Some geometric properties of generalized modular sequence space derived by the generalized de la Vallée Poussin mean

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R E S E A R C H

Open Access

Some geometric properties of generalized

modular sequence space derived by the

generalized de la Vallée-Poussin mean

Abdul Latif

1*

_{, Chirasak Mongkolkeha}

2

_{and Wutiphol Sintunavarat}

3

*_{Correspondence: [email protected]} 1_{Department of Mathematics, King}

Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia Full list of author information is available at the end of the article

Abstract

In this paper, we deﬁne a generalized modular sequence space by using the generalized de la Vallée-Poussin mean with a generalized Riesz transformation. Moreover, we investigate the property (β) and the uniform Opial property which is equipped with the Luxemburg norm. Finally, we show that this space has the ﬁxed point property.

Keywords: generalized Cesàro sequence spaces; property (β); uniform Opial property; Vallée-Poussin; generalized Riesz transformations; ﬁxed point property

1 Introduction

A number of mathematicians are studying the geometric properties of Banach spaces, be-cause such properties were identiﬁed as an important characteristic of the Banach spaces. For example, if Banach spaces have some geometric properties such as the uniformly

ro-tund,P-convexity,Q-convexity, Banach-Saks property, then they are reﬂexive spaces. The

investigations of the metric geometry of Banach spaces date back to , when Radon [] introduced the Kadec-Klee property (sometimes called the Radon-Riesz property, or property (H)), and later, when Riesz [, ] showed that the classicalLp-spaces,  <p<∞,

have the Kadec-Klee property. Although the spaceL[, ] (with Lebesgue measure) fails to

have the Kadec-Klee property. In , Clarkson [] introduced the notion of the uniform

convexity property (UC) or the uniform rotund property (UR) of Banach spaces, and it was

shown thatLpwith  <p<∞are examples of such space. In , Opial [] introduced a

new property which was called the Opial property and proved that the sequence spaceslp

( <p<∞) have this property butLp[,π] (p= ,  <p<∞) do not have it. In , Huﬀ [] introduced the nearly uniform convexity for Banach spaces and he also proved that ev-ery nearly uniformly convex Banach space is reﬂexive and it has the uniformly Kadec-Klee

property (UKK). In , Rolewicz [] deﬁned the drop property and property (β) and the

characterization of property (β), which is proved in []. In , Kutzarova [] deﬁned and

studied k-nearly uniformly Banach spaces. In , Prus [] introduced the notion of the uniform Opial property. There are many papers about the geometrical properties of se-quence spaces. In , Suantai [, ] deﬁned the generalized Cesàro sese-quence space

with a bounded sequencep= (pk) of positive real numbers. In , Şimşeket al.[]

in-troduced a new modular sequence space which is more general than the Cesàro sequence

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space defined by Shiue [] and the generalized Cesàro sequence space defined by Suan-tai. In , Mongkolkeha and Kumam [] defined the generalized Cesàro sequence space ces(p)(q) for a bounded sequencep= (pk) withpk≥ for allk∈Nandq= (qk) of positive

real numbers. Recently, Şimşeket al.[, ] deﬁned it by the modular sequence space

with de la Vallée-Poussin’s mean and studied some geometric properties in these spaces. Some examples of the geometry of sequence spaces and their generalizations have been extensively studied in [–].

The main purpose of this paper is to investigate the property (β) and the uniform Opial

property equipped with the Luxemburg norm of the new modular sequence space, which is deﬁned by using the generalized de la Vallée-Poussin mean with generalized Riesz trans-formation. Furthermore, we show that this space has the ﬁxed point property.

2 Preliminaries and notations

Letlbe the space of all real sequences. For ≤p<∞, the Cesàro sequence space (cesp,

for short) of Shue is deﬁned by

cesp=

x∈l: ∞

k=



k k

i=

_x₍_i₎p

<∞

equipped with the norm

x=

_∞

k=



k k

i=

x(i)

p_p

. (.)

