then from equation (.), equation (.), equation

(.) and using Lemma ., we have



hr_k_∈_I

r

fdLu_,_Lvpk _≤

C hr_k_∈_I

r

fdLu_,m_Xu k

pk

+C  hr _k_∈_I

r

fdmX_ku,mX_kvpk

+C  hr

k∈Ir

fdmX_kv,Lvpk

< Cε for allu,v≥uand for allr≥r. (.)

Now using the fact that the modulus function is monotone and for a suitable choice of ε> , we get

dLu,Lv<ε for allu,v≥u,

i.e., (Lu_{) is a Cauchy sequence in}_L₍_R_{). Since}_L₍_R_{) is complete, so there exists}_ξ _∈_L₍_R₎

such thatLu_→_ξ_as_u_{→ ∞}_{. Keeping}_u_{ﬁxed and letting}_v_{→ ∞}_{in equation (.), we get}



hr k∈Ir

fdLu,ξpk

< Cε foru≥uand forr≥r. (.)

Hence, from equation (.), equation (.) and equation (.), we have



hr_k_∈_I

r

fdmXk,ξ

pk_≤

C  hr _k_∈_I

r

fdmXk,mX_ku

pk

+C  hr_k_∈_I

r

fdmXu

k ,L upk

+C  hr_k_∈_I

r

fdLu_,_ξpk

< Cε+ Cε≡ε for allr≥r,

which implies _h r

k∈Ir(f(d(

m_X

k,ξ)))pk → asr→ ∞,i.e.,X= (Xk)∈NθF(m(p),f) and hence the sequence spaceNF

θ(m(p),f) is a complete metric space.

Theorem . Letθ = (kr)be a lacunary sequence and X = (Xk)be a sequence of fuzzy

numbers.Then:

(i) Xk→ξ(NθF(m(p),f))impliesXk→ξ(SFθ(m)).

(ii) Iff is a bounded modulus function andXk→ξ(SFθ(m)),thenXk→ξ(NθF(m(p),f)).

Proof Easy, so omitted.

Theorem . Letθ= (kr)be a lacunary sequence,and let X= (Xk)be a sequence of fuzzy

numbers.Then:

(i) WF₍m_,_f₎_⊆_NF

θ(m,f)if and only iflim infrqr> .

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where

WFm,f=

(Xk) :



n

k=

fdmXk,ξ

→as n→ ∞

and

NθF

m,f=

(Xk) :



hr_k_∈_I

r

fdmXk,ξ

→as r→ ∞

.

Proof (i) Assume thatlim infrqr> . Then there existsδ>  such thatqr≥ +δfor

suﬃ-ciently larger.

LetXk→ξ(WF(m,f)). Then



n

k=

fdmXk,ξ

→ asn→ ∞.

Consider



hr k∈Ir

fdmXk,ξ

= 

hr kr

k=

fdmXk,ξ

– 

hr kr–

k=

fdmXk,ξ

≥ kr

hr



kr kr

k=

fdmXk,ξ

–kr–

hr



kr–

k=

fdmXk,ξ

.

Sincehr=kr–kr–, we have

kr

hr

= kr

kr–kr– = qr

qr– 

=  + 

qr– ≤

 + δ=

δ+  δ

and

kr–

hr

= kr–

kr–kr–

= 

qr– 

≤

δ.

SinceX∈WF(m,f). So both the terms _k r

kr

k=f(d(mXk,ξ)) and_k_r_–

kr–

k=f(d(mXk,

ξ)) converge to  asr→ ∞and hence_h_r_k_r_∈_I_rf(d(mXk,ξ)) converges to  asr→ ∞,

i.e.,X∈NθF(m,f) and henceWF(m,f)⊆NθF(m,f).

Conversely, letXk→ξ(WF(m,f)) implyXk→ξ(NθF(m,f)) butlim infrqr= .

Sinceθis a lacunary sequence, we can select a subsequence (kr(j)) ofθsatisfying

kr(j)

kr(j)– <  +

j and

kr(j)–

kr(j–)

>j, wherer(j)≥r(j– ) + .

