# Some Geometric Properties of Lacunary Sequence Spaces Related to Fixed Point Property

Authors: Chirasak Mongkolkeha; Poom Kumam
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

The main purpose of this paper is considering the lacunary sequence spaces defined by Karakaya (2007), by proving the property (β) and Uniform Opial property.


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## Body

## 1. Introduction

Let(X,∥·∥) be a real Banach space and let B(X) (resp., S(X)) be a closed unit ball (resp., the unit sphere) of X. For any subset A of X, we denote by conv(A) the convex hull of A. The Banach space X is uniformly convex (UC), if for each ɛ>0 there exists δ>0 such that for x,y∈S(X) the inequality ∥x-y∥>ɛ implies ∥(x+y)/2∥<1-δ (see [1]). A Banach space X has the property (β) if for each ɛ>0 there exists δ>0 such that 1<∥x∥<1+δ implies α(conv(B(X)∪{x})∖B(X))<ɛ, where α(A) denotes the Kuratowski measure noncompactness of a subset A of X defined as the infimum of all ɛ>0 such that A can be covered by a finite union of sets of diameter less than ɛ. The following characterization of the property (β) is very useful (see [2]): A Banach space X has the property (β) if and only if for each ɛ>0 there exists δ>0 such that for each element x∈B(X) and each sequence (xn) in B(X) with sep(xn)≥ɛ there is an index k for which ∥(x+xk)/2∥<1-δ where sep(xn)=inf⁡{∥xn-xm∥:n≠m}>ɛ. A Banach space X is nearly uniformly convex (NUC) if for each ɛ>0 and every sequence (xn) in B(X) with sep(xn)≥ɛ, there exists δ∈(0,1) such that conv(xn)∩(1-δ)B(X)≠∅. Define for any x∉B(X) the drop D(x,B(X)) determined by x by D(x,B(X))=conv(B(X)∪{x}). A Banach space X has the drop property (write (D)) if for every closed set C disjoint with B(X) there exists an element x∈C such that D(x,B(X))∩C={x}. A point x∈S(X) is an H-point of B(X) if for any sequence (xn) in X such that ∥xn∥→1 as n→∞, the week convergence of (xn) to x implies that ∥xn-x∥→0 as n→∞. If every point in S(X) is an H-point of B(X), then X is said to have the property (H). A Banach space is said to have the uniform Kadec-Klee property (abbreviated as (UKK)) if for every ɛ>0 there exists δ>0 such that for every sequence (xn) in S(X) with sep(xn)≥ɛ and xn→wx as n→∞, we have ∥x∥<1-δ. Every (UKK) Banach space has H-property (see [3]). The following implications are true in any Banach spaces,(1.1)where(Rfx) denotes the property of reflexivity (see [3–6]). A Banach space X is said to have the Opial property (see [7]) if every sequence (xn) weakly convergent to x0 satisfies
(1.2)lim⁡n→∞inf⁡‖xn-x0‖≤lim⁡n→∞inf⁡‖xn-x‖,
for every x∈X. Opial proved in [7] that the sequence space lp(1<p<∞) have this property but Lp[0,π](p≠2,1<p<∞) do not have it. A Banach space X is said to have the uniform Opial property (see [8]), if for each ɛ>0 there exists τ>0 such that for any weakly null sequence (xn) in S(X) and x∈X with ∥x∥≥ɛ there holds
(1.3)1+τ≤lim⁡n→∞inf⁡‖xn+x‖.
For example, the space in [9, 10] has the uniform Opial property. The Opial property is important because Banach spaces with this property have the weak fixed point property (see [11]) and the geometric property involving fixed point theory can be found, for example, in [9, 12–14].For a bounded subsetA⊂X, the set measure of noncompactness was defined in [15] by
(1.4)α(A)=inf⁡{ɛ>0:Acanbecoveredbyfinitelymanysetsofdiameter≤ɛ}.
The ball measure of noncompactness was defined in [16, 17] by
(1.5)β(A)=inf⁡{ɛ>0:Acanbecoveredbyfinitelymanyballsofdiameter≤ɛ}.
