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

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.

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 𝜀>0 there exists 𝛿>0 such that for 𝑥,𝑦∈𝑆(𝑋) the inequality ‖𝑥−𝑦‖>𝜀 implies ‖(𝑥+𝑦)/2‖<1−𝛿 (see [1]). A Banach space 𝑋 has the property (𝛽) if for each 𝜀>0 there exists 𝛿>0 such that 1<‖𝑥‖<1+𝛿 implies 𝛼(conv(𝐵(𝑋)∪{𝑥})⧵𝐵(𝑋))<𝜀, where 𝛼(𝐴) denotes the Kuratowski measure noncompactness of a subset 𝐴 of 𝑋 defined as the infimum of all 𝜀>0 such that 𝐴 can be covered by a finite union of sets of diameter less than 𝜀. 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 𝜀>0 there exists 𝛿>0 such that for each element 𝑥∈𝐵(𝑋) and each sequence (𝑥_𝑛) in 𝐵(𝑋) with sep(𝑥_𝑛)≥𝜀 there is an index 𝑘 for which ‖(𝑥+𝑥_𝑘)/2‖<1−𝛿 where sep(𝑥_𝑛)=inf{‖𝑥_𝑛−𝑥_𝑚‖∶𝑛≠𝑚}>𝜀. A Banach space 𝑋 is nearly uniformly convex (NUC) if for each 𝜀>0 and every sequence (𝑥_𝑛) in 𝐵(𝑋) with sep(𝑥_𝑛)≥𝜀, there exists 𝛿∈(0,1) such that conv(𝑥_𝑛)∩(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 (𝑥_𝑛) in 𝑋 such that ‖𝑥_𝑛‖→1 as 𝑛→∞, the week convergence of (𝑥_𝑛) to 𝑥 implies that ‖𝑥_𝑛−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 𝜀>0 there exists 𝛿>0 such that for every sequence (𝑥_𝑛) in 𝑆(𝑋) with sep(𝑥_𝑛)≥𝜀 and 𝑥−→_𝑛^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 (𝑥_𝑛) weakly convergent to 𝑥₀ satisfies lim_𝑛→∞inf𝑥_𝑛−𝑥₀≤lim_𝑛→∞inf𝑥_𝑛−𝑥,(1.2) for every 𝑥∈𝑋. Opial proved in [7] that the sequence space 𝑙_𝑝 (1<𝑝<∞) have this property but 𝐿_𝑝[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 𝜀>0 there exists 𝜏>0 such that for any weakly null sequence (𝑥_𝑛) in 𝑆(𝑋) and 𝑥∈𝑋 with ‖𝑥‖≥𝜀 there holds 1+𝜏≤lim_𝑛→∞inf𝑥_𝑛+𝑥.(1.3) 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 subset 𝐴⊂𝑋, the set measure of noncompactness was defined in [15] by 𝛼(𝐴)=inf𝜀>0∶𝐴canbecoveredbyfinitelymanysetsofdiameter≤𝜀.(1.4) The ball measure of noncompactness was defined in [16, 17] by 𝛽(𝐴)=inf𝜀>0∶𝐴canbecoveredbyfinitelymanyballsofdiameter≤𝜀.(1.5) 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 𝜀>0 define that Δ(𝜀)=inf{1−inf[‖𝑥‖∶𝑥∈𝐴]:𝐴 is closed convex subset of 𝐵(𝑋) with 𝛽(𝐴)≥𝜀}. The function Δ is called the modulus of noncompact convexity (see [16]). A Banach space 𝑋 is said to have property (𝐿) if lim𝜀_→1⁻Δ(𝜀)=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 space 𝑋, 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 modular 𝜌 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 modular 𝜌 on 𝑋, the space 𝑋_𝜌=𝑥∈𝑋∶𝜌(𝜆𝑥)⟶0as𝜆⟶0⁺(1.6) is called the modular space.

A sequence (𝑥_𝑛) in 𝑋_𝜌 is called modular convergent to 𝑥∈𝑋_𝜌 if there exists a 𝜆>0 such that 𝜌(𝜆(𝑥_𝑛−𝑥))→0 as 𝑛→∞.

