research article a new difference sequence set of order ... · in this section we introduce a new...

Research ArticleA New Difference Sequence Set of Order 𝛼 andIts Geometrical Properties

Mikail Et1 and Vatan Karakaya2

1 Department of Mathematics, Firat University, 23119, Elazğ, Turkey2Department of Mathematical Engineering, Yildiz Technical University, Davutpasa Campus, Esenler, 34750 Istanbul, Turkey

Correspondence should be addressed to Vatan Karakaya; [email protected]

Received 15 June 2014; Accepted 19 August 2014; Published 11 September 2014

Academic Editor: Cristina Pignotti

Copyright © 2014 M. Et and V. Karakaya. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

We introduce a new class of sequences named as 𝑚𝛼(Δ𝑟

, 𝜙, 𝑝) and, for this space, we study some inclusion relations, topologicalproperties, and geometrical properties such as order continuous, the Fatou property, and the Banach-Saks property of type 𝑝.

1. Introduction, Definitions, and Preliminaries

By𝑤we denote the space of all complex (or real) sequences. If𝑥 ∈ 𝑤, then we simply write 𝑥 = (𝑥

𝑘) instead of 𝑥 = (𝑥

𝑘)∞

𝑘=0.

We will write ℓ∞, 𝑐, and 𝑐

0for the sequence spaces of all

bounded, convergent, and null sequences, respectively. Alsoby ℓ1and ℓ𝑝we denote the spaces of all absolutely summable

and 𝑝-absolutely summable series, respectively.The notion of difference sequence spaces was generalized

by Et and Çolak [1] such as𝑋(Δ𝑟) = {𝑥 = (𝑥𝑘) : Δ𝑟

𝑥 ∈ 𝑋}, for𝑋 = ℓ

∞, 𝑐, and 𝑐

0. They showed that these sequence spaces

are 𝐵𝐾-spaces with the norm

‖𝑥‖Δ=

𝑟

∑

𝑖=1

𝑥𝑖

+Δ𝑟

𝑥∞, (1)

where 𝑟 ∈ N, Δ0𝑥 = 𝑥, Δ𝑥 = (𝑥𝑘− 𝑥𝑘+1), Δ𝑟𝑥 = (Δ𝑟𝑥

𝑘) =

(Δ𝑟−1

𝑥𝑘−Δ𝑟−1

𝑥𝑘+1), andΔ𝑟𝑥

𝑘= ∑𝑟

V=0 (−1)V(𝑟

V ) 𝑥𝑘+V. Recentlydifference sequences and related concepts have been studiedin ([2–13]) and by many others.

Let 𝐸 be a sequence space. Then 𝐸 is called

(i) solid (ornormal) if (𝛼𝑘𝑥𝑘) ∈ 𝐸 for all sequences (𝛼

𝑘) of

scalars with |𝛼𝑘| ≤ 1 for all 𝑘 ∈ N, whenever (𝑥

𝑘) ∈ 𝐸,

(ii) symmetric if (𝑥𝑘) ∈ 𝐸 implies (𝑥

𝜋(𝑘)) ∈ 𝐸, where 𝜋 is

a permutation of N,

(iii) monotone provided 𝐸 contains the canaonical preim-ages of all its step spaces,

(iv) sequence algebra if 𝑥 ⋅ 𝑦 ∈ 𝐸, whenever 𝑥, 𝑦 ∈ 𝐸.

It is well known that if 𝐸 is normal then 𝐸 is monotone.Throughout this paper 𝜑

𝑠denotes the class of all subsets

of N; those do not contain more than 𝑠 elements. Let (𝜙𝑛)

be a nondecreasing sequence of positive numbers such that𝑛𝜙𝑛+1≤ (𝑛+ 1)𝜙

𝑛for all 𝑛 ∈ N. The class of all sequences (𝜙

𝑛)

is denoted by Φ. The sequence space 𝑚(𝜙) was introducedby Sargent [14] and he studied some of its properties andobtained some relations with the space ℓ

𝑝. Later on it was

investigated by Tripathy and Sen [15] and Tripathy andMahanta [16].

