Some Geometric Properties of Generalized Cesàro–Musielak–Orlicz Sequence Spaces

Abstract

A generalized Cesàro–Musielak–Orlicz sequence space $Ces_\Phi (q)$ equipped with the Luxemberg norm is introduced. It is proved that $Ces_\Phi (q)$ is a Banach space and also criteria for the coordinatewise uniformly Kadec–Klee property and the uniform Opial property are obtained.

The authors are very much thankful to the anonymous reviewers for their valuable comments, which improved the presentation of the paper. First author is grateful to CSIR, New Delhi, Government of India for the research fellowship with award no. 09/081(0988)/2009-EMR-I during this work.

1 Introduction

In fixed point theory, geometrical properties of Banach space, such as Kadec–Klee property, Opial property, and their several generalizations play fundamental role. In particular, the Opial property of a Banach space has its applications in differential equations and integral equations, etc. On the other hand the Kadec–Klee property has several applications in Ergodic theory and many other branches of analysis [22].

In recent times, the theory of Cesàro–Orlicz sequence spaces and Musielak–Orlicz sequence spaces and their geometric properties has been studied extensively. Some topological properties like absolute continuity, order continuity, separability, completeness, and relations between norm and modular as well as some geometrical properties like Fatou property, monotonicity, Kadec–Klee property, uniform Opial property, rotundity, local rotundity, property-$\beta$ etc. are studied in [2–4, 6, 8, 13, 20, 21]. Recently, Khan (see [15, 16]) introduced Riesz–Musielak–Orlicz sequence spaces and studied some geometric properties of this space. Quite recently, Mongkolkeha, and Kumam [17] studied $(H)$-property and uniform Opial property of generalized Cesàro sequence spaces. Some topological properties of sequence spaces defined by using Orlicz function are also studied in [1, 5, 25]. This motivated us to introduce generalized Cesàro–Musielak–Orlicz sequence spaces, which include the well known Cesàro, generalized Cesàro [24], Cesàro-Orlicz, Cesàro–Musielak–Orlicz sequence spaces etc. in particular cases. In this paper, we have made an attempt to study some of the geometric properties in generalized Cesàro–Musielak–Orlicz sequence spaces.

Throughout the paper, we denote $\mathbb {N}$, $\mathbb {R}$ and $\mathbb {R^{+}}$ as the set of natural numbers, real numbers, and nonnegative real numbers, respectively. Let $(X, \Vert .\Vert )$ 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. A sequence $(x_l)\subset X$ is said to be $\varepsilon$-separated sequence if separation of the sequence $(x_l)$ denoted by $sep(x_l)=\inf \{\Vert x_l-x_m\Vert : l\ne m\}>\varepsilon$ for some $\varepsilon >0$ [11].

A Banach space $X$ is said to have the Kadec–Klee property, denoted by $(H)$, if weakly convergent sequence on the unit sphere is strongly convergent, i.e., convergent in norm [12]. A Banach space $X$ is said to possess coordinatewise Kadec–Klee property, denoted by ($H_c$) [7], if $x\in X$ and every sequence $(x_l)\subset X$ such that

$$\begin{aligned} \Vert x_l\Vert \rightarrow \Vert x\Vert \,\,\text {and} \,\,x_l(i)\rightarrow x(i) \,\,\text {for each} \,\,i,\,\, \text {then}\,\, \Vert x_l-x\Vert \rightarrow 0. \end{aligned}$$

It is known that $X\in (H_c)$ implies $X\in (H)$, because weak convergence in $X$ implies the coordinatewise convergence. A Banach space $X$ has the coordinatewise uniformly Kadec–Klee property, denoted by ($UKK_c$) [27], if for every $\varepsilon >0$ there exists a $\delta >0$ such that

$$\begin{aligned} (x_l)\subset B(X), sep(x_l)\ge \varepsilon , \Vert x_l\Vert \rightarrow \Vert x\Vert \,\,\text {and}\,\, x_l(i)\rightarrow x(i)\,\, \text {for each}\,\, i\,\, \text {implies}\,\, \Vert x\Vert \le 1-\delta . \end{aligned}$$

It is known that the property $(UKK_c)$ implies property $(H_c)$.

A Banach space $X$ is said to have the Opial property [23] if for every weakly null sequence $(x_l)\subset X$ and every nonzero $x\in X$, we have

$$\begin{aligned} \displaystyle \liminf _{l\rightarrow \infty }\Vert x_l\Vert <\displaystyle \liminf _{l\rightarrow \infty }\Vert x_l+x\Vert . \end{aligned}$$

A Banach space $X$ is said to have the uniform Opial property [23] if for each $\varepsilon >0$ there exists $\mu >0$ such that for any weakly null sequence $(x_l)$ in $S(X)$ and $x\in X$ with $\Vert x\Vert \ge \varepsilon$ the following inequality hold:

$$\begin{aligned} 1+ \mu \le \displaystyle \liminf _{l\rightarrow \infty }\Vert x_l+x\Vert . \end{aligned}$$

In any Banach space $X$ an Opial property is important because it ensures that $X$ has a weak fixed point property [9]. Opial in [19] has shown that the space $L_p[0, 2\pi ]$ $(p\ne 2, 1<p<\infty )$ does not have this property, but the Lebesgue sequence space $l_p (1< p<\infty )$ has.

