# Some geometric properties of a new modular space defined by Zweier operator

## Abstract

In this paper, we define the modular space ${\mathcal{Z}}_{\sigma }\left(s,p\right)$ by using the Zweier operator and a modular. Then, we consider it equipped with the Luxemburg norm and also examine the uniform Opial property and property β. Finally, we show that this space has the fixed point property.

MSC:40A05, 46A45, 46B20.

## Dedication

Dedicated to Professor Hari M Srivastava

## 1 Introduction

In literature, there are many papers about the geometrical properties of different sequence spaces such as . Opial  introduced the Opial property and proved that the sequence spaces ${\ell }_{p}$ ($1) have this property but ${L}_{p}\left[0,2\pi \right]$ ($p\ne 2$, $1) does not have it. Franchetti  showed that any infinite dimensional Banach space has an equivalent norm that satisfies the Opial property. Later, Prus  introduced and investigated the uniform Opial property for Banach spaces. The Opial property is important because Banach spaces with this property have the weak fixed point property.

## 2 Definition and preliminaries

Let $\left(X,\parallel \cdot \parallel \right)$ be a real Banach space and let $S\left(X\right)$ (resp. $B\left(X\right)$) be the unit sphere (resp. the unit ball) of X. A Banach space X has the Opial property if for any weakly null sequence $\left\{{x}_{n}\right\}$ in X and any x in $X\setminus \left\{0\right\}$, the inequality ${lim}_{n\to \mathrm{\infty }}inf\parallel x\parallel <{lim}_{n\to \mathrm{\infty }}inf\parallel {x}_{n}+x\parallel$ holds. We say that X has the uniform Opial property if for any $\epsilon >0$ there exists $r>0$ such that for any $x\in X$ with $\parallel x\parallel \ge \epsilon$ and any weakly null sequence $\left\{{x}_{n}\right\}$ in the unit sphere of X, the inequality $1+r\le {lim}_{n\to \mathrm{\infty }}inf\parallel {x}_{n}+x\parallel$ holds.

For a bounded set $A\subset X$, the ball-measure of noncompactness was defined by . The function Δ defined by is called the modulus of noncompact convexity. A Banach space X is said to have property $\left(L\right)$, if ${lim}_{\epsilon \to {1}^{-}}\mathrm{\Delta }\left(\epsilon \right)=1$. This property is an important concept in the fixed point theory and a Banach space X possesses property $\left(L\right)$ if and only if it is reflexive and has the uniform Opial property.

A Banach space X is said to satisfy the weak fixed point property if every nonempty weakly compact convex subset C and every nonexpansive mapping $T:C\to C\left(\parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\mathrm{\forall }x,y\in C\right)$ have a fixed point, that is, there exists $x\in C$ such that $T\left(x\right)=x$. Property $\left(L\right)$ and the fixed point property were also studied by Goebel and Kirk , Toledano et al. , Benavides , Benavides and Phothi . A Banach space X is said to have property $\left(H\right)$ if every weakly convergent sequence on the unit sphere is convergent in norm. Clarkson  introduced the uniform convexity, and it is known that the uniform convexity implies the reflexivity of Banach spaces. Huff  introduced the concept of nearly uniform convexity of Banach spaces. A Banach space X is called uniformly convex (UC) if for each $\epsilon >0$, there is $\delta >0$ such that for $x,y\in S\left(X\right)$, the inequality $\parallel x-y\parallel >\epsilon$ implies that $\parallel \frac{1}{2}\left(x+y\right)\parallel <1-\delta$. For any $x\notin B\left(X\right)$, the drop determined by x is the set $D\left(x,B\left(X\right)\right)=conv\left(\left\{x\right\}\cup B\left(X\right)\right)$. A Banach space X has the drop property $\left(D\right)$ if for every closed set C disjoint with $B\left(X\right)$, there exists an element $x\in C$ such that $D\left(x,B\left(X\right)\right)\cap C=\left\{x\right\}$. Rolewicz  showed that the Banach space X is reflexive if X has the drop property. Later, Montesinos  extended this result and proved that X has the drop property if and only if X is reflexive and has property $\left(H\right)$. A sequence $\left\{{x}_{n}\right\}$ is said to be ε-separated sequence for some $\epsilon >0$ if

$\mathit{sep}\left({x}_{n}\right)=inf\left\{\parallel {x}_{n}-{x}_{m}\parallel :n\ne m\right\}>\epsilon .$

A Banach space X is called nearly uniformly convex (NUC) if for every $\epsilon >0$, there exists $\delta \in \left(0,1\right)$ such that for every sequence $\left({x}_{n}\right)\subseteq B\left(X\right)$ with $\mathit{sep}\left({x}_{n}\right)>\epsilon$, we have $conv\left({x}_{n}\right)\cap \left(\left(1-\delta \right)B\left(X\right)\right)\ne \mathrm{\varnothing }$. Huff  proved that every (NUC) Banach space is reflexive and has property $\left(H\right)$. A Banach space X has property $\left(\beta \right)$ if and only if for each $\epsilon >0$, there exists $\delta >0$ such that for each element $x\in B\left(X\right)$ and each sequence $\left({x}_{n}\right)$ in $B\left(X\right)$ with $\mathit{sep}\left({x}_{n}\right)\ge \epsilon$, there is an index k for which $\parallel \frac{x+{x}_{k}}{2}\parallel <1-\delta$.