A modular 𝜌 is said to satisfy the Δ₂-𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛(𝜌∈Δ₂) if for any 𝜀>0 there exist constants 𝐾≥2 and 𝑎>0 such that 𝜌(2𝑢)≤𝐾𝜌(𝑢)+𝜀(1.7) for all 𝑢∈𝑋_𝜌 with 𝜌(𝑢)≤𝑎.

If 𝜌 satisfies the Δ₂-𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 for any 𝑎>0 with 𝐾≥2 dependent on 𝑎, we say that 𝜌 satisfies the strong Δ₂-𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛(𝜌∈Δ^𝑠₂).

By a lacunary sequence 𝜃=(𝑘_𝑟), where 𝑘₀=0, we will mean an increasing sequence of nonnegative integers with 𝑘_𝑟−𝑘_𝑟−1→∞ as 𝑟→∞. The intervals determined by 𝜃 will be denote by 𝐼_𝑟=(𝑘_𝑟−1,𝑘_𝑟]. We write ℎ_𝑟=𝑘_𝑟−𝑘_𝑟−1 and the ratio 𝑘_𝑟/𝑘_𝑟−1, will be denoted by 𝑞_𝑟. The space of lacunary strongly convergent sequence 𝑁_𝜃 was defined by Freedman et al. [19] as 𝑁_𝜃=𝑥=𝑥_𝑘:lim_𝑟→∞1ℎ_{𝑟𝑘∈𝐼𝑟}𝑥_𝑘−𝑙=0,forsome𝑙.(1.8) 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.

Let 𝑤 be the space of all real sequences. Let 𝑝=(𝑝_𝑟) 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: 𝑙𝑝,𝜃=𝑥=(𝑥(𝑖))∶^∞_𝑟=11ℎ_{𝑟𝑖∈𝐼𝑟}|𝑥(𝑖)|^𝑝𝑟<∞(1.9) and paranorm on 𝑙(𝑝,𝜃) is given by ‖𝑥‖_{𝑙(𝑝,𝜃)}=^∞_𝑟=11ℎ_{𝑟𝑖∈𝐼𝑟}|𝑥(𝑖)|^{𝑝𝑟1/𝑀},(1.10) where 𝑀=sup_𝑟𝑝_𝑟. If 𝑝_𝑟=𝑝 for all 𝑟∈ℕ, we will use the notation 𝑙_𝑝(𝜃) in place of 𝑙(𝑝,𝜃). The norm on 𝑙_𝑝(𝜃) is given by ‖𝑥‖_{𝑙𝑝(𝜃)}=^∞_𝑟=11ℎ_{𝑟𝑖∈𝐼𝑟}|𝑥(𝑖)|^𝑝1/𝑝.(1.11) By using the properties of lacunary sequence in the space 𝑙(𝑝,𝜃), we get the following sequences. If 𝜃=(2^𝑟), then 𝑙(𝑝,𝜃)=ces(𝑝). If 𝜃=(2^𝑟) and 𝑝_𝑟=𝑝 for all 𝑟∈ℕ, then 𝑙(𝑝,𝜃)=ces_𝑝. For 𝑥∈𝑙(𝑝,𝜃) defined the modular on 𝑙(𝑝,𝜃) by 𝜚(𝑥)=^∞_𝑟=11ℎ_{𝑟𝑖∈𝐼𝑟}|𝑥(𝑖)|^𝑝𝑟.(1.12) It is easy to see that if lim_𝑟→∞sup𝑝_𝑟<∞ then 𝜚∈Δ^𝑠₂. The Luxembourg norm on 𝑙(𝑝,𝜃) is defined by ‖𝑥‖𝑥=inf𝜀>0∶𝜚𝜀≤1.(1.13) The Luxembourg norm on 𝑙_𝑝(𝜃) can be reduced to a usual norm on 𝑙_𝑝(𝜃) [24], that is, ‖𝑥‖=‖𝑥‖_{𝑙𝑝(𝜃)}.(1.14)

Throughout this paper, we assume that lim_𝑟→∞inf𝑝_𝑟>1 and lim_𝑟→∞sup𝑝_𝑟<∞ and for 𝑥∈𝑤, 𝑖∈ℕ, we denote 𝑒_𝑖=0,0,…^{𝑖−1times0𝑥0𝑓𝑒37},0,1,0,0,0,…,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),…).(1.15) The following results are very important for our consideration.