The generalized Cesàro sequence spaceces(p) forp= (pk) a bounded sequence of positive

real numbers withpk≥ for allk∈Nof Suantai [, ] is deﬁned by

ces₍_p₎=x∈l:(λx) <∞for someλ>  , where

(x) = ∞

k=



k k

i=

x(i)

pk

equipped with the Luxemburg norm

x=inf

ε>  :

x

ε

≤

.

In the case whenpk=p, ≤p<∞for allk∈N, the generalized Cesàro sequence space

ces(p)is nothing but the Cesàro sequence spacecespand the Luxemburg norm is expressed

by (.).

Let∧= (λk) be a nondecreasing sequence of positive real numbers tending to inﬁnity

and letλ=  andλk+≤λk+ . The generalized de la Vallée-Poussin means of a sequence

x= (xk) is deﬁned as follows:

tk(x) = 

λk

j∈Ik

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The modular sequence spaceV(λ;p) of Şimşeket al.[, ] is deﬁned by de la

Vallée-Poussin’s mean, namely

V(λ;p) =

x∈l:(τx) <∞for someτ>  ,

where

(x) = ∞

k=



λk

i∈Ik x(i)

pk

x=inf

τ>  :

x

τ

≤

.

Letq= (qk) be a sequence of positive real numbers andQk=k_i_=qi. Then the Riesz trans-formation ofx= (xk) is deﬁned as

tk= 

Qk k

i=

qixi. (.)

In , Mursaleenet al.[] has modiﬁed the deﬁnition of weighted statistical

conver-gence due to Karakaya and Chishti [], they showed that the deﬁnition must be as follows: A sequencex= (xk) is weighted statistically convergent (orSN¯-convergent) toLif, for every

ε> ,

lim

k→∞ 

Qki≤Qk:qi|xi–L| ≥ε = ,

where Qk=

k

i=qi→ ∞ask→ ∞. In the same year, Mongkolkeha and Kumam []

deﬁned the generalized Cesàro sequence spaceces(p)(q) for a bounded sequencep= (pk)

withpk≥ for allk∈Nandq= (qk) of positive real numbers by

ces₍_p₎(q) =x∈l:(λx) <∞for someλ>  ,

where

(x) = ∞

k=



Qk k

i=

qix(i)

pk

andQk=k_i_=qi withQk=k_i_=qi→ ∞ask→ ∞. Thus, we see thatpk=p, ≤p<

∞for allk∈N; thences(p)(q) reduces tocesp(q) deﬁned by Khan []. Recently, Belen

and Mohiuddine [] generalized the concept of weighted statistical convergence due to

Mursaleenet al.for a nondecreasing sequence (λk) of positive real numbers tending to

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numbers be such thatq>  andQλ_k=

i∈Ikqi→ ∞ask→ ∞and

σk:=



Qλk

i∈Ik

qi, (.)

whereIk= [k–λk+ ,k].

A sequencex= (xk) is weightedλ-statistically convergent (orSN¯λ-convergent) toLif for everyε> ,

lim

n→∞



Qλk

i≤Qλk:qi|xi–L| ≥ε = .

Now, we deﬁne the new generalized modular sequence space forp= (pk) a bounded

sequence of positive real numbers withpk≥ for allk∈Nand (qk) is a sequence of positive real numbers such thatq> . LetQλk=

i∈Ikqi→ ∞ask→ ∞by

V(λ;p,q) =

x∈l:(τx) <∞for someτ>  , (.)

where

(x) = ∞

k=



Qλk

i∈Ik qix(i)

pk

x=inf

τ>  :

x

τ

≤

,

whenQλk=

i∈IkqiandIk= [k–λk+ ,k] fork≥.

By applying the reasoning of Remark . in [], if we take λk=k for allk≥, then

the weighted generalized modular sequence spaceV(λ;p,q) becomes the spaceces(p)(q).