Let us recall that a sequence {V(𝑖)}∞𝑖=1

in a Banach space𝑋is called 𝑆𝑐ℎ𝑎𝑢𝑑𝑒𝑟 𝑏𝑎𝑠𝑖𝑠 of 𝑋 (or 𝑏𝑎𝑠𝑖𝑠 for short) if for each𝑥 ∈ 𝑋 there exists a unique sequence {𝜆(𝑖)}∞

𝑖=1of scalars such

that 𝑥 = ∑∞𝑖=1𝜆(𝑖)V(𝑖); that is, lim

𝑛→∞∑𝑛

𝑖=1𝜆(𝑖)V(𝑖) = 𝑥.

A sequence space 𝑋 with a linear topology is called a 𝐾-𝑠𝑝𝑎𝑐𝑒 if each of the projection maps 𝑃

𝑖: 𝑋 → C defined

by 𝑃𝑖(𝑥) = 𝑥(𝑖) for 𝑥 = (𝑥(𝑖))∞

𝑖=1∈ 𝑋 is continuous for each

natural 𝑖. A Fréchet space is a complete metric linear spaceand themetric is generated by an𝐹-norm and a Fréchet spacewhich is a𝐾-space is called an 𝐹𝐾-space; that is, a𝐾-space𝑋is called an 𝐹𝐾-space if 𝑋 is a complete linear metric space.In other words,𝑋 is an 𝐹𝐾-space if𝑋 is a Fréchet space with

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 278907, 4 pageshttp://dx.doi.org/10.1155/2014/278907

2 Abstract and Applied Analysis

continuous coordinate projections. All the sequence spacesmentioned above are 𝐹𝐾 spaces except the space 𝑐

00.

An 𝐹𝐾-space 𝑋 which contains the space 𝑐00

is said tohave the 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝐴𝐾 if for every sequence {𝑥(𝑖)} ∈ 𝑋, 𝑥 =∑∞

𝑖=1𝑥(𝑖)𝑒(𝑖) where 𝑒(𝑖) = (0, 0, . . . , 1(𝑖th-place), 0, 0, . . .).A Banach space𝑋 is said to be aKöthe sequence space (see

[17, 18]) if𝑋 is a subspace of 𝑤 such that

(i) if 𝑥 ∈ 𝑤, 𝑦 ∈ 𝑋 and |𝑥(𝑖)| ≤ |𝑦(𝑖)| for all 𝑖 ∈ N, then𝑥 ∈ 𝑋 and ‖𝑥‖ ≤ ‖𝑦‖;

(ii) there exists an element 𝑥 ∈ 𝑋 such that 𝑥(𝑖) > 0 forall 𝑖 ∈ N.

We say that 𝑥 ∈ 𝑋 is order continuous if for any sequence(𝑥𝑛) in𝑋 such that𝑥

𝑛(𝑖) ≤ |𝑥(𝑖)| for each 𝑖 ∈ N and𝑥

𝑛(𝑖) → 0

(𝑛 → ∞) we have that ‖𝑥𝑛‖ → 0 holds.

A Köthe sequence space𝑋 is said to be order continuousif all sequences in𝑋 are order continuous. It is easy to see that𝑥 ∈ 𝑋 is order continuous if and only if ‖(0, 0, . . . , 0, 𝑥(𝑛 +1), 𝑥(𝑛 + 2), . . .)‖ → 0 as 𝑛 → ∞.

A Köthe sequence space 𝑋 is said to have the𝐹𝑎𝑡𝑜𝑢 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦 if, for any real sequence 𝑥 and any {𝑥

𝑛} in

𝑋 such that 𝑥𝑛↑ 𝑥 coordinatewisely and sup

𝑛‖𝑥𝑛‖ < ∞, we

have that 𝑥 ∈ 𝑋 and ‖𝑥𝑛‖ → ‖𝑥‖.

