On H-property and uniform Opial property of generalized cesàro sequence spaces

Abstract

In this article, we define the generalized cesàro sequence spaces ces_(p)(q) and consider it equipped with the Luxemburg norm. We show that the spaces ces_(p)(q) has the H-property and Uniform Opial property. The results of this article, we improve and extend some results of Petrot and Suantai.

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. 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 X is said to have the Opial property (see [1]), if every weakly null sequence (x _n ) in X satisfies

$$\underset{n\to \infty }{\mathsf{\text{lim}}}\mathsf{\text{inf}}\left|\right|x_n\left|\right|\le \underset{n\to \infty }{\mathsf{\text{lim}}}\mathsf{\text{inf}}\left|\right|x_n-x\left|\right|,$$

for every x ∈ X \{0}. Opial proved in [1] 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 [2]), 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

$$1+\tau \le \underset{n\to \infty }{\mathsf{\text{lim}}}\mathsf{\text{inf}}\left|\right|x_n+x\left|\right|.$$

For example, the space in [3–5] have the uniform Opial property.

Let l⁰ be the space of all real sequences. For 1 ≤ p < ∞, the Cesàro sequence space (ces _p , for short) is defined by

$$\mathsf{\text{ce}}{\mathsf{\text{s}}}_p=\left\{x\in l^0:\sum _{n=1}^{\infty }{\left(\frac{1}{n}\sum _{i=0}^n\left|x\left(i\right)\right|\right)}^p<\infty \right\}$$

equipped with the norm

$$\left|\right|x\left|\right|={\left({\sum _{n=1}^{\infty }\left(\frac{1}{n}\sum _{i=1}^n\left|x\left(i\right)\right|\right)}^p\right)}^{\frac{1}{p}}$$

(1.1)

This space was first introduced by Shiue [6]. It is useful in the theory of matrix operators and others (see [7, 8]). Suantai [9, 10] defined the generalized Cesàro sequence space ces_(p)when p = (p _k ) is a bounded sequence of positive real numbers with p _k ≥ 1 for all k ∈ ℕ by

$$\mathsf{\text{ce}}{\mathsf{\text{s}}}_{\left(p\right)}=\left\{x\in l^0:\varrho \left(\lambda x\right)<\infty \mathsf{\text{for some }}\lambda >0\right\},$$

where

$$\varrho \left(x\right)=\sum _{n=1}^{\infty }{\left(\frac{1}{n}\sum _{i=1}^k\left|x\left(i\right)\right|\right)}^{p_n}$$

equipped with the Luxemburg norm

$$\left|\right|x\left|\right|=\mathsf{\text{inf}}\left\{\epsilon >0:\varrho \left(\frac{x}{\epsilon }\right)\le 1\right\}.$$

In the case when p _k = p, 1 ≤ p < ∞ for all k ∈ ℕ, the generalized Cesàro sequence space ces_(p)is the Cesàro sequence space ces _p and the Luxemburg norm is expressed by the formula (1.1). Khan [11] defined the generalized Cesàro sequence space for 1 ≤ p < ∞ with q = q _k is a bounded sequence of positive real numbers by

$$\mathsf{\text{ce}}{\mathsf{\text{s}}}_p\left(q\right)=\left\{x\in l^0:{\left(\sum _{k=1}^{\infty }{\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_{i}x\left(i\right)\right|\right)}^p\right)}^{1/p}<\infty \right\},$$

where $Q_k=\sum _{k=1}^n{q}_k,$n ∈ ℕ. If q _k = 1 for all k ∈ ℕ, then ces _p (q) reduces to ces _p .

In this article, we define the generalized Cesàro sequence space for a bounded sequence p = (p _k ) and q = q _k of positive real numbers with p _k ≥ 1 and q _k ≥ 1 for all k ∈ ℕ by

$$\mathsf{\text{ce}}{\mathsf{\text{s}}}_{\left(p\right)}\left(q\right)=\left\{x\in l^0:\varrho \left(\lambda x\right)<\infty \mathsf{\text{for some }}\lambda >0\right\},$$

where

$$\varrho \left(x\right)=\sum _{k=1}^{\infty }{\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_{i}x\left(i\right)\right|\right)}^{p_k}$$

with $Q_k=\sum _{k=1}^n{q}_k$ and consider ces_(p)(q) equipped with the Luxemburg norm

$$\left|\right|x\left|\right|=\mathsf{\text{inf}}\left\{\epsilon >0:\varrho \left(\frac{x}{\epsilon }\right)\le 1\right\}.$$

