# Some geometric properties of the metric space $V\left[\lambda ,p\right]$

## Abstract

In this study, we consider the space $V\left[\lambda ,p\right]$ with an invariant metric. Then, we examine some geometric properties of the linear metric space $V\left[\lambda ,p\right]$ such as property (β), property (H) and k-NUC property.

MSC:40A05, 46A45, 46B20.

## 1 Introduction

Let X be a vector space over the scalar field of real numbers and d be an invariant metric on X. We denote ${B}_{d}\left(X\right)$ and ${S}_{d}\left(X\right)$ as follows:

Let $\left(X,d\right)$ be a linear metric space and ${B}_{d}\left(X\right)$ (resp., ${S}_{d}\left(X\right)$) be a closed unit ball (resp., the unit sphere) of X. A linear metric space $\left(X,d\right)$ has property (β) if and only if for each $r>0$ and $\epsilon >0$, there exists $\delta >0$ such that for each element $x\in {B}_{d}\left(0,r\right)$ and each sequence $\left({x}_{n}\right)$ in ${B}_{d}\left(0,r\right)$ with $sep\left({x}_{n}\right)\ge \epsilon$, there is an index k for which $d\left(\frac{x+{x}_{k}}{2},\mathbf{0}\right)\le 1-\delta$, where $sep\left({x}_{n}\right)=inf\left\{d\left({x}_{n},{x}_{m}\right):n\ne m\right\}>\epsilon$ [1]. If for each $x\in {S}_{d}\left(0,r\right)$ and $\left({x}_{n}\right)\subset {S}_{d}\left(0,r\right)$, ${x}_{n}\stackrel{w}{\to }x$ implies ${x}_{n}\to x$, a linear metric space $\left(X,d\right)$ is said to have property (H). Let $k\ge 2$ be an integer. A linear metric space $\left(X,d\right)$ is said to be k-nearly uniform convex (k-NUC) if for every $\epsilon >0$ and $r>0$, there exists $\delta >0$ such that for any sequence $\left({x}_{n}\right)\subset {B}_{d}\left(0,r\right)$ with $sep\left({x}_{n}\right)\ge \epsilon$, there are ${s}_{1},{s}_{2},\dots ,{s}_{k}$ such that $d\left(\frac{{x}_{{s}_{1}}+{x}_{{s}_{2}}+\cdots +{x}_{{s}_{k}}}{k},\mathbf{0}\right)\le r-\delta$ [2]. These properties have been studied by Mongkolkeha and Pumam [3], Sanhan and Suantai [4], Cui et al. [5] and Cui and Hudzik [6].

Ahuja et al. [7] introduced the notions of strict convexity and U.C.I (uniform convexity) in linear metric spaces which are generalizations of the corresponding concepts in linear normed spaces. Later, Sastry and Naidu [8] introduced the notions of U.C.II and U.C.III in linear metric spaces and showed that these three forms are not always equivalent. Further, Junde et al. [9, 10] showed that if a linear metric space is complete and U.C.I, then it is reflexive.

In summability theory, de la Vallée-Poussin mean was first used to define the $\left(V,\lambda \right)$-summability by Leindler [11]. $\left(V,\lambda \right)$-summable sequences have been studied by many authors including Et et al. [12, 13], Savas [1418], Savas and Malkowsky [19] and Şimsek et al. [20, 21]. Let $\mathrm{\Lambda }=\left({\lambda }_{k}\right)$ be a nondecreasing sequence of positive real numbers tending to infinity and let ${\lambda }_{1}=1$ and ${\lambda }_{k+1}\le {\lambda }_{k}+1$. The generalized de la Vallée-Poussin mean is defined by ${t}_{n}\left(x\right)=\frac{1}{{\lambda }_{n}}{\sum }_{k\in {I}_{n}}{x}_{k}$, where ${I}_{n}=\left[n-{\lambda }_{n}+1,n\right]$ for $n=1,2,\dots$ . A sequence $x=\left({x}_{k}\right)$ is said to be $\left(V,\lambda \right)$-summable to a number if ${t}_{n}\left(x\right)\to \ell$ as $n\to \mathrm{\infty }$. If ${\lambda }_{n}=n$, then $\left(V,\lambda \right)$-summability is reduced to Cesàro summability.