A Banach space 𝑋 is said to have the 𝐵𝑎𝑛𝑎𝑐ℎ-𝑆𝑎𝑘𝑠 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦 if every bounded sequence {𝑥

𝑛} in 𝑋

admits a subsequence {𝑧𝑛} such that the sequence {𝑡

𝑘(𝑧)} is

convergent in𝑋 with respect to the norm, where

𝑡𝑘(𝑧) =

1

𝑘

(𝑧1+ 𝑧2+ ⋅ ⋅ ⋅ + 𝑧

𝑘) , ∀𝑘 ∈ N. (2)

Some of recent works on geometric properties ofsequence space can be found in the following list ([19–21]).

2. Inclusion and Topological Properties ofthe Space 𝑚

𝛼(Δ𝑟

,𝜙,𝑝)

In this section we introduce a new class of sequences andestablish some inclusion relations. Also we show that thisspace is not perfect and normal.

Let 𝑟 be a fixed positive integer,𝛼 ∈ (0, 1] any real number,and 𝑝 a positive real number such that 1 ≤ 𝑝 < ∞. Now wedefine the sequence space𝑚

𝛼(Δ𝑟

, 𝜙, 𝑝) as

𝑚𝛼(Δ𝑟

, 𝜙, 𝑝) = {𝑥 ∈ 𝑤 : sup𝑠≥1,𝜎∈𝜑

𝑠

1

𝜙𝛼

𝑠

∑

𝑛∈𝜎

Δ𝑟

𝑥𝑛

𝑝

< ∞} . (3)

In the case 𝑝 = 1, we will write 𝑚𝛼(Δ𝑟

, 𝜙) instead of𝑚𝛼(Δ𝑟

, 𝜙, 𝑝) and in the special case 𝑚 = 0 and 𝑝 = 1 wewill write𝑚

𝛼(𝜙) instead of𝑚

𝛼(Δ𝑟

, 𝜙, 𝑝).The proof of each of the following results is straightfor-

ward, so we choose to state these results without proof.

Theorem 1. Let 𝜙 ∈ Φ, 𝛼 ∈ (0, 1], and let 𝑝 be a positivereal number such that 1 ≤ 𝑝 < ∞. Then the sequence space𝑚𝛼(Δ𝑟

, 𝜙, 𝑝) is a 𝐵𝐾-space normed by

‖𝑥‖ =

𝑟

∑

𝑖=1

𝑥𝑖

+ sup𝑠≥1,𝜎∈𝜑

𝑠

1

𝜙𝛼

𝑠

(∑

𝑛∈𝜎

Δ𝑟

𝑥𝑛

𝑝

)

1/𝑝

. (4)

Theorem 2. Let 𝜙 ∈ Φ, 𝛼 ∈ (0, 1], and let 𝑝 be a positive realnumber such that 1 ≤ 𝑝 < ∞; then𝑚

𝛼(Δ𝑟

, 𝜙) ⊂ 𝑚𝛼(Δ𝑟

, 𝜙, 𝑝).

Theorem 3. Let 𝛼 and 𝛽 be fixed real numbers such that 0 <𝛼 ≤ 𝛽 ≤ 1 and 𝑝 a positive real number such that 1 ≤ 𝑝 < ∞;then𝑚

𝛼(Δ𝑟

, 𝜙, 𝑝) ⊂ 𝑚𝛽(Δ𝑟

, 𝜙, 𝑝).

Theorem 4. Let 𝛼 and 𝛽 be fixed real numbers such that 0 <𝛼 ≤ 𝛽 ≤ 1 and 𝑝 a positive real number such that 1 ≤ 𝑝 <∞. For any two sequences (𝜙

𝑠) and (𝜓

𝑠) of real numbers such

that 𝜙, 𝜓 ∈ Φ. Then𝑚𝛼(Δ𝑟


, 𝜓, 𝑝) if and only ifsup𝑠≥1(𝜙𝛼

𝑠/𝜓𝛽

𝑠) < ∞.

