Some geometric properties of the metric space V
          [
          λ
          ,
          p
          ]

Muhammed Çinar; Murat Karakaş; Mikail Et

doi:10.1186/1029-242X-2013-28

Research
Open Access

Some geometric properties of the metric space $V [λ, p]$

Muhammed Çinar¹,
Murat Karakaş²Email author and
Mikail Et³

Journal of Inequalities and Applications20132013:28

https://doi.org/10.1186/1029-242X-2013-28

Received: 18 September 2012
Accepted: 26 December 2012
Published: 22 January 2013

Abstract

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

MSC:40A05, 46A45, 46B20.

Keywords

linear metric space
Luxemburg norm
de la Vallée-Poussin mean
k-NUC property
property (H)

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} (X)

and

S_{d} (X)

as follows:

Let $(X, d)$ be a linear metric space and $B_{d} (X)$ (resp., $S_{d} (X)$ ) be a closed unit ball (resp., the unit sphere) of X. A linear metric space $(X, d)$ has property (β) if and only if for each $r > 0$ and $ε > 0$ , there exists $δ > 0$ such that for each element $x \in B_{d} (0, r)$ and each sequence $(x_{n})$ in $B_{d} (0, r)$ with $sep (x_{n}) \geq ε$ , there is an index k for which $d (\frac{x + x_{k}}{2}, 0) \leq 1 - δ$ , where $sep (x_{n}) = inf {d (x_{n}, x_{m}) : n \neq m} > ε$ [1]. If for each $x \in S_{d} (0, r)$ and $(x_{n}) \subset S_{d} (0, r)$ , $x_{n} \overset{w}{\to} x$ implies $x_{n} \to x$ , a linear metric space $(X, d)$ is said to have property (H). Let $k \geq 2$ be an integer. A linear metric space $(X, d)$ is said to be k-nearly uniform convex (k-NUC) if for every $ε > 0$ and $r > 0$ , there exists $δ > 0$ such that for any sequence $(x_{n}) \subset B_{d} (0, r)$ with $sep (x_{n}) \geq ε$ , there are $s_{1}, s_{2}, \dots, s_{k}$ such that $d (\frac{x_{s_{1}} + x_{s_{2}} + \dots + x_{s_{k}}}{k}, 0) \leq r - δ$ [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 $(V, λ)$ -summability by Leindler [11]. $(V, λ)$ -summable sequences have been studied by many authors including Et et al. [12, 13], Savas [14–18], Savas and Malkowsky [19] and Şimsek et al. [20, 21]. Let $Λ = (λ_{k})$ be a nondecreasing sequence of positive real numbers tending to infinity and let $λ_{1} = 1$ and $λ_{k + 1} \leq λ_{k} + 1$ . The generalized de la Vallée-Poussin mean is defined by $t_{n} (x) = \frac{1}{λ_{n}} \sum_{k \in I_{n}} x_{k}$ , where $I_{n} = [n - λ_{n} + 1, n]$ for $n = 1, 2, \dots$ . A sequence $x = (x_{k})$ is said to be $(V, λ)$ -summable to a number ℓ if $t_{n} (x) \to ℓ$ as $n \to \infty$ . If $λ_{n} = n$ , then $(V, λ)$ -summability is reduced to Cesàro summability.

Let w be the space of all real sequences. Let

p = (p_{k})

be a bounded sequence of positive real numbers. Şimşek et al. [20] defined the space

V [λ, p]

as follows:

V [λ, p] = {x = (x_{k}) \in ω : \sum_{k = 1}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | x_{j} |)}^{p_{k}} < \infty} .

If

λ_{k} = k

, then

V [λ, p] = ces (p)

[22]. If

λ_{k} = k

and

p_{k} = p

for all

k \in N

, then

V [λ, p] = {ces}_{p}

[23]. Paranorm on

V [λ, p]

is given by

h (x) = {(\sum_{k = 1}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | x_{j} |)}^{p_{k}})}^{\frac{1}{M}},

where

M = max {1, H}

and

H = sup p_{k}

. If

p_{k} = p

for all

k \in N

, the notation

V_{p} (λ)

is used in place of

V [λ, p]

and the norm on

V_{p} (λ)

is as follows:

{∥ x ∥}_{V_{p} (λ)} = {(\sum_{k = 1}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | x_{j} |)}^{p})}^{\frac{1}{p}} .