Deﬁne

mXk=

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Now we prove that the sequenceXfor whichm_X

k, which is deﬁned as above, is strongly

m_{-summable but not lacunary strongly}m_{-summable. Since}



hr(j)_k_∈_I r(j)

fdmXk, 

=f() forr=r(j)

and



hr_k_∈_I

r

fdmXk, 

=  forr=r(j),

which impliesXk(NθF(m,f)), but ifnis suﬃciently large integer andjis the unique

integer withkr(j)–<n≤kr(j+)–, then, sincen→ ∞impliesj→ ∞, we have



n

k=

fdmXk, 

≤ kr(j)

kr(j)– ≤

kr(j)–+hr(j)

kr(j)– ≤ 

j → asn→ ∞,

i.e.,Xk→(WF(m,f)). This shows thatWF(m,f) /∈NθF(m,f), which leads to a

con-tradiction. Therefore it proves thatlim infqr> .

(ii) (Suﬃciency) Supposelim sup_rqr<∞. Then there exists a constantA>  such that

qr<Afor allr∈N. LetXk→ξ(NθF(m,f)). Solimr→∞hr

k∈Irf(d(

m_X

k,ξ)) = .

Letε> . Then we can ﬁndR>  andK>  such that

sup

r≥R



hr _r_∈_I

r

fdmXk,ξ

<ε

and



hr _i_∈_I

j

fdmXk,ξ

<K for allj= , , . . . .

Then ifnis any integer withkr–<n≤kr, wherer>R, then we can write



n

k=

fdmXk,ξ

≤ 

kr–

kr

k=

fdmXk,ξ

= 

kr– _I



fdmXk,ξ

+

I

fdmXk,ξ

+· · ·

+

Ir

fdmXk,ξ

, whereIr= (kr–,kr]

= k

kr– 

h _I



fdmXk,ξ

+k–k

kr– 

h _I



fdmXk,ξ

+· · ·

+kR–kR–

kr– 

hR _I

R

fdmXk,ξ

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+kr–kr–

kr– 

hr _I

r

fdmXk,ξ

≤ kr

kr–

sup

≤i≤R



hi _i

fdmXk,ξ

+kr–kR

kr–

sup

i≥R



hi _i

fdmXk,ξ

<K kr kr–

+ε

qr–

kR

kr–

<K kr kr–

+εqr<K

kr

kr– +Aε,

which implies _nn_k_=f(d(mXk,ξ))→ asn→ ∞and it follows thatXk→ξ(WF(m,

f)).

(Necessity) Suppose that Xk → ξ(WθF(m,f)) implies Xk → ξ(NF(m,f)) but

lim sup_rqr=∞.

Sinceθis any lacunary sequence, we can select a subsequence (kr(j)) ofθsatisfyingqr(j)>j and deﬁne a bounded diﬀerence sequenceX= (Xk) by

mXk=

 ifkr(j)–<k≤kr(j)–for somej= , , . . . ,

 otherwise.

Letε>  be given. Then we have ifr=r(j), then



hr(j) _I r(j)

fdmXk, 

=kr(j)––kr(j)–

kr(j)–kr(j)–

f() = kr(j)–

kr(j)–kr(j)–

f() < f()

j– ,

and ifr=r(j), then



hr(j) _I r(j)

fdmXk, 

= ,

which implies thatXk→(NθF(m,f)).

For the above sequence and fork= , , . . . ,kr(j),



kr(j)

k=

fdmXk, 

≥ 

kr(j)

(kr(j)– kr(j)–)f() =

 – 

qr(j)

f() >

 –

j

f(),

and fork= , , . . . , kr(j)–,

 kr(j)–

kr(j)–

k=

fdmXk, 

≥ kr(j)– kr(j)–

f() =f()  .

It proves thatX= (Xk) /∈(WF(m,f)) and hence the result follows immediately.

Theorem . Letθ = (kr)be a lacunary sequence and X = (Xk)be a sequence of fuzzy

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(i) SF

θ(m)⊆SF(m)if and only iflim suprqr<∞.

(ii) SF(m)⊆SθF(m)if and only iflim infrqr> .

Proof The proof can be established following the technique used in Theorem ..

Theorem . Ifθ= (k_r)is a lacunary reﬁnement ofθ = (kr)and Xk→ξ(SFθ(m)),then Xk→ξ(SFθ(m)).