The functions α and β are called the Kuratowski measure of noncompactness and the Hausdorff measure of noncompactness in X, respectively. We can associate these functions with the notions of the set-contraction and ball contraction (see [18]). These notions are very useful tools to study nonlinear operator propblems (see [8, 18]). For each ɛ>0 define that Δ(ɛ)=inf⁡{1-inf⁡[∥x∥:x∈A]:A is closed convex subset of B(X) with β(A)≥ɛ}. The function Δ is called the modulus of noncompact convexity (see [16]). A Banach space X is said to have property (L) if lim⁡ɛ→1-Δ(ɛ)=1. It has been proved in [8] that property (L) is a useful tool in the fixed point theory and that a Banach space X has property (L) if and only if it is reflexive and has the uniform Opial property.For a real vector spaceX, a function ρ:X→[0,∞] is called a modular if it satisfies the following conditions:(i)
ρ(x)=0 if and only if x=0;(ii)
ρ(αx)=ρ(x) for all scalar α with |α|=1;(iii)
ρ(αx+βy)≤ρ(x)+ρ(y), for all x,y∈X and all α,β≥0 with α+β=1;the modularρ is called convex if(iv)
ρ(αx+βy)≤αρ(x)+βρ(y), for all x,y∈X and all α,β≥0 with α+β=1.For modularρ on X, the space
(1.6)Xρ={x∈X:ρ(λx)⟶0asλ⟶0+}
is called the modular space.A sequence(xn) in Xρ is called modular convergentto x∈Xρ if there exists a λ>0 such that ρ(λ(xn-x))→0 as n→∞.A modularρ is said to satisfy the Δ2-condition(ρ∈Δ2) if for any ɛ>0 there exist constants K≥2 and a>0 such that
(1.7)ρ(2u)≤Kρ(u)+ɛ
for all u∈Xρ with ρ(u)≤a.Ifρ satisfies the Δ2-condition for any a>0 with K≥2 dependent on a, we say that ρ satisfies the strongΔ2-condition(ρ∈Δ2s).By a lacunary sequenceθ=(kr), where k0=0, we will mean an increasing sequence of nonnegative integers with kr-kr-1→∞ as r→∞. The intervals determined by θ will be denote by Ir=(kr-1,kr]. We write hr=kr-kr-1 and the ratio kr/kr-1, will be denoted by qr. The space of lacunary strongly convergent sequence Nθ was defined by Freedman et al. [19] as
(1.8)Nθ={x=(xk):lim⁡r→∞1hr∑k∈Ir|xk-l|=0,forsomel}.
It is well known that there is very closed connection between the space of lacunary strongly convergent sequence and the space of strongly Cesaro summability sequences. This connection can be found in [18–23], because a lot of these connection, a lot of geometric property of Cesaro sequence spaces can generalize the lacunary sequence spaces.Letw be the space of all real sequences. Let p=(pr) be a bounded sequence of the positive real numbers. In 2007, Karakaya [24] introduced the new sequence spaces l(p,θ) involving lacunary sequence as follows:
(1.9)l(p,θ)={x=(x(i)):∑r=1∞(1hr∑i∈Ir|x(i)|)pr<∞}
and paranorm on l(p,θ) is given by
(1.10)‖x‖l(p,θ)=(∑r=1∞(1hr∑i∈Ir|x(i)|)pr)1/M,
where M=sup⁡rpr. If pr=p for all r∈ℕ, we will use the notation lp(θ) in place of l(p,θ). The norm on lp(θ) is given by
(1.11)‖x‖lp(θ)=(∑r=1∞(1hr∑i∈Ir|x(i)|)p)1/p.
By using the properties of lacunary sequence in the space l(p,θ), we get the following sequences. If θ=(2r), then l(p,θ)=ces(p). If θ=(2r) and pr=p for all r∈ℕ, then l(p,θ)=cesp. For x∈l(p,θ) defined the modular on l(p,θ) by
(1.12)ϱ(x)=∑r=1∞(1hr∑i∈Ir|x(i)|)pr.
It is easy to see that if lim⁡r→∞⁡sup⁡pr<∞ then ϱ∈Δ2s. The Luxembourg norm on l(p,θ) is defined by
(1.13)‖x‖=inf⁡{ɛ>0:ϱ(xɛ)≤1}.
The Luxembourg norm on lp(θ) can be reduced to a usual norm on lp(θ) [24], that is,
(1.14)‖x‖=‖x‖lp(θ).Throughout this paper, we assume thatlim⁡r→∞inf⁡pr>1 and lim⁡r→∞sup⁡pr<∞ and for x∈w, i∈ℕ, we denote
(1.15)ei=(0,0,…,0︷,i-1times1,0,0,0,…),x∣i=(x(1),x(2),x(3),…,x(i),0,0,0,…),x∣N-i=(0,0,0,…,x(i+1),x(i+2),…).
The following results are very important for our consideration.Lemma 1.1 (see [25, Lemma  2.1]).
