On Inclusion Relations between Some Sequence Spaces

4 downloads 0 Views 2MB Size Report
Jul 10, 2016 - In this case, we write − lim = or → ( ) and denotes the set of all statistically convergent sequences. A sequence ...
Hindawi Publishing Corporation International Journal of Analysis Volume 2016, Article ID 7283527, 4 pages http://dx.doi.org/10.1155/2016/7283527

Research Article On Inclusion Relations between Some Sequence Spaces R. Çolak, Ç. A. BektaG, H. AltJnok, and S. Ercan Department of Mathematics, Firat University, 23119 Elˆazı˘g, Turkey Correspondence should be addressed to C ¸ . A. Bektas¸; [email protected] Received 15 April 2016; Accepted 10 July 2016 Academic Editor: Julien Salomon Copyright © 2016 R. C ¸ olak et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We determine the relations between the classes 𝑆̂𝜆 of almost 𝜆-statistically convergent sequences and the relations between the ̂ 𝜆] of strongly almost (𝑉, 𝜆)-summable sequences for various sequences 𝜆, 𝜇 in the class Λ. Furthermore we also give the classes [𝑉, ̂ 𝜆] of strongly almost (𝑉, 𝜆)-sumrelations between the classes 𝑆̂𝜆 of almost 𝜆-statistically convergent sequences and the classes [𝑉, mable sequences for various sequences 𝜆, 𝜇 ∈ Λ.

1. Introduction A sequence 𝑥 = (𝑥𝑘 ) of real (or complex) numbers is said to be statistically convergent to the number 𝐿 if for every 𝜀 > 0 lim

1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨{𝑘 ≤ 𝑛 : 󵄨󵄨󵄨𝑥𝑘 − 𝐿󵄨󵄨󵄨 ≥ 𝜀}󵄨󵄨󵄨 = 0.

𝑛→∞ 𝑛

1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨{𝑘 ≤ 𝑛 : 󵄨󵄨󵄨𝑥𝑘+𝑚 − 𝐿󵄨󵄨󵄨 ≥ 𝜀}󵄨󵄨󵄨 = 0 uniformly in 𝑚. (2) 𝑛→∞ 𝑛 󵄨 lim

̂ ̂ and 𝑆̂ denotes In this case, we write 𝑆−lim 𝑥 = 𝐿 or 𝑥𝑘 → 𝐿(𝑆) the set of all almost statistically convergent sequences [1]. Let 𝜆 = (𝜆 𝑛 ) be a nondecreasing sequence of positive real numbers tending to ∞ such that 𝜆 1 = 1.

(3)

The set of all such sequences will be denoted by Λ. The generalized de la Vall´ee-Poussin mean is defined by 𝑡𝑛 (𝑥) =

1 ∑𝑥 , 𝜆 𝑛 𝑘∈𝐼 𝑘 𝑛

where 𝐼𝑛 = [𝑛 − 𝜆 𝑛 + 1, 𝑛].

𝑡𝑛 (𝑥) 󳨀→ 𝐿 as 𝑛 󳨀→ ∞.

(1)

In this case, we write 𝑆 − lim 𝑥 = 𝐿 or 𝑥𝑘 → 𝐿(𝑆) and 𝑆 denotes the set of all statistically convergent sequences. A sequence 𝑥 = (𝑥𝑘 ) of real (or complex) numbers is said to be almost statistically convergent to the number 𝐿 if for every 𝜀 > 0

𝜆 𝑛+1 ≤ 𝜆 𝑛 + 1,

A sequence 𝑥 = (𝑥𝑘 ) is said to be (𝑉, 𝜆)-summable to a number 𝐿 (see [2]) if

(4)

(5)

If 𝜆 𝑛 = 𝑛 for each 𝑛 ∈ N, then (𝑉, 𝜆)-summability reduces to (𝐶, 1)-summability. We write [𝐶, 1] 1 𝑛 󵄨󵄨 󵄨 ∑ 󵄨󵄨𝑥𝑘 − 𝐿󵄨󵄨󵄨 = 0 for some 𝐿} , 𝑛→∞ 𝑛 𝑘=1

= {𝑥 = (𝑥𝑘 ) : lim [𝑉, 𝜆]

(6)

{ } 1 󵄨 󵄨 = {𝑥 = (𝑥𝑘 ) : lim ∑ 󵄨󵄨󵄨𝑥𝑘 − 𝐿󵄨󵄨󵄨 = 0 for some 𝐿} 𝑛→∞ 𝜆 𝑛 𝑘∈𝐼𝑛 { } for the sets of sequences 𝑥 = (𝑥𝑘 ) which are strongly Ces`aro summable and strongly (𝑉, 𝜆)-summable to 𝐿; that is, 𝑥𝑘 → 𝐿[𝐶, 1] and 𝑥𝑘 → 𝐿[𝑉, 𝜆], respectively.