Lemma 1.1 (see [25, Lemma  2.1]). If 𝜌∈Δ^𝑠₂, then for any 𝐿>0 and 𝜀>0, there exists 𝛿=𝛿(𝐿,𝜀)>0 such that 𝜌(𝑢+𝑣)−𝜌(𝑢)<𝜀,(1.16) whenever 𝑢,𝑣∈𝑋_𝜌 with 𝜌(𝑢)≤𝐿, and 𝜌(𝑣)≤𝛿.

Lemma 1.2 (see [25, Lemma  2.3]). Convergence in norm and in modular are equivalent in 𝑋_𝜌 if 𝜌∈Δ₂.

Lemma 1.3 (see [25, Corollary  2.2, Lemma  2.3]). If 𝜌∈Δ₂, then for any sequence (𝑥_𝑛) in 𝑋_𝜌, ‖𝑥_𝑛‖→0 if and only if 𝜌(𝑥_𝑛)→0 as 𝑛→∞.

Lemma 1.4 (see [25, Lemma  2.4]). If 𝜌∈Δ^𝑠₂, then for any 𝜀>0 there exists 𝛿=𝛿(𝜀)>0 such that ‖𝑥‖≥1+𝛿 whenever 𝜌(𝑥)≥1+𝜀.

Lemma 1.5 (see [24, Lemma  2.3]). The functional 𝜚 is a convex modular on 𝑙(𝑝,𝜃).

Lemma 1.6 (see [24, Lemma  2.5]). (i) For any 𝑥∈𝑙(𝑝,𝜃), if ‖𝑥‖<1, then 𝜚(𝑥)≤‖𝑥‖.
(ii) For any 𝑥∈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 any 𝑥∈𝑙(𝑝,𝜃), there exists 𝑘₀∈ℕ and 𝜆∈(0,1) such that 𝜚(𝑥^𝑘/2)≤((1−𝜆)/2)𝜚(𝑥^𝑘) for all 𝑘∈ℕ with 𝑘≥𝑘₀, where 𝑥^𝑘=0,0,…,^{𝑘−10𝑥0𝑓𝑒37}0,𝑥(𝑘),𝑥(𝑘+1),𝑥(𝑘+2),….(2.1)

Proof. Let 𝑘∈ℕ be fixed. So there exist 𝑟_𝑘∈ℕ such that 𝑘∈𝐼_𝑟𝑘. Let 𝛼 be a real number such that 1<𝛼≤lim_𝑟→∞inf𝑝_𝑟, then there exists 𝑘₀∈ℕ such that 𝛼<𝑝_𝑟𝑘 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 𝜚𝑥^𝑘2=^∞_{𝑟=𝑟𝑘}1ℎ_{𝑟𝑖∈𝐼𝑟}𝑥(𝑖)2^𝑝𝑟=^∞_{𝑟=𝑟𝑘}12^𝑝𝑟1ℎ_{𝑟𝑖∈𝐼𝑟}|𝑥(𝑖)|^𝑝𝑟≤12^𝛼∞_{𝑟=𝑟𝑘}1ℎ_{𝑟𝑖∈𝐼𝑟}|𝑥(𝑖)|^𝑝𝑟≤1−𝜆2𝜚𝑥^𝑘.(2.2)

Lemma 2.2. For any 𝑥∈𝑙(𝑝,𝜃) and 𝜀∈(0,1) there exists 𝛿∈(0,1) such that 𝜚(𝑥)≤1−𝜀 implies ‖𝑥‖≤1−𝛿.

Proof. Suppose that the lemma does not hold, then there exist 𝜀>0 and 𝑥_𝑛∈𝑙(𝑝,𝜃) such that 𝜚(𝑥_𝑛)≤1−𝜀 and 1/2≤‖𝑥_𝑛‖↗1. Let 𝑎_𝑛=(1/‖𝑥_𝑛‖)−1. Then 𝑎_𝑛→0 as 𝑛→∞. Let 𝐿=sup{𝜚(2𝑥_𝑛)∶𝑛∈ℕ}. Since 𝜚∈Δ^𝑠₂ there exists 𝐾≥2 such that 𝜚(2𝑢)≤𝐾𝜚(𝑢)+1,(2.3) for every 𝑢∈𝑙(𝑝,𝜃) with 𝜚(𝑢)<1. By (2.3), we have 𝜚(2𝑥_𝑛)≤𝐾𝜚(𝑥_𝑛)+1≤𝐾+1 for all 𝑛∈ℕ. Hence 0<𝐿<∞. By Lemmas 1.5 and 1.6(ii), we have 𝑥1=𝜚^𝑛𝑥_𝑛=𝜚2𝑎_𝑛𝑥_𝑛+1−𝑎_𝑛𝑥_𝑛≤𝑎_𝑛𝜚2𝑥_𝑛+1−𝑎_𝑛𝜚𝑥_𝑛≤𝑎_𝑛𝐿+(1−𝜀)⟶1−𝜀,(2.4) which is a contradiction.