Proof. Let 𝑥 ∈ 𝑚𝛼(Δ𝑟

, 𝜙, 𝑝) and sup𝑠≥1(𝜙𝛼

𝑠/𝜓𝛽

𝑠) < ∞. Then

sup𝑠≥1,𝜎∈𝜑

𝑠

1

𝜙𝛼

𝑠

∑

𝑛∈𝜎

Δ𝑟

𝑥𝑛

𝑝

< ∞ (5)

and there exists a positive number𝐾 such that 𝜙𝛼𝑠≤ 𝐾𝜓

𝛽

𝑠and

so that 1/𝜓𝛽𝑠≤ 𝐾/𝜙

𝛼

𝑠for all 𝑠. Therefore for all 𝑠 we have

1

𝜓𝛽

𝑠

∑

𝑛∈𝜎

Δ𝑟

𝑥𝑛

𝑝

≤

𝐾

𝜙𝛼

𝑠

∑

𝑛∈𝜎

Δ𝑟

𝑥𝑛

𝑝

. (6)

Now taking supremum over 𝑠 ≥ 1 and 𝜎 ∈ 𝜑𝑠we get


𝑠

1

𝜓𝛽

𝑠

∑

𝑛∈𝜎

Δ𝑟

𝑥𝑛

𝑝

≤ sup𝑠≥1,𝜎∈𝜑

𝑠

𝐾

𝜙𝛼

𝑠

∑

𝑛∈𝜎

Δ𝑟

𝑥𝑛

𝑝

(7)

and so 𝑥 ∈ 𝑚𝛽(Δ𝑟

, 𝜓, 𝑝).Conversely let 𝑚

𝛼(Δ𝑟


, 𝜓, 𝑝) and supposethat sup

𝑠≥1(𝜙𝛼

𝑠/𝜓𝛽

𝑠) = ∞. Then there exists an increasing

sequence (𝑠𝑖) of naturals numbers such that lim

𝑖(𝜙𝛼

𝑠𝑖

/𝜓𝛽

𝑠𝑖

) =

∞. Let 𝐵 ∈ R+, where R+ is the set of positive real numbers;then there exists 𝑖

0∈ N such that 𝜙𝛼

𝑠𝑖

/𝜓𝛽

𝑠𝑖

> 𝐵 for all 𝑠𝑖≥ 𝑖0.

Hence 𝜙𝛼𝑠𝑖

> 𝐵𝜓𝛽

𝑠𝑖

and so 1/𝜓𝛽𝑠𝑖

> 𝐵/𝜙𝛼

𝑠𝑖

. Then we can write

1

𝜓𝛽

𝑠𝑖

∑

𝑛∈𝜎

Δ𝑟

𝑥𝑛

𝑝

>

𝐵

𝜙𝛼

𝑠𝑖

∑

𝑛∈𝜎

Δ𝑟

𝑥𝑛

𝑝

(8)

for all 𝑠𝑖≥ 𝑖0. Now taking supremum over 𝑠

𝑖≥ 𝑖0and 𝜎 ∈ 𝜑

𝑠

we get

sup𝑠𝑖≥𝑖0,𝜎∈𝜑𝑠

1

𝜓𝛽

𝑠𝑖

∑

𝑛∈𝜎

Δ𝑟

𝑥𝑛

𝑝

> sup𝑠𝑖≥𝑖0,𝜎∈𝜑𝑠

𝐵

𝜙𝛼

𝑠𝑖

∑

𝑛∈𝜎

Δ𝑟

𝑥𝑛

𝑝

. (9)

Since (9) holds for all 𝐵 ∈ R+ (we may take the number 𝐵sufficiently large), we have

sup𝑠𝑖≥𝑖0,𝜎∈𝜑𝑠

1

𝜓𝛽

𝑠𝑖

∑

𝑛∈𝜎

Δ𝑟

𝑥𝑛

𝑝

= ∞ (10)

when 𝑥 ∈ 𝑚𝛼(Δ𝑟

, 𝜙, 𝑝) with

0 < sup𝑠≥1,𝜎∈𝜑

𝑠

1

𝜙𝛼

𝑠

∑

𝑛∈𝜎

Δ𝑟

𝑥𝑛

𝑝

< ∞. (11)

Hence 𝑥 ∉ 𝑚𝛽(Δ𝑟

, 𝜓, 𝑝). This contradicts to 𝑚𝛼(Δ𝑟

, 𝜙, 𝑝) ⊂

𝑚𝛽(Δ𝑟

, 𝜓, 𝑝). Hence sup𝑠≥1(𝜙𝛼

𝑠/𝜓𝛽

𝑠) < ∞.

