Some Geometric Properties of Generalized Cesàro-Musielak-Orlicz Spaces Equipped with the Amemiya Norm

Abstract

A generalized Cesàro-Musielak-Orlicz sequence space C e s _Φ (q) endowed with the Amemiya norm is introduced. Criteria for the coordinatewise uniformly Kadec-Klee property and the uniform Opial property of the space C e s _Φ (q) with respect to the Amemiya norm are obtained.

1 Introduction

The study of geometric properties of Banach spaces such as Kadec-Klee property, Opial property, and their several generalizations play very important role in metric fixed point theory. In particular, the Opial property of a Banach space has a great importance in the fixed point theory, differential equation, and integral equations. On the other hand, the Kadec-Klee property has several applications in Ergodic theory and many other branches [23].

Recently, several authors are interested in studying the geometric properties of Cesàro, Cesàro-Orlicz, and Musielak-Orlicz sequence spaces due to their several applications in various branches of mathematical analysis. Some topological properties such as order continuity, separability, completeness, and relations between norm and modular as well as some geometric properties such as the Fatou property, monotonicity, Kadec-Klee property, uniform Opial property, rotundity, local rotundity, etc. are discussed in [2, 3, 6, 11, 20, 21]. Khan [13, 14] has introduced the Riesz-Musielak-Orlicz sequence space and studied some geometric properties of it. Recently, Mongkolkeha and Kumam [16] studied (H)-property and uniform Opial property for the generalized Cesàro sequence space C e s _(p)(q). Quite recently, Manna and Srivastava [15] introduced the generalized Cesàro-Musielak-Orlicz sequence space C e s _Φ(q), which include the well-known Cesàro [25], generalized Cesàro [21, 22], Cesàro-Orlicz [2, 20], Cesàro-Musielak-Orlicz [26] sequence spaces, etc. as in particular cases, and studied coordinatewise uniformly Kadec-Klee property and uniform Opial property for these spaces equipped with the Luxemberg norm. In this paper, we continue our study by investigating these properties in generalized Cesàro-Musielak-Orlicz sequence space with respect to the Amemiya norm.

Throughout this paper, we denote by $\mathbb {N}, \mathbb {R}$, and $\mathbb {R}^{+}$ the set of natural numbers, of reals, and of nonnegative reals, respectively. Let (X,∥.∥) be a Banach space and l ⁰ be the space of all real sequences $x=(x(i))_{i=1}^{\infty }$. Let S(X) and B(X) denote the unit sphere and closed unit ball, respectively.

Let (E,∥.∥ _E ) be a real normed linear subspace of l ⁰. E is said to be a normed sequence lattice [12] if it satisfies the following two conditions:

• (i) For any x∈E and y∈l ⁰ such that |y(k)|≤|x(k)| for every $k\in \mathbb {N}$, then y∈E and ∥y∥ _E ≤∥x∥ _E .

• (ii) There exists a sequence $x=(x(k))_{k=1}^{\infty }\in E$ such that x(k)>0 for all $k\in \mathbb {N}$.

A normed sequence lattice (E,∥.∥ _E ) with complete norm is called Banach sequence lattice [12].

Note

In many literatures, Banach sequence lattice E is also called Köthe sequence space [2, 20].

An element x∈E is said to be order continuous if for any sequence $(x_{l})\subset E_{+}$, where $x_{l}=(x_{l}(i))_{i=1}^{\infty }, l\in \mathbb {N}$ such that |x(i)|≥x _l (i)↘0, i.e., x _l (i) decreases to zero as $l\rightarrow \infty$ for each $i\in \mathbb {N}$, implies that $\|x_{l}\|_{E}\rightarrow 0$. The set of all order-continuous elements in E is denoted by E _a . A Banach sequence lattice E is said to be order continuous if E _a =E. It is known that E is order continuous if and only if [2]

$$\|(0, 0, \ldots, x(i+1), x(i+2), \ldots)\|_{E}\rightarrow 0 \text{ as } i\rightarrow\infty \text{ for any } x\in E. $$

A sequence $(x_{l})\subset X$ is said to be ε-separated sequence if the separation of the sequence (x _l ), defined by $\text {sep}(x_{l})=\inf \{\|x_{l}-x_{m}\|: l\neq m\}$ is >ε for some ε>0 [9].

A Banach space X is said to have the Kadec-Klee property, denoted by (H), if each weakly convergent sequence on the unit sphere is strongly convergent, i.e., convergent in norm [10]. A Banach space X is said to possess coordinatewise Kadec-Klee property, denoted by (H _c ) [7], if x∈X and every sequence $(x_{l})\subset X$ such that

$$\|x_{l}\|\rightarrow \|x\| \text{~and ~}x_{l}(i)\rightarrow x(i) \text{~for each~} i\in \mathbb{N}, \text{~then~} \|x_{l}-x\|\rightarrow 0. $$

It is known that X∈(H _c ) implies X∈(H), because the weak convergence in any Köthe sequence space X implies the coordinatewise convergence (see also [7]). A Banach space X has the coordinatewise uniformly Kadec-Klee property, denoted by (UKK _c ) [27], if for every ε>0 there exists a δ>0 such that

$$(x_{l})\subset B(X), ~{\text{sep}}(x_{l})\geq\varepsilon, \|x_{l}\|\rightarrow \|x\| \text{~and~} x_{l}(i)\rightarrow x(i) \text{~for each~} i\in \mathbb{N}, \text{~implies~} \|x\|\leq1-\delta. $$

It is well known that the property (UKK _c ) implies property (H _c ).