If we take qk =  for allk≥, then the weighted generalized modular sequence space

V(λ;p,q) becomes the spaceV(λ;p). Also, if we takeλk=kandqk=  for allk≥, then

the weighted generalized modular sequence spaceV(λ;p,q) becomes the spaceces(p). Let (X, · ) be a real Banach space and letB(X) (resp.,S(X)) be a closed unit ball (resp., the unit sphere) ofX. A pointx∈S(X) is anH-pointofB(X) if for any sequence (xn) in

X such thatxn → asn→ ∞, the weak convergence of (xn) toximplies thatxn–

x → asn→ ∞. If every point inS(X) is anH-pointofB(X), thenX is said to have

the property (H). A Banach spaceXhas the property (β) if for eachε>  there exists

δ>  such that  <x<  +δimpliesα(conv(B(X)∪ {x})\B(X)) <ε, whereα(A) denotes

the Kuratowski measure noncompactness of a subsetAofX deﬁned as the inﬁmum of

allε>  such thatAcan be covered by a ﬁnite union of sets of diameter less thanε. The

following characterization of the property (β) is very useful (see []): A Banach space

Xhas the property (β) if and only if for eachε>  there existsδ>  such that for each elementx∈B(X) and for each sequence (xn) inB(X) withsep(xn)≥εthere is an index

kfor whichx+xk

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sep(xn)≥ε, there existsδ∈(, ) such thatconv(xn)∩( –δ)B(X)=∅. A Banach spaceX

is said to have the Opial property (see []) if every sequence (xn) weakly convergent tox

satisﬁes

lim

n→∞infxn–x ≤nlim→∞infxn–x,

for every x∈X. Opial proved in [] that the sequence spaces lp ( <p<∞) have this

property butLp[,π] (p= ,  <p<∞) do not have it. A Banach spaceXis said to have

the uniform Opial property (see []), if for eachε>  there existsτ >  such that for any weakly null sequence (xn) inS(X) andx∈Xwithx>εthe following holds:

 +τ≤ lim

n→∞infxn–x.

For example, the spaces in [, ] have the uniform Opial property. The ball-measure of noncompactness was deﬁned in [, ] by

β(A) =inf{ε>  :Acan be covered by ﬁnitely many balls of diameter≤ε}.

A Banach spaceXis said to have property (L) iflimε→–(ε) = , where(ε) =inf{ – inf[x: x ∈A] : Ais closed convex subset ofB(X) withβ(A) ≥ε}. The function is called the modulus of noncompact convexity (see []). It has been proved in [] that

property (L) is a useful tool in ﬁxed point theory and that a Banach spaceXhas property

(L) if and only if it is reﬂexive and has the uniform Opial property.

Throughout this paper, we assume thatlim_k_→∞infpk>  andlimk→∞suppk<∞and for x∈l,i∈N, we denote

ei=

i–

, , . . . , , , , , , . . .,

x|i=

x(),x(),x(), . . . ,x(i), , , , . . .,

x|N–i=

, , , . . . ,x(i+ ),x(i+ ), . . ..

In addition, we putM=sup_kpkfor allk≥.

First, we start with a brief recollection of basic concepts and facts in modular space.

For a real vector spaceX, a functionρ:X→[,∞] is called amodularif it satisﬁes the

following conditions:

(i) ρ(x) = if and only ifx= ;

(ii) ρ(αx) =ρ(x)for all scalarαwith|α|= ;

(iii) ρ(αx+βy)≤ρ(x) +ρ(y), for allx,y∈Xand allα,β≥withα+β= . The modularρis called convex if

(iv) ρ(αx+βy)≤αρ(x) +βρ(y), for allx,y∈Xand allα,β≥withα+β= .

For modularρonX, the space

Xρ=

x∈X:ρ(λx)→ asλ→+

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A sequence (xn) inXρis calledmodular convergenttox∈Xρif there exists aλ>  such

thatρ(λ(xn–x))→ asn→ ∞.