Theorem 2.3. The space 𝑙(𝑝,𝜃) is Banach spaces with respect to the Luxemburg norm.

Proof. Let (𝑥_𝑛)=(𝑥_𝑛(𝑖)) be a Cauchy sequence in 𝑙(𝑝,𝜃) and 𝜀∈(0,1). Thus there exists 𝑁∈ℕ such that ‖𝑥_𝑛−𝑥_𝑚‖<𝜀^𝑀 for all 𝑛,𝑚≥𝑁. By Lemma 1.6 (i), we have 𝜚𝑥_𝑛−𝑥_𝑚≤𝑥_𝑛−𝑥_𝑚<𝜀^𝑀∀𝑛,𝑚≥𝑁.(2.5) That is, ^∞_𝑟=11ℎ_{𝑟𝑖∈𝐼𝑟}𝑥_𝑛(𝑖)−𝑥_𝑚(𝑖)^𝑝𝑟<𝜀^𝑀∀𝑛,𝑚≥𝑁.(2.6) For fixed 𝑟, we get that 𝑥_𝑛(𝑖)−𝑥_𝑚(𝑖)<𝜀∀𝑛,𝑚≥𝑁.(2.7) Thus let (𝑥_𝑛(𝑖)) be a Cauchy sequence in ℝ for all 𝑖∈ℕ. Since ℝ is complete, then there exists 𝑥(𝑖)∈ℝ such that 𝑥_𝑚(𝑖)→𝑥(𝑖) as 𝑚→∞ for all 𝑖∈ℕ. Thus for fixed 𝑟, we have 𝑥_𝑛(𝑖)−𝑥(𝑖)<𝜀as𝑚⟶∞,∀𝑛≥𝑁.(2.8) This implies that, for all 𝑛≥𝑁, 𝜚𝑥_𝑛−𝑥_𝑚⟶𝜚𝑥_𝑛−𝑥as𝑚⟶∞.(2.9) This means that, for all 𝑛≥𝑁, ^∞_𝑟=11ℎ_{𝑟𝑖∈𝐼𝑟}𝑥_𝑛(𝑖)−𝑥_𝑚(𝑖)^𝑝𝑟⟶^∞_𝑟=11ℎ_{𝑟𝑖∈𝐼𝑟}𝑥_𝑛(𝑖)−𝑥(𝑖)^𝑝𝑟(2.10) as 𝑚→∞. By (2.6), we have 𝜚𝑥_𝑛−𝑥≤𝑥_𝑛−𝑥_𝑚<𝜀^𝑀≤𝜀∀𝑛≥𝑁.(2.11) This implies that 𝑥_𝑛→𝑥 as 𝑛→∞. So we have 𝑥_𝑁−𝑥∈𝑙(𝑝,𝜃). By the linearity of the sequence space 𝑙(𝑝,𝜃), we have 𝑥=(𝑥_𝑁−𝑥)+𝑥_𝑁∈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 space 𝑙(𝑝,𝜃) has property (𝛽).