A map $\varphi : \mathbb {R}\rightarrow [0, \infty ]$ is said to be an Orlicz function if it is an even, convex, left continuous on $[0, \infty )$, $\varphi (0)=0$, not identically zero and $\varphi (u)\rightarrow \infty$ as $u\rightarrow \infty$. A sequence $\Phi =(\varphi _n)$ of Orlicz functions $\varphi _n$ is called Musielak–Orlicz function [18]. For a Musielak–Orlicz function $\Phi$, the complementary function $\Psi =(\psi _n)$ of $\Phi$ is defined in the sense of Young as

$$\begin{aligned} \psi _n(u)=\displaystyle \sup _{v\ge 0}\{~|u|v-\varphi _n(v)\}\quad \text {for all}\,\, u\in \mathbb {R} \,\,\text {and}\,\, n\in \mathbb {N}. \end{aligned}$$

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

$$\begin{aligned} I_\Phi (x)= \displaystyle \sum _{n=1}^{\infty }\varphi _n\Big (|x(n)|\Big ) \,\,\text {and} \end{aligned}$$

the linear space $l_\Phi =\{x\in l^0: I_\Phi (rx)<\infty$ for some $r>0\}$ is called Musielak–Orlicz sequence space. The space $l_\Phi$ equipped with functional $|||x|||_{\Phi }^L$ defined by

$$\begin{aligned} |||x|||_{\Phi }^L= \inf \Big \{r> 0 : I_\Phi \Big (\frac{x}{r}\Big ) \le 1 \Big \} \end{aligned}$$

becomes a Banach space. This functional $|||x|||_{\Phi }^L$ is called Luxemberg norm and the corresponding Musielak–Orlicz sequence space is denoted by $l_\Phi ^L$. For the details about Musielak–Orlicz sequence spaces and their geometric properties we refer to the articles [3, 10, 13, 18]. The subspace of $l_\Phi$ defined as

$$\begin{aligned} \Big \{x=(x(n))\in l^0: \forall r>0 \,\,\exists n_r\in \mathbb {N} \,\,\text {such that}\,\, \displaystyle \sum _{n=n_r}^{\infty }\varphi _n\Big (r|x(n)|\Big )<\infty \Big \}, \end{aligned}$$

equipped with the Luxemberg norm induced from $l_\Phi$ is denoted by $h_\Phi ^L$.

A Musielak–Orlicz function $\Phi$ is said to satisfy the $\delta _2^0$-condition denoted by $\Phi \in \delta _2^0$ if there are positive constants $a, K$, a natural $m$ and a sequence $(c_n)$ of positive numbers such that $(c_n)_{n=m}^{\infty }\in l_1$ and the inequality

$$\begin{aligned} {} \varphi _n(2u)\le K \varphi _n(u) + c_n \end{aligned}$$

(1)

holds for every $n\in \mathbb {N}$ whenever $\varphi _n(u)\le a$. If a Musielak–Orlicz function $\Phi$ satisfies $\delta _2^0$-condition with $m=1$, then $\Phi$ is said to satisfy $\delta _2$-condition [10, 18].

For any Musielak–Orlicz function $\Phi$, $h_\Phi$ coincides with $l_\Phi$ if and only if $\Phi$ satisfies $\delta _2^0$-condition [10].

A Musielak–Orlicz function $\Phi =(\varphi _n)_{n=1}^{\infty }$ satisfies the condition $(*)$ [13] if for any $\varepsilon \in (0, 1)$ there is a $\delta >0$ such that

$$\begin{aligned} {} \varphi _n(u)< 1-\varepsilon \,\, \text {implies}\,\, \varphi _n((1+\delta )u)\le 1 \,\,\text {for all}\,\, n\in \mathbb {N}\,\, \text {and}\,\, u\ge 0. \end{aligned}$$

(2)

A Musielak–Orlicz function $\Phi$ is to said to vanishes only at zero, which is denoted by $\Phi >0$ if $\varphi _n(u)>0$ for any $n\in \mathbb {N}$ and $u>0$.

2 Class $Ces_\Phi (q)$

Let $q=(q_n)_{n=1}^{\infty }$ be a sequence of real numbers with $q_k\ge 1$ for $k\in \mathbb {N}$, and $Q_n=\displaystyle \sum _{k=1}^{n}q_k$. We introduce the Riesz weighted mean map $R^q$ on $l^0$ as $R^q: l^0\rightarrow [0, \infty )$ such that $x\rightarrow R^q x$, where

$$\begin{aligned} R^q x=&~(R^q x(n))_{n=1}^{\infty }, \text {with}\,\, R^q x(n)=\frac{1}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x(k)| \,\,\text {for each}\,\, n=1, 2, \ldots \\&\quad \text {and}\,\, x\in l^0. \end{aligned}$$

Using this Riesz weighted mean map and a Musielak–Orlicz function $\Phi =(\varphi _n)$, we define on $l^0$ a functional $\sigma _\Phi (x)$ by

$$\begin{aligned} \sigma _\Phi (x)= I_\Phi (R^q x)=\displaystyle \sum _{n=1}^{\infty }\varphi _n\Big (\frac{1}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x(k)|\Big ). \end{aligned}$$

Since $\Phi$ is convex, so it is easy to verify that $\sigma _\Phi (x)$ is a convex modular on $l^0$(for definition see [18]), i.e., it satisfies $\sigma _\Phi (x)=0$ if and only if $x=0$, $\sigma _\Phi (-x)=\sigma _\Phi (x)$, $\sigma _\Phi (\gamma x+ \delta y)\le \gamma \sigma _\Phi (x)+ \delta \sigma _\Phi (y)$ whenever $x, y\in l^0$ and $\gamma ,\delta \ge 0$ with $\gamma + \delta =1$.