A modularρis said to satisfy the-condition(ρ∈) if for anyε>  there exist con-stantsK≥ anda>  such that

ρ(u)≤Kρ(u) +ε

for allu∈Xρwithρ(u)≤a.

Ifρsatisﬁes the-conditionfor anya>  withK≥ dependent ona, we say thatρ

has thestrong-condition(ρ∈s).

Lemma .[, Lemma .] Ifρ∈s

,then for any L> andε> ,there existsδ=δ(L,ε) > such that

ρ(u+v) –ρ(u)<ε,

whenever u,v∈Xρwithρ(u)≤L,andρ(v)≤δ.

Lemma .[, Lemma .] Convergences in norm and in modular sense are equivalent in Xρifρ∈.

Lemma .[, Lemma .] Ifρ∈s_,then for anyε> there existsδ=δ(ε) > such thatx ≥ +δwheneverρ(x)≥ +ε.

3 Main results

In this section, we prove the property (β) and uniform Opial property in a generalized

modular sequence spaceV(λ;p,q). Finally, we show that this space has the ﬁxed point

property. First we shall give some results which are very important for our consideration.

Proposition . The functionalis a convex modular on V(λ;p,q).

Proof Letx,y∈V(λ;p,q). It is obvious that(x) =  if and only ifx=  and(αx) =(x)

for scalarαwith|α|= . Letα≥,β≥ withα+β= . By the convexity of the function

t→ |t|pk_{, for all}_k_∈_N_{, we have}

(αx+βy) = ∞

k=



Qλk

i∈Ik

αqix(i) +βqiy(i)

pk

≤ ∞

k=

α 

Qλk

i∈Ik

qix(i)+β 

Qλk

i∈Ik qiy(i)

pk

≤α

∞

k=



Qλ_k

i∈Ik qix(i)

pk

+β

∞

k=



Qλ_k

i∈Ik qiy(i)

pk

=α(x) +β(y).

Proposition . For x∈V(λ;p,q),the modular on V(λ;p,q)satisﬁes the following

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(i) if <a< ,thenaM₍x

a)≤(x)and(ax)≤a(x); (ii) ifa> ,then(x)≤aM₍x

a); (iii) ifa≥,then(x)≤a(x)≤(ax). Proof (i) Let  <a< . Then we have

(x) = ∞

k=



Qλk

i∈Ik qix(i)

pk

= ∞

k=

a Qλk

i∈Ik qix(i)

a pk

= ∞

k=

apk



Qλk

i∈Ik qix(i)

a pk

≥ ∞

k=

aM



Qλ_k

i∈Ik qix(i)

a pk

=aM

∞

k=



Qλk

i∈Ik qi

x(_ai)pk

=aM

x a

.

By the convexity of modular, we have(ax)≤a(x), so (i) is obtained. (ii) Leta> . Then

(x) = ∞

k=



Qλk

i∈Ik qix(i)

pk

= ∞

k=

apk



Qλk

i∈Ik qix(i)

a pk

≤aM

∞

k=



Qλk

i∈Ik qix(i)

a pk

=aM

x a

.

Hence (ii) is satisﬁed. (iii) follows from the convexity of.

Following the line of the proof in [, , ], we get the following results.

Proposition . For any x∈V(λ;p,q),we have

(i) ifx< ,then(x)≤ x; (ii) ifx> ,then(x)≥ x; (iii) x= if and only if(x) = ; (iv) x< if and only if(x) < ; (v) x> if and only if(x) > .

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(i) if <a< andx>a,then(x) >aM_; (ii) ifa≥andx<a,then(x) <aM_.

Proposition . Let(xn)be a sequence in V(λ;p,q).

(i) Ifxn →asn→ ∞,then(xn)→asn→ ∞. (ii) If(xn)→asn→ ∞,thenxn →asn→ ∞.