Proof. Let 𝜀>0 and (𝑥_𝑛)⊂𝐵(𝑙(𝑝,𝜃)) with sep(𝑥_𝑛)≥𝜀. For each 𝑘∈ℕ, there exist 𝑟_𝑘∈ℕ such that 𝑘 is a minimal element in 𝐼_𝑟𝑘. Let 𝑥^𝑘_𝑛=0,0,…,^{𝑘−10𝑥0𝑓𝑒37}0,𝑥_𝑛(𝑘),𝑥_𝑛(𝑘+1),𝑥_𝑛(𝑘+2),….(2.12) Since for each 𝑖∈ℕ, (𝑥_𝑛(𝑖))^∞_𝑛=1 is bounded. By using the diagonal method, we have that for each 𝑘∈ℕ we can find subsequence (𝑥_𝑛𝑗) of (𝑥_𝑛) such that (𝑥_𝑛𝑗(𝑖)) converges for each 𝑖∈ℕ. Therefore, for any 𝑘∈ℕ there exists an increasing sequence (𝑡_𝑘) such that sep((𝑥^𝑘_𝑛𝑗)_{𝑗>𝑡𝑘})≥𝜀. Hence for each 𝑘∈ℕ there exists sequence of positive integers (𝑠_𝑘)^∞_𝑘=1 with 𝑠₁<𝑠₂<𝑠₃<⋯ such that ‖𝑥^𝑘_𝑠𝑘‖≥𝜀/2, and since 𝜚∈Δ^𝑠₂, by Lemma 1.3 we may assume that there exists 𝜂>0 such that 𝜚(𝑥^𝑘_𝑠𝑘)≥𝜂 for all 𝑘∈ℕ, that is, ^∞_{𝑟=𝑟𝑘}1ℎ_{𝑟𝑖∈𝐼𝑟}𝑥^𝑘_𝑠𝑘(𝑖)^𝑝𝑟≥𝜂(2.13) for all 𝑘∈ℕ. On the other hand by Lemma 2.1, there exist 𝑘₀∈ℕ and 𝜆∈(0,1) such that 𝜚𝑢^𝑘2≤1−𝜆2𝜚𝑢^𝑘(2.14) 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), 𝜚𝑦≤1−𝜆𝜂4⟹𝑦≤1−𝛿.