We now introduce the space $Ces_\Phi (q)$ as follows:

$$\begin{aligned} Ces_\Phi (q)= \{x\in l^0: R^q x \in l_\Phi \}= \{x\in l^0: \sigma _\Phi (rx)<\infty \,\, \text {for some}\,\, r>0\}. \end{aligned}$$

Clearly, it is a linear space and also forms a normed linear space under the norm $\Vert x\Vert _{\Phi }^L=|||R^q x|||_{\Phi }^L$ introduced with the help of the norm on $l_\Phi$. We call $Ces_\Phi (q)$ as the generalized Cesàro–Musielak–Orlicz sequence space.

The generalized class $Ces_\Phi (q)$ include the following classes in particular cases:

• (i) When $q_n=1$, $n=1, 2, \ldots$, the $Ces_\Phi (q)$ reduces to the Cesàro–Musielak–Orlicz sequence space $ces_\Phi$ studied by Wangkeeree [26], where

  $$\begin{aligned} ces_\Phi =\Big \{x\in l^0 : \displaystyle \sum _{n=1}^{\infty }\varphi _n\Big (\frac{r}{n}\displaystyle \sum _{k=1}^{n}|x(k)|\Big )<\infty \,\, \text {for some}\,\, r>0\Big \}, \end{aligned}$$

• (ii) For $\varphi _n=\varphi$, $\forall n$ the $ces_\Phi$ becomes well-known Cesàro–Orlicz sequence space $ces_\varphi$ studied recently by Cui et al. [2], Foralewski et al. [6], Petrot and Suantai [20],

• (iii) For $\varphi _n(x)=|x|^{p_n}$, $p_n\ge 1$ $\forall n$ the $Ces_\Phi (q)$ reduces to the sequence space $Ces_{(p)}(q)$ studied by Mongkolkeha and Kumam [17] and when $\varphi _n(x)=|x|^{p_n}$ with $p_n=p\ge 1$ $\forall n$ then $Ces_\Phi (q)$ reduces to the sequence space $Ces_{p}(q)$ studied by Khan [14].

We consider the subspace $(Ces_\Phi ^L(q))_a$ of $Ces_\Phi (q)$ as

$$\begin{aligned} ({Ces_\Phi (q)})_a=\bigg \{x\in Ces_\Phi (q) :\forall r>0~~\exists n_r~ \text {such that} \displaystyle \sum _{n=n_r}^{\infty }\varphi _n\Big (\frac{r}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x(k)|\Big )<\infty \bigg \}. \end{aligned}$$

In this article, we have introduced the generalized Cesàro–Musielak–Orlicz sequence space and have established the completeness property of the space and also obtained criteria for some geometric properties like coordinatewise Uniform Kadec–Klee property, uniform Opial property with respect to the Luxemberg norm.

Notations:

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

$x|_i=(x(1), x(2), x(3), \ldots , x(i), 0, 0, \ldots )$, called the truncation of $x$ at $i$,

$x|_{\mathbb {N}-i}=(0, 0, 0, \ldots , 0, x(i+1), x(i+2), \ldots )$,

$x|_I=\{x=(x(i))\in l^0: x(i)\ne 0 \,\text {for all}\, i\in I\subseteq \mathbb {N} \text{ and } x(i)=0 \,\text {for all}\, i\in \mathbb {N}\setminus I\}$,

For simplifying notations, we write $Ces_\Phi ^L(q)=(Ces_\Phi (q), \Vert .\Vert _{\Phi }^L)$.

3 Main Results

This section contains main results of our work.

Theorem 1

Let $\Phi$ be a Musielak–Orlicz function. Then the following statements are true:

• (i) $(Ces_\Phi (q), \Vert .\Vert _\Phi ^L)$ is a Banach space,

• (ii) $(Ces_\Phi ^L(q))_a$ is a closed subspace of $Ces_\Phi ^L(q)$,

• (iii) if $\Phi$ satisfies $\delta _2$-condition then $(Ces_\Phi ^L(q))_a=Ces_\Phi ^L(q)$.

Proof

Let $(x^s)_{s=1}^{\infty }$ be a Cauchy sequence in $Ces_\Phi ^L(q)$, where $x^s=(x^s(k))_{k=1}^{\infty }$ and $\varepsilon >0$ be given. Then there exists a natural number $T$ such that for every $\varepsilon >0$ one can find $r_\varepsilon$ with $r_\varepsilon <\varepsilon$, we have

$$ \sigma _{\Phi }\bigg (\frac{x^s-x^t}{r_\varepsilon }\bigg )\le 1\,\,\text {for all} \,\,s, t\ge T. $$

By definition of $\sigma _{\Phi }$ for each $l\in \mathbb {N}$, we have

$$\begin{aligned} {} \displaystyle \sum _{n=1}^{l}\varphi _n\bigg (\frac{1}{r_\varepsilon Q_n}\displaystyle \sum _{k=1}^{n}q_k|x^s(k)-x^t(k)|\bigg )\le 1 \,\,\text {for all}\,\, s, t\ge T, \end{aligned}$$