Let w be the space of all real sequences. Let $p=\left({p}_{k}\right)$ be a bounded sequence of positive real numbers. Şimşek et al. [20] defined the space $V\left[\lambda ,p\right]$ as follows:

$V\left[\lambda ,p\right]=\left\{x=\left({x}_{k}\right)\in \omega :\sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|{x}_{j}|\right)}^{{p}_{k}}<\mathrm{\infty }\right\}.$

If ${\lambda }_{k}=k$, then $V\left[\lambda ,p\right]=ces\left(p\right)$ [22]. If ${\lambda }_{k}=k$ and ${p}_{k}=p$ for all $k\in \mathrm{N}$, then $V\left[\lambda ,p\right]={ces}_{p}$ [23]. Paranorm on $V\left[\lambda ,p\right]$ is given by

$h\left(x\right)={\left(\sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|{x}_{j}|\right)}^{{p}_{k}}\right)}^{\frac{1}{M}},$

where $M=max\left\{1,H\right\}$ and $H=sup{p}_{k}$. If ${p}_{k}=p$ for all $k\in \mathrm{N}$, the notation ${V}_{p}\left(\lambda \right)$ is used in place of $V\left[\lambda ,p\right]$ and the norm on ${V}_{p}\left(\lambda \right)$ is as follows:

${\parallel x\parallel }_{{V}_{p}\left(\lambda \right)}={\left(\sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|{x}_{j}|\right)}^{p}\right)}^{\frac{1}{p}}.$

$\rho :{V}_{\rho }\left[\lambda ,p\right]\to \left[0,\mathrm{\infty }\right]$, $\rho \left(x\right)=\left({\sum }_{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}{\sum }_{j\in {I}_{k}}|{x}_{j}|\right)}^{{p}_{k}}\right)$ is a modular on ${V}_{\rho }\left[\lambda ,p\right]$ and the Luxemburg norm on ${V}_{\rho }\left[\lambda ,p\right]$ is defined by ${\parallel x\parallel }_{L}=inf\left\{\sigma >0:\rho \left(\frac{x}{\sigma }\right)\le 1\right\}$ for all $x\in {V}_{\rho }\left[\lambda ,p\right]$. The Amemiya norm on the space ${V}_{\rho }\left[\lambda ,p\right]$ can be similarly introduced as follows:

## 2 Main results

In this part of the paper, our main purpose is to define a metric on $V\left[\lambda ,p\right]$ and show that $V\left[\lambda ,p\right]$ possesses property (β), property (H) and k-NUC property. Let $p=\left({p}_{k}\right)$ be a bounded sequence of real numbers with ${p}_{k}>1$ for all $k\in \mathrm{N}$. The mapping $d\left(x,y\right)={\left({\sum }_{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}{\sum }_{j\in {I}_{k}}|x\left(j\right)-y\left(j\right)|\right)}^{{p}_{k}}\right)}^{1/H}$ is a metric on the space $V\left[\lambda ,p\right]$, where $M=max\left(1,H=sup{p}_{k}\right)$ and $m=inf{p}_{k}$ since the function ${|t|}^{p}$ is convex for $p>1$. First, we will show that the space $V\left[\lambda ,p\right]$ has property (β) under the above metric. To do this, we need the following two lemmas. To prove these lemmas, we use the technique given in Sanhan and Mongkolkeha [1].

Lemma 2.1 Let $y,z\in \left(V\left[\lambda ,p\right],d\right)$. If $\beta \in \left(0,1\right)$, then

${\left(d\left(y+z,\mathbf{0}\right)\right)}^{M}\le {\left(d\left(y,\mathbf{0}\right)\right)}^{M}+{2}^{M}\beta {\left(d\left(y,\mathbf{0}\right)\right)}^{M}+\frac{{2}^{M}}{{\beta }^{M-1}}{\left(d\left(z,\mathbf{0}\right)\right)}^{M}.$

Proof Let $y,z\in \left(V\left[\lambda ,p\right],d\right)$ and $0<\beta <1$. Then