2

International Journal of Analysis Proof. (i) Suppose that 𝜆 𝑛 ≤ 𝜇𝑛 for all 𝑛 ∈ N𝑛𝑜 and let (10) be satisfied. Then 𝐼𝑛 ⊂ 𝐽𝑛 so that for 𝜀 > 0 we may write

Savaˇs [1] defined the following sequence space: { ̂ 𝜆] = 𝑥 = (𝑥𝑘 ) : lim 1 ∑ 󵄨󵄨󵄨𝑥𝑘+𝑚 − 𝐿󵄨󵄨󵄨 [𝑉, 󵄨 󵄨 { 𝑛→∞ 𝜆 𝑛 𝑘∈𝐼𝑛 {

(7)

} = 0 for some 𝐿, uniformly in 𝑚} }

lim

1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨{𝑘 ∈ 𝐼𝑛 : 󵄨󵄨󵄨𝑥𝑘 − 𝐿󵄨󵄨󵄨 ≥ 𝜀}󵄨󵄨󵄨 = 0, 𝑛

(8)

where 𝐼𝑛 = [𝑛−𝜆 𝑛 +1, 𝑛]. In this case we write 𝑆𝜆 −lim 𝑥 = 𝐿 or 𝑥𝑘 → 𝐿(𝑆𝜆 ), and 𝑆𝜆 = {𝑥 = (𝑥𝑘 ) : 𝑆𝜆 − lim 𝑥 = 𝐿 for some 𝐿}. The sequence 𝑥 = (𝑥𝑘 ) is said to be 𝜆-almost statistically convergent if there is a complex number 𝐿 such that lim

1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨{𝑘 ∈ 𝐼𝑛 : 󵄨󵄨󵄨𝑥𝑘+𝑚 − 𝐿󵄨󵄨󵄨 ≥ 𝜀}󵄨󵄨󵄨 = 0,

𝑛→∞ 𝜆

𝑛

1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨{𝑘 ∈ 𝐽𝑛 : 󵄨󵄨󵄨𝑥𝑘+𝑚 − 𝐿󵄨󵄨󵄨 ≥ 𝜀}󵄨󵄨󵄨 𝜇𝑛 󵄨 𝜆 1 󵄨󵄨 󵄨 󵄨 󵄨 ≥ 𝑛 󵄨{𝑘 ∈ 𝐼𝑛 : 󵄨󵄨󵄨𝑥𝑘+𝑚 − 𝐿󵄨󵄨󵄨 ≥ 𝜀}󵄨󵄨󵄨 𝜇𝑛 𝜆 𝑛 󵄨

1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨{𝑘 ∈ 𝐽𝑛 : 󵄨󵄨󵄨𝑥𝑘+𝑚 − 𝐿󵄨󵄨󵄨 ≥ 𝜀}󵄨󵄨󵄨 𝜇𝑛 󵄨 =

(9)

In this case, we write 𝑆̂𝜆 − lim 𝑥 = 𝐿 or 𝑥𝑘 → 𝐿(𝑆̂𝜆 ) and 𝑆̂𝜆 denotes the set of all 𝜆-almost statistically convergent sequences. If we choose 𝜆 𝑛 = 𝑛 for all 𝑛, then 𝜆-almost statistical convergence reduces to almost statistical convergence [1].