(3)

which implies that for each $l\ge n\ge 1$

$$\begin{aligned} {} \varphi _n\bigg (\frac{1}{r_\varepsilon Q_n}\displaystyle \sum _{k=1}^{n}q_k|x^s(k)-x^t(k)|\bigg )\le 1 \,\,\text {for all}\,\, s, t\ge T. \end{aligned}$$

(4)

Let $p_n$ be the corresponding kernel of the Orlicz function $\varphi _n$ for each $n$. We choose a constant $s_0>0$ and $\gamma >1$ such that $\gamma \frac{s_0}{2}p_n{(\frac{s_0}{2})}\ge 1$, for each $n\in \mathbb {N}$ (which is follows from $\varphi _n(\frac{s_0}{2})=\int _{0}^{\frac{s_0}{2}}p_n(t)\mathrm{{d}}t$ and $s_0>0$).

By the integral representation of $\varphi _n$ for each $n$, we have

$$\begin{aligned} {} \frac{1}{r_\varepsilon Q_n}\displaystyle \sum _{k=1}^{n}q_k|x^s(k)-x^t(k)|\le \gamma s_0 \,\,\text {for each}\,\, n\in \mathbb {N}\,\, \text {and for all}\,\, s, t\ge T. \end{aligned}$$

(5)

Otherwise, one can find a natural $n$ with $\frac{1}{r_\varepsilon Q_n}\displaystyle \sum _{k=1}^{n}q_k|x^s(k)-x^t(k)|> \gamma s_0$ such that

$$ {\varphi _n}{\bigg (\displaystyle \sum _{k=1}^{n}\frac{q_k|x^s(k)-x^t(k)|}{r_\varepsilon Q_n}\bigg )}\ge \int \limits _{\frac{\gamma s_0}{2}}^{\displaystyle \sum _{k=1}^{n}\frac{q_k|x^s(k)-x^t(k)|}{r_\varepsilon Q_n}}p_n(t)\mathrm{{d}}t>\frac{\gamma s_0}{2}p_n{(\frac{s_0}{2})}, $$

which contradicts (4). Hence from (5), we have $(x^s{(k)})_{s=1}^{\infty }$ is a Cauchy sequence of real numbers for each $k$ and hence converges for each $k$. Suppose for each $k\in \mathbb {N}$, $\displaystyle \lim _{t \rightarrow \infty }x^{t}(k)=x(k)$. Taking $t\rightarrow \infty$ in (3), we obtain for each $l\in \mathbb {N}$

$$ \displaystyle \sum _{n=1}^{l}\varphi _n\bigg (\frac{1}{r_\varepsilon Q_n}\displaystyle \sum _{k=1}^{n}q_k|x^s(k)-x(k)|\bigg )\le 1 \,\,\text {for all}\,\, s\ge T, $$

which implies that $\sigma _{\Phi }\big (\frac{x^s-x}{r_\varepsilon }\big )\le 1~~\text {for all}\, s\ge T$, i.e., $\Vert x^s-x\Vert _\Phi ^L\le r_\varepsilon < \varepsilon$ for all $s\ge T$. Therefore $x^s\rightarrow x$ in $\Vert .\Vert _\Phi ^L$ as $s\rightarrow \infty$. We omit the verification of $x\in Ces_\Phi ^L(q)$ as it is easy to obtain. This finishes the proof of part $(i)$.

• (ii) Clearly $({Ces_\Phi ^L(q)})_a$ is a subspace ${Ces_\Phi ^L(q)}$. It is sufficient to show that $({Ces_\Phi ^L(q)})_a$ is a closed subspace of $Ces_\Phi ^L(q)$. For this, let $x_i=(x_i(k))_{k=1}^{\infty }\in ({Ces_\Phi ^L(q)})_a$ for each $i\in \mathbb {N}$ and $\Vert x-x_i\Vert _\Phi ^L\rightarrow 0$ as $i\rightarrow \infty$ and $x\in {Ces_\Phi ^L(q)}$. We show that $x\in ({Ces_\Phi ^L(q)})_a$. By the equivalent definition of norm and modular convergence, we have $\sigma _\Phi (r(x-x_i))\rightarrow 0$ as $i\rightarrow \infty$ for all $r>0$. So for all $r>0$ there exists $J\in \mathbb {N}$ such that $\sigma _\Phi (2r(x-x_J))< 1$. Since $x_J\in ({Ces_\Phi ^L(q)})_a$ so there exists $n_J$ such that $\displaystyle \sum _{n=n_J}^{\infty }\varphi _n\Big (\frac{2r}{Q_n}\displaystyle \sum _{k=1}^{n}|q_kx_J(k)|\Big )<\infty$ $\forall r>0$. We choose $n_r=n_J$, then we have

  $$\begin{aligned}&\displaystyle \sum _{n=n_J}^{\infty }\varphi _n\Big (\frac{r}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x(k)|\Big )\\&\le \displaystyle \sum _{n=n_J}^{\infty }\varphi _n\Big (\frac{r}{2Q_n}\displaystyle \sum _{k=1}^{n}2q_k|x(k)-x_J(k)|+ \frac{r}{2Q_n}\displaystyle \sum _{k=1}^{n}2q_k|x_J(k)|\Big ) \\&\le \frac{1}{2}\displaystyle \sum _{n=n_J}^{\infty }\varphi _n\Big (\frac{2r}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x(k)-x_J(k)|\Big ) + \frac{1}{2}\displaystyle \sum _{n=n_J}^{\infty }\varphi _n\Big (\frac{2r}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x_J(k)|\Big )\\&\le \frac{1}{2} \sigma _\Phi (2r(x-x_J)) + \frac{1}{2}\displaystyle \sum _{n=n_J}^{\infty }\varphi _n\Big (\frac{2r}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x_J(k)|\Big )<\infty . \end{aligned}$$