ρ : V_{ρ} [λ, p] \to [0, \infty]

,

ρ (x) = (\sum_{k = 1}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | x_{j} |)}^{p_{k}})

is a modular on

V_{ρ} [λ, p]

and the Luxemburg norm on

V_{ρ} [λ, p]

is defined by

{∥ x ∥}_{L} = inf {σ > 0 : ρ (\frac{x}{σ}) \leq 1}

for all

x \in V_{ρ} [λ, p]

. The Amemiya norm on the space

V_{ρ} [λ, p]

can be similarly introduced as follows:

{∥ x ∥}_{A} = inf_{σ > 0} \frac{1}{σ} (1 + ρ (σ x)) for all x \in V_{ρ} [λ, p] .

2 Main results

In this part of the paper, our main purpose is to define a metric on $V [λ, p]$ and show that $V [λ, p]$ possesses property (β), property (H) and k-NUC property. Let $p = (p_{k})$ be a bounded sequence of real numbers with $p_{k} > 1$ for all $k \in N$ . The mapping $d (x, y) = {(\sum_{k = 1}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | x (j) - y (j) |)}^{p_{k}})}^{1 / H}$ is a metric on the space $V [λ, p]$ , where $M = max (1, H = sup p_{k})$ and $m = inf p_{k}$ since the function ${| t |}^{p}$ is convex for $p > 1$ . First, we will show that the space $V [λ, p]$ 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 (V [λ, p], d)

. If

β \in (0, 1)

, then

{(d (y + z, 0))}^{M} \leq {(d (y, 0))}^{M} + 2^{M} β {(d (y, 0))}^{M} + \frac{2^{M}}{β^{M - 1}} {(d (z, 0))}^{M} .

Proof Let

y, z \in (V [λ, p], d)

and

0 < β < 1

. Then

\begin{array}{rcl} {(d (y + z, 0))}^{M} & = & \sum_{k = 1}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | y (j) + z (j) |)}^{p_{k}} \\ \leq & \sum_{k = 1}^{\infty} {((1 - β) \frac{1}{λ_{k}} \sum_{j \in I_{k}} | y (j) | + β \frac{1}{λ_{k}} \sum_{j \in I_{k}} | y (j) + \frac{z (j)}{β} |)}^{p_{k}} \\ \leq & (1 - β) \sum_{k = 1}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | y (j) |)}^{p_{k}} + β \sum_{k = 1}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | y (j) + \frac{z (j)}{β} |)}^{p_{k}} \\ \leq & \sum_{k = 1}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | y (j) |)}^{p_{k}} + 2^{M} β \sum_{k = 1}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | y (j) |)}^{p_{k}} \\ + 2^{M} \sum_{k = 1}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | \frac{z (j)}{β} |)}^{p_{k}} \\ \leq & \sum_{k = 1}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | y (j) |)}^{p_{k}} + 2^{M} β \sum_{k = 1}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | y (j) |)}^{p_{k}} \\ + \frac{2^{M}}{β^{M - 1}} \sum_{k = 1}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | z (j) |)}^{p_{k}} \\ = & {(d (y, 0))}^{M} + 2^{M} β {(d (y, 0))}^{M} + \frac{2^{M}}{β^{M - 1}} {(d (z, 0))}^{M} . \end{array}

□

Lemma 2.2 Let

y, z \in (V [λ, p], d)

. Then for any

ε > 0

and

L > 0

, there exists

δ > 0

such that

| {(d (y + z, 0))}^{M} - {(d (y, 0))}^{M} | < ε,

where ${(d (y, 0))}^{M} \leq L$ and ${(d (z, 0))}^{M} \leq δ$ .

Proof Let

ε > 0

and

L > 0

. For

β = \frac{ε}{2^{M + 1} (L + ε)}

, we take

δ = \frac{ε β^{M - 1}}{2^{M + 1}}

. From Lemma 2.1, we have

\begin{array}{rcl} {(d (y + z, 0))}^{M} & \leq & {(d (y, 0))}^{M} + 2^{M} β {(d (y, 0))}^{M} + \frac{2^{M}}{β^{M - 1}} {(d (z, 0))}^{M} \\ \leq & {(d (y, 0))}^{M} + 2^{M} β L + \frac{2^{M}}{β^{M - 1}} δ \\ \leq & {(d (y, 0))}^{M} + 2^{M} \frac{ε}{2^{M + 1}} \frac{L}{L + ε} + \frac{2^{M}}{β^{M - 1}} \frac{ε β^{M - 1}}{2^{M + 1}} \\ \leq & {(d (y, 0))}^{M} + \frac{ε}{2} + \frac{ε}{2} \\ \leq & {(d (y, 0))}^{M} + ε \end{array}