Theorem . Letβ= (li)be a lacunary reﬁnement of the lacunary sequenceθ= (ki).Let

Ii= (ki–,ki]and Ji= (li–,li],i= , , , . . . ,be the corresponding intervals ofθandβ,

respec-tively,and hi=ki–ki–and gi=li–li–,respectively.If there existsδ> such that

gj

hi≥

δ for every Jj⊆Ii,

then Xk→ξ(SθF(m))implies Xk→ξ(SFβ(m)).

Proof Letε>  be given,Xk→ξ(SFθ(m)) and for everyJj, we can ﬁndIisuch thatJj⊆Ii,

then we have



gj

k∈Jj:d

mXk,ξ

≥ε= hi

gj



hi

k∈Jj:d

mXk,ξ

≥ε

≤hi

gj



hi

_k_∈_I_i_:_d

mXk,ξ

≥ε (asJj⊆Ii)

≤ 

δ 

hi

k∈Ii:d

mXk,ξ

≥ε,

which implies thatXk→ξ(SFβ(m)).

Theorem . Letβ= (li)andθ= (ki)be two lacunary sequences.Let Ii= (ki–,ki]and Ji=

(li–,li],i= , , , . . . ,be the corresponding intervals ofθandβ,respectively,and Iij=Ii∩Jj,

i,j= , , , . . . ,and hi=ki–ki–and gi=li–li–.If there existsδ> such that

|Iij|

hi ≥

δ for every i,j= , , , . . .provided Ii,j=∅,

then Xk→ξ(SθF(m))implies Xk→ξ(SFβ(m)).

Proof Letα=β∪θ. Thenαis a lacunary reﬁnement of both the lacunary sequencesβ andθ. The interval sequence ofαis{Iij=Ij∩Jj:Iij=φ}.

Letε>  be given,Xk→ξ(SθF(m)), and for everyIi,j, we can ﬁndIisuch thatIij⊆Iiand Iij=∅. Then we have



|Iij|

k∈Iij:d

mXk,ξ

≥ε= hi

|Iij|



hi

k∈Iij:d

mXk,ξ

≥ε

≤ hi

|Iij|



hi

k∈Ii:d

mXk,ξ

≥ε (asIij⊆Ii)

≤ 

δ 

hi

k∈Ii:d

mXk,ξ

≥ε,

(11)

Sinceαis a lacunary reﬁnement ofβ, so by Theorem . it follows thatXk→ξ(SαF(m))

impliesXk→ξ(SFβ(m)) and hence the result follows.

Theorem . Letβ = (li)andθ = (ki)be two lacunary sequences. Let Ii= (ki–,ki],Ji=

(li–,li],i= , , , . . . ,and Iij=Ii∩Jj,i,j= , , , . . . .If there existsδ> such that

|Iij|

(hi+gj)≥

δ for every i,j= , , , . . . provided Ii,j=∅,

then Xk→ξ(SθF(m))if and only if Xk→ξ(SFβ(m)).

Deﬁnition . Let β = (li) and θ = (kr) be two lacunary sequences. Let Ji= (li–,li],

Ii= (ki–,ki],i= , , . . . . There exist a sequenceXand a fuzzy numberLsuch thatXk→

ξ(SF

β(m)) andXkξ(SFθ(m)) if and only if there exist (sr), (tr)⊆Nandδ>  which

sat-isfy the following conditions:

(i) Jsr∩Itr=φ;

(ii) lim_r_→∞|Jsr|

|Itr|=∞;

(iii) |Jsr∩Itr|

|Itr| ≥δ,r= , , . . ..

Proof Let there exist a sequenceX= (Xk) and a fuzzy numberξsuch thatXk→ξ(SβF(m))

andXkξ(SFθ(m)). Then we can ﬁnd a subsequence (tr)⊆N,ε>  andδ>  such that



|Itr|

k∈Itr:d

mXk,ξ

≥ε≥δ, r= , , . . . .

For eachtr, there exist a positive integersrand a whole numberursuch that

Jsr+j∩Itr=φ for everyj= , , , . . . ,r= , , . . . .