Thus, we see that p _k = p, 1 ≤ p < ∞ for all k ∈ ℕ, then ces_(p)(q) reduces to ces _p (q) and if q _k = 1 for all k ∈ ℕ, then ces_(p)(q) reduces to ces_(p). Throughout this article, for x ∈ l⁰, i ∈ ℕ, we denote

$$\begin{array}{c}{e}_i=\left(\stackrel{i-1}{\overbrace{0,0,\dots ,0,}}1,0,0,0,\dots \right),\hfill \\ x{|}_i=\left(x\left(1\right),x\left(2\right),x\left(3\right),\dots ,x\left(i\right),0,0,0,\dots \right),\hfill \\ x{|}_{\mathbb{N}-i}=\left(0,0,0,\dots ,x\left(i+1\right),x\left(i+2\right),\dots \right),\hfill \end{array}$$

and M = sup _k p _k with p _k > 1 for all k ∈ ℕ. First, we start with a brief recollection of basic concepts and facts in modular space. 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

$$X_{\rho }=\left\{x\in X:\rho \left(\lambda x\right)\to 0\mathsf{\text{as}}\lambda \to 0^{+}\right\}$$

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 a constants K ≥ 2 and a > 0 such that

$$\rho \left(2u\right)\le K\rho \left(u\right)+\epsilon$$

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 the strong Δ₂-condition$\left(\rho \in {\text{Δ}}_2^s\right)$.

Lemma 1.1. [[12], Lemma 2.1] If$\rho \in {\text{Δ}}_2^s$, then for any L > 0 and ε > 0, there exists δ = δ(L, ε) > 0 such that

$$|\rho \left(u+v\right)-\rho \left(u\right)|<\epsilon ,$$

whenever u, v ∈ X _ρ with ρ(u) ≤ L, and ρ(v) ≤ δ.

Lemma 1.2. [[12], Lemma 2.3] Convergences in norm and in modular are equivalent in X _ρ if ρ ∈ Δ₂.

Lemma 1.3. [[12], Lemma 2.4] If$\rho \in {\text{Δ}}_2^s$, then for any ε > 0 there exists δ = δ(ε) > 0 such that || x || ≥ 1 + δ, whenever ρ(x) ≥ 1 + ε.

2. Main results

In this section, we prove the property H and uniform Opial property in generalized Cesàro sequence space ces_(p)(q). First, we give some results which are very important for our con-sideration.

Proposition 2.1. The functional ϱ is a convex modular on ces_(p)(q).

Proof. Let x, y ∈ ces_(p)(q). It is obvious that ϱ(x) = 0 if and only if x = 0 and ϱ(αx) = ϱ(x) for scalar α with |α| = 1. Let α ≥ 0, β ≥ 0 with α + β = 1. By the convexity of the function $t\mapsto |t{|}^{p_k}$, for all k ∈ ℕ, we have

$$\begin{array}{cc}\hfill \varrho \left(\alpha x+\beta y\right)& =\sum _{k=1}^{\infty }{\left(\frac{1}{Q_k}\sum _{i=1}^k|\alpha q_{i}x\left(i\right)+\beta q_{i}y\left(i\right)|\right)}^{p_k}\hfill \\ \le \sum _{k=1}^{\infty }{\left(\alpha \frac{1}{Q_k}\sum _{i=1}^k\left|q_{i}x\left(i\right)\right|+\beta \frac{1}{Q_k}\sum _{i=1}^k\left|q_{i}y\left(i\right)\right|\right)}^{p_k}\hfill \\ \le \alpha \sum _{k=1}^{\infty }{\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_{i}x\left(i\right)\right|\right)}^{p_k}+\beta \sum _{k=1}^{\infty }{\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_{i}y\left(i\right)\right|\right)}^{p_k}\hfill \\ =\alpha \varrho \left(x\right)+\beta \varrho \left(y\right).\hfill \end{array}$$

□

Proposition 2.2. For x ∈ ces_(p)(q), the modular ϱ on ces_(p)(q) satisfies the following properties:

• (i) if 0 < a < 1, then $a^M\varrho \left(\frac{x}{a}\right)\le \varrho \left(x\right)$ and ϱ(ax) ≤ aϱ(x);

• (ii) if a > 1, then ϱ(x) ≤ $a^M\varrho \left(\frac{x}{a}\right)$;

• (iii) if a ≥ 1, then ϱ(x) ≤ aϱ(x) ≤ ϱ(ax).