$\begin{array}{rcl}{\left(d\left(y+z,\mathbf{0}\right)\right)}^{M}& =& \sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|y\left(j\right)+z\left(j\right)|\right)}^{{p}_{k}}\\ \le & \sum _{k=1}^{\mathrm{\infty }}{\left(\left(1-\beta \right)\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|y\left(j\right)|+\beta \frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|y\left(j\right)+\frac{z\left(j\right)}{\beta }|\right)}^{{p}_{k}}\\ \le & \left(1-\beta \right)\sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|y\left(j\right)|\right)}^{{p}_{k}}+\beta \sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|y\left(j\right)+\frac{z\left(j\right)}{\beta }|\right)}^{{p}_{k}}\\ \le & \sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|y\left(j\right)|\right)}^{{p}_{k}}+{2}^{M}\beta \sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|y\left(j\right)|\right)}^{{p}_{k}}\\ +{2}^{M}\sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|\frac{z\left(j\right)}{\beta }|\right)}^{{p}_{k}}\\ \le & \sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|y\left(j\right)|\right)}^{{p}_{k}}+{2}^{M}\beta \sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|y\left(j\right)|\right)}^{{p}_{k}}\\ +\frac{{2}^{M}}{{\beta }^{M-1}}\sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|z\left(j\right)|\right)}^{{p}_{k}}\\ =& {\left(d\left(y,\mathbf{0}\right)\right)}^{M}+{2}^{M}\beta {\left(d\left(y,\mathbf{0}\right)\right)}^{M}+\frac{{2}^{M}}{{\beta }^{M-1}}{\left(d\left(z,\mathbf{0}\right)\right)}^{M}.\end{array}$

□

Lemma 2.2 Let $y,z\in \left(V\left[\lambda ,p\right],d\right)$. Then for any $\epsilon >0$ and $L>0$, there exists $\delta >0$ such that

$|{\left(d\left(y+z,\mathbf{0}\right)\right)}^{M}-{\left(d\left(y,\mathbf{0}\right)\right)}^{M}|<\epsilon ,$

where ${\left(d\left(y,\mathbf{0}\right)\right)}^{M}\le L$ and ${\left(d\left(z,\mathbf{0}\right)\right)}^{M}\le \delta$.

Proof Let $\epsilon >0$ and $L>0$. For $\beta =\frac{\epsilon }{{2}^{M+1}\left(L+\epsilon \right)}$, we take $\delta =\frac{\epsilon {\beta }^{M-1}}{{2}^{M+1}}$. From Lemma 2.1, we have

$\begin{array}{rcl}{\left(d\left(y+z,\mathbf{0}\right)\right)}^{M}& \le & {\left(d\left(y,\mathbf{0}\right)\right)}^{M}+{2}^{M}\beta {\left(d\left(y,\mathbf{0}\right)\right)}^{M}+\frac{{2}^{M}}{{\beta }^{M-1}}{\left(d\left(z,\mathbf{0}\right)\right)}^{M}\\ \le & {\left(d\left(y,\mathbf{0}\right)\right)}^{M}+{2}^{M}\beta L+\frac{{2}^{M}}{{\beta }^{M-1}}\delta \\ \le & {\left(d\left(y,\mathbf{0}\right)\right)}^{M}+{2}^{M}\frac{\epsilon }{{2}^{M+1}}\frac{L}{L+\epsilon }+\frac{{2}^{M}}{{\beta }^{M-1}}\frac{\epsilon {\beta }^{M-1}}{{2}^{M+1}}\\ \le & {\left(d\left(y,\mathbf{0}\right)\right)}^{M}+\frac{\epsilon }{2}+\frac{\epsilon }{2}\\ \le & {\left(d\left(y,\mathbf{0}\right)\right)}^{M}+\epsilon \end{array}$
(2.1)