Lemma . For any x∈V(λ;p,q), there exist j∈Nandγ ∈(, )such that (x

j

)≤ –γ

 (xj)for all j∈Nwith j≥j,where

xj=

j–

, , . . . , ,k–λ_k+≤i≤jx(i),x(j+ ),x(j+ ), . . .

andλkcorresponding to Ikfor k≥.

Proof Letj∈Nbe ﬁxed. So there existkj∈Nsuch thatj∈Ikj. Letαbe a real number such

that  <α≤lim_k_→∞infpk, then there existsj∈Nsuch thatα<pkj for allj≥j. Choose

γ ∈(, ) to be a real such that ( )

α_≤–γ

 . Then for eachx∈V(λ;p,q) andj≥j, we have

xj



= ∞

k=kj



Qλk

i∈Ik qix(i)



pk

= ∞

k=kj

 

pk



Qλk

i∈Ik qix(i)

pk

≤

 

α∞

k=kj



Qλk

i∈Ik qix(i)

pk

≤ –γ



xj.

Lemma . For any x∈V(λ;p,q)andε∈(, )there existsδ∈(, )such that(x)≤ –ε

impliesx ≤ –δ.

Proof For a proof of this lemma, we apply and follow the line of the proof of

Theo-rem .() in []. Suppose that the lemma does not hold, then there existε>  andxn∈

V(λ;p,q) such that(xn)≤ –εand_≤ xn . Letan=_xn– . Thenan→ asn→

∞. LetL=sup{(xn) :n∈N}. Bysupkpk<∞,i.e.,∈s, there existsK≥ such that

(u)≤K(u) + , (.)

for every u∈l(p,θ) with(u) < . By (.), we have(xn)≤K(xn) + ≤K+  for all n∈N. Hence  <L<∞. By Proposition . and Proposition .(iii), we have

 =

xn

=anxn+ ( –an)xn≤an(xn) + ( –an)(xn) ≤anL+ ( –ε)→ –ε,

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Theorem . The space V(λ;p,q)is a Banach space with respect to the Luxemburg norm.

Proof Let (xn) = (xn(i)) be a Cauchy sequence inV(λ;p,q) andε∈(, ). Thus there exists

N∈Nsuch thatxn–xm<εfor alln,m≥N. By Proposition .(i), we have

(xn–xm)≤ xn–xm<ε for alln,m≥N. (.)

That is,

∞

k=



Qλk

i∈Ik

qixn(i) –xm(i) pk

<ε for alln,m≥N. (.)

For ﬁxedk, we see that

qixn(i) –qixm(i)<ε for alln,m≥N.

Thus, let (qixn(i)) be a Cauchy sequence inRfor alli∈N. SinceRis complete, for each

i≥, there existsx(i)∈Rsuch thatqixm(i)→qix(i) asm→ ∞. Thus for ﬁxedkand for

eachi∈Ik, we have

qixn(i) –x(i)<ε asm→ ∞, for alln≥N. This implies that

(xn–xm)→(xn–x) asm→ ∞. (.)

That is,

∞

k=



Qλk

i∈Ik

qixn(i) –xm(i)

pk

→ ∞

k=



Qλk

i∈Ik

qixn(i) –x(i)

pk

(.)

asm→ ∞. By (.), we have

(xn–x)≤ xn–x<ε for alln≥N,

and hencexn→xasn→ ∞. So we havexn–x∈V(λ;p,q). Since (xn)∈V(λ;p,q) and the

linearity of the sequence spaceV(λ;p,q), we getx=xn– (xn–x)∈V(λ;p,q). Therefore

the sequence spaceV(λ;p,q) is a Banach space, with respect to the Luxemburg norm, and

the proof is complete.

Theorem . The space V(λ;p,q)has property(β).

Proof Letε>  and (xn)⊂B(V(λ;p,q)) withsep(xn)≥ε. For eachj∈N, there existkj∈N

such thatj∈Ikj. Let

xj_n=

j–

, , . . . , ,k–λk+≤i≤jxn(i),xn(j+ ),xn(j+ ), . . .