Abstract and Applied Analysis 3

The following results are derivable easily from Theorem4.

Corollary 5. Let 𝛼 and 𝛽 be fixed real numbers such that 0 <𝛼 ≤ 𝛽 ≤ 1 and 𝑝 a positive real number such that 1 ≤ 𝑝 < ∞.For any two sequences (𝜙

𝑠) and (𝜓

𝑠) of real numbers such that

𝜙, 𝜓 ∈ Φ. Then one has

(i) 𝑚𝛼(Δ𝑟

, 𝜙, 𝑝) = 𝑚𝛽(Δ𝑟

, 𝜓, 𝑝) if and only if 0 <inf𝑠≥1(𝜙𝛼

𝑠/𝜓𝛽

𝑠) < sup

𝑠≥1(𝜙𝛼

𝑠/𝜓𝛽

𝑠) < ∞,

(ii) 𝑚𝛼(Δ𝑟

, 𝜙, 𝑝) = 𝑚𝛼(Δ𝑟

, 𝜓, 𝑝) if and only if 0 <inf𝑠≥1(𝜙𝛼

𝑠/𝜓𝛼

𝑠) < sup

𝑠≥1(𝜙𝛼

𝑠/𝜓𝛼

𝑠) < ∞,

(iii) 𝑚𝛼(Δ𝑟

, 𝜙, 𝑝) = 𝑚𝛽(Δ𝑟

, 𝜙, 𝑝) if and only if 0 <inf𝑠≥1(𝜙𝛼

𝑠/𝜙𝛽

𝑠) < sup

𝑠≥1(𝜙𝛼

𝑠/𝜙𝛽

𝑠) < ∞.

Theorem 6. Consider 𝑚𝛼(Δ𝑟−1

, 𝜙, 𝑝) ⊂ 𝑚𝛼(Δ𝑟

, 𝜙, 𝑝) and theinclusion is strict.

Proof. It follows from Minkowski’s inequality. To show theinclusion is strict, let 𝜙

𝑛= 1 for all 𝑛 ∈ N, 𝛼 = 1, 𝑝 = 1,

and 𝑥 = (𝑘𝑟−1); then 𝑥 ∈ 𝑚𝛼(Δ𝑟

, 𝜙, 𝑝) \ 𝑚𝛼(Δ𝑟−1

, 𝜙, 𝑝).

Theorem 7. The sequence space 𝑚𝛼(𝜙) is solid and hence

monotone, but the sequence space𝑚𝛼(Δ𝑟

, 𝜙, 𝑝) is neither solidnor symmetric and sequence algebra for𝑚 ≥ 1.

Proof. Let 𝑥 ∈ 𝑚𝛼(𝜙) and 𝑦 = (𝑦

𝑛) be sequences such that

|𝑥𝑛| ≤ |𝑦𝑛| for each 𝑛 ∈ N. Then we get


𝑠

1

𝜙𝛼

𝑠

∑

𝑛∈𝜎

𝑥𝑛

≤ sup𝑠≥1,𝜎∈𝜑

𝑠

1

𝜙𝛼

𝑠

∑

𝑛∈𝜎

𝑦𝑛

. (12)

Hence 𝑚𝛼(𝜙) is solid and hence monotone. To show the

space 𝑚𝛼(Δ𝑟

, 𝜙, 𝑝) is normal, let 𝜙𝑛= 1, for all 𝑛 ∈ N,

𝛼 = 1, 𝑝 = 1, and 𝑥 = (𝑘𝑟−1), then 𝑥 ∈ 𝑚𝛼(Δ𝑟

, 𝜙, 𝑝); but(𝛼𝑘𝑥𝑘) ∉ 𝑚

𝛼(Δ𝑟

, 𝜙, 𝑝) when 𝛼𝑘= (−1)

𝑘 for all 𝑘 ∈ N. Hence𝑚𝛼(Δ𝑟

, 𝜙, 𝑝) is not solid. The other cases can be proved onconsidering similar examples.