and

$\begin{array}{rcl}{\left(d\left(y,\mathbf{0}\right)\right)}^{M}& \le & {\left(d\left(y+z,\mathbf{0}\right)\right)}^{M}+{2}^{M}\beta {\left(d\left(y+z,\mathbf{0}\right)\right)}^{M}+\frac{{2}^{M}}{{\beta }^{M-1}}{\left(d\left(-z,\mathbf{0}\right)\right)}^{M}\\ \le & {\left(d\left(y+z,\mathbf{0}\right)\right)}^{M}+{2}^{M}\beta \left({\left(d\left(y,\mathbf{0}\right)\right)}^{M}+\epsilon \right)+\frac{{2}^{M}}{{\beta }^{M-1}}\delta \\ \le & {\left(d\left(y+z,\mathbf{0}\right)\right)}^{M}+{2}^{M}\beta \left(L+\epsilon \right)+\frac{{2}^{M}}{{\beta }^{M-1}}\frac{\epsilon {\beta }^{M-1}}{{2}^{M+1}}\\ =& {\left(d\left(y+z,\mathbf{0}\right)\right)}^{M}+{2}^{M}\frac{\epsilon }{{2}^{M+1}\left(L+\epsilon \right)}\left(L+\epsilon \right)+\frac{\epsilon }{2}\\ =& {\left(d\left(y+z,\mathbf{0}\right)\right)}^{M}+\frac{\epsilon }{2}+\frac{\epsilon }{2}\\ =& {\left(d\left(y+z,\mathbf{0}\right)\right)}^{M}+\epsilon .\end{array}$
(2.2)

From (2.1) and (2.2), we obtain that $|{\left(d\left(y+z,\mathbf{0}\right)\right)}^{M}-{\left(d\left(y,\mathbf{0}\right)\right)}^{M}|<\epsilon$. □

Theorem 2.3 The space $\left(V\left[\lambda ,p\right],d\right)$ has property (β).

Proof Let $\epsilon >0$ and $\left({x}_{n}\right)\subset B\left(V\left[\lambda ,p\right],d\right)$ such that $sep\left({x}_{n}\right)\ge \epsilon$ and $x\in B\left(V\left[\lambda ,p\right],d\right)$. We take ${y}^{N}=\left(0,0,\dots ,0,{\sum }_{k=1}^{N}y\left(k\right),y\left(N+1\right),y\left(N+2\right),\dots \right)$. By using the diagonal method, we can find a subsequence $\left({x}_{{n}_{r}}\right)$ of $\left({x}_{n}\right)$ for each $N\in \mathrm{N}$ such that $\left({x}_{{n}_{r}}\left(k\right)\right)$ converges for each $k\in \mathrm{N}$ with $1\le k\le N$, since ${\left({x}_{n}\left(k\right)\right)}_{k=1}^{\mathrm{\infty }}$ is bounded for each $k\in \mathrm{N}$. Therefore, there is ${t}_{N}\in \mathrm{N}$ for each $N\in \mathrm{N}$ such that $sep\left({\left({x}_{n}^{N}\right)}_{r>{t}_{N}}\right)\ge \epsilon$. So, there is a sequence of positive integers ${\left({t}_{N}\right)}_{N=1}^{\mathrm{\infty }}$ with ${t}_{1}<{t}_{2}<{t}_{3}\cdots$ such that $d\left({x}_{{t}_{N}}^{N},\mathbf{0}\right)\ge \frac{\epsilon }{2}$ for all $N\in \mathrm{N}$. Then there exists $\kappa >0$ such that for all $N\in \mathrm{N}$,

$\sum _{k=N}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|{x}_{{t}_{N}}|\right)}^{{p}_{k}}\ge \kappa .$
(2.3)

By Lemma 2.2, there exists ${\delta }_{0}$ such that

$|{\left(d\left(y+z,\mathbf{0}\right)\right)}^{M}-{\left(d\left(y,\mathbf{0}\right)\right)}^{M}|<\frac{\kappa }{{2}^{m}},$
(2.4)

where ${\left(d\left(y,\mathbf{0}\right)\right)}^{M}<{j}^{M}$ and ${\left(d\left(z,\mathbf{0}\right)\right)}^{M}\le {\delta }_{0}$. There exists ${N}_{1}\in \mathrm{N}$ such that ${\left(d\left({x}^{{N}_{1}},\mathbf{0}\right)\right)}^{M}\le {\delta }_{0}$ if $x\in B\left(V\left[\lambda ,p\right]\right)$ and ${\left(d\left(x,\mathbf{0}\right)\right)}^{M}\le {\delta }_{0}$. Let us take $y={x}_{{t}_{{N}_{1}}}^{{N}_{1}}$ and $z={x}^{{N}_{1}}$. Hence, we have