For a real vector space X, a function $\rho :X\to \left[0,\mathrm{\infty }\right]$ is called a modular if it satisfies the following conditions:

1. (i)

$\rho \left(x\right)=0$ if and only if $x=0$,

2. (ii)

$\rho \left(\alpha x\right)=\rho \left(x\right)$ for all scalar α with $|\alpha |=1$,

3. (iii)

$\rho \left(\alpha x+\beta y\right)\le \rho \left(x\right)+\rho \left(y\right)$ for all $x,y\in X$ and all $\alpha ,\beta \ge 0$ with $\alpha +\beta =1$.

The modular ρ is called convex if

1. (iv)

$\rho \left(\alpha x+\beta y\right)\le \alpha \rho \left(x\right)+\beta \rho \left(y\right)$ for all $x,y\in X$ and all $\alpha ,\beta \ge 0$ with $\alpha +\beta =1$.

For any modular ρ on X, the space

is called a modular space. In general, the modular is not subadditive and thus it does not behave as a norm or a distance. But we can associate the modular with an F-norm. A functional $\parallel \cdot \parallel :X\to \left[0,\mathrm{\infty }\right]$ defines an F-norm if and only if

1. (i)

$\parallel x\parallel =0⇔x=0$,

2. (ii)

$\parallel \alpha x\parallel =\parallel x\parallel$ whenever $|\alpha |=1$,

3. (iii)

$\parallel x+y\parallel \le \parallel x\parallel +\parallel y\parallel$,

4. (iv)

if ${\alpha }_{n}\to \alpha$ and $\parallel {x}_{n}-x\parallel \to 0$, then $\parallel {\alpha }_{n}{x}_{n}-\alpha x\parallel \to 0$.

F-norm defines a distance on X by $d\left(x,y\right)=\parallel x-y\parallel$. If the linear metric space $\left(X,d\right)$ is complete, then it is called an F-space. The modular space ${X}_{\rho }$ can be equipped with the following F-norm:

$\parallel x\parallel =inf\left\{\alpha >0:\rho \left(\frac{x}{\alpha }\right)\le \alpha \right\}.$

If the modular ρ is convex, then the equality $\parallel x\parallel =inf\left\{\alpha >0:\rho \left(\frac{x}{\alpha }\right)\le 1\right\}$ defines a norm which is called the Luxemburg norm.

A modular ρ is said to satisfy the ${\delta }_{2}$-condition if for any $\epsilon >0$, there exist constants $K\ge 2$, $a>0$ such that $\rho \left(2u\right)\le K\rho \left(u\right)+\epsilon$ for all $u\in {X}_{\rho }$ with $\rho \left(u\right)\le a$. If ρ provides the ${\delta }_{2}$-condition for any $a>0$ with $K\ge 2$ dependent on a, then ρ provides the strong ${\delta }_{2}$-condition (briefly $\rho \in {\delta }_{2}^{s}$).

Let us denote by ${\ell }^{0}$ the space of all real sequences. The Cesàro sequence spaces

${\mathit{Ces}}_{p}=\left\{x\in {\ell }^{0}:\sum _{n=1}^{\mathrm{\infty }}{\left({n}^{-1}\sum _{i=1}^{n}|{x}_{i}|\right)}^{p}<\mathrm{\infty }\right\},\phantom{\rule{1em}{0ex}}1\le p<\mathrm{\infty },$

and

${\mathit{Ces}}_{\mathrm{\infty }}=\left\{x\in {\ell }^{0}:\underset{n}{sup}{n}^{-1}\sum _{i=1}^{n}|{x}_{i}|<\mathrm{\infty }\right\},$

were introduced by Shiue . Jagers  determined the Köthe duals of the sequence space ${\mathit{Ces}}_{p}$ ($1). It can be shown that the inclusion ${\ell }_{p}\subset {\mathit{Ces}}_{p}$ is strict for $1 although it does not hold for $p=1$. Also, Suantai  defined the generalized Cesàro sequence space by

where $\rho \left(x\right)={\sum }_{n=1}^{\mathrm{\infty }}{\left(\frac{1}{n}{\sum }_{i=1}^{n}|x\left(i\right)|\right)}^{{p}_{n}}$. If $p=\left({p}_{n}\right)$ is bounded, then

$\mathit{ces}\left(p\right)=\left\{x=\left({x}_{k}\right):\sum _{n=1}^{\mathrm{\infty }}{\left({n}^{-1}\sum _{i=1}^{n}|x\left(i\right)|\right)}^{{p}_{n}}<\mathrm{\infty }\right\}.$

The sequence space $C\left(s,p\right)$ was defined by Bilgin  as follows:

$C\left(s,p\right)=\left\{x=\left({x}_{k}\right):\sum _{r=0}^{\mathrm{\infty }}{\left({2}^{-r}\sum _{r}{k}^{-s}|{x}_{k}|\right)}^{{p}_{r}}<\mathrm{\infty },s\ge 0\right\}$

for $p=\left({p}_{r}\right)$ with $inf{p}_{r}>0$, where ${\sum }_{r}$ denotes a sum over the ranges ${2}^{r}\le k<{2}^{r+1}$. The special case of $C\left(s,p\right)$ for $s=0$ is the space