  Since $r$ is arbitrary, we have $x\in ({Ces_\Phi ^L(q)})_a$. This completes the proof.

• (iii) We need to show here only the inclusion ${Ces_\Phi ^L(q)}\subset ({Ces_\Phi ^L(q)})_a$. Let $x\in {Ces_\Phi ^L(q)}$. Then for some $t>0$, $\sigma _\Phi (t x)<\infty$, i.e., $\displaystyle \sum _{n=1}^{\infty }\varphi _n\Big (\frac{t}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x(k)|\Big )<\infty$. We show that for any $r>0$ there exists a $n_r\in \mathbb {N}$ such that

  $$\begin{aligned} \displaystyle \sum _{n=n_r}^{\infty }\varphi _n\Big (\frac{r}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x(k)|\Big )<\infty . \end{aligned}$$

  If $r\in [0, t]$ then it is easily follows from

  $$\begin{aligned} \displaystyle \sum _{n=n_r}^{\infty }\varphi _n\Big (\frac{r}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x(k)|\Big )\le \displaystyle \sum _{n=n_r}^{\infty }\varphi _n\Big (\frac{t}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x(k)|\Big )<\infty . \end{aligned}$$

Now, we fix $t$ and choose $r>t$. Since $x\in {Ces_\Phi ^L(q)}$, i.e., for some $t>0$, $\sigma _\Phi (t x)<\infty$, so there exists $n_r$ and a constant $a$ such that

$$\begin{aligned} \displaystyle \sum _{n=n_r}^{\infty }\varphi _n\Big (\frac{t}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x(k)|\Big )<\frac{a}{2}. \end{aligned}$$

Therefore for each $n\ge n_r$, we have

$$\begin{aligned} \varphi _n\Big (\frac{t}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x(k)|\Big )<\frac{a}{2}. \end{aligned}$$

Choose a sequence $(c_n)_{n=1}^{\infty }$ of positive real numbers such that $\displaystyle \sum _{n=1}^{\infty }c_n<\infty$. So for a given $\varepsilon >0$, there exists a $n_r$ such that $\displaystyle \sum _{n=n_r}^{\infty }c_n<\frac{\varepsilon }{2}$. Let $u=\frac{t}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x(k)|$, $K>0$ be a constant and $a$ is chosen above. Since $r>t$ so there is a $l\in \mathbb {N}$ such that $r\le 2^lt$. Applying $\delta _2$-condition for all $n\ge n_r$, we have

$$\begin{aligned} \varphi _n\Big (\frac{r}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x(k)|\Big )&\le \varphi _n\Big (\frac{2^lt}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x(k)|\Big )\le K^l\varphi _n\Big (\frac{t}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x(k)|\Big )\\&\quad \,+ \Big (\displaystyle \sum _{i=0}^{l-1}K^i\Big ) c_n \end{aligned}$$

Taking summation on both sides over $n\ge n_r$, we obtain

$$\begin{aligned} \displaystyle \sum _{n=n_r}^{\infty }\varphi _n\Big (\frac{r}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x(k)|\Big )\le K^l\displaystyle \sum _{n=n_r}^{\infty }\varphi _n\Big (\frac{t}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x(k)|\Big )+ \Big (\displaystyle \sum _{i=0}^{l-1}K^i\Big ) \displaystyle \sum _{n=n_r}^{\infty }c_n< \infty . \end{aligned}$$

Hence $x\in ({Ces_\Phi ^L(q)})_a$.

We assume in the rest of this work that Musielak–Orlicz function $\Phi =(\varphi _n)$ with all $\varphi _n$ being finitely valued. The following known lemmas are useful in the sequel:

Lemma 1

Let $x\in ({Ces_\Phi ^L(q)})_a$ be an arbitrary element. Then $\Vert x\Vert _\Phi ^L=1$ if and only if $\sigma _\Phi (x)=1$.

Proof

The proof will run on the parallel lines of the proof of Lemma 2.1 in [2].

Lemma 2

Suppose $\Phi \in \delta _2$ and $\Phi >0$. Then for any sequence $(x_l)$ in ${Ces_\Phi ^L(q)}$, $\Vert x_l\Vert _\Phi ^L\rightarrow 0$ if and only if $\sigma _\Phi (x_l)\rightarrow 0$.

Proof

For the proof of this lemma see [7, 13].

Lemma 3

If $\Phi \in \delta _2$, i.e., (1), then for any $x\in {Ces_\Phi ^L(q)}$,

$$\begin{aligned} \Vert x\Vert _\Phi ^L=1 \,\,\text {if and only if}\,\,\sigma _\Phi (x)=1. \end{aligned}$$

Proof

Since $\Phi \in \delta _2$ implies ${Ces_\Phi ^L(q)}=({Ces_\Phi ^L(q)})_a$. The proof follows from Lemma 1.