Theorem 8. Consider

ℓ𝑝(Δ𝑟

) ⊂ 𝑚𝛼(Δ𝑟

, 𝜙, 𝑝) ⊂ ℓ∞(Δ𝑟

) . (13)

Proof. It is omitted.

Theorem 9. If 0 < 𝑝 < 𝑞, then𝑚𝛼(Δ𝑟

, 𝜙, 𝑝) ⊂ 𝑚𝛼(Δ𝑟

, 𝜙, 𝑞).

Proof. Proof follows from the following inequality:

(

𝑛

∑

𝑘=1

𝑥𝑘

𝑞

)

1/𝑞

< (

𝑛

∑

𝑘=1

𝑥𝑘

𝑞

)

1/𝑞

, (0 < 𝑝 < 𝑞) . (14)

3. Geometrical Properties ofthe Space 𝑚

𝛼(Δ𝑟

,𝜙,𝑝)

In this section, we study some geometrical properties of thespace 𝑚

𝛼(Δ𝑟

, 𝜙, 𝑝). Some of these geometrical properties are

the order continuous, the Fatou property, and the Banach-Saks property of type 𝑝. Let us start with the followingtheorem.

Theorem 10. The space𝑚𝛼(Δ𝑟

, 𝜙, 𝑝) is order continuous.

Proof. To prove this theorem, we have to show that 𝑚𝛼(Δ𝑟

,

𝜙, 𝑝) is an 𝐴𝐾-space. It is easy to see that 𝑚𝛼(Δ𝑟

, 𝜙, 𝑝)

contains 𝑐00

which is the space of real sequences whichhave only a finite number of nonzero coordinates. By usingdefinition of 𝐴𝐾-properties, we have that 𝑥 = {𝑥(𝑖)} ∈𝑚𝛼(Δ𝑟

, 𝜙, 𝑝) has a unique representation 𝑥 = ∑∞𝑖=1𝑥(𝑖)𝑒(𝑖);

that is, ‖𝑥 − 𝑥[𝑗]‖𝑚𝛼(Δ𝑟,𝜙,𝑝)

= ‖(0, 0, . . . , 𝑥(𝑗), 𝑥(𝑗 + 1),

. . . )‖𝑚𝛼(Δ𝑟,𝜙,𝑝)

→ 0 as 𝑗 → ∞, which means that 𝑚𝛼(Δ𝑟

, 𝜙,

𝑝) has 𝐴𝐾. Hence, since 𝐵𝐾-space 𝑚𝛼(Δ𝑟

, 𝜙, 𝑝) containing𝑐00has 𝐴𝐾-property, the space𝑚

𝛼(Δ𝑟

, 𝜙, 𝑝) is order continu-ous.

Theorem 11. The space𝑚𝛼(Δ𝑟

, 𝜙, 𝑝) has the Fatou property.

Proof. Let 𝑥 be any real sequence from (𝑤)+

and {𝑥𝑛}

any nondecreasing sequence of nonnegative elements from𝑚𝛼(Δ𝑟

, 𝜙, 𝑝) such that 𝑥𝑛(𝑖) → 𝑥(𝑖) as 𝑛 → ∞ coordinate-

wisely and sup𝑛‖𝑥𝑛‖𝑚𝛼(Δ𝑟,𝜙,𝑝)

< ∞.Let us denote 𝜏 = sup

𝑛‖𝑥𝑛‖𝑚𝛼(Δ𝑟,𝜙,𝑝)