(2.1)

and

\begin{array}{rcl} {(d (y, 0))}^{M} & \leq & {(d (y + z, 0))}^{M} + 2^{M} β {(d (y + z, 0))}^{M} + \frac{2^{M}}{β^{M - 1}} {(d (- z, 0))}^{M} \\ \leq & {(d (y + z, 0))}^{M} + 2^{M} β ({(d (y, 0))}^{M} + ε) + \frac{2^{M}}{β^{M - 1}} δ \\ \leq & {(d (y + z, 0))}^{M} + 2^{M} β (L + ε) + \frac{2^{M}}{β^{M - 1}} \frac{ε β^{M - 1}}{2^{M + 1}} \\ = & {(d (y + z, 0))}^{M} + 2^{M} \frac{ε}{2^{M + 1} (L + ε)} (L + ε) + \frac{ε}{2} \\ = & {(d (y + z, 0))}^{M} + \frac{ε}{2} + \frac{ε}{2} \\ = & {(d (y + z, 0))}^{M} + ε . \end{array}

(2.2)

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

Theorem 2.3 The space $(V [λ, p], d)$ has property (β).

Proof Let

ε > 0

and

(x_{n}) \subset B (V [λ, p], d)

such that

sep (x_{n}) \geq ε

and

x \in B (V [λ, p], d)

. We take

y^{N} = (0, 0, \dots, 0, \sum_{k = 1}^{N} y (k), y (N + 1), y (N + 2), \dots)

. By using the diagonal method, we can find a subsequence

(x_{n_{r}})

of

(x_{n})

for each

N \in N

such that

(x_{n_{r}} (k))

converges for each

k \in N

with

1 \leq k \leq N

, since

{(x_{n} (k))}_{k = 1}^{\infty}

is bounded for each

k \in N

. Therefore, there is

t_{N} \in N

for each

N \in N

such that

sep ({(x_{n}^{N})}_{r > t_{N}}) \geq ε

. So, there is a sequence of positive integers

{(t_{N})}_{N = 1}^{\infty}

with

t_{1} < t_{2} < t_{3} \dots

such that

d (x_{t_{N}}^{N}, 0) \geq \frac{ε}{2}

for all

N \in N

. Then there exists

κ > 0

such that for all

N \in N

,

\sum_{k = N}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | x_{t_{N}} |)}^{p_{k}} \geq κ .

(2.3)

By Lemma 2.2, there exists

δ_{0}

such that

| {(d (y + z, 0))}^{M} - {(d (y, 0))}^{M} | < \frac{κ}{2^{m}},

(2.4)

where

{(d (y, 0))}^{M} < j^{M}

and

{(d (z, 0))}^{M} \leq δ_{0}

. There exists

N_{1} \in N

such that

{(d (x^{N_{1}}, 0))}^{M} \leq δ_{0}

if

x \in B (V [λ, p])

and

{(d (x, 0))}^{M} \leq δ_{0}

. Let us take

y = x_{t_{N_{1}}}^{N_{1}}

and

z = x^{N_{1}}

. Hence, we have

\sum_{k = N_{1}}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | \frac{x (j) + x_{t_{N_{1}}} (j)}{2} |)}^{p_{k}} \leq \sum_{k = N_{1}}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | \frac{x_{t_{N_{1}}} (j)}{2} |)}^{p_{k}} + \frac{κ}{2^{m}} .

(2.5)