Lemma 4

Let $\Phi \in \delta _2$, i.e., (1) and satisfies the condition $(*)$, i.e., (2). Then for any $x\in {Ces_\Phi ^L(q)}$ and every $\varepsilon \in (0,1)$ there exists $\delta (\varepsilon )\in (0, 1)$ such that $\sigma _\Phi (x)\le 1-\varepsilon$ implies $\Vert x\Vert _\Phi ^L\le 1-\delta$.

Proof

The proof of this lemma will be in a way similar to that of the proof of Lemma $9$ in [13].

Lemma 5

[13] Let $(X, \Vert .\Vert )$ be normed space. If $f:X\rightarrow \mathbb {R}$ is a convex function in the set $K(0, 1)=\{x\in X: \Vert x\Vert \le 1\}$ and $|f(x)|\le M$ for all $x\in K(0, 1)$ and some $M>0$ then $f$ is almost uniformly continuous in $K(0, 1)$; i.e., for all $d\in (0,1)$ and $\varepsilon >0$ there exists a $\delta >0$ such that $\Vert y\Vert \le d$ and $\Vert x-y\Vert <\delta$ implies $|f(x)-f(y)|<\varepsilon$ for all $x, y \in K(0, 1)$.

Lemma 6

Let $\Phi \in \delta _2$, i.e., (1), $\Phi >0$ and satisfies the condition $(*)$, i.e., (2). Then for each $d\in (0, 1)$ and $\varepsilon >0$ there exists $\delta =\delta (d, \varepsilon )>0$ such that $\sigma _\Phi (x)\le d$, $\sigma _\Phi (y)\le \delta$ imply

$$\begin{aligned} |\sigma _\Phi (x+y)-\sigma _\Phi (x)|<\varepsilon \text{ for } \text{ any } x, y\in {Ces_\Phi ^L(q)}. \end{aligned}$$

(6)

Proof

Since $\Phi \in \delta _2$ and satisfies condition $(*)$, so by Lemma 4, there exists $d_1\in (0,1)$ such that $\Vert x\Vert _\Phi ^L\le d_1$. Also by Lemma 2, we find a $\delta >0$ such that for every $\delta _1>0$, $\sigma _\Phi (y)\le \delta$ implies $\Vert y\Vert _\Phi ^L\le \delta _1$ for any $y\in {Ces_\Phi ^L(q)}$. So, if $\sigma _\Phi (x)\le d$ and $\sigma _\Phi (y)\le \delta$ then $\Vert x\Vert _\Phi ^L\le d_1$ and $\Vert y\Vert _\Phi ^L\le \delta _1$. Hence by Lemma 5, we have $|\sigma _\Phi (x+y)-\sigma _\Phi (x)|<\varepsilon$ because the functional $\sigma _\Phi$ satisfies all the assumptions of $f$ defined in Lemma 5.

Lemma 7

Let $\Phi \in \delta _2$, i.e., (1) and satisfies the condition $(*)$, i.e., (2) and $\Phi >0$. Then for any $x\in {Ces_\Phi ^L(q)}$ and any $\varepsilon >0$ there exists $\delta =\delta (\varepsilon )>0$ such that $\sigma _\Phi (x)\ge 1+\varepsilon$ implies $\Vert x\Vert _\Phi ^L\ge 1+\delta$.

Proof

The proof of this lemma is parallel to the proof of the Lemma 4 in [3].

Theorem 2

Let $\Phi >0$ be a Musielak–Orlicz function satisfying condition $\delta _2$, i.e., (1) and $(*)$, i.e., (2). Then sequence space ${Ces_\Phi ^L(q)}$ has the ${\text {UKK}}_c$-property.

Proof

Since $\Phi >0$ and it satisfies the condition $\delta _2$, so by Lemma 2, for a given $\varepsilon >0$ there exist a $\eta >0$, we have

$$\begin{aligned} {} \Vert x\Vert _\Phi ^L\ge \frac{\varepsilon }{4}\Rightarrow \sigma _\Phi (x)\ge \eta . \end{aligned}$$

(7)

With this $\eta >0$, by Lemma 4, one can find a $\delta \in (0, 1)$ such that

$$\begin{aligned} {} \Vert x\Vert _\Phi ^L> 1-\delta \Rightarrow \sigma _\Phi (x)> 1-\eta . \end{aligned}$$

(8)

Let $(x_l)\subset B(Ces_\Phi ^L(q))$, $\Vert x_l\Vert _\Phi ^L\rightarrow \Vert x\Vert _\Phi ^L$, $x_l(i)\rightarrow x(i)$ for all $i\in \mathbb {N}$ and $sep(x_l)\ge \varepsilon$. We show that there exists a $\delta >0$ such that $\Vert x\Vert _\Phi ^L\le 1-\delta$. If possible, let $\Vert x\Vert _\Phi ^L> 1-\delta$. Then one can select a finite set $I=\{1, 2, \ldots , N-1\}$ on which $\Vert x|_I\Vert _\Phi ^L> 1-\delta$. Since $x_l(i)\rightarrow x(i)$ for each $i\in \mathbb {N}$, so we obtain $x_l\rightarrow x$ uniformly on $I$. Consequently, by assumption $\Vert x_l\Vert _\Phi ^L\rightarrow \Vert x\Vert _\Phi ^L$ there exists $l_N\in \mathbb {N}$ such that

$$\begin{aligned} \Vert x_l|_I\Vert _\Phi ^L> 1-\delta \,\,\text {and}\,\, \Vert (x_l-x_m)|_{I}\Vert _\Phi ^L\le \frac{\varepsilon }{2} \,\,\text {for all}\,\, l, m\ge l_N. \end{aligned}$$