.Then, since the sup-remum is homogeneous, we have

1

𝜏

(

𝑟

∑

𝑖=1

𝑥𝑛(𝑖)+ sup𝑠≥1,𝜎∈𝜑

𝑠

1

𝜙𝛼

𝑠

(∑

𝑖∈𝜎

Δ𝑟

𝑥𝑛(𝑖)

𝑝

)

1/𝑝

)

=

𝑟

∑

𝑖=1

𝑥𝑛(𝑖)

𝜏


𝑠

1

𝜙𝛼

𝑠

(∑

𝑛∈𝜎

Δ𝑟

𝑥𝑛(𝑖)

𝜏

𝑝

)

1/𝑝

≤

𝑟

∑

𝑖=1

𝑥𝑛(𝑖)

𝑥𝑛

𝑚𝛼(Δ𝑟,𝜙,𝑝)


𝑠

1

𝜙𝛼

𝑠

(∑

𝑛∈𝜎

Δ𝑟

𝑥𝑛(𝑖)

𝑥𝑛


𝑝

)

1/𝑝

=

1

𝑥𝑛


𝑥𝑛


= 1.

(15)

Moreover, by the assumptions that {𝑥𝑛} is nondecreasing

and convergent to 𝑥 coordinatewisely and by the Beppo-Levitheorem, we have

1

𝜏

lim𝑛→∞

(

𝑟

∑

𝑖=1

𝑥𝑛(𝑖)+ sup𝑠≥1,𝜎∈𝜑

𝑠

1

𝜙𝛼

𝑠

(∑

𝑖∈𝜎

Δ𝑟

𝑥𝑛(𝑖)

𝑝

)

1/𝑝

)

=

𝑟

∑

𝑖=1

𝑥 (𝑖)

𝜏


𝑠

1

𝜙𝛼

𝑠

(∑

𝑖∈𝜎

Δ𝑟

𝑥 (𝑖)

𝜏

𝑝

)

1/𝑝

=

𝑥

𝑠

𝑚𝛼(Δ𝑟,𝜙,𝑝))

≤ 1,

(16)

4 Abstract and Applied Analysis

whence

‖𝑥‖𝑚𝛼(Δ𝑟,𝜙,𝑝)))

≤ 𝜏 = sup𝑛

𝑥𝑛


= lim𝑛→∞

𝑥𝑛


< ∞.

(17)

Therefore, 𝑥 ∈ 𝑚𝛼(Δ𝑟

, 𝜙, 𝑝)). On the other hand, since0 ≤ 𝑥

𝑛≤ 𝑥 for any natural number 𝑛 and the

sequence {𝑥𝑛} is nondecreasing, we obtain that the sequence

{‖𝑥𝑛‖𝑚𝛼(Δ𝑟,𝜙,𝑝))

} is bounded from above by ‖𝑥‖𝑚𝛼(Δ𝑟,𝜙,𝑝))

.As a result, lim

𝑛→∞‖𝑥𝑛‖𝑚𝛼(Δ𝑟,𝜙,𝑝))

≤ ‖𝑥‖𝑚𝛼(Δ𝑟,𝜙,𝑝))

, whichtogether with the opposite inequality proved already yieldsthat ‖𝑥‖


= lim𝑛‖𝑥𝑛‖𝑚𝛼(Δ𝑟,𝜙,𝑝))

.

Theorem 12. The space 𝑚𝛼(Δ𝑟

, 𝜙, 𝑝)) has the Banach-Saksproperty of the type 𝑝.

Proof. It can be proved with standard technic.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

References

[1] M. Et and R. Çolak, “On some generalized difference sequencespaces,” Soochow Journal of Mathematics, vol. 21, no. 4, pp. 377–386, 1995.

[2] B. Altay and F. Basar, “On the fine spectrum of the differenceoperator Δ on 𝑐

0and and 𝑐,” Information Science, vol. 168, no.

1–4, pp. 217–224, 2004.[3] Ç. A. Bektas, M. Et, and R. Çolak, “Generalized difference

sequence spaces and their dual spaces,” Journal of MathematicalAnalysis and Applications, vol. 292, no. 2, pp. 423–432, 2004.