$\mathit{Ces}\left(p\right)=\left\{x=\left({x}_{k}\right):\sum _{r=0}^{\mathrm{\infty }}{\left({2}^{-r}\sum _{r}|{x}_{k}|\right)}^{{p}_{r}}<\mathrm{\infty }\right\}$

which was introduced by Lim . Also, the inclusion $\mathit{Ces}\left(p\right)\subseteq C\left(s,p\right)$ holds. A paranorm on $C\left(s,p\right)$ is given by

$\rho \left(x\right)={\left(\sum _{r=0}^{\mathrm{\infty }}{\left({2}^{-r}\sum _{r}{k}^{-s}|{x}_{k}|\right)}^{{p}_{r}}\right)}^{1/M}$

for $M=max\left(1,H\right)$ and $H=sup{p}_{r}<\mathrm{\infty }$.

The Z-transform of a sequence $x=\left({x}_{k}\right)$ is defined by ${\left(Zx\right)}_{n}={y}_{n}=\alpha {x}_{n}+\left(1-\alpha \right){x}_{n-1}$ by using the Zweier operator

where is the field of all complex or real numbers. The Zweier operator was studied by Şengönül and Kayaduman .

Now we introduce a new modular sequence space ${\mathcal{Z}}_{\sigma }\left(s,p\right)$ by

where $\sigma \left(x\right)={\sum }_{r=0}^{\mathrm{\infty }}{\left({2}^{-r}{\sum }_{r}{k}^{-s}|\alpha {x}_{k}+\left(1-\alpha \right){x}_{k-1}|\right)}^{{p}_{r}}<\mathrm{\infty }$ and $s\ge 0$. If we take $\alpha =1$, then ${\mathcal{Z}}_{\sigma }\left(s,p\right)=C\left(s,p\right)$; if $\alpha =1$ and $s=0$, then ${\mathcal{Z}}_{\sigma }\left(s,p\right)=\mathit{Ces}\left(p\right)$. It can be easily seen that $\sigma :{\mathcal{Z}}_{\sigma }\left(s,p\right)\to \left[0,\mathrm{\infty }\right]$ is a modular on ${\mathcal{Z}}_{\sigma }\left(s,p\right)$. We define the Luxemburg norm on the sequence space ${\mathcal{Z}}_{\sigma }\left(s,p\right)$ as follows:

$\parallel x\parallel =inf\left\{t>0:\sigma \left(\frac{x}{t}\right)\le 1\right\},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in {\mathcal{Z}}_{\sigma }\left(s,p\right).$

It is easy to see that the space ${\mathcal{Z}}_{\sigma }\left(s,p\right)$ is a Banach space with respect to the Luxemburg norm.

Throughout the paper, suppose that $p=\left({p}_{r}\right)$ is bounded with ${p}_{r}>1$ for all $r\in \mathbb{N}$ and for $i\in \mathbb{N}$ and $x\in {\ell }^{0}$. In addition, we will require the following inequalities:

$|{a}_{k}+{b}_{k}{|}^{{p}_{k}}\le C\left(|{a}_{k}{|}^{{p}_{k}}+|{b}_{k}{|}^{{p}_{k}}\right),\phantom{\rule{2em}{0ex}}|{a}_{k}+{b}_{k}{|}^{{t}_{k}}\le |{a}_{k}{|}^{{t}_{k}}+|{b}_{k}{|}^{{t}_{k}},$

where ${t}_{k}=\frac{{p}_{k}}{M}\le 1$ and $C=max\left\{1,{2}^{H-1}\right\}$ with $H=sup{p}_{k}$.

## 3 Main results

Since ${\ell }_{p}$ is reflexive and convex, $\ell \left(p\right)$-type spaces have many useful applications, and it is natural to consider a geometric structure of these spaces. From this point of view, we generalized the space $C\left(s,p\right)$ by using the Zweier operator and then obtained the equality ${\mathcal{Z}}_{\sigma }\left(s,p\right)=\mathit{Ces}\left(p\right)$, that is, it was seen that the structure of the space $\mathit{Ces}\left(p\right)$ was preserved. In this section, our goal is to investigate a geometric structure of the modular space ${\mathcal{Z}}_{\sigma }\left(s,p\right)$ related to the fixed point theory. For this, we will examine property $\left(\beta \right)$ and the uniform Opial property for ${\mathcal{Z}}_{\sigma }\left(s,p\right)$. Finally, we will give some fixed point results. To do this, we need some results which are important in our opinion.

Lemma 3.1 

If $\sigma \in {\delta }_{2}^{s}$, then for any $L>0$ and $\epsilon >0$, there exists $\delta >0$ such that

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

where $u,v\in {X}_{\sigma }$ with $\sigma \left(u\right)\le L$ and $\sigma \left(v\right)\le \delta$.

Lemma 3.2 

If $\sigma \in {\delta }_{2}^{s}$, convergence in norm and in modular are equivalent in ${X}_{\sigma }$.

Lemma 3.3 

If $\sigma \in {\delta }_{2}^{s}$, then for any $\epsilon >0$, there exists $\delta =\delta \left(\epsilon \right)>0$ such that $\parallel x\parallel \ge 1+\delta$ implies $\sigma \left(x\right)\ge 1+\epsilon$.

