nach oben

Journal of Inequalities and Applications

Erschienen in:

Open Access 01.12.2013 | Research

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

verfasst von: Muhammed Çinar, Murat Karakaş, Mikail Et

Erschienen in: Journal of Inequalities and Applications | Ausgabe 1/2013

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Patentsuche

Aus

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.

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.

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:

https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-28/MediaObjects/13660_2012_Article_470_Equa_HTML.gif

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})

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

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} .

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}})

(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}

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}})

(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

https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-28/MediaObjects/13660_2012_Article_470_Equj_HTML.gif

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).

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

Vorheriger Artikel Continuity of the Bessel wavelet transform on certain Beurling-type function spaces

Nächster Artikel Notes on Greub-Rheinboldt inequalities

Sanhan W, Mongkolkeha C: A locally uniform rotund and property ( β ) of generalized Cesàro

(ces (p), d)

metric space. Int. J. Math. Anal. 2011, 5(9–12):549–558.MATHMathSciNet

Junde W, Narang TD: H -property, normal structure and fixed points of nonexpansive mappings in metric linear spaces. Acta Math. Vietnam. 2000, 25(1):13–18.MATHMathSciNet

Mongkolkeha C, Kumam P: Some geometric properties of lacunary sequence spaces related to fixed point property. Abstr. Appl. Anal. 2011., 2011: Article ID 903736

Sanhan W, Suantai S: On k -nearly uniform convex property in generalized Cesàro sequence spaces. Int. J. Math. Math. Sci. 2003, 57: 3599–3607.MathSciNetCrossRef

Cui Y, Hudzik H, Ping W: On k -nearly uniform convexity in Orlicz spaces. Perspectives in mathematical analysis. Rev. R. Acad. Cienc. Exactas Fís. Nat. 2000, 94(4):461–466. (Spanish)MATHMathSciNet

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

Ahuja GC, Narang TD, Trehan S: Best approximation on convex sets in metric linear spaces. Math. Nachr. 1977, 78: 125–130. 10.1002/mana.19770780110MATHMathSciNetCrossRef

Sastry KPR, Naidu SVR: Convexity conditions in metric linear spaces. Math. Semin. Notes 1979, 7: 235–251.MATHMathSciNet

Junde W, Lianchang C: Reflexivity of uniform convexity in metric linear spaces and its applications. Adv. Math. (China) 1994, 23: 439–444.

10.

Junde W, Dehai Y, Wenbo Q: The uniform convexity and reflexivity in metric linear spaces. Math. Appl., Chin. Ser. 1995, 8: 322–324.MATH

11.

Leindler L: Über die verallgemeinerte de la Vallee-Poussinsche summierbarkeit allgemeiner Orthogonalreihen. Acta Math. Acad. Sci. Hung. 1965, 16: 375–387. 10.1007/BF01904844MATHMathSciNetCrossRef

12.

Et M: Spaces of Cesàro difference sequences of order r defined by a modulus function in a locally convex space. Taiwan. J. Math. 2006, 10: 865–879.MATHMathSciNet

13.

Güngör M, Et M, Altin Y:Strongly

(V_{σ}, λ, q)

-summable sequences defined by Orlicz functions. Appl. Math. Comput. 2004, 157: 561–571. 10.1016/j.amc.2003.08.051MATHMathSciNetCrossRef

14.

Savas E, Savas R: Some λ -sequence spaces defined by Orlicz functions. Indian J. Pure Appl. Math. 2003, 34: 1673–1680.MATHMathSciNet

15.

Savas E: Strong almost convergence and almost λ -statistical convergence. Hokkaido Math. J. 2000, 29(3):531–536.MATHMathSciNetCrossRef

16.

Savas E: On asymptotically λ -statistical equivalent sequences of fuzzy numbers. New Math. Nat. Comput. 2007, 3(3):301–306. 10.1142/S1793005707000781MATHMathSciNetCrossRef

17.

Savas E:On

\bar{λ}

-statistically convergent double sequences of fuzzy numbers. J. Inequal. Appl. 2008., 2008: Article ID 147827

18.

Savas E:

\bar{λ}

-double sequence spaces of fuzzy real numbers defined by Orlicz function. Math. Commun. 2009, 14(2):287–297.MATHMathSciNet

19.

Savas E, Malkowsky E: Some λ -sequence spaces defined by a modulus. Arch. Math. 2000, 36(3):219–228.MATHMathSciNet

20.

Simsek N, Savas 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: 768–779.MATHMathSciNet

21.

Simsek N: On some geometric properties of sequence space defined by de la Vallée-Poussin mean. J. Comput. Anal. Appl. 2011, 13: 565–573.MATHMathSciNet

22.

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

23.

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

Titel: Some geometric properties of the metric space
verfasst von: Muhammed Çinar
Murat Karakaş
Mikail Et
Publikationsdatum: 01.12.2013
Verlag: Springer International Publishing
Erschienen in: Journal of Inequalities and Applications / Ausgabe 1/2013
Elektronische ISSN: 1029-242X
DOI: https://doi.org/10.1186/1029-242X-2013-28

Springer Professional

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

Abstract

Competing interests

Authors’ contributions

1 Introduction

2 Main results

Competing interests

Authors’ contributions

Premium Partner

Springer Professional

Abstract

Competing interests

Authors’ contributions

1 Introduction

2 Main results

Competing interests

Authors’ contributions

Weitere Artikel der Ausgabe 1/2013

Existence of positive solutions for third-order boundary value problems with integral boundary conditions on time scales

On inequalities of subgroups and the structure of finite groups

Volume inequalities for -Minkowski combination of convex bodies

Hausdorff measure of noncompactness in some sequence spaces of a triple band matrix

New sharp bounds for logarithmic mean and identric mean

Bounds for the ratio of two gamma functions: from Wendel’s asymptotic relation to Elezović-Giordano-Pečarić’s theorem

Premium Partner