From (2.3), (2.4), (2.5) and by using the convexity of the function

f (t) = {| t |}^{p_{k}}

for all

k \in N

, we obtain that

\begin{array}{rcl} {(d (\frac{y + z}{2}, 0))}^{M} & = & \sum_{k = 1}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | \frac{x (j) + x_{t_{N_{1}}} (j)}{2} |)}^{p_{k}} \\ = & \sum_{k = 1}^{N_{1} - 1} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | \frac{x (j) + x_{t_{N_{1}}} (j)}{2} |)}^{p_{k}} + \sum_{k = N_{1}}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | \frac{x (j) + x_{t_{N_{1}}} (j)}{2} |)}^{p_{k}} \\ \leq & \sum_{k = 1}^{N_{1} - 1} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | \frac{x (j) + x_{t_{N_{1}}} (j)}{2} |)}^{p_{k}} + \sum_{k = N_{1}}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | \frac{x_{t_{N_{1}}} (k)}{2} |)}^{p_{k}} + \frac{κ}{2^{m}} \\ \leq & \frac{1}{2} \sum_{k = 1}^{N_{1} - 1} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | x (j) |)}^{p_{k}} + \frac{1}{2} \sum_{k = 1}^{N_{1} - 1} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | x_{t_{N_{1}}} (j) |)}^{p_{k}} \\ + \frac{1}{2^{m}} \sum_{k = N_{1}}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | x_{t_{N_{1}}} (j) |)}^{p_{k}} + \frac{κ}{2^{m}} \\ \leq & \frac{1}{2} \sum_{k = 1}^{N_{1} - 1} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | x (j) |)}^{p_{k}} + \frac{1}{2} \sum_{k = 1}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | x_{t_{N_{1}}} (j) |)}^{p_{k}} \\ - \frac{2^{m} - 2}{2^{m + 1}} \sum_{k = N_{1}}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | x_{t_{N_{1}}} (j) |)}^{p_{k}} + \frac{κ}{2^{m}} \\ < & \frac{j^{M}}{2} + \frac{j^{M}}{2} - \frac{2^{m} - 2}{2^{m + 1}} κ + \frac{κ}{2^{m}} \\ = & j^{M} - \frac{κ}{2} . \end{array}

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

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

Theorem 2.4 The space $V [λ, p]$ is k-NUC for any integer $k \geq 2$ .

Proof Let

ε > 0

and

(x_{n}) \subset B_{d} (V [λ, p])

with

sep (x_{n}) \geq ε

. For each

m \in N

, let

x_{n}^{m} = (0, 0, \dots, x_{n} (m), x_{n} (m + 1), \dots) .

(2.6)

Since the sequence

{(x_{n} (i))}_{i = 1}^{\infty}

is bounded for each

i \in N

, by using the diagonal method, we can find a subsequence

(x_{n_{l}})

of

(x_{n})

such that

(x_{n_{l}} (k))

converges for each

k \in N

. Therefore, there is an increasing sequence

t_{m}

with

sep ({(x_{n_{l}}^{m})}_{l > t_{m}}) \geq ε

. Hence, there exists a sequence of positive integers

{(r_{m})}_{m = 1}^{\infty}

with

r_{1} < r_{2} < r_{3} < \dots

such that

d (x_{r_{m}}^{m}, 0) \geq \frac{ε}{2}

for all

m \in N

. Then there is

ζ > 0

such that

\sum_{k = m}^{\infty} {(\frac{1}{λ_{k}} \sum_{j \in I_{k}} | x_{r_{m}} |)}^{p_{k}} \geq ζ .

(2.7)

Let

α > 0

such that

1 < α < {lim}_{k \to \infty} inf p_{k}

. Let

ε_{1} = \frac{n^{α - 1} - 1}{(n - 1) n^{α}} \frac{ζ}{2}

for

k \geq 2

. From Lemma 2.2, there is a

δ > 0

such that

| {(d (y + z, 0))}^{M} - {(d (y, 0))}^{M} | < ε_{1},

(2.8)

where

{(d (y, 0))}^{M} < r^{M}

and

{(d (z, 0))}^{M} \leq δ

. Then there exist positive integers

m_{i}

(

i = 1, 2, \dots, n - 1

) with

m_{1} < m_{2} < \dots < m_{n - 1}

such that

d (x_{i}^{m_{i}}, 0) \leq δ

. Now, define

m_{n} = m_{n - 1} + 1

. Then we have

d (x_{r_{m_{n}}}^{m_{n}}, 0) \geq ζ

for all

m \in N

. For

1 \leq i \leq 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} (u) = {| u |}^{p_{i}}

(

i \in N

), we obtain

Thus, we have $d (\frac{x_{s_{1}} (j) + x_{s_{2}} (j) + \dots + x_{s_{n}} (j)}{n}, 0) < {(r^{M} - (\frac{n^{α - 1} - 1}{n^{α}}) \frac{ζ}{2})}^{1 / M} < r - δ$ for $δ \in (0, r - {(r^{M} - (\frac{n^{α - 1} - 1}{n^{α}}) \frac{ζ}{2})}^{1 / M})$ . Hence, $(V [λ, p], d)$ 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 $(V [λ, p], d)$ has property (H).

Declarations

Authors’ Affiliations

(1)

Department of Mathematics, Muş Alparslan University, Muş, 49100, Turkey

(2)

Department of Statistics, Bitlis Eren University, Bitlis, 13000, Turkey

(3)

Department of Mathematics, Firat University, Elazig, 23119, Turkey

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Some geometric properties of the metric space V [ λ , p ]

Some geometric properties of the metric space $V [λ, p]$