A Banach space X is said to have the Opial property [24] if for every weakly null sequence $(x_{l})\subset X$ and every nonzero x∈X, we have

$$\liminf\limits_{l\rightarrow\infty}\|x_{l}\|<\liminf_{l\rightarrow\infty}\|x_{l}+x\|. $$

A Banach space X is said to have the uniform Opial property [24] if for each ε>0 there exists μ>0 such that for any weakly null sequence (x _l ) in S(X) and x∈X with ∥x∥≥ε, the following inequality holds:

$$1+ \mu\leq \liminf\limits_{l\rightarrow\infty}\|x_{l}+x\|. $$

In any Banach space X the Opial property is important because it ensures that X has the weak fixed point property [8]. Opial in [18] has shown that the space $L_{p}[0, 2\pi ] (p\neq 2, 1<p<\infty )$ does not have this property but the Lebesgue sequence space $l_{p} (1< p<\infty )$ has.

For a real vector space X, a functional $\varrho :X\rightarrow [0,\infty ]$ is called a modular, if for arbitrary x,y∈X, the following conditions hold:

• (i) ϱ(x)=0 if and only if x=0,

• (ii) ϱ(−x)=ϱ(x),

• (iii) ϱ(α x+β y)≤ϱ(x)+ϱ(y) for α,β≥0,α+β=1.

If instead of (iii), there holds

• (iii)′ϱ(α x+β y)≤α ϱ(x)+β ϱ(y) for α,β≥0,α+β=1, then the modular ϱ is called convex.

For any modular ϱ on X, the modular space generated by the modular ϱ is denoted by X _ϱ and is defined as

$$X_{\varrho}=\{x\in X: ~~ \varrho(\lambda x)\rightarrow 0~ \text{as~} \lambda\rightarrow0^{+}\}.$$

It is shown by Orlicz [19] that the modular space X _ϱ is equivalent to the set $X_{\varrho }^{\ast }=\{x\in X: ~~ \varrho (\lambda x)<\infty ~ \text {for some ~} \lambda >0\}$ in the case of convex modular ϱ.

A sequence (x _n ) of elements of X _ϱ is called modular convergent to x∈X _ϱ if there exists a λ>0 such that $\varrho (\lambda (x_{n}-x))\rightarrow$ as $n\rightarrow \infty$.

In particular, in the case that ϱ is a convex modular, X _ϱ becomes a normed linear space with the norm ∥.∥ _ϱ induced by the convex modular ϱ defined by

$$\|x\|_{\varrho}^{L}=\inf \left\{ r>0 : \varrho\left(\frac {x} {r}\right) \leq 1 \right\} \text{~for~} x\in X_{\varrho}. $$

In the case of convex modular ϱ, it is shown in [17] that the functional

$$\|x\|_{\varrho}^{A}=\inf\limits_{k>0}\frac{1}{k}\left\{1+ \varrho(kx)\right\} $$

defines a norm on X _ϱ and the relation $\|x\|_{\varrho }^{L}\leq \|x\|_{\varrho }^{A}\leq 2\|x\|_{\varrho }^{L}$ holds for every x∈X _ϱ . The norms $\|x\|_{\varrho }^{L}$ and $\|x\|_{\varrho }^{A}$ defined on the modular space X _ϱ are called the Luxemberg norm and the Amemiya norm, respectively (see [17]).

A map $\varphi : \mathbb {R}\rightarrow [0, \infty ]$ is said to be an Orlicz function if it is an even, convex, left continuous on $[0, \infty )$, φ(0)=0, not identically zero, and $\varphi (u)\rightarrow \infty$ as $u\rightarrow \infty$. A sequence $\varPhi =(\varphi _{n})_{n=1}^{\infty }$ of Orlicz functions φ _n is called a Musielak-Orlicz function [17]. A Musielak-Orlicz function $\varPhi =(\varphi _{n})_{n=1}^{\infty }$ is said to satisfy condition $(\infty _{1})$ if for each $n\in \mathbb {N}$, we have

$$(\infty_{1}):\quad \lim\limits_{u\rightarrow +\infty}\frac{\varphi_{n}(u)}{u}= +\infty. $$

For any Musielak-Orlicz function Φ, the complementary function Ψ=(ψ _n ) of Φ is defined in the sense of Young as

$$\psi_{n}(u)=\sup\limits_{v\geq0}\{~|u|v-\varphi_{n}(v)\}\quad \text{for all~} u\in \mathbb{R} \text{~and~} n\in \mathbb{N}. $$

Given any Musielak-Orlicz function Φ and $x=(x(n))_{n=1}^{\infty }\in l^{0}$, a convex modular $I_{\varPhi }: l^{0}\rightarrow [0, \infty ]$ is defined by

$$I_{\varPhi}(x)= \sum\limits_{n=1}^{\infty}\varphi_{n}\left(|x(n)|\right) $$

and the linear space $l_{\varPhi }=\{x\in l^{0}: I_{\varPhi }(rx)<\infty$ for some r>0} is called a Musielak-Orlicz sequence space. The sequence spaces $(l_{\varPhi }, \|x\|_{I_{\varPhi }}^{L})$ and $(l_{\varPhi }, \|x\|_{I_{\varPhi }}^{A})$ are Banach spaces. The set of all k>0 such that $\|x\|_{I_{\varPhi }}^{A}=\frac {1}{k}(1+ I_{\varPhi }(kx))$ is attained for a fixed $x\in l_{\varPhi }^{A}$ is denoted by K(x). Moreover, it is known that for any $x\in l_{\varPhi }^{A}$, there exists a k>0 such that $\|x\|_{I_{\varPhi }}^{A}=\frac {1}{k}(1+ I_{\varPhi }(k x))$ whenever $\frac {\varphi _{n}(u)}{u}\rightarrow \infty$ as $u\rightarrow \infty$ for each $n\in \mathbb {N}$ (see [5]). For the details about Musielak-Orlicz sequence spaces and their geometric properties, we refer to [1, 3, 11, 17].