Using Eq. (8), first one of the above inequalities implies that $\sigma _\Phi (x_l|_I)> 1-\eta$ for $l\ge l_N$. Since $sep(x_l)\ge \varepsilon$, i.e., $\Vert x_l-x_m\Vert _\Phi ^L\ge \varepsilon$, so second one of the above inequalities implies that $\Vert (x_l-x_m)|_{\mathbb {N}-I}\Vert _\Phi ^L$ $\ge \frac{\varepsilon }{2}$ for $l, m\ge l_N, l\ne m$. Hence for $N\in \mathbb {N}$ there exists a $l_N$ such that $\Vert x_{l_N}|_{\mathbb {N}-I}\Vert _\Phi ^L\ge \frac{\varepsilon }{4}$. Without loss of generality, we assume that $\Vert x_l|_{\mathbb {N}-I}\Vert _\Phi ^L\ge \frac{\varepsilon }{4}$ for all $l, N\in \mathbb {N}$. Therefore by (7), we have $\sigma _\Phi (x_l|_{\mathbb {N}-I})\ge \eta$.

By the integral representation of Musielak–Orlicz function $\Phi$, we have $\varphi _n(u+v)\ge \varphi _n(u)+ \varphi _n(v)$ for each $n$ and all $u, v\in \mathbb {R^+}$. Using this, we obtain $\sigma _\Phi (x_l|_{I})+\sigma _\Phi (x_l|_{\mathbb {N}-I})\le \sigma _\Phi (x_l)\le 1$. This implies that $\sigma _\Phi (x_l|_{\mathbb {N}-I})\le 1-\sigma _\Phi (x_l|_{I})< 1-(1-\eta )=\eta$, i.e., $\sigma _\Phi (x_l|_{\mathbb {N}-I})<\eta$, which contradicts to the fact that $\sigma _\Phi (x_l|_{\mathbb {N}-I})\ge \eta$. This finishes the proof.

Theorem 3

Let $\Phi >0$ be a Musielak–Orlicz function satisfying condition $\delta _2$, i.e., (1) and $(*)$, i.e., (2). Then $Ces_\Phi ^L(q)$ has the uniform Opial property.

Proof

Let $(x_l)\subset S(Ces_\Phi ^L(q))$ be any weakly null sequence and $\varepsilon >0$ be given. We show that for any $\varepsilon >0$ there is a $\mu >0$ such that

$$\begin{aligned} \displaystyle \liminf _{l\rightarrow \infty }\Vert x_l+x\Vert _\Phi ^L\ge 1+\mu , \end{aligned}$$

for each $x\in Ces_\Phi ^L(q)$ satisfying $\Vert x\Vert _\Phi ^L\ge \varepsilon$. Since $\Phi \in \delta _2$ and $\Phi >0$, so by Lemma 2, for each $\varepsilon >0$ there is a number $\delta \in (0, 1)$ such that for each $x\in Ces_\Phi ^L(q)$, we have $\sigma _\Phi (x)\ge \delta$. Since $\Phi$ $(>0)$ satisfies the condition $\delta _2$, and the condition $(*)$, so by Lemma 6 for any $\varepsilon >0$, there exists $\delta _1\in (0, \delta )$ such that $\sigma _\Phi (u)\le 1$, $\sigma _\Phi (v)\le \delta _1$ imply

$$\begin{aligned} {} |\sigma _\Phi (u+v)-\sigma _\Phi (u)|<\frac{\delta }{6} \,\,\text {for any}\,\, u, v\in Ces_\Phi ^L(q). \end{aligned}$$

(9)

Since $\sigma _\Phi (x)<\infty$, so there is a number $n_0\in \mathbb {N}$ such that

$$\begin{aligned} {} \displaystyle \sum _{n=n_0 + 1}^{\infty }\varphi _n\Big (\frac{1}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x(k)|\Big )\le \frac{\delta _1}{6}. \end{aligned}$$

(10)

From Eq. (10) it follows that

$$\begin{aligned} \delta&\le \displaystyle \sum _{n=1}^{n_0}\varphi _n\Big (\frac{1}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x(k)|\Big ) + \displaystyle \sum _{n=n_0+1}^{\infty }\varphi _n\Big (\frac{1}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x(k)|\Big ) \\&\le \displaystyle \sum _{n=1}^{n_0}\varphi _n\Big (\frac{1}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x(k)|\Big ) + \frac{\delta _1}{6}, \end{aligned}$$

which implies $\displaystyle \sum _{n=1}^{n_0}\varphi _n\Big (\frac{1}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x(k)|\Big )\ge \delta -\frac{\delta _1}{6}>\delta -\frac{\delta }{6}=\frac{5\delta }{6}.$ Since $x_l\rightarrow 0$ weakly, i.e., $x_l(i)\rightarrow 0$ for each $i$, so there exists a $l_0$ such that for all $l\ge l_0$, the last inequality yields

$$\begin{aligned} {} \displaystyle \sum _{n= 1}^{n_0}\varphi _n\Big (\frac{1}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x_l(k)+ x(k)|\Big )\ge \frac{5\delta }{6}. \end{aligned}$$