(2.15) Since again 𝜚∈Δ^𝑠₂, by Lemma 1.1, there exists 𝛿₀ such that 𝜚(𝑢+𝑣)−𝜚(𝑢)<𝜆𝜂4,(2.16) 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(𝑥^𝑘)≤𝛿₀. We put 𝑢=𝑥^𝑘_𝑠𝑘 and 𝑣=𝑥^𝑘, 𝜚𝑢2=^∞_{𝑟=𝑟𝑘}1ℎ_{𝑟𝑖∈𝐼𝑟}𝑥_𝑠𝑘(𝑖)2^𝑝𝑟𝑣<1,𝜚2=^∞_{𝑟=𝑟𝑘}1ℎ_{𝑟𝑖∈𝐼𝑟}𝑥(𝑖)2^𝑝𝑟<𝛿₀.(2.17) From (2.14) and (2.16), we have ^∞_{𝑟=𝑟𝑘}1ℎ_{𝑟𝑖∈𝐼𝑟}𝑥(𝑖)+𝑥_𝑠𝑘(𝑖)2^𝑝𝑟=𝜚𝑢+𝑣2𝑢≤𝜚2+𝜆𝜂4≤1−𝜆2𝜚(𝑢)+𝜆𝜂4.(2.18) By (2.13), (2.16), (2.18), and convexity of function 𝑓(𝑡)=|𝑡|^𝑝𝑟, for all 𝑟∈ℕ, we have 𝜚𝑥+𝑥_𝑠𝑘2=^∞_𝑟=11ℎ_{𝑟𝑖∈𝐼𝑟}𝑥(𝑖)+𝑥_𝑠𝑘(𝑖)2^𝑝𝑟=^𝑟𝑘−1_𝑟=11ℎ_{𝑟𝑖∈𝐼𝑟}𝑥(𝑖)+𝑥_𝑠𝑘(𝑖)2^𝑝𝑟+^∞_{𝑟=𝑟𝑘}1ℎ_{𝑟𝑖∈𝐼𝑟}𝑥(𝑖)+𝑥_𝑠𝑘(𝑖)2^𝑝𝑟≤12^𝑟𝑘−1_𝑟=11ℎ_{𝑟𝑖∈𝐼𝑟}|𝑥(𝑖)|^𝑝𝑟+^𝑟𝑘−1_𝑟=11ℎ_{𝑟𝑖∈𝐼𝑟}𝑥_𝑠𝑘(𝑖)^𝑝𝑟+^∞_{𝑟=𝑟𝑘}1ℎ_{𝑟𝑖∈𝐼𝑟}𝑥_𝑠𝑘(𝑖)2^𝑝𝑟+𝜆𝜂4≤12^𝑟𝑘−1_𝑟=11ℎ_{𝑟𝑖∈𝐼𝑟}|𝑥(𝑖)|^𝑝𝑟+^𝑟𝑘−1_𝑟=11ℎ_{𝑟𝑖∈𝐼𝑟}𝑥_𝑠𝑘(𝑖)^𝑝𝑟+1−𝜆2^∞_{𝑟=𝑟𝑘}1ℎ_{𝑟𝑖∈𝐼𝑟}𝑥_𝑠𝑘(𝑖)^𝑝𝑟+𝜆𝜂4=12^𝑟𝑘−1_𝑟=11ℎ_{𝑟𝑖∈𝐼𝑟}|𝑥(𝑖)|^𝑝𝑟+12^𝑟𝑘−1_𝑟=11ℎ_{𝑟𝑖∈𝐼𝑟}𝑥_𝑠𝑘(𝑖)^𝑝𝑟+1−𝜆2^∞_{𝑟=𝑟𝑘}1ℎ_{𝑟𝑖∈𝐼𝑟}𝑥_𝑠𝑘(𝑖)^𝑝𝑟+𝜆𝜂4=12^𝑟𝑘−1_𝑟=11ℎ_{𝑟𝑖∈𝐼𝑟}|𝑥(𝑖)|^𝑝𝑟+12^∞_𝑟=11ℎ_{𝑟𝑖∈𝐼𝑟}𝑥_𝑠𝑘(𝑖)^𝑝𝑟−𝜆2^∞_{𝑟=𝑟𝑘}1ℎ_{𝑟𝑖∈𝐼𝑟}𝑥_𝑠𝑘(𝑖)^𝑝𝑟+𝜆𝜂4≤12+12−𝜆𝜂2+𝜆𝜂4=1−𝜆𝜂4.(2.19) So it follows from (2.15) that 𝑥+𝑥_𝑠𝑘2≤1−𝛿.(2.20) Therefore, the space 𝑙(𝑝,𝜃) has property (𝛽).