A Musielak-Orlicz function Φ satisfies the ${\delta _{2}^{0}}$-condition, denoted by $\varPhi \in {\delta _{2}^{0}}$, if there are positive constants a,K, a natural number m, and a sequence $(c_{n})_{n=1}^{\infty }$ of positive numbers such that $(c_{n})_{n=m}^{\infty }\in l_{1}$ and the inequality

$$ \varphi_{n}(2u)\leq K \varphi_{n}(u) + c_{n} $$

(1)

holds for every $n\in \mathbb {N}$ and $u\in \mathbb {R}$ whenever φ _n (u)≤a. If a Musielak-Orlicz function Φ satisfies ${\delta _{2}^{0}}$-condition with m=1, then Φ is said to satisfy δ ₂-condition [17].

A Musielak-Orlicz function $\varPhi =(\varphi _{n})_{n=1}^{\infty }$ is said to vanish only at zero, which is denoted by Φ>0 if φ _n (u)>0 for any $n\in \mathbb {N}$ and u>0.

2 Class C e s _Φ (q)

Let x∈l ⁰ and $\varPhi =(\varphi _{n})_{n=1}^{\infty }$ be a Musielak-Orlicz function. Let $q=(q_{n})_{n=1}^{\infty }, q_{n}\geq 1\ \forall \ n\in \mathbb {N}$ be a sequence of real numbers such that $Q_{n}={\sum }_{k=1}^{n}q_{k}$. The sequence space C e s _Φ (q), being studied in [15] and is defined as follows:

$${Ces}_{\varPhi}(q)= \{x\in l^{0}: R^{q} x \in l_{\varPhi}\}= \{x\in l^{0}: \sigma_{\varPhi}(rx)<\infty \text{~for some~} r>0\}, $$

where $\sigma _{\varPhi }(x)= I_{\varPhi }(R^{q} x)={\sum }_{n=1}^{\infty }\varphi _{n}(\frac {1}{Q_{n}}{\sum }_{k=1}^{n}q_{k}|x(k)|)$ and R ^q is a generalized Cesàro means map on l ⁰ defined as

$$R^{q} x= (R^{q} x(n))_{n=1}^{\infty}, \text{~with~} R^{q} x(n)=\frac{1}{Q_{n}}\sum\limits_{k=1}^{n}q_{k}|x(k)|\text{~ for each~} n=1, 2, \ldots $$

Clearly, C e s _Φ (q) is a linear space and also becomes a normed linear space under the norms $\|x\|_{\sigma _{\varPhi }}^{L}=\|R^{q}x\|_{I_{\varPhi }}^{L}$ and $\|x\|_{\sigma _{\varPhi }}^{A}=\|R^{q}x\|_{I_{\varPhi }}^{A}$ introduced with the help of the norms on l _Φ . The space C e s _Φ (q) will be called generalized Cesàro-Musielak-Orlicz sequence space. For simplifying notation, we write $\|x\|_{\varPhi }^{L}$ and $\|x\|_{\varPhi }^{A}$ instead of $\|x\|_{\sigma _{\varPhi }}^{L}$ and $\|x\|_{\sigma _{\varPhi }}^{A}$, respectively.

The class C e s _Φ (q) includes the following classes in particular cases:

• (i) For q _n =1, n=1,2,…, the Ces _Φ (q) reduces to Cesàro-Musielak-Orlicz sequence space ces _Φ studied by Wangkeeree [26], where

  $$\text{ces}_{\varPhi}=\left\{x\in l^{0} : \sum\limits_{n=1}^{\infty}\varphi_{n}\left(\frac{r}{n}\sum\limits_{k=1}^{n}|x(k)|\right)<\infty \text{~for some~} r>0\right\}. $$

• (ii) For φ _n =φ for any n, the space ces _Φ becomes the well-known Cesàro-Orlicz sequence space ces _φ , studied recently by Petrot and Suantai [20], Foralewski et al. [6], and Cui et al. [2].

• (iii) For φ _n (x)=|x|^pn,p _n ≥1 for all n, the space C e s _Φ (q) reduces to the sequence space C e s _(p)(q) studied by Mongkolkeha et al. [16], and for φ _n (x)=|x|^pn with p _n =p≥1 for all n, then C e s _Φ (q) reduces to the sequence space C e s _p (q) studied by Khan [13].

Notation

For any x∈l ⁰ and $i\in \mathbb {N}$, we shall use the following notations throughout the paper:

$$\begin{array}{@{}rcl@{}} x|_{i}&=&(x(1), x(2), x(3), \ldots, x(i), 0, 0, \ldots), \text{~called the truncation of~} x \text{~at~} i,\\ x|_{\mathbb{N}-i}&=&(0, 0, 0, \ldots, 0, x(i+1), x(i+2), \ldots),\\ x|_{I}&=&\{x\in l^{0}: x(i)\neq 0 \text{~for all~} i\in I\subseteq \mathbb{N} \text{~and~} x(i)=0 \text{~for all~} i\in \mathbb{N}\setminus I\} \text{~and~} \\ \text{supp}~ x&=&\{i\in \mathbb{N}: x(i)\neq 0\}.\\ \end{array} $$

For simplifying notation, we write $\text {Ces}_{\varPhi }^{A}(q)=(\text {Ces}_{\varPhi }(q), \|.\|_{\varPhi }^{A})$.

3 Main results

We assume throughout that the sequence space ${Ces}_{\varPhi }^{A}(q)$ is nontrivial, i.e., ${Ces}_{\varPhi }^{A}(q) \neq \{0\}$. It is easy to observe that the space ${Ces}_{\varPhi }^{A}(q)$ belongs to the class of normed sequence lattice.

Theorem 1

The space $\text {Ces}_{\varPhi }^{A}(q)=(\text {Ces}_{\varPhi }(q), \|.\|_{\varPhi }^{A})$ is a Banach space.