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whereλkcorresponds toIkfork≥. This is so since for eachi∈N, (xn(i))∞n=is bounded.

By using the diagonal method, we see that for eachj∈Nwe can ﬁnd a subsequence (xnl)

of (xn) such that (xnl(i)) converges for eachi∈N. Therefore, for anyj∈Nthere exists an increasing sequence (tj) such thatsep((xjnl)l>tj)≥ε. Hence for eachj∈Nthere exists

a sequence of positive integers (sj)∞_j_= withs<s<s<· · · such thatxjsj ≥ε_ and, since

∈s_, by Lemma . we may assume that there existsη>  such that(xjsj)≥ηfor all j∈N, that is,

∞

k=kj



Qλk

i∈Ik

qixj_sj(i)

pk

≥η (.)

for allj∈N. On the other hand, by Lemma ., there existj∈Nandγ ∈(, ) such that

uj



≤ –γ



uj (.)

for all u∈V(λ;p,q) andj≥j. From Lemma ., there existsδ>  such that for any

y∈V(λ;p,q)

(y)≤ –γ η

 ⇒ y ≤ –δ. (.)

Since again∈s_, by Lemma ., there existsδsuch that

(u+v) –(u)<γ η

 , (.)

whenever(u)≤ and(v)≤δ. Sincex∈B(V(λ;p,q)), we have(x)≤. Then there

existsj≥jsuch that(xj)≤δ. We putu=x

j

sjandv=xj,

u  = ∞

k=kj



Qλk

i∈Ik

qixsj(i)



pk<  and

v  = ∞

k=kj



Qλk

i∈Ik qix(i)



pk<δ.

From (.) and (.), we have

∞

k=kk



Qλk

i∈Ik

qix(i) +xsj(i)



pk=

u+v

 ≤ u 

+γ η



≤ –γ 

(u)+γ η

 . (.)

By (.), (.), (.), and convexity of the functionf(t) =|t|pk_{, for all}_k∈N_{, we have}

_x₊_x sj  = ∞ k= 

Qλk

i∈Ik

qix(i) +xsj(i)

 pk = kj– k= 

Qλk

i∈Ik

qix(i) +xsj(i)



pk+

∞

k=kj



Qλk

i∈Ik

qix(i) +xsj(i)



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≤  kj– k= 

Qλk

i∈Ik qix(i)

pk + kj– k= 

Qλk

i∈Ik

qixsj(i)

pk

+ ∞

k=kj



Qλk

i∈Ik

qixsj(i)



pk+γ η  ≤  kj– k= 

Qλk

i∈Ik qix(i)

pk + kj– k= 

Qλk

i∈Ik

qixsj(i)

pj

+ –γ

 ∞

k=kj



Qλk

i∈Ik

qixsj(i)

pk

+γ η

 =  kj– k= 

Qλk

i∈Ik qix(i)

pk +  kj– k= 

Qλk

i∈Ik

qixsj(i)

pk

+ –γ

 ∞

k=kj



Qλk

i∈Ik

qixsj(i)

pk

+γ η

 =  kj– k= 

Qλn

i∈In qix(i)

pk +  ∞ k= 

Qλk

i∈Ik

qixsj(i)

pk

–γ

 ∞

k=kj



Qλk

i∈Ik

qixsj(i)

pj

+γ η

 ≤ +  – γ η  + γ η 

=  –γ η  .

So it follows from (.) that

x+_xsj≤ –δ.

Therefore, the spaceV(λ;p,q) has property (β).

By the facts that property (β) implies (NUC), and (NUC) implies property (UKK),

prop-erty (H), and reﬂexivity (see [–]). The following results are obtained directly from

Theorem ..

Corollary . The space V(λ;p)has property(β).

Corollary . The space V(λ;p,q)is nearly uniform convexity and reﬂexive.

Corollary . The space V(λ;p,q)has property(UKK)and property(H).