[4] N. L. Braha, “On asymptotically Δ𝑚-lacunary statistical equiva-lent sequences,”AppliedMathematics andComputation, vol. 219,no. 1, pp. 280–288, 2012.

[5] R. Çolak andM. Et, “On some difference sequence sets and theirtopological properties,” Bulletin of the Malaysian MathematicalSciences Society, vol. 28, no. 2, pp. 125–130, 2005.

[6] M. Et, “Strongly almost summable difference sequences of order𝑚 defined by a modulus,” Studia Scientiarum MathematicarumHungarica, vol. 40, no. 4, pp. 463–476, 2003.

[7] M.Güngor andM.Et, “Δ𝑟-strongly almost summable sequencesdefined by Orlicz functions,” Indian Journal of Pure and AppliedMathematics, vol. 34, no. 8, pp. 1141–1151, 2003.

[8] M. Karakas, M. Et, and V. Karakaya, “Some geometric prop-erties of a new difference sequence space involving lacunarysequences,” Acta Mathematica Scientia, vol. 33, no. 6, pp. 1711–1720, 2013.

[9] M. Mursaleen, R. Çolak, and M. Et, “Some geometric inequali-ties in a new banach sequence space,” Journal of Inequalities andApplications, vol. 2007, Article ID 86757, 6 pages, 2007.

[10] H. Polat, V. Karakaya, and N. Şimşek, “Difference sequencespaces derived by using a generalized weighted mean,” AppliedMathematics Letters, vol. 24, no. 5, pp. 608–614, 2011.

[11] B. C. Tripathy, “On a class of difference sequences related to the𝑝-normed space 𝑙p,” Demonstratio Mathematica, vol. 36, no. 4,pp. 867–872, 2003.

[12] S. A. Mohiuddine, K. Raj, and A. Alotaibi, “Some paranormeddouble difference sequence spaces for Orlicz functions andbounded-regular matrices,” Abstract and Applied Analysis, vol.2014, Article ID 419064, 10 pages, 2014.

[13] S. A. Mohiuddine, K. Raj, and A. Alotaibi, “On some classes ofdouble difference sequences of interval numbers,” Abstract andApplied Analysis, vol. 2014, Article ID 516956, 8 pages, 2014.

[14] W. L. C. Sargent, “Some sequence spaces related to the l𝑝spaces,”

Journal of the London Mathematical Society. Second Series, vol.35, pp. 161–171, 1960.

[15] B. C. Tripathy andM. Sen, “On a new class of sequences relatedto the space 𝐿

𝑝,” Tamkang Journal of Mathematics, vol. 33, no. 2,

pp. 167–171, 2002.[16] B. C. Tripathy and S. Mahanta, “On a class of sequences related

to the 𝑙p space defined by Orlicz functions,” Soochow Journal ofMathematics, vol. 29, no. 4, pp. 379–391, 2003.

[17] Y. A. Cui and H. Hudzik, “On the Banach-Saks and weakBanach-Saks properties of some Banach sequence spaces,” ActaScientiarumMathematicarum, vol. 65, no. 1-2, pp. 179–187, 1999.

[18] J. Diestel, Sequence and Series in Banach Spaces, vol. 92 ofGraduate Texts in Mathematics, Springer, Berlin, Germany,1984.

[19] V. Karakaya and F. Altun, “On some geometric properties of anew paranormed sequence space,” Journal of Function Spaces,vol. 2014, Article ID 685382, 8 pages, 2014.

[20] H. Hudzik, V. Karakaya, and M. Mursaleen, “Banach-Saks typeand GGurarii modulus of convexity of some Banach sequencespaces,” Abstract and Applied Analysis, vol. 2014, Article ID427382, 9 pages, 2014.

[21] M. Et, M. Karakaş, and V. Karakaya, “Some geometric proper-ties of a new difference sequence space defined by de la Vallée-Poussinmean,”AppliedMathematics andComputation, vol. 234,pp. 237–244, 2014.

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