By the facts presented in the introduction, following results are obtained directly from Theorem 2.4.

Corollary 2.5. The space 𝑙_𝑝(𝜃) has property (𝛽).

Corollary 2.6. The space 𝑙(𝑝,𝜃) is nearly uniform convexity, has drop property, and is reflexive.

Corollary 2.7. The space 𝑙(𝑝,𝜃) has property (UKK).

Corollary 2.8 (see [24, Theorem  2.9]). The space 𝑙(𝑝,𝜃) has property (𝐻).

Corollary 2.9. The space 𝑙_𝑝(𝜃) is nearly uniform convexity, has drop property, and is reflexive.

Corollary 2.10. The space 𝑙_𝑝(𝜃) has property (UKK) and (𝐻).

Theorem 2.11. The space 𝑙(𝑝,𝜃) has uniform Opial property.

Proof. Take any 𝜀>0 and 𝑥∈𝑙(𝑝,𝜃) with ‖𝑥‖≥𝜀. Let (𝑥_𝑛) be weakly null sequence in 𝑆(𝑙(𝑝,𝜃)). By lim_𝑟→∞sup𝑝_𝑟<∞, that is, 𝜚∈Δ^𝑠₂, hence by Lemma 1.2 there exists 𝛿∈(0,1) independent of 𝑥 such that 𝜚(𝑥)>𝛿. Also, by 𝜚∈Δ^𝑠₂, Lemma 1.1 asserts that there exists 𝛿₁∈(0,𝛿) such that 𝜚𝑦<𝛿𝑦+𝑧−𝜚4,(2.21) whenever 𝜚(𝑦)≤1 and 𝜚(𝑧)≤𝛿₁. Choose 𝑟₀∈ℕ such that ^∞_𝑟=𝑟0+11ℎ_{𝑟𝑖∈𝐼𝑟}|𝑥(𝑖)|^𝑝𝑟<𝛿₁4.(2.22) So, we have 𝛿<^𝑟0_𝑟=11ℎ_{𝑟𝑖∈𝐼𝑟}|𝑥(𝑖)|^𝑝𝑟+^∞_𝑟=𝑟0+11ℎ_{𝑟𝑖∈𝐼𝑟}|𝑥(𝑖)|^𝑝𝑟≤^𝑟0_𝑟=11ℎ_{𝑟𝑖∈𝐼𝑟}|𝑥(𝑖)|^𝑝𝑟+𝛿₁4,(2.23) which implies that ^𝑟0_𝑟=11ℎ_{𝑟𝑖∈𝐼𝑟}|𝑥(𝑖)|^𝑝𝑟𝛿>𝛿−¹4𝛿>𝛿−4=3𝛿4.(2.24) Since 𝑥−→_𝑛^𝑤0, then there exists 𝑛₀∈ℕ such that 3𝛿4≤^𝑟0_𝑟=11ℎ_{𝑟𝑖∈𝐼𝑟}𝑥_𝑛(𝑖)+𝑥(𝑖)^𝑝𝑟(2.25) for all 𝑛>𝑛₀, since weak convergence implies coordinatewise convergence. Again, by 𝑥−→_𝑛^𝑤0, then there exists 𝑛₁∈ℕ such that 𝑥_𝑛∣𝑘0𝛿<1−1−4^1/𝑀(2.26) for all 𝑛>𝑛₁ where 𝑘₀ is a minimal element in 𝐼_𝑟0+1 and 𝑀∈ℕ with 𝑝_𝑟≤𝑀 for all 𝑟∈ℕ. Hence, by the triangle inequality of the norm, we get 𝑥_{𝑛∣0ℕ−𝑘}>𝛿1−4^1/𝑀.(2.27) It follows by the definition of ‖⋅‖ that we have 𝑥1<𝜚^{𝑛∣0ℕ−𝑘}(1−(𝛿/4))^1/𝑀=^∞_𝑟=𝑟0+11/ℎ_𝑟∑_{𝑖∈𝐼𝑟}𝑥_𝑛(𝑖)(1−(𝛿/4))^{1/𝑀𝑝𝑟}≤1(1−(𝛿/4))^{1/𝑀𝑀∞}_𝑟=𝑟0+11ℎ_{𝑟𝑖∈𝐼𝑟}𝑥_𝑛(𝑖)^𝑝𝑟(2.28) Which implies that ^∞_𝑟=𝑟0+11ℎ_{𝑟𝑖∈𝐼𝑟}𝑥_𝑛(𝑖)^𝑝𝑟𝛿>1−4(2.29) for all 𝑛>𝑛₁. By inequality (2.21), (2.25), and (2.29), it yields for any 𝑛>𝑛₁ that 𝜚𝑥_𝑛+𝑥=^𝑟0_𝑟=11ℎ_{𝑟𝑖∈𝐼𝑟}𝑥_𝑛(𝑖)+𝑥(𝑖)^𝑝𝑟+^∞_𝑟=𝑟0+11ℎ_{𝑟𝑖∈𝐼𝑟}𝑥_𝑛(𝑖)+𝑥(𝑖)^𝑝𝑟≥3𝛿4+^∞_𝑟=𝑟0+11ℎ_{𝑟𝑖∈𝐼𝑟}𝑥_𝑛(𝑖)^𝑝𝑟−𝛿4≥3𝛿4+𝛿1−4−𝛿4𝛿≥1+4.(2.30) Since 𝜚∈Δ^𝑠₂ and by Lemma 1.4 there exists 𝜏 depending on 𝛿 only such that ‖𝑥_𝑛+𝑥‖≥1+𝜏, which implies that lim_𝑛→∞inf‖𝑥_𝑛+𝑥‖≥1+𝜏, hence the prove is complete.

By the facts presented in the introduction and the reflexivity of 𝑙(𝑝,𝜃), we get the following results.

Corollary 2.12. The space 𝑙_𝑝(𝜃) has uniform Opial property.

Corollary 2.13. The space 𝑙(𝑝,𝜃) has property (𝐿) and the fixed point property.

Corollary 2.14. The space 𝑙_𝑝(𝜃) has property (𝐿) and the fixed point property.

Acknowledgments

The authors are grateful to the referees for their valuable comments. They also wish to thank the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission for financial support (under the Project no. 54000267). Mr. Chirasak Mongkolkeha was supported by the Thailand Research Fund through the Royal Golden Jubilee Program under Grant PHD/0029/2553 for Ph.D. program at KMUTT, Thailand. The second author was supported by the Commission on Higher Education, the Thailand Research Fund, and the King Mongkuts University of Technology Thonburi (Grant no. MRG5380044).