Corollary . The space V(λ;p)is nearly uniform convexity and reﬂexive.

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Next, we will prove the uniform Opial property for the spaceV(λ;p,q).

Theorem . The space V(λ;p,q)has the uniform Opial property.

Proof Take anyε>  andx∈V(λ;p,q) withx ≥ε. Let (xn) be a weakly null sequence in

S(V(λ;p,q)). By∈s, hence by Lemma . there existsδ∈(, ) independent ofxsuch that(x) >δ. Also, by∈s_and Lemma ., one may assert that there existsδ∈(,δ) such that

(y+z) –(y)<δ

 (.)

whenever(y)≤ and(z)≤δ. Choosek∈Nsuch that ∞

k=k+



Qλk

i∈Ik qix(i)

pk

<δ

. (.)

So, we have

δ<

k

k=



Qλ_k

i∈Ik qix(i)

pk

+ ∞

k=k+



Qλ_k

i∈Ik qix(i)

pk

≤

k

k=



Qλk

i∈Ik qix(i)

pk

+δ

, (.)

which implies that

k

k=



Qλk

i∈Ik qix(i)

pk

>δ–δ 

>δ–δ 

=δ

. (.)

Sincexn w

→, there existsn∈Nsuch that

δ

 ≤

k

k=



Qλk

i∈Ik

qixn(i) +x(i)

pk

(.)

for all n>n, since weak convergence implies coordinatewise convergence. Again, by

xn→w , there existsn∈Nsuch that

xn|_ko<  –

 –δ 



M

(.)

for alln>n. Hence, by the triangle inequality of the norm, we get

xn|N–ko>

 –δ 



M

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It follows from Proposition .(ii) that

≤

_x n|N–ko

( –δ

) 

M

= ∞

k=k+



Qλk

i∈Ikqi|xn(i)|

( –δ_)M

pk

≤



( –δ_)M

M ∞

k=k+



Qλ_k

i∈Ik

qixn(i)

pk

(.)

implies that

∞

k=k+



Qλ_k

i∈Ik

qixn(i)

pk

≥ –δ

 (.)

for alln>n. By inequality (.), (.), (.), and (.), we have for anyn>n

(xn+x) =

k

k=



Qλk

i∈Ik

qixn(i) +x(i)

pk

+ ∞

k=k+



Qλk

i∈Ik

qixn(i) +x(i)

pk

>

k

k=



Qλ_k

i∈Ik

qixn(i) +x(i)

pk

+ ∞

k=k+



Qλ_k

i∈Ik

qixn(i) +x(i)

pk

≥δ

 +

∞

k=k+



Qλk

i∈Ik

qixn(i) pk

–δ



≥δ

 +

 – δ



– δ



≥ +δ .

Since∈s_and by Lemma . there existsτ depending onδonly such thatxn+x ≥

 +τ, which implies thatlimn→∞infxn+x ≥ +τ, hence the proof is complete.

Corollary . The space V(λ;p)has the uniform Opial property.

Corollary .[, Theorem .] The spaceces(p)has the uniform Opial property.

Corollary .[, Theorem ] For any <p<∞,the spaceces_phas the uniform Opial property.

Corollary . The space V(λ;p,q)has property(L)and the ﬁxed point property.

Corollary . The space V(λ;p)has property(L)and the ﬁxed point property.

Competing interests

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Authors’ contributions

All authors contributed equally and signiﬁcantly in writing this paper. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.}2_{Department of}

Mathematics, Faculty of Liberal Arts and Science, Kasetsart University, Kamphaeng-Saen Campus, Nakhonpathom, 73140, Thailand.3_{Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Rangsit}

Center, Pathumthani, 12121, Thailand.

Acknowledgements

This article was funded by the Deanship of Scientiﬁc Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR for the technical and ﬁnancial support.

Received: 24 June 2014 Accepted: 8 September 2014 Published:29 Sep 2014

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10.1186/1029-242X-2014-375