Some geometric properties of a new modular space defined by Zweier operator

Et et al. Fixed Point Theory and Applications 2013, 2013:165http://www.fixedpointtheoryandapplications.com/content/2013/1/165

RESEARCH Open Access

Some geometric properties of a newmodularspace defined by Zweier operatorMikail Et1, Murat Karakas2* and Muhammed Çınar3

Dedicated to Professor Hari M Srivastava

*Correspondence:[email protected] of Statistics, Bitlis ErenUniversity, Bitlis, 13000, TurkeyFull list of author information isavailable at the end of the article

AbstractIn this paper, we define the modular spaceZσ (s,p) by using the Zweier operator anda modular. Then, we consider it equipped with the Luxemburg norm and alsoexamine the uniform Opial property and property β . Finally, we show that this spacehas the fixed point property.MSC: 40A05; 46A45; 46B20

Keywords: Zweier operator; Luxemburg norm; modular space; uniform Opialproperty; property (β)

1 IntroductionIn literature, there aremany papers about the geometrical properties of different sequencespaces such as [–]. Opial [] introduced the Opial property and proved that the se-quence spaces �p ( < p < ∞) have this property but Lp[, π ] (p �= , < p < ∞) does nothave it. Franchetti [] showed that any infinite dimensional Banach space has an equiv-alent norm that satisfies the Opial property. Later, Prus [] introduced and investigatedthe uniform Opial property for Banach spaces. The Opial property is important becauseBanach spaces with this property have the weak fixed point property.

2 Definition and preliminariesLet (X,‖ · ‖) be a real Banach space and let S(X) (resp. B(X)) be the unit sphere (resp. theunit ball) of X. A Banach space X has the Opial property if for any weakly null sequence{xn} in X and any x in X \ {}, the inequality limn→∞ inf‖x‖ < limn→∞ inf‖xn + x‖ holds.We say that X has the uniform Opial property if for any ε > there exists r > such thatfor any x ∈ X with ‖x‖ ≥ ε and any weakly null sequence {xn} in the unit sphere of X, theinequality + r ≤ limn→∞ inf‖xn + x‖ holds.For a bounded set A ⊂ X, the ball-measure of noncompactness was defined by β(A) =

inf{ε > : A can be covered by finitely many balls with diameter ≤ ε}. The function� de-fined by �(ε) = inf{ – inf(‖x‖ : x ∈ A) : A is closed convex subset of B(X) with β(A) ≤ ε}is called the modulus of noncompact convexity. A Banach space X is said to have prop-erty (L), if limε→– �(ε) = . This property is an important concept in the fixed point theoryand a Banach spaceX possesses property (L) if and only if it is reflexive and has the uniformOpial property.

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A Banach space X is said to satisfy the weak fixed point property if every nonemptyweakly compact convex subsetC and every nonexpansivemappingT : C → C(‖Tx–Ty‖ ≤‖x – y‖,∀x, y ∈ C) have a fixed point, that is, there exists x ∈ C such that T(x) = x. Prop-erty (L) and the fixed point property were also studied by Goebel and Kirk [], Toledanoet al. [], Benavides [], Benavides and Phothi []. A Banach space X is said to haveproperty (H) if every weakly convergent sequence on the unit sphere is convergent innorm. Clarkson [] introduced the uniform convexity, and it is known that the uniformconvexity implies the reflexivity of Banach spaces. Huff [] introduced the concept ofnearly uniform convexity of Banach spaces. A Banach space X is called uniformly convex(UC) if for each ε > , there is δ > such that for x, y ∈ S(X), the inequality ‖x – y‖ > ε

implies that ‖ (x + y)‖ < – δ. For any x /∈ B(X), the drop determined by x is the set

D(x,B(X)) = conv({x} ∪ B(X)). A Banach space X has the drop property (D) if for everyclosed setC disjointwithB(X), there exists an element x ∈ C such thatD(x,B(X))∩C = {x}.Rolewicz [] showed that the Banach spaceX is reflexive ifX has the drop property. Later,Montesinos [] extended this result and proved that X has the drop property if and onlyif X is reflexive and has property (H). A sequence {xn} is said to be ε-separated sequencefor some ε > if

sep(xn) = inf{‖xn – xm‖ : n �=m

}> ε.