Ifρ∈Δ2s, then for any L>0 and ɛ>0, there exists δ=δ(L,ɛ)>0 such that
(1.16)|ρ(u+v)-ρ(u)|<ɛ,
whenever u,v∈Xρ with ρ(u)≤L, and ρ(v)≤δ.Lemma 1.2 (see [25, Lemma  2.3]).
Convergence in norm and in modular are equivalent inXρ if ρ∈Δ2.Lemma 1.3 (see [25, Corollary  2.2, Lemma  2.3]).
Ifρ∈Δ2, then for any sequence (xn) in Xρ, ∥xn∥→0 if and only if ρ(xn)→0 as n→∞.Lemma 1.4 (see [25, Lemma  2.4]).
Ifρ∈Δ2s, then for any ɛ>0 there exists δ=δ(ɛ)>0 such that ∥x∥≥1+δ whenever ρ(x)≥1+ɛ.Lemma 1.5 (see [24, Lemma  2.3]).
The functionalϱ is a convex modular on l(p,θ).Lemma 1.6 (see [24, Lemma  2.5]).
(i) For anyx∈l(p,θ), if ∥x∥<1, then ϱ(x)≤∥x∥.
(ii) For anyx∈l(p,θ), ∥x∥=1 if and only if ϱ(x)=1.


## 2. The Main Results

In this section, we prove the property(β) and uniform Opial property in lacunary sequence and connect to the fixed point property. First we shall give some results which are very important for our consideration.Lemma 2.1.
For anyx∈l(p,θ), there exists k0∈ℕ and λ∈(0,1) such that ϱ(xk/2)≤((1-λ)/2)ϱ(xk) for all k∈ℕ with k≥k0, where
(2.1)xk=(0,0,…,0,︷k-1x(k),x(k+1),x(k+2),…).Proof.
Letk∈ℕ be fixed. So there exist rk∈ℕ such that k∈Irk. Let α be a real number such that 1<α≤lim⁡r→∞inf⁡pr, then there exists k0∈ℕ such that α<prk for all k≥k0. Choose λ∈(0,1) as a real such that (1/2)α≤(1-λ)/2. Then for each x∈l(p,θ) and k≥k0, we have
(2.2)ϱ(xk2)=∑r=rk∞(1hr∑i∈Ir|x(i)2|)pr=∑r=rk∞(12)pr(1hr∑i∈Ir|x(i)|)pr≤(12)α∑r=rk∞(1hr∑i∈Ir|x(i)|)pr≤1-λ2ϱ(xk).Lemma 2.2.
For anyx∈l(p,θ) and ɛ∈(0,1) there exists δ∈(0,1) such that ϱ(x)≤1-ɛ implies ∥x∥≤1-δ.Proof.
Suppose that the lemma does not hold, then there existɛ>0 and xn∈l(p,θ) such that ϱ(xn)≤1-ɛ and 1/2≤∥xn∥↗1. Let an=(1/∥xn∥)-1. Then an→0 as n→∞. Let L=sup⁡{ϱ(2xn):n∈ℕ}. Since ϱ∈Δ2s there exists K≥2 such that
(2.3)ϱ(2u)≤Kϱ(u)+1,
for every u∈l(p,θ) with ϱ(u)<1. By (2.3), we have ϱ(2xn)≤Kϱ(xn)+1≤K+1 for all n∈ℕ. Hence 0<L<∞. By Lemmas 1.5 and 1.6(ii), we have
(2.4)1=ϱ(xn‖xn‖)=ϱ(2anxn+(1-an)xn)≤anϱ(2xn)+(1-an)ϱ(xn)≤anL+(1-ɛ)⟶1-ɛ,
which is a contradiction.Theorem 2.3.
The spacel(p,θ) is Banach spaces with respect to the Luxemburg norm.Proof.
Let(xn)=(xn(i)) be a Cauchy sequence in l(p,θ) and ɛ∈(0,1). Thus there exists N∈ℕ such that ∥xn-xm∥<ɛM for all n,m≥N. By Lemma 1.6 (i), we have
(2.5)ϱ(xn-xm)≤‖xn-xm‖<ɛM∀n,m≥N.
That is,
(2.6)∑r=1∞(1hr∑i∈Ir|xn(i)-xm(i)|)pr<ɛM∀n,m≥N.
For fixed r, we get that
(2.7)|xn(i)-xm(i)|<ɛ∀n,m≥N.
Thus let (xn(i)) be a Cauchy sequence in ℝ for all i∈ℕ. Since ℝ is complete, then there exists x(i)∈ℝ such that xm(i)→x(i) as m→∞ for all i∈ℕ. Thus for fixed r, we have
(2.8)|xn(i)-x(i)|<ɛasm⟶∞,∀n≥N.