Proof

It is shown in [15, Theorem 1 (i)] that ${Ces}_{\varPhi }^{L}(q)=({Ces}_{\varPhi }(q), \|.\|_{\varPhi }^{L})$ is a Banach space. But the norms $\|.\|_{\varPhi }^{L}$ and $\|.\|_{\varPhi }^{A}$ are equivalent, so the proof follows easily. □

Theorem 2

Let $({Ces_{\varPhi }^{A}(q)})_{a}=\{x\in Ces_{\varPhi }^{A}(q) : \sigma _{\varPhi }(rx)<\infty , \text { for all } r>0\}$ . Then the following statements are true:

• (i) $({Ces_{\varPhi }^{A}(q)})_{a}$ is a closed subspace of ${Ces_{\varPhi }^{A}(q)}$,

• (ii) $({Ces_{\varPhi }^{A}(q)})_{a}\subseteq \{x\in Ces_{\varPhi }^{A}(q): \|x-x|_{j}\|_{\varPhi }^{A}\rightarrow 0\}$,

• (iii) if Φ is a Musielak-Orlicz function satisfying the condition δ ₂ then $({Ces_{\varPhi }^{A}(q)})_{a} ={Ces_{\varPhi }^{A}(q)}$.

Proof

• (i) Clearly $({Ces}_{\varPhi }^{A}(q))_{a}$ is a subspace of ${{Ces}_{\varPhi }^{A}(q)}$. It is required to show that $({Ces_{\varPhi }^{A}(q)})_{a}$ is closed in ${{Ces}_{\varPhi }^{A}(q)}$. For this, let $x\in \overline {({{Ces}_{\varPhi }^{A}(q)})_{a}}$, the closure of $({{Ces}_{\varPhi }^{A}(q)})_{a}$. So there exists $x_{i}=(x_{i}(k))_{k=1}^{\infty }\in ({{Ces}_{\varPhi }^{A}(q)})_{a}$ for each $i\in \mathbb {N}$ such that $\|x-x_{i}\|_{\varPhi }^{A}\rightarrow 0$ as $i\rightarrow \infty$. We prove that $x\in ({{Ces}_{\varPhi }^{A}(q)})_{a}$. By the equivalence definition of norm and modular convergence, we have $\sigma _{\varPhi }(r(x-x_{i}))\rightarrow 0$ as $i\rightarrow \infty$ for all r>0. So for all r>0, there exists $J\in \mathbb {N}$ such that σ _Φ (2r(x−x _J ))<1. Since $x_{J}\in ({{Ces}_{\varPhi }^{A}(q)})_{a}$, so we have $\sigma _{\varPhi }(2rx_{J})<\infty$ for all r>0. Now, consider

  $$\begin{array}{@{}rcl@{}} \sum\limits_{n=1}^{\infty}\varphi_{n}\left(\frac{r}{Q_{n}}\sum\limits_{k=1}^{n}q_{k}|x(k)|\right) &\leq&\sum\limits_{n=1}^{\infty}\varphi_{n}\left(\frac{r}{2Q_{n}}\sum\limits_{k=1}^{n}(2q_{k}|x(k)-x_{J}(k)|)\right.\\ &&\left.+ \frac{r}{2Q_{n}}\sum\limits_{k=1}^{n}2q_{k}|x_{J}(k)|\right)\\ & \leq& \frac{1}{2} \sigma_{\varPhi}(2r(x-x_{J})) + \frac{1}{2}\sigma_{\varPhi}(2rx_{J})<\infty. \end{array} $$

  Since r is arbitrary, so we have $x\in ({\text {Ces}_{\varPhi }^{A}(q)})_{a}$.

• (ii) Let $A=\{x\in {{Ces}}_{\varPhi }^{A}(q): \|x-x|_{j}\|_{\varPhi }^{A}\rightarrow 0\}$, $x\in ({Ces}_{\varPhi }^{A}(q))_{a}$ and 𝜖>0. Since $x\in ({Ces}_{\varPhi }^{A}(q))_{a}$, so there exists $j_{0}\in \mathbb {N}$ such that

  $$\sigma_{\varPhi}\left(\frac{x-x|_{j}}{\epsilon}\right)= \sum\limits_{n=j+1}^{\infty} \varphi_{n}\left(\frac{1}{\epsilon Q_{n}}\sum\limits_{k=j+1}^{n}|q_{k}x(k)|\right)< \epsilon $$

  for all j>j ₀. Hence, by the definition of norm $\|.\|_{\varPhi }^{A}$, we have

  $$\left\|x-x|_{j}\right\|_{\varPhi}^{A}\leq \epsilon\left(1+ \sigma_{\varPhi}\left(\frac{x-x|_{j}}{\epsilon}\right)\right)<\epsilon(1+\epsilon) $$

  for all j>j ₀. Since 𝜖 is arbitrary, we have $\|x-x|_{j}\|_{\varPhi }^{A}\rightarrow 0$ as $j\rightarrow \infty$. So x∈A.

• (iii) We show only the inclusion ${{Ces}_{\varPhi }^{A}(q)}\subset ({{Ces}_{\varPhi }^{A}(q)})_{a}$ because the other inclusion is always true. Let $x\in {{Ces}_{\varPhi }^{A}(q)}$. Then for some t>0, $\sigma _{\varPhi }(t x)<\infty$, i.e., ${\sum }_{n=1}^{\infty }\varphi _{n}(\frac {t}{Q_{n}}{\sum }_{k=1}^{n}q_{k}|x(k)|)<\infty$. We show that for any r>0

  $$\sum\limits_{n=1}^{\infty}\varphi_{n}\left(\frac{r}{Q_{n}}\sum\limits_{k=1}^{n}|q_{k}x(k)|\right)<\infty $$

  holds. If r∈[0,t], t is fixed, then it follows easily because

  $$\sum\limits_{n=1}^{\infty}\varphi_{n}\left(\frac{r}{Q_{n}}\sum\limits_{k=1}^{n}q_{k}|x(k)|\right)\leq \sum\limits_{n=1}^{\infty}\varphi_{n}\left(\frac{t}{Q_{n}}\sum\limits_{k=1}^{n}q_{k}|x(k)|\right)<\infty. $$