A Banach space X is called nearly uniformly convex (NUC) if for every ε > , there existsδ ∈ (, ) such that for every sequence (xn) ⊆ B(X) with sep(xn) > ε, we have conv(xn) ∩(( – δ)B(X)) �= ∅. Huff [] proved that every (NUC) Banach space is reflexive and hasproperty (H). A Banach space X has property (β) if and only if for each ε > , there existsδ > such that for each element x ∈ B(X) and each sequence (xn) in B(X) with sep(xn) ≥ ε,there is an index k for which ‖ x+xk

‖ < – δ.For a real vector space X, a function ρ : X → [,∞] is called a modular if it satisfies the

following conditions:(i) ρ(x) = if and only if x = ,(ii) ρ(αx) = ρ(x) for all scalar α with |α| = ,(iii) ρ(αx + βy) ≤ ρ(x) + ρ(y) for all x, y ∈ X and all α,β ≥ with α + β = .

The modular ρ is called convex if(iv) ρ(αx + βy) ≤ αρ(x) + βρ(y) for all x, y ∈ X and all α,β ≥ with α + β = .

For any modular ρ on X, the space

Xρ ={x ∈ X : ρ(σx) < ∞ for some σ >

}is called a modular space. In general, the modular is not subadditive and thus it does notbehave as a norm or a distance. But we can associate themodular with an F-norm. A func-tional ‖ · ‖ : X → [,∞] defines an F-norm if and only if

(i) ‖x‖ = ⇔ x = ,(ii) ‖αx‖ = ‖x‖ whenever |α| = ,(iii) ‖x + y‖ ≤ ‖x‖ + ‖y‖,(iv) if αn → α and ‖xn – x‖ → , then ‖αnxn – αx‖ → .



F-norm defines a distance on X by d(x, y) = ‖x – y‖. If the linear metric space (X,d) iscomplete, then it is called an F-space. The modular space Xρ can be equipped with thefollowing F-norm:

‖x‖ = inf

{α > : ρ

(xα

)≤ α

}.

If the modular ρ is convex, then the equality ‖x‖ = inf{α > : ρ( xα) ≤ } defines a norm

which is called the Luxemburg norm.A modular ρ is said to satisfy the δ-condition if for any ε > , there exist constants

K ≥ , a > such that ρ(u) ≤ Kρ(u) + ε for all u ∈ Xρ with ρ(u) ≤ a. If ρ providesthe δ-condition for any a > with K ≥ dependent on a, then ρ provides the strongδ-condition (briefly ρ ∈ δs).Let us denote by � the space of all real sequences. The Cesàro sequence spaces

Cesp ={x ∈ � :

∞∑n=

(n–

n∑i=

|xi|)p

< ∞}, ≤ p < ∞,

and

Ces∞ ={x ∈ � : sup

nn–

n∑i=

|xi| < ∞},

were introduced by Shiue []. Jagers [] determined the Köthe duals of the sequencespace Cesp ( < p <∞). It can be shown that the inclusion �p ⊂ Cesp is strict for < p <∞although it does not hold for p = . Also, Suantai [] defined the generalized Cesàro se-quence space by

ces(p) ={x ∈ � : ρ(λx) < ∞ for some λ >

},

where ρ(x) =∑∞

n=( n∑n

i= |x(i)|)pn . If p = (pn) is bounded, then

ces(p) ={x = (xk) :

∞∑n=

(n–

n∑i=

∣∣x(i)∣∣)pn

< ∞}.

The sequence space C(s,p) was defined by Bilgin [] as follows:

C(s,p) ={x = (xk) :

∞∑r=

(–r

∑rk–s|xk|

)pr< ∞, s ≥

}

for p = (pr) with infpr > , where∑

r denotes a sum over the ranges r ≤ k < r+. Thespecial case of C(s,p) for s = is the space

Ces(p) ={x = (xk) :

∞∑r=

(–r

∑r

|xk|)pr

<∞}



which was introduced by Lim []. Also, the inclusion Ces(p) ⊆ C(s,p) holds. A paranormon C(s,p) is given by

ρ(x) =( ∞∑

r=

(–r

∑rk–s|xk|

)pr)/M

forM = max(,H) and H = suppr <∞.The Z-transform of a sequence x = (xk) is defined by (Zx)n = yn = αxn + ( – α)xn– by

using the Zweier operator

Z = (znk) =

⎧⎪⎪⎨⎪⎪⎩

α, k = n, – α, k = n – ,, otherwise

for n,k ∈N and α ∈F\{},

where F is the field of all complex or real numbers. The Zweier operator was studied byŞengönül and Kayaduman [].Now we introduce a new modular sequence space Zσ (s,p) by

Zσ (s,p) ={x ∈ � : σ (tx) <∞, for some t >

},

where σ (x) =∑∞

r=(–r∑

r k–s|αxk + ( – α)xk–|)pr < ∞ and s ≥ . If we take α = , thenZσ (s,p) = C(s,p); if α = and s = , then Zσ (s,p) = Ces(p). It can be easily seen thatσ : Zσ (s,p) → [,∞] is a modular on Zσ (s,p). We define the Luxemburg norm on thesequence space Zσ (s,p) as follows:

‖x‖ = inf

{t > : σ

(xt

)≤

}, ∀x ∈Zσ (s,p).