Then we can write

δ≤



|Itr|

k∈Itr:d

mXk,ξ

≥ε

=



|Itr|

k∈ _n

j=

Jsr+j

∩Itr:d

mXk,ξ

≥ε

=



|Itr|

ur

j=

k∈Jsr+j∩Itr:d

mXk,ξ

≥ε

⇒ δ≤



|Itr|

j=,ur

k∈Jsr+j∩Itr :d

mXk,ξ

≥ε

+



|Itr|

<j<ur

mXk,ξ

≥ε

=



|Itr|

j=,ur

_k_∈_J_s_r+_j_∩_I_t_r :dmXk,ξ

≥ε

+



|Itr|

<j<ur

k∈Jsr+j:d

mXk,ξ

(12)

=



|Itr|

j=,ur

mXk,ξ

≥ε

+ <j<ur

_|J

sr+j|

|Itr|



|Jsr+j|

k∈Jsr+j:d

mXk,ξ

≥ε. (.)

SinceXk→ξ(SβF(m)), so for suﬃciently largesr+j, we have



|Jsr+j|

k∈Jsr+j:d

mXk,ξ

≥ε<δ

. (.)

It can be seen that_<_j_<_u_r(|J_|sr_I+j|

tr|)≤. Hence, by using equation (.) in equation (.), we get

δ≤



|Itr|

k∈Itr:d

mXk,ξ

≥ε

≤



|Itr|

j=,ur

k∈Jsr+j∩Itr:d

mXk,ξ

≥ε+δ .

This implies that (_|_I tr|)

j=,ur|{k∈Jsr+j∩Itr:d(mXk,ξ)≥ε}| ≥_δ, which implies that at

least one of the following two inequalities holds:



|Itr|

k∈Jsr∩Itr:d

mXk,ξ

≥ε≥ δ

 (.)

or



|Itr|

k∈Jsr+ur∩Itr:d

mXk,ξ

≥ε≥ δ

. (.)

Suppose that equation (.) holds. Since|{k∈Jsr∩Itr:d(mXk,ξ)≥ε}| ≤ |Jsr∩Itr|, so using equation (.), we conclude that δ_≤|Jsr∩Itr|

|I_tr| , which proves (iii).

For suchsr,trchosen in the proof of (iii), from equation (.), we have

δ ≤



|Itr|

k∈Jsr∩Itr:d

mXk,ξ

≥ε

≤ _|J

sr|

|Itr|



|Jsr|

k∈Jsr∩Itr:d

mXk,ξ

≥ε. (.)

Since ( 

|Jsr|)|{k∈Jsr ∩Itr :d(

m_X

k,ξ)≥ε}| → asr→ ∞, from equation (.) we have

|J_sr|

|Itr|→ ∞asr→ ∞, which implies that condition (ii) is satisﬁed.

Conversely, let for the two lacunary sequencesβ= (li) andθ= (ki) there exist sequences

(sr), (tr)⊆Nandδ>  which satisfy the above three conditions. Deﬁne

mXk=

 ifk∈Jsr∩Itr,  otherwise.

Then, for any  <ε< , ifj=srfor anyr= , , , . . . ,_|_J_r_||{k∈Jr:d(mXk, )≥ε}|=  and

ifj=sr for somer, _J

sr|{k∈Jsr :d(

m_X

k, )≥ε}|=|Jsr_|_J∩Itr|

sr| ≤

|Itr|

(13)

But (_|_I

tr|)|{k∈Itr :d(

m_X

k, )≥ε}|= |Jsr_|_I∩Itr|

tr| ≥δforr= , , . . . , which impliesXk

(SFθ(m)).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

PDS in this paper introduced the space of lacunary strongly(mp)-summable sequences of fuzzy numbers by using a modulus function, deﬁned a suitable metric for the completeness property and gave the idea to introduce the relation between two arbitrary lacunary sequences. SM proved the relation between the spaces of lacunary strongly

m_{-summable sequences using a modulus function and}m_{-summable sequences using a modulus function of fuzzy} numbers and does some inclusion relations between any two arbitrary lacunary sequences. All authors read and approved the ﬁnal manuscript.

Received: 12 May 2013 Accepted: 16 October 2013 Published:25 Nov 2013

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Cite this article as:Srivastava and Mohanta:Generalized lacunarym_{-statistically convergent sequences of fuzzy}