Proof. (i) Let 0 < a < 1. Then we have

$$\begin{array}{cc}\hfill \varrho \left(x\right)& =\sum _{k=1}^{\infty }{\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_{i}x\left(i\right)\right|\right)}^{p_k}\hfill \\ =\sum _{k=1}^{\infty }{\left(\frac{a}{Q_k}\sum _{i=1}^k\left|\frac{q_{i}x\left(i\right)}{a}\right|\right)}^{p_k}\hfill \\ =\sum _{k=1}^{\infty }{a^{p_k}\left(\frac{1}{Q_k}\sum _{i=1}^k\left|\frac{q_{i}x\left(i\right)}{a}\right|\right)}^{p_k}\hfill \\ \ge \sum _{k=1}^{\infty }{a^M\left(\frac{1}{Q_k}\sum _{i=1}^k\left|\frac{q_{i}x\left(i\right)}{a}\right|\right)}^{p_k}\hfill \\ =a^M\sum _{k=1}^{\infty }{\left(\frac{1}{Q_k}\sum _{i=1}^k\left|\frac{q_{i}x\left(i\right)}{a}\right|\right)}^{p_k}\hfill \\ =a^M\varrho \left(\frac{x}{a}\right).\hfill \end{array}$$

By convexity of modular ϱ, we have ϱ(ax) ≤ aϱ(x), so (i) is obtained.

(ii) Let a > 1. Then

$$\begin{array}{cc}\hfill \varrho \left(x\right)& =\sum _{k=1}^{\infty }{\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_{i}x\left(i\right)\right|\right)}^{p_k}\hfill \\ =\sum _{k=1}^{\infty }{a^{p_k}\left(\frac{1}{Q_k}\sum _{i=1}^k\left|\frac{q_{i}x\left(i\right)}{a}\right|\right)}^{p_k}\hfill \\ \le a^M\sum _{k=1}^{\infty }{\left(\frac{1}{Q_k}\sum _{i=1}^k\left|\frac{q_{i}x\left(i\right)}{a}\right|\right)}^{p_k}\hfill \\ =a^M\varrho \left(\frac{x}{a}\right).\hfill \end{array}$$

Hence (ii) is satisfies. (iii) follows from the convexity of ϱ. □

Proposition 2.3. For any x ∈ ces_(p)(q), we have

• (i) if ||x|| < 1, then ϱ(x) ≤ ||x||;

• (ii) if ||x|| > 1, then ϱ(x) ≥ ||x||;

• (iii) ||x|| = 1 if and only if ϱ(x) = 1;

• (iv) ||x|| < 1 if and only if ϱ(x) < 1;

• (v) ||x|| > 1 if and only if ϱ(x) > 1.

Proof. (i) Let ε > 0 be such that 0 < ε < 1 - ||x||, so ||x|| + ε < 1. By the definition of ||·||, then there exits λ > 0 such that ||x|| + ε > λ and $\varrho \left(\frac{x}{\lambda }\right)\le 1$. By (i) and (iii) of Proposition 2.2, we have

$$\begin{array}{cc}\hfill \varrho \left(x\right)& \le \varrho \left(\frac{\left(\left|\right|x\left|\right|+\epsilon \right)}{\lambda }x\right)\hfill \\ =\varrho \left(\left(\left|\right|x\left|\right|+\epsilon \right)\frac{x}{\lambda }\right)\hfill \\ \le \left(\left|\right|x\left|\right|+\epsilon \right)\varrho \left(\frac{x}{\lambda }\right)\hfill \\ \le \left|\right|x\left|\right|+\epsilon ,\hfill \end{array}$$

which implies that ϱ(x) ≤ ||x||. Hence (i) is satisfies.