Now choose r>t. Since $x\in {{Ces}_{\varPhi }^{A}(q)}$, i.e., for some $t>0, \sigma _{\varPhi }(t x)<\infty$, there exists a finite positive constant a such that

$$\sum\limits_{n=1}^{\infty}\varphi_{n}\left(\frac{t}{Q_{n}}\sum\limits_{k=1}^{n}q_{k}|x(k)|\right)\leq a.$$

Therefore for each n≥1, we have

$$\varphi_{n}\left(\frac{t}{Q_{n}}\sum\limits_{k=1}^{n}q_{k}|x(k)|\right)\leq a.$$

Since $\varPhi =(\varphi _{n})_{n=1}^{\infty }$ satisfies the δ ₂-condition, so by definition there are positive constants a and K and a sequence $(c_{n})_{n=1}^{\infty }$ of positive numbers such that $(c_{n})_{n=1}^{\infty }\in l_{1}$ and the inequality

$$\varphi_{n}(2u)\leq K \varphi_{n}(u) + c_{n}$$

holds for every $n\in \mathbb {N}$ and $u\in \mathbb {R}$ whenever φ _n (u)≤a. Let $u=\frac {t}{Q_{n}}{\sum }_{k=1}^{n}q_{k}|x(k)|, K>0$ be a constant and a be chosen as above. Since r>t, there exists $l\in \mathbb {N}$ such that r≤2^l t. Now applying the δ ₂-condition for all n≥1, we have

$$\begin{array}{@{}rcl@{}}\varphi_{n}\left(\frac{r}{Q_{n}}\sum\limits_{k=1}^{n}q_{k}|x(k)|\right) &\leq& \varphi_{n}\left(\frac{2^{l}t}{Q_{n}}\sum\limits_{k=1}^{n}q_{k}|x(k)|\right)+ c_{n} \\ & \leq& K^{l}\varphi_{n}\left(\frac{t}{Q_{n}}\sum\limits_{k=1}^{n}q_{k}|x(k)|\right) + \left(\sum\limits_{i=0}^{l-1}K^{i}\right) c_{n}. \end{array} $$

Taking summation in both sides over n≥1, we obtain

$$\sum\limits_{n=1}^{\infty}\varphi_{n}\left(\frac{r}{Q_{n}}\sum\limits_{k=1}^{n}q_{k}|x(k)|\right)\leq K^{l}\sum\limits_{n=1}^{\infty}\varphi_{n}\left(\frac{t}{Q_{n}}\sum\limits_{k=1}^{n}q_{k}|x(k)|\right)+ \left(\sum\limits_{i=0}^{l-1}K^{i}\right) \sum\limits_{n=1}^{\infty}c_{n}< \infty. $$

Hence, $x\in ({\text {Ces}_{\varPhi }^{A}(q)})_{a}$. □

We will assume in the rest of the paper that the Musielak-Orlicz function $\varPhi =(\varphi _{n})_{n=1}^{\infty }$ with $\varphi _{n}(u)<\infty$ for each $n\in \mathbb {N}, u\in \mathbb {R}$. The following lemmas are useful to prove our result.

Lemma 1

Suppose Φ∈δ ₂ and Φ>0. Then for any $(x_{l})\subset {\text {Ces}_{\varPhi }^{A}(q)}$ , where $x_{l}=(x_{l}(i))_{i=1}^{\infty }, l\in \mathbb {N}, \|x_{l}\|_{\varPhi }^{A}\rightarrow 0$ if and only if $\sigma _{\varPhi }(x_{l})\rightarrow 0$.

Proof

See [7, 11]. □

It is noted that, for a fixed $x\in {\text {Ces}}_{\varPhi }^{A}(q)$, the set K(x) defined earlier (see Section 1) has the form $K(x)=\{k>0: \frac {1}{k}(1+ \sigma _{\varPhi }(kx))=\|x\|_{\varPhi }^{A}\}$.

Lemma 2

Let $x\in {\text {Ces}_{\varPhi }^{A}(q)}$ be given and x≠0. If K(x)=∅, then $\|x\|_{\varPhi }^{A}={\sum }_{n=1}^{\infty }\lambda _{n} R^{q} x(n)$ , where $\lambda _{n}= \lim _{u\rightarrow \infty } \frac {\varphi _{n}(u)}{u}$ and $R^{q} x(n)= \frac {1}{Q_{n}}{\sum }_{i=1}^{n}q_{i}|x(i)|, n\in \mathbb {N}$.

Proof

Let $f(k)=\frac {1}{k}(1+ \sigma _{\varPhi }(k x))$, where $\sigma _{\varPhi }(x)={\sum }_{n=1}^{\infty }\varphi _{n}(\frac {1}{Q_{n}}{\sum }_{i=1}^{n}q_{i}|x(i)|)={\sum }_{n=1}^{\infty }\varphi _{n}(R^{q} x(n))$. Since f(k) is continuous and K(x)=∅, so we have $\|x\|_{\varPhi }^{A}= \lim _{k\rightarrow \infty } f(k) = \lim _{k\rightarrow \infty }\frac {\sigma _{\varPhi }(kx)}{k}$. Then $\lambda _{n}= \lim _{u\rightarrow \infty } \frac {\varphi _{n}(u)}{u}$ is finite for all n∈supp x. If not, there exists a n ₀∈supp x such that

$$\|x\|_{\varPhi}^{A}=\lim\limits_{k\rightarrow\infty}\frac{\sigma_{\varPhi}(kx)}{k}\geq\lim\limits_{k\rightarrow\infty}\frac{\varphi_{n_{0}}(k R^{q}x(n_{0}))}{k R^{q}x(n_{0})} R^{q}x(n_{0})=\infty. $$

So we have

$$\|x\|_{\varPhi}^{A}=\lim_{k\rightarrow\infty}\frac{\sigma_{\varPhi}(kx)}{k}= \lim_{k\rightarrow\infty}\sum\limits_{n=1}^{\infty}\frac{\varphi_{n}(kR^{q}x(n))}{kR^{q}x(n)}R^{q}x(n)=\sum\limits_{n=1}^{\infty}\lambda_{n} R^{q}x(n). $$

□

Lemma 3

Let $x\in {Ces_{\varPhi }^{A}(q)}$ be given and x≠0. If $\varPhi =(\varphi _{n})_{n=1}^{\infty }$ is a Musielak-Orlicz function satisfying condition $(\infty _{1})$ , then K(x)≠∅.