It is easy to see that the space Zσ (s,p) is a Banach space with respect to the Luxemburgnorm.Throughout the paper, suppose that p = (pr) is bounded with pr > for all r ∈N and

ei =( i–︷︸︸︷,, . . . , , , , , , . . .

),

x|i =(x(),x(),x(), . . . ,x(i), , , , . . .

),

x|N–i =(,, , . . . ,x(i + ),x(i + ), . . .

),

for i ∈ N and x ∈ �. In addition, we will require the following inequalities:

|ak + bk|pk ≤ C(|ak|pk + |bk|pk

), |ak + bk|tk ≤ |ak|tk + |bk|tk ,

where tk = pkM ≤ and C = max{, H–} with H = suppk .

3 Main resultsSince �p is reflexive and convex, �(p)-type spaces have many useful applications, and itis natural to consider a geometric structure of these spaces. From this point of view, we



generalized the space C(s,p) by using the Zweier operator and then obtained the equalityZσ (s,p) = Ces(p), that is, it was seen that the structure of the spaceCes(p) was preserved. Inthis section, our goal is to investigate a geometric structure of the modular space Zσ (s,p)related to the fixed point theory. For this, we will examine property (β) and the uniformOpial property for Zσ (s,p). Finally, we will give some fixed point results. To do this, weneed some results which are important in our opinion.

Lemma . [] If σ ∈ δs, then for any L > and ε > , there exists δ > such that

∣∣σ (u + v) – σ (u)∣∣ < ε,

where u, v ∈ Xσ with σ (u)≤ L and σ (v)≤ δ.

Lemma . [] If σ ∈ δs, convergence in norm and in modular are equivalent in Xσ .

Lemma . [] If σ ∈ δs, then for any ε > , there exists δ = δ(ε) > such that ‖x‖ ≥ + δ

implies σ (x)≥ + ε.

Now we give the following two lemmas without proof.

Lemma . If ‖x‖L < for any x ∈Zσ (s,p), then σ (x)≤ ‖x‖L.

Lemma . For any x ∈Zσ (s,p), ‖x‖L = if and only if σ (x) = .

Lemma . If lim infpr > , then for any x ∈Zσ (s,p), there exist k ∈N and μ ∈ (, ) suchthat

σ

(xk

)≤ –μ

σ(xk

)

for all k ∈ N with k ≥ k, where xk = (k–︷︸︸︷

,, . . . , ,∑

r≤i≤k |x(i)|,x(k + ),x(k + ), . . .) andr ≤ k < r+.

Proof Let k ∈ N be fixed. Then there exists rk ∈N such that k ∈ Irk . Let γ be a real number < γ ≤ lim infpr , and so there exists k ∈ N such that γ < prk for all k ≥ k. Choose μ ∈(, ) such that ( )

γ ≤ –μ

. Therefore, we have

σ

(xk

)=

∞∑r=

(–r

∑rk–s

∣∣∣∣αx(k) + ( – α)x(k – )

∣∣∣∣)pr

=∞∑r=

(

)pr(–r

∑rk–s

∣∣αx(k) + ( – α)x(k – )∣∣)pr

≤(

)γ ∞∑r=

(–r

∑rk–s

∣∣αx(k) + ( – α)x(k – )∣∣)pr

< –μ

σ(xk

)for each x ∈Zσ (s,p) and k ≥ k. �



Lemma . If σ ∈ δs, then for any ε ∈ (, ), there exists δ ∈ (, ) such that σ (x) ≤ – ε

implies ‖x‖ ≤ – δ.

Proof Suppose that lemma does not hold. So, there exist ε > and xn ∈ Zσ (s,p) suchthat σ (xn) < – ε and

≤ ‖xn‖ → . Take sn = ‖xn‖– , and so sn → as n → ∞. Let

P = sup{σ (xn) : n ∈ N}. There exists D ≥ such that

σ (u)≤ Dσ (u) + (.)

for every u ∈Zσ (s,p) with σ (u) < , since σ ∈ δs. We have

σ (xn) ≤ Dσ (xn) + <D +

for all n ∈N by (.). Therefore, < P < ∞ and from Lemma . we have

= σ

(xn

‖xn‖)= σ

(snxn + ( – sn)xn

)≤ snσ (xn) + ( – sn)σ (xn)

≤ snP + ( – ε) → ( – ε).