(ii) Let ε > 0 such that $0<\epsilon <\frac{\left|\right|x\left|\right|-1}{\left|\right|x\left|\right|}$, then 0 < (1 - ε)||x|| ≤ ||x||. By definition of ||.|| and Proposition 2.2(i), we have $1<\varrho \left(\frac{x}{\left(1-\epsilon \right)\left|\right|x\left|\right|}\right)<\frac{x}{\left(1-\epsilon \right)\left|\right|x\left|\right|}\varrho \left(x\right)$, so (1 - ε)||x|| < ϱ(x) for all $\epsilon \in \left(0,\frac{\left|\right|x\left|\right|-1}{\left|\right|x\left|\right|}\right)$ which implies that ||x|| ≤ ϱ(x).

(iii) Assume that ||x|| = 1. Let ε > 0 then there exits λ > 0 such that 1 + ε > λ > ||x|| and $\varrho \left(\frac{x}{\lambda }\right)\le 1$. By Proposition 2.2(ii), we have $\varrho \left(x\right)\le {\lambda }^M\varrho \left(\frac{x}{\lambda }\right)\le {\lambda }^M<{\left(1+\epsilon \right)}^M$, so ${\left(\varrho \left(x\right)\right)}^{\frac{1}{M}}<1+\epsilon$ for all ε > 0 which implies that ϱ(x) ≤ 1. If ϱ(x) < 1, let a ∈ (0, 1) such that ϱ(x) < a^M < 1. From Proposition 2.2(i), we have $\varrho \left(\frac{x}{a}\right)\le \frac{1}{a^M}\varrho \left(x\right)<1$. Hence ||x|| ≤ a < 1, which is contradiction. Thus, we have ϱ(x) = 1.

Conversely, assume that ϱ(x) = 1. By definition of ||·||, we conclude that ||x|| ≤ 1. If ||x|| < 1, then we have by (i) that ϱ(x) ≤ ||x|| < 1, which is contradiction, so we obtain that ||x|| = 1. (iv) follows from (i) and (iii), (v) follows from (iii) and (iv). □

Proposition 2.4. For any x ∈ ces_(p)(q), we have

• (i) if 0 < a < 1 and ||x|| > a, then ϱ(x) > a^M;

• (ii) if a ≥ 1 and ||x|| < a, then ϱ(x) < a^M .

Proof. (i) Let 0 < a < 1 and ||x|| > a. Then $\left|\right|\frac{x}{a}\left|\right|>1$, by Proposition 2.3(v), we have $\varrho \left(\frac{x}{a}\right)>1$. Hence by Proposition 2.2(i), we have $\varrho \left(x\right)\ge a^M\varrho \left(\frac{x}{a}\right)>a^M$, so we obtain (i).

(ii) Suppose a ≥ 1 and ||x|| < a. Then $\left|\right|\frac{x}{a}\left|\right|<1$, by Proposition 2.3(iv), we have $\varrho \left(\frac{x}{a}\right)<1$.

If a = 1, it is obvious that ϱ(x) < 1 = a^M . If a > 1, then by Proposition 2.2(ii), we obtain that $\varrho \left(x\right)\le a^M\varrho \left(\frac{x}{a}\right)<a^M$. □

Proposition 2.5. Let (x _n ) be a sequence in ces_(p)(q).

• (i) If ||x _n || → 1 as n → ∞, then ϱ(x _n ) → 1 as n → ∞.

• (ii) If ϱ(x _n ) → 0 as n → ∞, then ||x _n || → 0 as n → ∞.

Proof. (i) Assume that ||x _n || → 1 as n → ∞. Let ε ∈ (0, 1). Then there exists N ∈ ℕ such that 1 - ε < ||x _n || < 1 + ε for all n ≥ N. By Proposition 2.4, we have (1 - ε) ^M < ϱ(x _n ) < (1 + ε) ^M for all n ≥ N, which implies that ϱ(x _n ) → 1 as n → ∞.

(ii) Suppose that ||x _n || ↛ 0 as n → ∞. Then there exists ε ∈ (0, 1) and a subsequence $\left(x_{n_k}\right)$ of (x _n ) such that $\left|\right|x_{n_k}\left|\right|>\epsilon$ for all k ∈ ℕ. By Proposition 2.4(i) we obtain $\varrho \left(x_{n_k}\right)>{\left(\epsilon \right)}^M$ for all k ∈ ℕ. This implies that ϱ(x _n ) ↛ 0 as n → ∞. □

Lemma 2.6. Let x ∈ ces_(p)(q) and (x _n ) ⊆ ces_(p)(q). If ϱ(x _n ) → ϱ(x) as n → ∞ and x _n (i) → x(i) as n → ∞ for all i ∈ ℕ, then x _n → x as n → ∞.