Now we give the following two lemmas without proof.

Lemma 3.4 If ${\parallel x\parallel }_{L}<1$ for any $x\in {\mathcal{Z}}_{\sigma }\left(s,p\right)$, then $\sigma \left(x\right)\le {\parallel x\parallel }_{L}$.

Lemma 3.5 For any $x\in {\mathcal{Z}}_{\sigma }\left(s,p\right)$, ${\parallel x\parallel }_{L}=1$ if and only if $\sigma \left(x\right)=1$.

Lemma 3.6 If $liminf{p}_{r}>1$, then for any $x\in {\mathcal{Z}}_{\sigma }\left(s,p\right)$, there exist ${k}_{0}\in \mathbb{N}$ and $\mu \in \left(0,1\right)$ such that

$\sigma \left(\frac{{x}^{k}}{2}\right)\le \frac{1-\mu }{2}\sigma \left({x}^{k}\right)$

for all $k\in \mathbb{N}$ with $k\ge {k}_{0}$, where and ${2}^{r}\le k<{2}^{r+1}$.

Proof Let $k\in \mathbb{N}$ be fixed. Then there exists ${r}_{k}\in \mathbb{N}$ such that $k\in {I}_{{r}_{k}}$. Let γ be a real number $1<\gamma \le liminf{p}_{r}$, and so there exists ${k}_{0}\in \mathbb{N}$ such that $\gamma <{p}_{{r}_{k}}$ for all $k\ge {k}_{0}$. Choose $\mu \in \left(0,1\right)$ such that ${\left(\frac{1}{2}\right)}^{\gamma }\le \frac{1-\mu }{2}$. Therefore, we have

$\begin{array}{rl}\sigma \left(\frac{{x}^{k}}{2}\right)& =\sum _{r=0}^{\mathrm{\infty }}{\left({2}^{-r}\sum _{r}{k}^{-s}|\frac{\alpha x\left(k\right)+\left(1-\alpha \right)x\left(k-1\right)}{2}|\right)}^{{p}_{r}}\\ =\sum _{r=0}^{\mathrm{\infty }}{\left(\frac{1}{2}\right)}^{{p}_{r}}{\left({2}^{-r}\sum _{r}{k}^{-s}|\alpha x\left(k\right)+\left(1-\alpha \right)x\left(k-1\right)|\right)}^{{p}_{r}}\\ \le {\left(\frac{1}{2}\right)}^{\gamma }\sum _{r=0}^{\mathrm{\infty }}{\left({2}^{-r}\sum _{r}{k}^{-s}|\alpha x\left(k\right)+\left(1-\alpha \right)x\left(k-1\right)|\right)}^{{p}_{r}}\\ <\frac{1-\mu }{2}\sigma \left({x}^{k}\right)\end{array}$

for each $x\in {\mathcal{Z}}_{\sigma }\left(s,p\right)$ and $k\ge {k}_{0}$. □

Lemma 3.7 If $\sigma \in {\delta }_{2}^{s}$, then for any $\epsilon \in \left(0,1\right)$, there exists $\delta \in \left(0,1\right)$ such that $\sigma \left(x\right)\le 1-\epsilon$ implies $\parallel x\parallel \le 1-\delta$.

Proof Suppose that lemma does not hold. So, there exist $\epsilon >0$ and ${x}_{n}\in {\mathcal{Z}}_{\sigma }\left(s,p\right)$ such that $\sigma \left({x}_{n}\right)<1-\epsilon$ and $\frac{1}{2}\le \parallel {x}_{n}\parallel \to 1$. Take ${s}_{n}=\frac{1}{\parallel {x}_{n}\parallel -1}$, and so ${s}_{n}\to 0$ as $n\to \mathrm{\infty }$. Let $P=sup\left\{\sigma \left(2{x}_{n}\right):n\in \mathbb{N}\right\}$. There exists $D\ge 2$ such that

$\sigma \left(2u\right)\le D\sigma \left(u\right)+1$
(3.2)

for every $u\in {\mathcal{Z}}_{\sigma }\left(s,p\right)$ with $\sigma \left(u\right)<1$, since $\sigma \in {\delta }_{2}^{s}$. We have

$\sigma \left(2{x}_{n}\right)\le D\sigma \left({x}_{n}\right)+1

for all $n\in \mathbb{N}$ by (3.1). Therefore, $0 and from Lemma 3.5 we have

$\begin{array}{rl}1& =\sigma \left(\frac{{x}_{n}}{\parallel {x}_{n}\parallel }\right)=\sigma \left(2{s}_{n}{x}_{n}+\left(1-{s}_{n}\right){x}_{n}\right)\\ \le {s}_{n}\sigma \left(2{x}_{n}\right)+\left(1-{s}_{n}\right)\sigma \left({x}_{n}\right)\\ \le {s}_{n}P+\left(1-\epsilon \right)\to \left(1-\epsilon \right).\end{array}$

This is a contradiction. So, the proof is complete. □

Theorem 3.8 The space ${\mathcal{Z}}_{\sigma }\left(s,p\right)$ has property $\left(\beta \right)$.