Proof

Suppose on contrary that K(x)=∅. Then by Lemma 2, we obtain $\lim \limits _{u\rightarrow \infty } \frac {\varphi _{n}(u)}{u} <\infty$ for each n∈supp x, a contradiction to the assumption that Φ satisfies the condition $(\infty _{1})$. □

Theorem 3

The sequence space $Ces_{\varPhi }^{A}(q)$ has the UKK _c -property whenever $\varPhi =(\varphi _{n})_{n=1}^{\infty }$ satisfying condition $(\infty _{1}), \varPhi \in \delta _{2}$ , i.e., ( 1 ) and Φ>0.

Proof

Let 𝜖>0 be given, $(x_{l})\subset B({Ces}_{\varPhi }^{A}(q)), x\in {Ces}_{\varPhi }^{A}(q), \|x_{l}\|_{\varPhi }^{A}\rightarrow \|x\|_{\varPhi }^{A}, x_{l}(i)\rightarrow x(i)$ for each $i\in \mathbb {N}$ and sep(x _l )≥𝜖. We prove that $\|x\|_{\varPhi }^{A}\leq 1-\delta$. It trivially holds when x=0. Let us assume that x≠0. Then using Lemma 3, we have K(x)≠∅, i.e., for each $x\in {Ces}_{\varPhi }^{A}(q)$, there exists a $k_{l}\in \mathbb {R_{+}}$ such that $\|x\|_{\varPhi }^{A}=\frac {1}{k_{l}}\left (1+\sigma _{\varPhi }(k_{l} x)\right )$. Since $x_{l}\rightarrow x$ in ${Ces}_{\varPhi }^{A}(q)$ weakly, it implies $x_{l}(i)\rightarrow x(i)$ for each $i\in \mathbb {N}$, so we may select a finite set I={1,2,3,…,N−1} for which $x_{l}\rightarrow x$ uniformly. So, there exists $l_{N}\in \mathbb {N}$ such that

$$ \|(x_{l}-x_{m})|_{I}\|_{\varPhi}^{A}\leq\frac{\epsilon}{2} ~\text{ for all~} l, m\geq l_{N} . $$

(2)

Since sep(x _l )≥𝜖, we have $\|x_{l}-x_{m}\|_{\varPhi }^{A}\geq \epsilon$ for l≠m by definition. This, together with (2), implies that $\|(x_{l}-x_{m})|_{\mathbb {N}-I}\|_{\varPhi }^{A}\geq \frac {\epsilon }{2}$ for l≠m and l,m≥l _N . Hence, for each $n\in \mathbb {N}$, there exists a l _N such that $\|x_{l_{N}}|_{\mathbb {N}-I}\|_{\varPhi }^{A}\geq \frac {\epsilon }{4}$. Without loss of generality, we may assume that $\|x_{l}|_{\mathbb {N}-I}\|_{\varPhi }^{A}\geq \frac {\epsilon }{4}$ for all $l, N\in \mathbb {N}$. Therefore, by Lemma 1, there exists δ ₁∈(0,𝜖) such that $\sigma _{\varPhi }(x_{l}|_{\mathbb {N}-I})\geq \delta _{1}$.

Since $x\in {{Ces}}_{\varPhi }^{A}(q)$ implies that $\|x- x|_{I}\|_{\varPhi }^{A}\rightarrow 0$ for sufficiently large N, there exists a $\frac {\delta _{1}}{2}>0$ such that $\|x|_{I}\|_{\varPhi }^{A}>\|x\|_{\varPhi }^{A}-\frac {\delta _{1}}{2}$. Also, since $x_{l}(i)\rightarrow x(i)$ for each i and $\|x_{l}\|_{\varPhi }^{A}\rightarrow \|x\|_{\varPhi }^{A}$, there exists $N_{0}\in \mathbb {N}$ such that

$$\|x_{l}|_{I}\|_{\varPhi}^{A}>\|x\|_{\varPhi}^{A}-\frac{\delta_{1}}{2} \text{~for~} l>N_{0}. $$

Since $\|x_{l}\|_{\varPhi }^{A}\leq 1$ implies k _l ≥1 for all $l\in \mathbb {N}$, so by the convexity of φ _n and the inequality φ _n (u+v)≥φ _n (u)+φ _n (v) for all $u, v\in \mathbb {R^{+}}$ for each $n\in \mathbb {N}$, we have