Proof. Let ε > 0 be given. Since $\varrho \left(x\right)={\sum }_{k=1}^{\infty }{\left(\frac{1}{Q_k}{\sum }_{i=1}^k\left|q_i{x}_n\left(i\right)\right|\right)}^{p_k}<\infty$, there exists k₀ ∈ ℕ such that

$$\sum _{k=k_0+1}^{\infty }{\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_i{x}_n\left(i\right)\right|\right)}^{p_k}<\frac{\epsilon }{3\cdot 2^{M+1}}.$$

(2.1)

Since $\varrho \left(x_n\right)-{\sum }_{k=1}^{k_0}{\left(\frac{1}{Q_k}{\sum }_{i=1}^k\left|q_i{x}_n\left(i\right)\right|\right)}^{p_k}\to \varrho \left(x\right)-{\sum }_{k=1}^{k_0}{\left(\frac{1}{Q_k}{\sum }_{i=1}^k\left|q_{i}x\left(i\right)\right|\right)}^{p_k}$ and x _n (i) → x(i) as n → ∞ for all i ∈ ℕ there exists n₀ ∈ ℕ such that

$$\varrho \left(x_n\right)-\sum _{k=1}^{k_0}{\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_i{x}_n\left(i\right)\right|\right)}^{p_k}<\varrho \left(x\right)-\sum _{k=1}^{k_0}{\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_{i}x\left(i\right)\right|\right)}^{p_k}+\frac{\epsilon }{3\cdot 2^M}$$

(2.2)

for all n ≥ n₀ and

$$\sum _{k=1}^{k_0}{\left(\frac{1}{Q_k}\sum _{i=1}^k|q_i{x}_n\left(i\right)-q_{i}x\left(i\right)|\right)}^{p_k}<\frac{\epsilon }{3},$$

(2.3)

for all n ≥ n₀. It follow from (2.1), (2.2), and (2.3), for all n ≥ n₀ we have

$$\begin{array}{cc}\hfill \varrho \left(x_n-x\right)& =\sum _{k=1}^{\infty }{\left(\frac{1}{Q_k}\sum _{i=1}^k|q_i{x}_n\left(i\right)-q_{i}x\left(i\right)|\right)}^{p_k}\hfill \\ =\sum _{k=1}^{k_0}{\left(\frac{1}{Q_k}\sum _{i=1}^k|q_i{x}_n\left(i\right)-q_{i}x\left(i\right)|\right)}^{p_k}+\sum _{k=k_0+1}^{\infty }{\left(\frac{1}{Q_k}\sum _{i=1}^k|q_i{x}_n\left(i\right)-q_{i}x\left(i\right)|\right)}^{p_k}\hfill \\ <\frac{\epsilon }{3}+2^M\left({\sum _{k=k_0+1}^{\infty }\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_i{x}_n\left(i\right))\right|\right)}^{p_k}+{\sum _{k=k_0+1}^{\infty }\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_{i}x\left(i\right))\right|\right)}^{p_k}\right)\hfill \\ =\frac{\epsilon }{3}+2^M\left(\varrho \left(x_n\right)-{\sum _{k=1}^{k_0}\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_i{x}_n\left(i\right)\right|\right)}^{p_k}+{\sum _{k=k_0+1}^{\infty }\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_{i}x\left(i\right))\right|\right)}^{p_k}\right)\hfill \\ <\frac{\epsilon }{3}+2^M\left(\varrho \left(x\right)-{\sum _{k=1}^{k_0}\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_{i}x\left(i\right)\right|\right)}^{p_k}+\frac{\epsilon }{3\cdot 2^M}+{\sum _{k=k_0+1}^{\infty }\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_{i}x\left(i\right))\right|\right)}^{p_k}\right)\hfill \\ =\frac{\epsilon }{3}+2^M\left({\sum _{k=k_0+1}^{\infty }\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_{i}x\left(i\right)\right|\right)}^{p_k}+\frac{\epsilon }{3\cdot 2^M}+{\sum _{k=k_0+1}^{\infty }\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_{i}x\left(i\right))\right|\right)}^{p_k}\right)\hfill \\ =\frac{\epsilon }{3}+2^M\left(2{\sum _{k=k_0+1}^{\infty }\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_{i}x\left(i\right)\right|\right)}^{p_k}+\frac{\epsilon }{3\cdot 2^M}\right)\hfill \\ <\frac{\epsilon }{3}+\frac{\epsilon }{3}+\frac{\epsilon }{3}\hfill \\ =\epsilon .\hfill \end{array}$$