Proof Let $\epsilon >0$ and $\left({x}_{n}\right)\subset B\left({\mathcal{Z}}_{\sigma }\left(s,p\right)\right)$ with $\mathit{sep}\left({x}_{n}\right)\ge \epsilon$ and $x\in B\left({\mathcal{Z}}_{\sigma }\left(s,p\right)\right)$. For each $l\in \mathbb{N}$, we can find ${r}_{k}\in \mathbb{N}$ such that ${2}^{{r}_{k}}\le l<{2}^{{r}_{k}+1}$. Let Since for each $i\in \mathbb{N}$, ${\left({x}_{n}\left(i\right)\right)}_{i=1}^{\mathrm{\infty }}$ is bounded, by using the diagonal method, we can find a subsequence $\left({x}_{{n}_{j}}\right)$ of $\left({x}_{n}\right)$ such that $\left({x}_{{n}_{j}}\left(i\right)\right)$ converges for each $i\in \mathbb{N}$ with $1\le i\le l$. Therefore, there exists an increasing sequence of positive integers ${t}_{l}$ such that $\mathit{sep}\left({\left({x}_{{n}_{j}}^{l}\right)}_{j\ge {t}_{l}}\right)\ge \epsilon$. Thus, there exists a sequence of positive integers ${\left({r}_{l}\right)}_{l=1}^{\mathrm{\infty }}$ with ${r}_{1}<{r}_{2}<\cdots$ such that $\parallel {x}_{{r}_{l}}^{l}\parallel \ge \frac{\epsilon }{2}$ for all $l\in \mathbb{N}$. Since $\sigma \in {\delta }_{2}^{s}$, there is $\eta >0$ such that

(3.4)

from Lemma 3.3. However, there exist ${k}_{0}\in \mathbb{N}$ and $\mu \in \left(0,1\right)$ such that

$\sigma \left(\frac{{v}^{k}}{2}\right)\le \frac{1-\mu }{2}\sigma \left({v}^{k}\right)$
(3.5)

for all $v\in {\mathcal{Z}}_{\sigma }\left(s,p\right)$ and $k\ge {k}_{0}$ by Lemma 3.6. There exists $\delta >0$ such that

$\sigma \left(y\right)\le 1-\frac{\mu \eta }{4}\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}\parallel y\parallel \le 1-\delta$
(3.6)

for any $y\in {\mathcal{Z}}_{\sigma }\left(s,p\right)$ by Lemma 3.7.

By Lemma 3.1, there exists ${\delta }_{0}$ such that

$|\sigma \left(u+v\right)-\sigma \left(u\right)|<\frac{\mu \eta }{4},$
(3.7)

where $\sigma \left(u\right)\le 1$ and $\sigma \left(v\right)\le {\delta }_{0}$. Hence, we get that $\sigma \left(x\right)\le 1$ since $x\in B\left({\mathcal{Z}}_{\sigma }\left(s,p\right)\right)$. Then there exists $k\ge {k}_{0}$ such that $\sigma \left({x}^{k}\right)\le {\delta }_{0}$. Let $u={x}_{{r}_{l}}^{l}$ and $v={x}^{l}$. Then

$\sigma \left(\frac{u}{2}\right)<1\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\sigma \left(\frac{v}{2}\right)<{\delta }_{0}.$

We obtain from (3.3) and (3.5) that

$\sigma \left(\frac{u+v}{2}\right)\le \sigma \left(\frac{u}{2}\right)+\frac{\mu \eta }{4}\le \frac{1-\mu }{2}\sigma \left(u\right)+\frac{\mu \eta }{4}.$
(3.8)

Choose ${s}_{i}={r}_{{l}_{i}}$. By the inequalities (3.2), (3.3), (3.6) and the convexity of the function $f\left(u\right)=|u{|}^{{p}_{r}}$, we have