$$\begin{array}{@{}rcl@{}} 1 &\geq&\|x_{l}\|_{\varPhi}^{A}\\ &=&\frac{1}{k_{l}}\left(1+\sum\limits_{n=1}^{N-1}\varphi_{n}\left(\frac{k_{l}}{Q_{n}}\sum\limits_{j=1}^{n}q_{j}|x_{l}(j)|\right) + \sum\limits_{n=N}^{\infty}\varphi_{n}\left(\frac{k_{l}}{Q_{n}}\sum\limits_{j=1}^{n}q_{j}|x_{l}(j)|\right)\right)\\ &=& \frac{1}{k_{l}}\left(1+\sum\limits_{n=1}^{N-1}\varphi_{n}\left(\frac{k_{l}}{Q_{n}}\sum\limits_{j=1}^{n}q_{j}|x_{l}(j)|\right)\right) + \frac{1}{k_{l}}\left( \sum\limits_{n=N}^{\infty}\varphi_{n}\left(\frac{k_{l}}{Q_{n}}\sum\limits_{j=1}^{N-1}q_{j}|x_{l}(j)| \right.\right.\\ &&\left.\left.+ \frac{k_{l}}{Q_{n}}\sum\limits_{j=N}^{n}q_{j}|x_{l}(j)|\right)\right)\\ &\geq& \frac{1}{k_{l}}\left(1+\sum\limits_{n=1}^{N-1}\varphi_{n}\left(\frac{k_{l}}{Q_{n}}\sum\limits_{j=1}^{n}q_{j}|x_{l}(j)|\right)\right) + \frac{1}{k_{l}} \sum\limits_{n=N}^{\infty}\varphi_{n}\left(\frac{k_{l}}{Q_{n}}\sum\limits_{j=1}^{N-1}q_{j}|x_{l}(j)|\right)\\ &&+ \frac{1}{k_{l}}\sum\limits_{n=N}^{\infty}\varphi_{n}\left(\frac{k_{l}}{Q_{n}}\sum\limits_{j=N}^{n}q_{j}|x_{l}(j)|\right)\\ &=&\frac{1}{k_{l}}\left(1+\sigma_{\varPhi}\left(k_{l} x_{l}|_{I}\right)\right) + \frac{1}{k_{l}}\sum\limits_{n=N}^{\infty}\varphi_{n}\left(\frac{k_{l}}{Q_{n}}\sum\limits_{j=N}^{n}q_{j}|x_{l}(j)|\right)\\ & \geq &\frac{1}{k_{l}}\left(1+\sigma_{\varPhi}\left(k_{l} x_{l}|_{I}\right)\right) + \sum\limits_{n=N}^{\infty}\varphi_{n}\left(\frac{1}{Q_{n}}\sum\limits_{j=N}^{n}q_{j}|x_{l}(j)|\right) ~~~(\text{since } k_{l}\geq 1)\\ &\geq&\|x_{l}|_{I}\|_{\varPhi}^{A} + \sigma_{\varPhi}\left(x_{l}|_{\mathbb{N}-I}\right)\\ & > &\|x\|_{\varPhi}^{A}- \frac{\delta_{1}}{2}+ \delta_{1}= \|x\|_{\varPhi}^{A} + \frac{\delta_{1}}{2} \text{~for~} l>N_{0}. \end{array} $$

Therefore, $\|x\|_{\varPhi }^{A}\leq 1- \frac {\delta _{1}}{2}$. Thus, ${Ces}_{\varPhi }^{A}(q)$ has the coordinatewise uniform Kadec-Klee property. □

Corollary 1

• (i) If φ _n =φ for all $n\in \mathbb {N}, q_{n}=1$ for $n\in \mathbb {N}$ and φ∈δ ₂ , then the Cesàro-Orlicz sequence space $ces_{\varphi }^{A}$ [ 20 ] has the (UKK _c ).

• (ii) If φ _n (u)=|u| ^pn for all $u\in \mathbb {R}, 1< p_{n}<\infty \ \forall n$ then $Ces{_{p}^{A}}(q)$ has the (UKK _c ).

Theorem 4

Let Φ>0 be a Musielak-Orlicz function satisfying conditions $(\infty _{1})$ and δ ₂ , i.e., ( 1 ). Then, $Ces_{\varPhi }^{A}(q)$ has the uniform Opial property.

Proof

Take any 𝜖>0 and $x\in {Ces}_{\varPhi }^{A}(q)$ with $\|x\|_{\varPhi }^{A}\geq \epsilon$. Let $(x_{l})\subset S({Ces}_{\varPhi }^{A}(q))$ be any weakly null sequence. We show that for every 𝜖>0, there is a μ>0 such that

$$\liminf\limits_{l\rightarrow\infty}\|x_{l}+x\|_{\varPhi}^{A}\geq 1+\mu $$

for each $x\in {Ces}_{\varPhi }^{A}(q)$. Since Φ∈δ ₂ and Φ>0, so by Lemma 1, there is a $\delta \in (0, \frac {4}{5})$ independent of x such that $\sigma _{\varPhi }(\frac {x}{2})\geq \delta$. Since Φ∈δ ₂ implies ${Ces}_{\varPhi }^{A}(q)=({Ces}_{\varPhi }^{A}(q))_{a}$ by Theorem 2 (iii), we have $\|x-x|_{n}\|_{\varPhi }^{A}\rightarrow 0$ as $n\rightarrow \infty$ (by Theorem 2(ii)). Therefore, for a given δ>0, there exists a natural number $N_{0}\in \mathbb {N}$ such that

$$\left\|x-x|_{n_{0}}\right\|_{\varPhi}^{A}=\left\|x|_{\mathbb{N}-n_{0}}\right\|_{\varPhi}^{A}<\frac{\delta}{8} $$

and

$$\sum\limits_{n=n_{0}+1}^{\infty}\varphi_{n}\left(\frac{1}{2Q_{n}}\sum\limits_{j=1}^{n}q_{j}|x(j)|\right)<\frac{\delta}{8}. $$

Since $\sigma _{\varPhi }(\frac {x}{2})\geq \delta$, it follows that

$$\begin{array}{@{}rcl@{}} \delta& \leq &\sum\limits_{n=1}^{n_{0}}\varphi_{n}\left(\frac{1}{2Q_{n}}\sum\limits_{j=1}^{n}q_{j}|x(j)|\right)+ \sum\limits_{n=n_{0}+1}^{\infty}\varphi_{n}\left(\frac{1}{2Q_{n}}\sum\limits_{j=1}^{n}q_{j}|x(j)|\right) \\ &< &\sum\limits_{n=1}^{n_{0}}\varphi_{n}\left(\frac{1}{2Q_{n}}\sum\limits_{j=1}^{n}q_{j}|x(j)|\right)+ \frac{\delta}{8}. \end{array} $$