$\sum _{k={N}_{1}}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|\frac{x\left(j\right)+{x}_{{t}_{{N}_{1}}}\left(j\right)}{2}|\right)}^{{p}_{k}}\le \sum _{k={N}_{1}}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|\frac{{x}_{{t}_{{N}_{1}}}\left(j\right)}{2}|\right)}^{{p}_{k}}+\frac{\kappa }{{2}^{m}}.$
(2.5)

From (2.3), (2.4), (2.5) and by using the convexity of the function $f\left(t\right)={|t|}^{{p}_{k}}$ for all $k\in \mathrm{N}$, we obtain that

$\begin{array}{rcl}{\left(d\left(\frac{y+z}{2},\mathbf{0}\right)\right)}^{M}& =& \sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|\frac{x\left(j\right)+{x}_{{t}_{{N}_{1}}}\left(j\right)}{2}|\right)}^{{p}_{k}}\\ =& \sum _{k=1}^{{N}_{1}-1}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|\frac{x\left(j\right)+{x}_{{t}_{{N}_{1}}}\left(j\right)}{2}|\right)}^{{p}_{k}}+\sum _{k={N}_{1}}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|\frac{x\left(j\right)+{x}_{{t}_{{N}_{1}}}\left(j\right)}{2}|\right)}^{{p}_{k}}\\ \le & \sum _{k=1}^{{N}_{1}-1}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|\frac{x\left(j\right)+{x}_{{t}_{{N}_{1}}}\left(j\right)}{2}|\right)}^{{p}_{k}}+\sum _{k={N}_{1}}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|\frac{{x}_{{t}_{{N}_{1}}}\left(k\right)}{2}|\right)}^{{p}_{k}}+\frac{\kappa }{{2}^{m}}\\ \le & \frac{1}{2}\sum _{k=1}^{{N}_{1}-1}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|x\left(j\right)|\right)}^{{p}_{k}}+\frac{1}{2}\sum _{k=1}^{{N}_{1}-1}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|{x}_{{t}_{{N}_{1}}}\left(j\right)|\right)}^{{p}_{k}}\\ +\frac{1}{{2}^{m}}\sum _{k={N}_{1}}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|{x}_{{t}_{{N}_{1}}}\left(j\right)|\right)}^{{p}_{k}}+\frac{\kappa }{{2}^{m}}\\ \le & \frac{1}{2}\sum _{k=1}^{{N}_{1}-1}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|x\left(j\right)|\right)}^{{p}_{k}}+\frac{1}{2}\sum _{k=1}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|{x}_{{t}_{{N}_{1}}}\left(j\right)|\right)}^{{p}_{k}}\\ -\frac{{2}^{m}-2}{{2}^{m+1}}\sum _{k={N}_{1}}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|{x}_{{t}_{{N}_{1}}}\left(j\right)|\right)}^{{p}_{k}}+\frac{\kappa }{{2}^{m}}\\ <& \frac{{j}^{M}}{2}+\frac{{j}^{M}}{2}-\frac{{2}^{m}-2}{{2}^{m+1}}\kappa +\frac{\kappa }{{2}^{m}}\\ =& {j}^{M}-\frac{\kappa }{2}.\end{array}$

Therefore, we have $d\left(\frac{y+z}{2},\mathbf{0}\right)<{\left({j}^{M}-\frac{\kappa }{2}\right)}^{1/M} whenever $\delta \in \left(0,j-{\left({j}^{M}-\frac{\kappa }{2}\right)}^{1/M}\right)$. Consequently, the space $\left(V\left[\lambda ,p\right],d\right)$ possesses property (β). □

Now, we will show that the space $\left(V\left[\lambda ,p\right],d\right)$ has k-NUC property.

Theorem 2.4 The space $V\left[\lambda ,p\right]$ is k-NUC for any integer $k\ge 2$.