This is a contradiction. So, the proof is complete. �

Theorem . The space Zσ (s,p) has property (β).

Proof Let ε > and (xn) ⊂ B(Zσ (s,p)) with sep(xn) ≥ ε and x ∈ B(Zσ (s,p)). For each l ∈N,we can find rk ∈N such that rk ≤ l < rk+. Let

xln =( l–︷︸︸︷,, . . . , ,

∑rk≤i≤l

∣∣x(i)∣∣,xn(l + ),xn(l + ), . . .).

Since for each i ∈N, (xn(i))∞i= is bounded, by using the diagonal method, we can find a sub-sequence (xnj ) of (xn) such that (xnj (i)) converges for each i ∈ N with ≤ i ≤ l. Therefore,there exists an increasing sequence of positive integers tl such that sep((xlnj )j≥tl ) ≥ ε. Thus,there exists a sequence of positive integers (rl)∞l= with r < r < · · · such that ‖xlrl‖ ≥ ε

forall l ∈N. Since σ ∈ δs, there is η > such that

σ(xlrl

) ≥ η for all l ∈N (.)

from Lemma .. However, there exist k ∈N and μ ∈ (, ) such that

σ

(vk

)≤ –μ

σ(vk

)(.)

for all v ∈Zσ (s,p) and k ≥ k by Lemma .. There exists δ > such that

σ (y)≤ –μη

⇒ ‖y‖ ≤ – δ (.)

for any y ∈Zσ (s,p) by Lemma ..



By Lemma ., there exists δ such that

∣∣σ (u + v) – σ (u)∣∣ < μη

, (.)

where σ (u)≤ and σ (v)≤ δ. Hence, we get that σ (x)≤ since x ∈ B(Zσ (s,p)). Then thereexists k ≥ k such that σ (xk) ≤ δ. Let u = xlrl and v = xl . Then

σ

(u

)< and σ

(v

)< δ.

We obtain from (.) and (.) that

σ

(u + v

)≤ σ

(u

)+

μη

≤ –μ

σ (u) + μη

. (.)

Choose si = rli . By the inequalities (.), (.), (.) and the convexity of the functionf (u) = |u|pr , we have

σ

(x + xsk

)=

∞∑r=

(–r

∑rk–s

∣∣∣∣α(x(k) + xsi (k)) + ( – α)(x(k – ) + xsi (k – ))

∣∣∣∣)pr

=rk–∑r=

(–r

∑rk–s

∣∣∣∣α(x(k) + xsi (k)) + ( – α)(x(k – ) + xsi (k – ))

∣∣∣∣)pr

+∞∑r=rk

(–r

∑rk–s

∣∣∣∣α(x(k) + xsi (k)) + ( – α)(x(k – ) + xsi (k – ))

∣∣∣∣)pr

≤

rk–∑r=

(–r

∑rk–s

∣∣αx(k) + ( – α)x(k – )∣∣)pr

+

rk–∑r=

(–r

∑rk–s

∣∣αxsi (k) + ( – α)xsi (k – )∣∣)pr

+∞∑r=rk

(–r

∑rk–s

∣∣∣∣αxsi (k) + ( – α)xsi (k – )

∣∣∣∣)pr

+μη

≤

rk–∑r=

(–r

∑rk–s

∣∣αx(k) + ( – α)x(k – )∣∣)pr

+

rk–∑r=

(–r

∑rk–s

∣∣αxsi (k) + ( – α)xsi (k – )∣∣)pr

+ –μ

∞∑r=rk

(–r

∑rk–s

∣∣∣∣αxsi (k) + ( – α)xsi (k – )

∣∣∣∣)pr

+μη

≤

rk–∑r=

(–r

∑rk–s

∣∣αx(k) + ( – α)x(k – )∣∣)pr

+

∞∑r=

(–r

∑rk–s

∣∣αxsi (k) + ( – α)xsi (k – )∣∣)pr



–μ

∞∑r=rk

(–r

∑rk–s

∣∣∣∣αxsi (k) + ( – α)xsi (k – )

∣∣∣∣)pr

+μη

≤ +–

μη

+

μη

= –μη

.