This gives ${\sum }_{n=1}^{n_{0}}\varphi _{n}(\frac {1}{2Q_{n}}{\sum }_{j=1}^{n}q_{j}|x(j)|)> \frac {7\delta }{8}$. Since $x_{l}\rightarrow 0$ weakly, it implies $x_{l}(i)\rightarrow 0$ as $l\rightarrow \infty$ for each i, so we have $\sigma _{\varPhi }(x_{l}|_{n_{0}})\rightarrow 0$ as $l\rightarrow \infty$. Hence, by Lemma 1, there exists a natural number l ₀ such that $\left \|x_{l}|_{n_{0}}\right \|_{\varPhi }^{A} <\frac {\delta }{8}$ for all l≥l ₀. This, together with $(x_{l})\subset S({Ces}_{\varPhi }^{A}(q))$, i.e., $\|x_{l}\|_{\varPhi }^{A}=1$, implies that

$$ \left\|x_{l}|_{\mathbb{N}-n_{0}}\right\|_{\varPhi}^{A} > 1-\frac{\delta}{8} \text{~~~for all~~~} l\geq l_{0}. $$

(3)

Now, for all l≥l ₀, we have

$$\begin{array}{@{}rcl@{}} \|x_{l}+x\|_{\varPhi}^{A} &=&\left\|(x_{l}+x)|_{n_{0}} + (x_{l}+x)|_{\mathbb{N}-n_{0}}\right\|_{\varPhi}^{A} \\ &\geq& \left\|(x_{l}+x)|_{n_{0}} + x_{l}|_{\mathbb{N}-n_{0}}\right\|_{\varPhi}^{A} -\frac{\delta}{8}\\ & \geq &\left\|x|_{n_{0}} + x_{l}|_{\mathbb{N}-n_{0}}\right\|_{\varPhi}^{A} -\frac{\delta}{8}-\frac{\delta}{8}=\big\|x|_{n_{0}} + x_{l}|_{\mathbb{N}-n_{0}}\big\|_{\varPhi}^{A} -\frac{\delta}{4}. \end{array} $$

Since Φ satisfies condition $(\infty _{1})$, so by Lemma 3, there exists k _l >0 such that for l≥l ₀, we have

$$\left\|x|_{n_{0}} + x_{l}|_{\mathbb{N}-n_{0}}\right\|_{\varPhi}^{A}=\frac{1}{k_{l}}\left(1+ \sigma_{\varPhi}\left(k_{l}\left(x|_{n_{0}} + x_{l}|_{\mathbb{N}-n_{0}}\right)\right)\right). $$

Now, using the fact that σ _Φ (u+v)≥σ _Φ (u)+σ _Φ (v), whenever $\text {supp}~u\cap \text {supp}~v=\emptyset$, we have

$$\begin{array}{@{}rcl@{}} \|x_{l}+x\|_{\varPhi}^{A} &\geq& \frac{1}{k_{l}}+ \frac{1}{k_{l}}\sigma_{\varPhi}\left(k_{l}x|_{n_{0}}\right) + \frac{1}{k_{l}}\sigma_{\varPhi}\left(k_{l}x_{l}|_{\mathbb{N}-n_{0}}\right)-\frac{\delta}{4} \\ &\geq& \left\|x_{l}|_{\mathbb{N}-n_{0}}\right\|_{\varPhi}^{A} + \frac{1}{k_{l}}\sigma_{\varPhi}\left(k_{l}x|_{n_{0}} \right) -\frac{\delta}{4}. \end{array} $$

(4)

Without loss of generality, we may assume that $k_{l}\geq \frac {1}{2}$ for all l because if $k_{l}<\frac {1}{2}$, then we have $\|x_{l}+x\|_{\varPhi }^{A}>2-\frac {\delta }{4}> 1+\delta$. Using the convexity of Φ, we have $\sigma _{\varPhi }(k_{l}x|_{n_{0}} )\geq 2k_{l} \sigma _{\varPhi }(\frac {1}{2}x|_{n_{0}} )$. Now using (3), from (4), we have

$$\begin{array}{@{}rcl@{}} \|x_{l}+x\|_{\varPhi}^{A} &\geq& \left\|x_{l}|_{\mathbb{N}-n_{0}}\right\|_{\varPhi}^{A} + 2 \sigma_{\varPhi}\left(\frac{1}{2}x|_{n_{0}}\right) -\frac{\delta}{4} \\ &>& \left\|x_{l}|_{\mathbb{N}-n_{0}}\right\|_{\varPhi}^{A} + 2\sum\limits_{n=1}^{n_{0}}\varphi_{n}\left(\frac{1}{2Q_{n}}\sum\limits_{j=1}^{n}q_{j}|x(j)|\right)-\frac{\delta}{4}\\ &\geq& 1-\frac{\delta}{8} +2.\frac{7\delta}{8}-\frac{\delta}{4}= 1+ \frac{11\delta}{8}. \end{array} $$

which implies that $\liminf _{n\rightarrow \infty }\|x_{l}+x\|_{\varPhi }^{A} \geq 1+\mu$, where μ depends upon δ. This completes the proof. □

Corollary 2

• Let q _n =1,n=1,2,… and φ _n (u)=|u| ^pn for all $u\in \mathbb {R}, 1< p_{n}<\infty \ \forall n$ . Then, Φ∈δ ₂ if and only if $\limsup _{n\rightarrow \infty }p_{n}<\infty$ . Therefore, $ces_{(p)}^{A}$ [ 21 ] has the uniform Opial property.

• If φ _n =φ ∀n,q _n =1 for n=1,2,… and φ∈δ ₂ , then the Cesàro-Orlicz sequence space $ces_{\varphi }^{A}$ [ 20 ] has the uniform Opial property.