This show that ϱ(x _n -x) → 0 as n → ∞. Hence, by Proposition 2.5(ii), we have ||x _n -x|| → 0 as → ∞. □

Theorem 2.7. The space ces_(p)(q) has the property (H).

Proof. Let x ∈ S(ces_(p)(q)) and (x _n ) ⊆ ces_(p)(q) such that ||x _n || → 1 and $x_n\stackrel{w}{\to }x$ as n → ∞. By Proposition 2.3(iii), we have ϱ(x) = 1, so it follow form Proposition 2.5(i), we get ϱ(x _n ) → ϱ(x) as n → ∞. Since the mapping π _i : ces_(p)(q) → ℝ defined by π _i (y) = y(i), is a continuous linear functional on ces_(p)(q), it follow that x _n (i) → x(i) as n → ∞ for all i ∈ ℕ. Thus by Lemma 2.6, we obtain x _n → x as n → ∞, and hence the space ces_(p)(q) has the property (H). □

Corollary 2.8. For any 1 < p < ∞, the space ces _p (q) has the property (H).

Corollary 2.9. [9, Theorem 2.6] The space ces_(p)has the property (H).

Corollary 2.10. For any 1 < p < ∞, the space ces _p has the property (H).

Theorem 2.11. The space ces_(p)(q) has uniform Opial property.

Proof. Take any ε > 0 and x ∈ ces_(p)(q) with ||x|| ≥ ε. Let (x _n ) be weakly null sequence in S(ces_(p)(q)). By sup _k p _k < ∞, i.e., $\varrho \in {\text{Δ}}_2^s$, hence by Lemma 1.2 there exists δ ∈ (0, 1) independent of x such that ϱ(x) > δ. Also, by $\varrho \in {\text{Δ}}_2^s$ and Lemma 1.1 asserts that there exists δ₁ ∈ (0, δ) such that

$$|\varrho \left(y+z\right)-\varrho \left(y\right)|<\frac{\delta }{4}$$

(2.4)

whenever, ϱ(y) ≤ 1 and ϱ(z) ≤ δ₁. Choose k₀ ∈ ℕ such that

$$\sum _{k=k_0+1}^{\infty }{\left(\frac{1}{Q_k}\sum _{i=k_0+1}^k\left|q_{i}x\left(i\right)\right|\right)}^{p_k}<\sum _{k=k_0+1}^{\infty }{\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_{i}x\left(i\right)\right|\right)}^{p_k}<\frac{{\delta }_1}{4}.$$

(2.5)

So, we have

$$\begin{array}{c}\delta <\sum _{k=1}^{k_0}{\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_{i}x\left(i\right)\right|\right)}^{p_k}+\sum _{k=k_0+1}^{\infty }{\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_{i}x\left(i\right)\right|\right)}^{p_k}\hfill \\ \le \sum _{k=1}^{k_0}{\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_{i}x\left(i\right)\right|\right)}^{p_k}+\frac{{\delta }_1}{4},\hfill \end{array}$$

(2.6)

which implies that

$$\begin{array}{cc}\hfill \sum _{k=1}^{k_0}{\left(\frac{1}{Q_k}\sum _{i=1}^k\left|q_{i}x\left(i\right)\right|\right)}^{p_k}& >\delta -\frac{{\delta }_1}{4}\hfill \\ >\delta -\frac{\delta }{4}\hfill \\ =\frac{3\delta }{4}.\hfill \end{array}$$

(2.7)

Since $x_n\stackrel{w}{\to }0$, then there exists n₀ ∈ ℕ such that

$$\frac{3\delta }{4}\le \sum _{k=1}^{k_0}{\left(\frac{1}{Q_k}\sum _{i=1}^k|q_i{x}_n\left(i\right)+q_{i}x\left(i\right)|\right)}^{p_k}$$