Proof Let $\epsilon >0$ and $\left({x}_{n}\right)\subset {B}_{d}\left(V\left[\lambda ,p\right]\right)$ with $sep\left({x}_{n}\right)\ge \epsilon$. For each $m\in \mathrm{N}$, let

${x}_{n}^{m}=\left(0,0,\dots ,{x}_{n}\left(m\right),{x}_{n}\left(m+1\right),\dots \right).$
(2.6)

Since the sequence ${\left({x}_{n}\left(i\right)\right)}_{i=1}^{\mathrm{\infty }}$ is bounded for each $i\in \mathrm{N}$, by using the diagonal method, we can find a subsequence $\left({x}_{{n}_{l}}\right)$ of $\left({x}_{n}\right)$ such that $\left({x}_{{n}_{l}}\left(k\right)\right)$ converges for each $k\in \mathrm{N}$. Therefore, there is an increasing sequence ${t}_{m}$ with $sep\left({\left({x}_{{n}_{l}}^{m}\right)}_{l>{t}_{m}}\right)\ge \epsilon$. Hence, there exists a sequence of positive integers ${\left({r}_{m}\right)}_{m=1}^{\mathrm{\infty }}$ with ${r}_{1}<{r}_{2}<{r}_{3}<\cdots$ such that $d\left({x}_{{r}_{m}}^{m},\mathbf{0}\right)\ge \frac{\epsilon }{2}$ for all $m\in \mathrm{N}$. Then there is $\zeta >0$ such that

$\sum _{k=m}^{\mathrm{\infty }}{\left(\frac{1}{{\lambda }_{k}}\sum _{j\in {I}_{k}}|{x}_{{r}_{m}}|\right)}^{{p}_{k}}\ge \zeta .$
(2.7)

Let $\alpha >0$ such that $1<\alpha <{lim}_{k\to \mathrm{\infty }}inf{p}_{k}$. Let ${\epsilon }_{1}=\frac{{n}^{\alpha -1}-1}{\left(n-1\right){n}^{\alpha }}\frac{\zeta }{2}$ for $k\ge 2$. From Lemma 2.2, there is a $\delta >0$ such that

$|{\left(d\left(y+z,\mathbf{0}\right)\right)}^{M}-{\left(d\left(y,\mathbf{0}\right)\right)}^{M}|<{\epsilon }_{1},$
(2.8)

where ${\left(d\left(y,\mathbf{0}\right)\right)}^{M}<{r}^{M}$ and ${\left(d\left(z,\mathbf{0}\right)\right)}^{M}\le \delta$. Then there exist positive integers ${m}_{i}$ ($i=1,2,\dots ,n-1$) with ${m}_{1}<{m}_{2}<\cdots <{m}_{n-1}$ such that $d\left({x}_{i}^{{m}_{i}},\mathbf{0}\right)\le \delta$. Now, define ${m}_{n}={m}_{n-1}+1$. Then we have $d\left({x}_{{r}_{{m}_{n}}}^{{m}_{n}},\mathbf{0}\right)\ge \zeta$ for all $m\in \mathrm{N}$. For $1\le i\le n-1$, let ${s}_{i}=i$ and ${s}_{n}={r}_{{m}_{n}}$. By using (2.6), (2.7), (2.8) and the convexity of the function ${f}_{i}\left(u\right)={|u|}^{{p}_{i}}$ ($i\in \mathrm{N}$), we obtain

Thus, we have $d\left(\frac{{x}_{{s}_{1}}\left(j\right)+{x}_{{s}_{2}}\left(j\right)+\cdots +{x}_{{s}_{n}}\left(j\right)}{n},\mathbf{0}\right)<{\left({r}^{M}-\left(\frac{{n}^{\alpha -1}-1}{{n}^{\alpha }}\right)\frac{\zeta }{2}\right)}^{1/M} for $\delta \in \left(0,r-{\left({r}^{M}-\left(\frac{{n}^{\alpha -1}-1}{{n}^{\alpha }}\right)\frac{\zeta }{2}\right)}^{1/M}\right)$. Hence, $\left(V\left[\lambda ,p\right],d\right)$ is k-NUC. □

Since k-NUC implies NUC and NUC implies property (H), by using the previous theorem, we can give the following result.

Corollary 2.5 The space $\left(V\left[\lambda ,p\right],d\right)$ has property (H).

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Correspondence to Murat Karakaş.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

MC, MK and ME have contributed to all parts of the article. All authors read and approved the final manuscript.

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Çinar, M., Karakaş, M. & Et, M. Some geometric properties of the metric space $V\left[\lambda ,p\right]$. J Inequal Appl 2013, 28 (2013). https://doi.org/10.1186/1029-242X-2013-28