This implies that, for all n≥N,
(2.9)ϱ(xn-xm)⟶ϱ(xn-x)asm⟶∞.
This means that, for all n≥N,
(2.10)∑r=1∞(1hr∑i∈Ir|xn(i)-xm(i)|)pr⟶∑r=1∞(1hr∑i∈Ir|xn(i)-x(i)|)pr
as m→∞. By (2.6), we have
(2.11)ϱ(xn-x)≤‖xn-xm‖<ɛM≤ɛ∀n≥N.
This implies that xn→x as n→∞. So we have xN-x∈l(p,θ). By the linearity of the sequence space l(p,θ), we have x=(xN-x)+xN∈l(p,θ). Therefore the sequence space l(p,θ) is Banach space, with respect to the Luxemburg norm and the proof is complete.Theorem 2.4.
The spacel(p,θ) has property (β).Proof.
Letɛ>0 and (xn)⊂B(l(p,θ)) with sep(xn)≥ɛ. For each k∈ℕ, there exist rk∈ℕ such that k is a minimal element in Irk. Let
(2.12)xnk=(0,0,…,0,︷k-1xn(k),xn(k+1),xn(k+2),…).
Since for each i∈ℕ, (xn(i))n=1∞ is bounded. By using the diagonal method, we have that for each k∈ℕ we can find subsequence (xnj) of (xn) such that (xnj(i)) converges for each i∈ℕ. Therefore, for any k∈ℕ there exists an increasing sequence (tk) such that sep((xnjk)j>tk)≥ɛ. Hence for each k∈ℕ there exists sequence of positive integers (sk)k=1∞ with s1<s2<s3<⋯ such that ∥xskk∥≥ɛ/2, and since ϱ∈Δ2s, by Lemma 1.3 we may assume that there exists η>0 such that ϱ(xskk)≥η for all k∈ℕ, that is,
(2.13)∑r=rk∞(1hr∑i∈Ir|xskk(i)|)pr≥η
for all k∈ℕ. On the other hand by Lemma 2.1, there exist k0∈ℕ and λ∈(0,1) such that
(2.14)ϱ(uk2)≤1-λ2ϱ(uk)
for all u∈l(p,θ) and k≥k0. From Lemma 2.2, there exist δ∈(0,1) such that for any y∈l(p,θ),
(2.15)ϱ(y)≤1-λη4⟹‖y‖≤1-δ.
Since again ϱ∈Δ2s, by Lemma 1.1, there exists δ0 such that
(2.16)|ϱ(u+v)-ϱ(u)|<λη4,
whenever ϱ(u)≤1 and ϱ(v)≤δ0. Since x∈B(l(p,θ)), we have that ϱ(x)≤1. Then there exits k≥k0 such that ϱ(xk)≤δ0. We put u=xskk and v=xk,
(2.17)ϱ(u2)=∑r=rk∞(1hr∑i∈Ir|xsk(i)2|)pr<1,ϱ(v2)=∑r=rk∞(1hr∑i∈Ir|x(i)2|)pr<δ0.
From (2.14) and (2.16), we have
(2.18)∑r=rk∞(1hr∑i∈Ir|x(i)+xsk(i)2|)pr=ϱ(u+v2)≤ϱ(u2)+λη4≤1-λ2(ϱ(u))+λη4.
By (2.13), (2.16), (2.18), and convexity of function f(t)=|t|pr, for all r∈ℕ, we have
(2.19)ϱ(x+xsk2)=∑r=1∞(1hr∑i∈Ir|x(i)+xsk(i)2|)pr=∑r=1rk-1(1hr∑i∈Ir|x(i)+xsk(i)2|)pr+∑r=rk∞(1hr∑i∈Ir|x(i)+xsk(i)2|)pr≤12(∑r=1rk-1(1hr∑i∈Ir|x(i)|)pr+∑r=1rk-1(1hr∑i∈Ir|xsk(i)|)pr)+∑r=rk∞(1hr∑i∈Ir|xsk(i)2|)pr+λη4≤12(∑r=1rk-1(1hr∑i∈Ir|x(i)|)pr+∑r=1rk-1(1hr∑i∈Ir|xsk(i)|)pr)+1-λ2∑r=rk∞(1hr∑i∈Ir|xsk(i)|)pr+λη4=12∑r=1rk-1(1hr∑i∈Ir|x(i)|)pr+12∑r=1rk-1(1hr∑i∈Ir|xsk(i)|)pr+1-λ2∑r=rk∞(1hr∑i∈Ir|xsk(i)|)pr+λη4=12∑r=1rk-1(1hr∑i∈Ir|x(i)|)pr+12∑r=1∞(1hr∑i∈Ir|xsk(i)|)pr-λ2∑r=rk∞(1hr∑i∈Ir|xsk(i)|)pr+λη4≤12+12-λη2+λη4=1-λη4.