$\begin{array}{rl}\sigma \left(\frac{x+{x}_{{s}_{k}}}{2}\right)=& \sum _{r=0}^{\mathrm{\infty }}{\left({2}^{-r}\sum _{r}{k}^{-s}|\frac{\alpha \left(x\left(k\right)+{x}_{{s}_{i}}\left(k\right)\right)+\left(1-\alpha \right)\left(x\left(k-1\right)+{x}_{{s}_{i}}\left(k-1\right)\right)}{2}|\right)}^{{p}_{r}}\\ =& \sum _{r=0}^{{r}_{k}-1}{\left({2}^{-r}\sum _{r}{k}^{-s}|\frac{\alpha \left(x\left(k\right)+{x}_{{s}_{i}}\left(k\right)\right)+\left(1-\alpha \right)\left(x\left(k-1\right)+{x}_{{s}_{i}}\left(k-1\right)\right)}{2}|\right)}^{{p}_{r}}\\ +\sum _{r={r}_{k}}^{\mathrm{\infty }}{\left({2}^{-r}\sum _{r}{k}^{-s}|\frac{\alpha \left(x\left(k\right)+{x}_{{s}_{i}}\left(k\right)\right)+\left(1-\alpha \right)\left(x\left(k-1\right)+{x}_{{s}_{i}}\left(k-1\right)\right)}{2}|\right)}^{{p}_{r}}\\ \le & \frac{1}{2}\sum _{r=0}^{{r}_{k}-1}{\left({2}^{-r}\sum _{r}{k}^{-s}|\alpha x\left(k\right)+\left(1-\alpha \right)x\left(k-1\right)|\right)}^{{p}_{r}}\\ +\frac{1}{2}\sum _{r=0}^{{r}_{k}-1}{\left({2}^{-r}\sum _{r}{k}^{-s}|\alpha {x}_{{s}_{i}}\left(k\right)+\left(1-\alpha \right){x}_{{s}_{i}}\left(k-1\right)|\right)}^{{p}_{r}}\\ +\sum _{r={r}_{k}}^{\mathrm{\infty }}{\left({2}^{-r}\sum _{r}{k}^{-s}|\frac{\alpha {x}_{{s}_{i}}\left(k\right)+\left(1-\alpha \right){x}_{{s}_{i}}\left(k-1\right)}{2}|\right)}^{{p}_{r}}+\frac{\mu \eta }{4}\\ \le & \frac{1}{2}\sum _{r=0}^{{r}_{k}-1}{\left({2}^{-r}\sum _{r}{k}^{-s}|\alpha x\left(k\right)+\left(1-\alpha \right)x\left(k-1\right)|\right)}^{{p}_{r}}\\ +\frac{1}{2}\sum _{r=0}^{{r}_{k}-1}{\left({2}^{-r}\sum _{r}{k}^{-s}|\alpha {x}_{{s}_{i}}\left(k\right)+\left(1-\alpha \right){x}_{{s}_{i}}\left(k-1\right)|\right)}^{{p}_{r}}\\ +\frac{1-\mu }{2}\sum _{r={r}_{k}}^{\mathrm{\infty }}{\left({2}^{-r}\sum _{r}{k}^{-s}|\frac{\alpha {x}_{{s}_{i}}\left(k\right)+\left(1-\alpha \right){x}_{{s}_{i}}\left(k-1\right)}{2}|\right)}^{{p}_{r}}+\frac{\mu \eta }{4}\\ \le & \frac{1}{2}\sum _{r=0}^{{r}_{k}-1}{\left({2}^{-r}\sum _{r}{k}^{-s}|\alpha x\left(k\right)+\left(1-\alpha \right)x\left(k-1\right)|\right)}^{{p}_{r}}\\ +\frac{1}{2}\sum _{r=0}^{\mathrm{\infty }}{\left({2}^{-r}\sum _{r}{k}^{-s}|\alpha {x}_{{s}_{i}}\left(k\right)+\left(1-\alpha \right){x}_{{s}_{i}}\left(k-1\right)|\right)}^{{p}_{r}}\\ -\frac{\mu }{2}\sum _{r={r}_{k}}^{\mathrm{\infty }}{\left({2}^{-r}\sum _{r}{k}^{-s}|\frac{\alpha {x}_{{s}_{i}}\left(k\right)+\left(1-\alpha \right){x}_{{s}_{i}}\left(k-1\right)}{2}|\right)}^{{p}_{r}}+\frac{\mu \eta }{4}\\ \le & \frac{1}{2}+\frac{1}{2}-\frac{\mu \eta }{2}+\frac{\mu \eta }{4}\\ =& 1-\frac{\mu \eta }{4}.\end{array}$

So, the inequality (3.4) implies that $\parallel \frac{x+{x}_{{s}_{k}}}{2}\parallel \le 1-\delta$. Consequently, the space ${\mathcal{Z}}_{\sigma }\left(s,p\right)$ possesses property $\left(\beta \right)$. □

Since property $\left(\beta \right)$ implies NUC, NUC implies property $\left(D\right)$ and property $\left(D\right)$ implies reflexivity, we can give the following result from Theorem 3.8.

Corollary 3.9 The space ${\mathcal{Z}}_{\sigma }\left(s,p\right)$ is nearly uniform convex, reflexive and also it has property $\left(D\right)$.

Theorem 3.10 The space ${\mathcal{Z}}_{\sigma }\left(s,p\right)$ has the uniform Opial property.

Proof Let $\epsilon >0$ and $x\in {\mathcal{Z}}_{\sigma }\left(s,p\right)$ be such that $\parallel x\parallel \ge \epsilon$ and $\left({x}_{n}\right)$ be a weakly null sequence in $S\left({\mathcal{Z}}_{\sigma }\left(s,p\right)\right)$. By $\sigma \in {\delta }_{2}^{s}$, there exists $\zeta \in \left(0,1\right)$ independent of x such that $\sigma \left(x\right)>\zeta$ by Lemma 3.2. Also since $\sigma \in {\delta }_{2}^{s}$, by Lemma 3.1, there is ${\zeta }_{1}\in \left(0,\zeta \right)$ such that

$|\sigma \left(y+z\right)-\sigma \left(y\right)|<\frac{\zeta }{4}$
(3.10)

whenever $\sigma \left(y\right)\le 1$ and $\sigma \left(z\right)\le {\zeta }_{1}$. Take ${r}_{0}\in \mathbb{N}$ such that

$\sum _{r={r}_{0}+1}^{\mathrm{\infty }}{\left({2}^{-r}\sum _{r}{k}^{-s}|\alpha x\left(k\right)+\left(1-\alpha \right)x\left(k-1\right)|\right)}^{{p}_{r}}<\frac{{\zeta }_{1}}{4}.$
(3.11)