So, the inequality (.) implies that ‖ x+xsk ‖ ≤ – δ. Consequently, the space Zσ (s,p)

possesses property (β). �

Since property (β) implies NUC, NUC implies property (D) and property (D) impliesreflexivity, we can give the following result from Theorem ..

Corollary . The space Zσ (s,p) is nearly uniform convex, reflexive and also it has prop-erty (D).

Theorem . The space Zσ (s,p) has the uniform Opial property.

Proof Let ε > and x ∈ Zσ (s,p) be such that ‖x‖ ≥ ε and (xn) be a weakly null sequencein S(Zσ (s,p)). By σ ∈ δs, there exists ζ ∈ (, ) independent of x such that σ (x) > ζ byLemma .. Also since σ ∈ δs, by Lemma ., there is ζ ∈ (, ζ ) such that

∣∣σ (y + z) – σ (y)∣∣ < ζ

(.)

whenever σ (y)≤ and σ (z)≤ ζ. Take r ∈ N such that

∞∑r=r+

(–r

∑rk–s

∣∣αx(k) + ( – α)x(k – )∣∣)pr

<ζ

. (.)

Hence, we have

ζ <r∑r=

(–r

∑rk–s

∣∣αx(k) + ( – α)x(k – )∣∣)pr

+∞∑

r=r+

(–r

∑rk–s

∣∣αx(k) + ( – α)x(k – )∣∣)pr

≤r∑r=

(–r

∑rk–s

∣∣αx(k) + ( – α)x(k – )∣∣)pr

+ζ

(.)

and this implies that

r∑r=

(–r

∑rk–s

∣∣αx(k) + ( – α)x(k – )∣∣)pr

> ζ –ζ

> ζ –ζ

=ζ. (.)



Since xn →w , by the inequality (.), there exists r ∈N such that

r∑r=

(–r

∑rk–s

∣∣α(xn(k) + x(k)

)+ ( – α)

(xn(k – ) + x(k – )

)∣∣)pr>ζ. (.)

Again, by xn →w , there is r > r such that for all r > r

‖xn|r ‖ < –( –

ζ

)/M, (.)

where pr ≤ M ∈N for all r ∈N. Therefore, we obtain that

‖xn|N–r ‖ >( –

ζ

)/M(.)

by the triangle inequality of the norm. It follows from the definition of the Luxemburgnorm that

≤ σ

( xn|N–r( – ζ

)/M

)

=∞∑

r=r+

(–r

∑r k–s|αxn(k) + ( – α)xn(k – )|

( – ζ

)/M

)pr

≤(

( – ζ

)/M

)M ∞∑r=r+

(–r

∑rk–s

∣∣αxn(k) + ( – α)xn(k – )∣∣)pr

(.)

and this implies that

∞∑r=r+

(–r

∑rk–s

∣∣αxn(k) + ( – α)xn(k – )∣∣)pr

≥ –ζ

. (.)

By (.), (.), (.), (.) and since xn →w ⇒ xn → (coordinatewise), we have forany r > r that

σ (xn + x) =r∑r=

(–r

∑rk–s

∣∣α(xn(k) + x(k)

)+ ( – α)

(xn(k – ) + x(k – )

)∣∣)pr

+∞∑

r=r+

(–r

∑rk–s

∣∣α(xn(k) + x(k)

)+ ( – α)

(xn(k – ) + x(k – )

)∣∣)pr

≥∞∑

r=r+

(–r

∑rk–s

∣∣α(xn(k) + x(k)

)+ ( – α)

(xn(k – ) + x(k – )

)∣∣)pr

–ζ

+ζ

≥ ζ

+( –

ζ

)–

ζ

= +ζ

.



Since σ ∈ δs, it follows from Lemma . that there is τ depending on ζ only such that‖xn + x‖ ≥ + τ . �

Corollary . The space Zσ (s,p) has property (L) and the fixed point property.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors read and approved the final manuscript.

Author details1Department of Mathematics, Firat University, Elaızg, 23119, Turkey. 2Department of Statistics, Bitlis Eren University, Bitlis,13000, Turkey. 3Department of Mathematics, Mus Alparslan University, Mus, 49100, Turkey.

Received: 14 December 2012 Accepted: 5 June 2013 Published: 25 June 2013

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doi:10.1186/1687-1812-2013-165Cite this article as: Et et al.: Some geometric properties of a newmodular space defined by Zweier operator. FixedPoint Theory and Applications 2013 2013:165.


Some geometric properties of a new modular space defined by Zweier operator

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