So it follows from (2.15) that
(2.20)‖x+xsk2‖≤1-δ.
Therefore, the space l(p,θ) has property (β).By the facts presented in the introduction, following results are obtained directly from Theorem2.4.Corollary 2.5.
The spacelp(θ) has property (β).Corollary 2.6.
The spacel(p,θ) is nearly uniform convexity, has drop property, and is reflexive.Corollary 2.7.
The spacel(p,θ) has property (UKK).Corollary 2.8 (see [24, Theorem  2.9]).
The spacel(p,θ) has property (H).Corollary 2.9.
The spacelp(θ) is nearly uniform convexity, has drop property, and is reflexive.Corollary 2.10.
The spacelp(θ) has property (UKK) and (H).Theorem 2.11.
The spacel(p,θ) has uniform Opial property.Proof.
Take anyɛ>0 and x∈l(p,θ) with ∥x∥≥ɛ. Let (xn) be weakly null sequence in S(l(p,θ)). By lim⁡r→∞⁡sup⁡pr<∞, that is, ϱ∈Δ2s, hence by Lemma 1.2 there exists δ∈(0,1) independent of x such that ϱ(x)>δ. Also, by ϱ∈Δ2s, Lemma 1.1 asserts that there exists δ1∈(0,δ) such that
(2.21)|ϱ(y+z)-ϱ(y)|<δ4,
whenever ϱ(y)≤1 and ϱ(z)≤δ1. Choose r0∈ℕ such that
(2.22)∑r=r0+1∞(1hr∑i∈Ir|x(i)|)pr<δ14.
So, we have
(2.23)δ<∑r=1r0(1hr∑i∈Ir|x(i)|)pr+∑r=r0+1∞(1hr∑i∈Ir|x(i)|)pr≤∑r=1r0(1hr∑i∈Ir|x(i)|)pr+δ14,
which implies that
(2.24)∑r=1r0(1hr∑i∈Ir|x(i)|)pr>δ-δ14>δ-δ4=3δ4.
Since xn→w0, then there exists n0∈ℕ such that
(2.25)3δ4≤∑r=1r0(1hr∑i∈Ir|xn(i)+x(i)|)pr
for all n>n0, since weak convergence implies coordinatewise convergence. Again, by xn→w0, then there exists n1∈ℕ such that
(2.26)‖xn∣k0‖<1-(1-δ4)1/M
for all n>n1 where k0 is a minimal element in Ir0+1 and M∈ℕ with pr≤M for all r∈ℕ. Hence, by the triangle inequality of the norm, we get
(2.27)‖xn∣N-k0‖>(1-δ4)1/M.
It follows by the definition of ∥·∥ that we have
(2.28)1<ϱ(xn∣N-k0(1-(δ/4))1/M)=∑r=r0+1∞((1/hr)∑i∈Ir|xn(i)|(1-(δ/4))1/M)pr≤(1(1-(δ/4))1/M)M∑r=r0+1∞(1hr∑i∈Ir|xn(i)|)pr
Which implies that
(2.29)∑r=r0+1∞(1hr∑i∈Ir|xn(i)|)pr>1-δ4
for all n>n1. By inequality (2.21), (2.25), and (2.29), it yields for any n>n1 that
(2.30)ϱ(xn+x)=∑r=1r0(1hr∑i∈Ir|xn(i)+x(i)|)pr+∑r=r0+1∞(1hr∑i∈Ir|xn(i)+x(i)|)pr≥3δ4+∑r=r0+1∞(1hr∑i∈Ir|xn(i)|)pr-δ4≥3δ4+(1-δ4)-δ4≥1+δ4.
Since ϱ∈Δ2s and by Lemma 1.4 there exists τ depending on δ only such that ∥xn+x∥≥1+τ, which implies that lim⁡n→∞⁡inf⁡∥xn+x∥≥1+τ, hence the prove is complete.By the facts presented in the introduction and the reflexivity ofl(p,θ), we get the following results.Corollary 2.12.
The spacelp(θ) has uniform Opial property.Corollary 2.13.
The spacel(p,θ) has property (L) and the fixed point property.Corollary 2.14.
The spacelp(θ) has property (L) and the fixed point property.



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