Hence, we have

$\begin{array}{rl}\zeta <& \sum _{r=1}^{{r}_{0}}{\left({2}^{-r}\sum _{r}{k}^{-s}|\alpha x\left(k\right)+\left(1-\alpha \right)x\left(k-1\right)|\right)}^{{p}_{r}}\\ +\sum _{r={r}_{0}+1}^{\mathrm{\infty }}{\left({2}^{-r}\sum _{r}{k}^{-s}|\alpha x\left(k\right)+\left(1-\alpha \right)x\left(k-1\right)|\right)}^{{p}_{r}}\\ \le & \sum _{r=1}^{{r}_{0}}{\left({2}^{-r}\sum _{r}{k}^{-s}|\alpha x\left(k\right)+\left(1-\alpha \right)x\left(k-1\right)|\right)}^{{p}_{r}}+\frac{{\zeta }_{1}}{4}\end{array}$
(3.12)

and this implies that

$\begin{array}{rl}\sum _{r=1}^{{r}_{0}}{\left({2}^{-r}\sum _{r}{k}^{-s}|\alpha x\left(k\right)+\left(1-\alpha \right)x\left(k-1\right)|\right)}^{{p}_{r}}& >\zeta -\frac{{\zeta }_{1}}{4}\\ >\zeta -\frac{\zeta }{4}\\ =\frac{3\zeta }{4}.\end{array}$
(3.13)

Since ${x}_{n}{\to }^{w}0$, by the inequality (3.10), there exists ${r}_{0}\in \mathbb{N}$ such that

$\sum _{r=1}^{{r}_{0}}{\left({2}^{-r}\sum _{r}{k}^{-s}|\alpha \left({x}_{n}\left(k\right)+x\left(k\right)\right)+\left(1-\alpha \right)\left({x}_{n}\left(k-1\right)+x\left(k-1\right)\right)|\right)}^{{p}_{r}}>\frac{3\zeta }{4}.$
(3.14)

Again, by ${x}_{n}{\to }^{w}0$, there is ${r}_{1}>{r}_{0}$ such that for all $r>{r}_{1}$

$\parallel {x}_{{n}_{|{r}_{0}}}\parallel <1-{\left(1-\frac{\zeta }{4}\right)}^{1/M},$
(3.15)

where ${p}_{r}\le M\in \mathbb{N}$ for all $r\in \mathbb{N}$. Therefore, we obtain that

$\parallel {x}_{{n}_{|\mathbb{N}-{r}_{0}}}\parallel >{\left(1-\frac{\zeta }{4}\right)}^{1/M}$
(3.16)

by the triangle inequality of the norm. It follows from the definition of the Luxemburg norm that

$\begin{array}{rl}1& \le \sigma \left(\frac{{x}_{{n}_{|\mathbb{N}-{r}_{0}}}}{{\left(1-\frac{\zeta }{4}\right)}^{1/M}}\right)\\ =\sum _{r={r}_{0}+1}^{\mathrm{\infty }}{\left(\frac{{2}^{-r}{\sum }_{r}{k}^{-s}|\alpha {x}_{n}\left(k\right)+\left(1-\alpha \right){x}_{n}\left(k-1\right)|}{{\left(1-\frac{\zeta }{4}\right)}^{1/M}}\right)}^{{p}_{r}}\\ \le {\left(\frac{1}{{\left(1-\frac{\zeta }{4}\right)}^{1/M}}\right)}^{M}\sum _{r={r}_{0}+1}^{\mathrm{\infty }}{\left({2}^{-r}\sum _{r}{k}^{-s}|\alpha {x}_{n}\left(k\right)+\left(1-\alpha \right){x}_{n}\left(k-1\right)|\right)}^{{p}_{r}}\end{array}$
(3.17)

and this implies that

$\sum _{r={r}_{0}+1}^{\mathrm{\infty }}{\left({2}^{-r}\sum _{r}{k}^{-s}|\alpha {x}_{n}\left(k\right)+\left(1-\alpha \right){x}_{n}\left(k-1\right)|\right)}^{{p}_{r}}\ge 1-\frac{\zeta }{4}.$
(3.18)

By (3.7), (3.8), (3.11), (3.15) and since ${x}_{n}{\to }^{w}0⇒{x}_{n}\to 0$ (coordinatewise), we have for any $r>{r}_{1}$ that

$\begin{array}{rl}\sigma \left({x}_{n}+x\right)=& \sum _{r=1}^{{r}_{0}}{\left({2}^{-r}\sum _{r}{k}^{-s}|\alpha \left({x}_{n}\left(k\right)+x\left(k\right)\right)+\left(1-\alpha \right)\left({x}_{n}\left(k-1\right)+x\left(k-1\right)\right)|\right)}^{{p}_{r}}\\ +\sum _{r={r}_{0}+1}^{\mathrm{\infty }}{\left({2}^{-r}\sum _{r}{k}^{-s}|\alpha \left({x}_{n}\left(k\right)+x\left(k\right)\right)+\left(1-\alpha \right)\left({x}_{n}\left(k-1\right)+x\left(k-1\right)\right)|\right)}^{{p}_{r}}\\ \ge & \sum _{r={r}_{0}+1}^{\mathrm{\infty }}{\left({2}^{-r}\sum _{r}{k}^{-s}|\alpha \left({x}_{n}\left(k\right)+x\left(k\right)\right)+\left(1-\alpha \right)\left({x}_{n}\left(k-1\right)+x\left(k-1\right)\right)|\right)}^{{p}_{r}}\\ -\frac{\zeta }{4}+\frac{3\zeta }{4}\\ \ge & \frac{3\zeta }{4}+\left(1-\frac{\zeta }{4}\right)-\frac{\zeta }{4}\\ =& 1+\frac{\zeta }{4}.\end{array}$

Since $\sigma \in {\delta }_{2}^{s}$, it follows from Lemma 3.3 that there is τ depending on ζ only such that $\parallel {x}_{n}+x\parallel \ge 1+\tau$. □

Corollary 3.11 The space ${\mathcal{Z}}_{\sigma }\left(s,p\right)$ has property $\left(L\right)$ and the fixed point property.