(2.8)

for all n > n₀, since weak convergence implies coordinatewise convergence. Again, by $x_n\stackrel{w}{\to }0$, then there exists n₁ ∈ ℕ such that

$$\left|\right|x_{n{|}_{k_o}}\left|\right|<1-{\left(1-\frac{\delta }{4}\right)}^{\frac{1}{M}}$$

(2.9)

for all n > n₁ where p _k ≤ M for all k ∈ ℕ. Hence, by the triangle inequality of the norm, we get

$$\left|\right|x_{n{|}_{\mathbb{N}-k_o}}\left|\right|>{\left(1-\frac{\delta }{4}\right)}^{\frac{1}{M}}.$$

(2.10)

It follows by the definition of || · ||, we have

$$\begin{array}{cc}\hfill 1& \le \varrho \left(\frac{x_{n{|}_{\mathbb{N}-k_o}}}{{\left(1-\frac{\delta }{4}\right)}^{\frac{1}{M}}}\right)\hfill \\ =\sum _{k=k_0+1}^{\infty }{\left(\frac{\frac{1}{Q_k}\sum _{i=k_0+1}^k\left|q_i{x}_n\left(i\right)\right|}{{\left(1-\frac{\delta }{4}\right)}^{\frac{1}{M}}}\right)}^{p_k}\hfill \\ \le {\left(\frac{1}{{\left(1-\frac{\delta }{4}\right)}^{\frac{1}{M}}}\right)}^M\sum _{k=k_0+1}^{\infty }{\left(\frac{1}{Q_k}\sum _{i=k_0+1}^k\left|q_i{x}_n\left(i\right)\right|\right)}^{p_k}\hfill \end{array}$$

(2.11)

implies that

$$\sum _{k=k_0+1}^{\infty }{\left(\frac{1}{Q_k}\sum _{i=k_0+1}^{\infty }\left|q_i{x}_n\left(i\right)\right|\right)}^{p_k}\ge 1-\frac{\delta }{4}$$

(2.12)

for all n > n₁. By inequality (2.4), (2.5), (2.8), and (2.12), yields for any n > n₁ that

$$\begin{array}{cc}\hfill \varrho \left(x_n+x\right)& =\sum _{k=1}^{k_0}{\left(\frac{1}{Q_k}\sum _{i=1}^k|q_i{x}_n\left(i\right)+q_{i}x\left(i\right)|\right)}^{p_k}+\sum _{k=k_0+1}^{\infty }{\left(\frac{1}{Q_k}\sum _{i=1}^k|q_i{x}_n\left(i\right)+q_{i}x\left(i\right)|\right)}^{p_k}\hfill \\ >\sum _{k=1}^{k_0}{\left(\frac{1}{Q_k}\sum _{i=1}^k|q_i{x}_n\left(i\right)+q_{i}x\left(i\right)|\right)}^{p_k}+\sum _{k=k_0+1}^{\infty }{\left(\frac{1}{Q_k}\sum _{i=k_0+1}^k|q_i{x}_n\left(i\right)+q_{i}x\left(i\right)|\right)}^{p_k}\hfill \\ \ge \frac{3\delta }{4}+\sum _{k=k_0+1}^{\infty }{\left(\frac{1}{Q_k}\sum _{i=k_0+1}^k\left|q_i{x}_n\left(i\right)\right|\right)}^{p_k}-\frac{\delta }{4}\hfill \\ \ge \frac{3\delta }{4}+\left(1-\frac{\delta }{4}\right)-\frac{\delta }{4}\hfill \\ \ge 1+\frac{\delta }{4}.\hfill \end{array}$$

Since $\varrho \in {\text{Δ}}_2^s$ and by Lemma 1.3 there exists τ depending on δ only such that || x _n +x || ≥ 1+τ, which implies that $\underset{n\to \infty }{\mathsf{\text{lim}}}\mathsf{\text{inf}}\left|\right|x_n+x\left|\right|\ge 1+\tau$, hence the proof is complete. □

Corollary 2.12. For any 1 < p < ∞, the space ces _p (q) has the uniform Opial property.

Corollary 2.13. [5, Theorem 2.6] The space ces_(p)has the uniform Opial property.

Corollary 2.14. [4, Theorem 2] For any 1 < p < ∞, the space ces _p has the uniform Opial property.