(11)

Also by $x_l\rightarrow 0$ weakly, we can choose an $n_0$ such that $\sigma _\Phi (x_l|_{n_0})\rightarrow 0$ as $l\rightarrow \infty$. So there exists a $l_1>l_0$ such that $\sigma _\Phi (x_l|_{n_0})\le \delta _1$ for all $l\ge l_1$. Since $(x_l)\subset S(Ces_\Phi ^L(q))$, i.e., $\Vert x_l\Vert _\Phi ^L=1$, so by Lemma 3, we have $\sigma _\Phi (x_l)=1$, which implies that there exists $n_0$ such that $\sigma _\Phi (x_l|_{\mathbb {N}-n_0})\le 1$. Now choose $u=x_l|_{\mathbb {N}-n_0}$ and $v=x_l|_{n_0}$. Then $u, v\in Ces_\Phi ^L(q)$, $\sigma _\Phi (u)\le 1$, $\sigma _\Phi (v)\le \delta _1$. So from (9), for all $l\ge l_1$ we have

$$\begin{aligned} \big |\sigma _\Phi (x_l|_{\mathbb {N}-n_0}+ x_l|_{n_0}\big )-\sigma _\Phi \big (x_l|_{\mathbb {N}-n_0}\big )\big |<\frac{\delta }{6}, \end{aligned}$$

which implies that $\sigma _\Phi (x_l)-\frac{\delta }{6}<\sigma _\Phi \big (x_l|_{\mathbb {N}-n_0}\big )$ for all $l\ge l_1$, i.e.,

$\displaystyle \sum _{n=n_0+1}^{\infty }\varphi _n\Big (\frac{1}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x_l(k)|\Big )$ $> 1-\frac{\delta }{6}$ for all $l\ge l_1$. Again, since $\sigma _\Phi \big (x_l|_{\mathbb {N}-n_0}\big )\le 1$ and $\sigma _\Phi \big (x|_{\mathbb {N}-n_0}\big )\le \frac{\delta _1}{6}<\delta _1$, so from the Eqs. (9) and (11), we obtain

$$\begin{aligned} \sigma _\Phi (x_l+x) =&\displaystyle \sum _{n=1}^{n_0}\varphi _n\Big (\frac{1}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x_l(k)+x(k)|\Big )\\&+ \displaystyle \sum _{n=n_0+1}^{\infty }\varphi _n\Big (\frac{1}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x_l(k)+x(k)|\Big ) \\&>\displaystyle \sum _{n=1}^{n_0}\varphi _n\Big (\frac{1}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x_l(k)+x(k)|\Big )\\&+ \displaystyle \sum _{n=n_0+1}^{\infty }\varphi _n\Big (\frac{1}{Q_n}\displaystyle \sum _{k=1}^{n}q_k|x_l(k)|\Big )-\frac{\delta }{6}\\&> \frac{5\delta }{6}+ \Big (1-\frac{\delta }{6}\Big )-\frac{\delta }{6}=1+ \frac{\delta }{2}. \end{aligned}$$

Since $\Phi \in \delta _2$ and satisfies the condition $(*)$ and $\Phi >0$, so by Lemma 7 there is a $\mu >0$ depending only on $\delta$ such that $\Vert x_l+x\Vert _\Phi ^L> 1+ \mu$. Hence $\displaystyle \liminf _{l\rightarrow \infty }\Vert x_l+x\Vert _\Phi ^L\ge 1+\mu$. This completes the proof.

Corollary 1

• (i) If $\varphi _n=\varphi$, $q_n=1$ $\forall n$ and $\Phi \in \delta _2$, then Cesàro–Orlicz sequence space $ces_\varphi ^L$ [20] has the uniform Opial property.

• (ii) Suppose $q_n=1$, $n=1, 2, \ldots$ and $\varphi _n(u)=|u|^{p_n}$ for all $u\in \mathbb {R}$, $1< p_n<\infty$ $\forall n$. Then it is easy to verify that $\Phi \in \delta _2$ if and only if $\displaystyle \limsup _{n\rightarrow \infty }p_n<\infty$. Therefore $ces_{(p)}^L$ [21] has the uniform Opial property.

• (iii) If $\varphi _n(u)=|u|^{p_n}$, $1\le p_n<\infty$ $\forall n$ and $\displaystyle \limsup _{n\rightarrow \infty }p_n<\infty$, then $Ces_{(p)}^L(q)$ has the uniform Opial property [17].

4 Conclusion

In this study, we have obtained geometric properties such as coordinatewise uniformly Kadec–Klee property and uniform Opial property in the generalized Cesàro–Musielak–Orlicz sequence spaces, which include the well known Cesàro [24], generalized Cesàro [21], Cesàro–Orlicz [2], Cesàro–Musielak–Orlicz [26] classes of sequences in particular cases with respect to the Luxemberg norm. In future, our plan is to obtain these results for a more generalized class of sequences with respect to both the Luxemberg and Amemiya norm.