## References

1. Cui Y, Hudzik H: Some geometric properties related to fixed point theory in Cesàro spaces. Collect. Math. 1999, 50(3):277–288.

2. Cui Y, Hudzik H: On the uniform Opial property in some modular sequence spaces. Funct. Approx. Comment. Math. 1998, XXVI: 93–102.

3. Karakaya V: Some geometric properties of sequence spaces involving lacunary sequence. J. Inequal. Appl. 2007., 2007: Article ID 81028

4. Savaş E, Karakaya V, Şimşek N:Some ${\ell }_{p}$-type new sequence spaces and their geometric properties. Abstr. Appl. Anal. 2009., 2009: Article ID 696971

5. Şimşek N, Savaş E, Karakaya V: Some geometric and topological properties of a new sequence space defined by de la Vallée-Poussin mean. J. Comput. Anal. Appl. 2010, 12(4):768–779.

6. Maligranda L, Petrot N, Suantai S: On the James constant and B -convexity of Cesàro and Cesàro-Orlicz sequences spaces. J. Math. Anal. Appl. 2007, 326(1):312–331. 10.1016/j.jmaa.2006.02.085

7. Mursaleen M, Çolak R, Et M: Some geometric inequalities in a new Banach sequence space. J. Inequal. Appl. 2007., 2007: Article ID 86757

8. Petrot N, Suantai S: On uniform Kadec-Klee properties and rotundity in generalized Cesàro sequence spaces. Int. J. Math. Sci. 2004, 2: 91–97.

9. Petrot N, Suantai S: Uniform Opial properties in generalized Cesàro sequence spaces. Nonlinear Anal. 2005, 63(8):1116–1125. 10.1016/j.na.2005.05.032

10. Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0

11. Franchetti C: Duality mapping and homeomorphisms in Banach theory. In Proceedings of Research Workshop on Banach Spaces Theory. University of Iowa Press, Iowa City; 1981.

12. Prus S: Banach spaces with uniform Opial property. Nonlinear Anal. 1992, 8: 697–704.

13. Goebel K, Kirk W: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.

14. Toledano JMA, Benavides TD, Acedo GL: Measureness of noncompactness. 99. In Metric Fixed Point Theory, Operator Theory: Advances and Applications. Birkhäuser, Basel; 1997.

15. Benavides TD: Weak uniform normal structure in direct sum spaces. Stud. Math. 1992, 103(37):293–299.

16. Benavides TD, Phothi S: The fixed point property under renorming in some classes of Banach spaces. Nonlinear Anal. 2010, 72(3):1409–1416.

17. Clarkson JA: Uniformly convex spaces. Trans. Am. Math. Soc. 1936, 40: 396–414. 10.1090/S0002-9947-1936-1501880-4

18. Huff R: Banach spaces which are nearly uniformly convex. Rocky Mt. J. Math. 1980, 10(4):743–749. 10.1216/RMJ-1980-10-4-743

19. Rolewicz S: On Δ-uniform convexity and drop property. Stud. Math. 1987, 87(2):181–191.

20. Montesinos V: Drop property equals reflexivity. Stud. Math. 1987, 87(1):93–100.

21. Shiue JS: On the Cesàro sequence space. Tamkang J. Math. 1970, 2: 19–25.

22. Jagers AA: A note on Cesàro sequence spaces. Nieuw Arch. Wiskd. 1974, 22(3):113–124.

23. Suantai S: On the H -property of some Banach sequence spaces. Arch. Math. 2003, 39: 309–316.

24. Bilgin T:The sequence space $C\left(s,p\right)$ and related matrix transformations. Punjab Univ. J. Math. 1997, 30: 67–77.

25. Lim KP: Matrix transformation in the Cesàro sequence spaces. Kyungpook Math. J. 1974, 14: 221–227.

26. Şengönül M, Kayaduman K:On the ${\mathcal{Z}}_{N}$-Nakano sequence space. Int. J. Math. Anal. 2010, 4(25–28):1363–1375.

## Author information

Authors

### Corresponding author

Correspondence to Murat Karakaş.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors read and approved the final manuscript.

## Rights and permissions

Reprints and Permissions

Et, M., Karakaş, M. & Çınar, M. Some geometric properties of a new modular space defined by Zweier operator. Fixed Point Theory Appl 2013, 165 (2013). https://doi.org/10.1186/1687-1812-2013-165

• Accepted:

• Published:

• DOI: https://doi.org/10.1186/1687-1812-2013-165

### Keywords

• Zweier operator
• Luxemburg norm
• modular space
• uniform Opial property
• property $\left(\beta \right)$ 