2. Main Results Throughout the paper, unless stated otherwise, by “for all 𝑛 ∈ N𝑛𝑜 ” we mean “for all 𝑛 ∈ N except finite numbers of positive integers” where N𝑛𝑜 = {𝑛𝑜 , 𝑛𝑜 + 1, 𝑛𝑜 + 2, . . .} for some 𝑛𝑜 ∈ N = {1, 2, 3, . . .}. Theorem 1. Let 𝜆 = (𝜆 𝑛 ) and 𝜇 = (𝜇𝑛 ) be two sequences in Λ such that 𝜆 𝑛 ≤ 𝜇𝑛 for all 𝑛 ∈ N𝑛𝑜 . Consider the following: (i) If 𝑛→∞

𝜆𝑛 >0 𝜇𝑛

(10)

1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨{𝑛 − 𝜇𝑛 + 1 ≤ 𝑘 ≤ 𝑛 − 𝜆 𝑛 : 󵄨󵄨󵄨𝑥𝑘+𝑚 − 𝐿󵄨󵄨󵄨 ≥ 𝜀}󵄨󵄨󵄨 𝜇𝑛 󵄨 +



1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨{𝑘 ∈ 𝐼𝑛 : 󵄨󵄨󵄨𝑥𝑘+𝑚 − 𝐿󵄨󵄨󵄨 ≥ 𝜀}󵄨󵄨󵄨 𝜇𝑛 󵄨

(14)

𝜇𝑛 − 𝜆 𝑛 1 󵄨󵄨 󵄨 󵄨 󵄨 + 󵄨{𝑘 ∈ 𝐼𝑛 : 󵄨󵄨󵄨𝑥𝑘+𝑚 − 𝐿󵄨󵄨󵄨 ≥ 𝜀}󵄨󵄨󵄨 𝜇𝑛 𝜆𝑛 󵄨

≤ (1 −

𝜆𝑛 1 󵄨󵄨 󵄨 󵄨 󵄨 )+ 󵄨{𝑘 ∈ 𝐼𝑛 : 󵄨󵄨󵄨𝑥𝑘+𝑚 − 𝐿󵄨󵄨󵄨 ≥ 𝜀}󵄨󵄨󵄨 𝜇𝑛 𝜆𝑛 󵄨

for all 𝑛 ∈ N𝑛𝑜 . Since lim𝑛 (𝜆 𝑛 /𝜇𝑛 ) = 1 by (11) and since 𝑥 = (𝑥𝑘 ) ∈ 𝑆̂𝜆 the first term and second term of right hand side of above inequality tend to 0 as 𝑛 → ∞ uniformly in 𝑚. This implies that (1/𝜇𝑛 )|{𝑘 ∈ 𝐽𝑛 : |𝑥𝑘+𝑚 − 𝐿| ≥ 𝜀}| → 0 as 𝑛 → ∞ uniformly in 𝑚. Therefore 𝑆̂𝜆 ⊆ 𝑆̂𝜇 . From Theorem 1 we have the following result. Corollary 2. Let 𝜆 = (𝜆 𝑛 ) and 𝜇 = (𝜇𝑛 ) be two sequences in Λ such that 𝜆 𝑛 ≤ 𝜇𝑛 for all 𝑛 ∈ N𝑛𝑜 . If (11) holds then 𝑆̂𝜆 = 𝑆̂𝜇 . If we take 𝜇 = (𝜇𝑛 ) = (𝑛) in Corollary 2 we have the following result. Corollary 3. Let 𝜆 = (𝜆 𝑛 ) ∈ Λ. If lim𝑛 (𝜆 𝑛 /𝑛) = 1 then we ̂ have 𝑆̂𝜆 = 𝑆.

then 𝑆̂𝜇 ⊆ 𝑆̂𝜆 . (ii) If 𝜆 lim 𝑛 𝑛→∞ 𝜇 𝑛 then 𝑆̂𝜆 ⊆ 𝑆̂𝜇 .

(13)

for all 𝑛 ∈ N𝑛𝑜 , where 𝐽𝑛 = [𝑛 − 𝜇𝑛 + 1, 𝑛]. Now taking the limit as 𝑛 → ∞ uniformly in 𝑚 in the last inequality and using (10) we get 𝑥𝑘 → 𝐿(𝑆̂𝜇 ) ⇒ 𝑥𝑘 → 𝐿(𝑆̂𝜆 ) so that 𝑆̂𝜇 ⊆ 𝑆̂𝜆 . (ii) Let (𝑥𝑘 ) ∈ 𝑆̂𝜆 and (11) be satisfied. Since 𝐼𝑛 ⊂ 𝐽𝑛 , for 𝜀 > 0, we may write

uniformly in 𝑚.

lim inf

(12)

and therefore we have

for the sets of sequences 𝑥 = (𝑥𝑘 ) which are strongly almost (𝑉, 𝜆)-summable to 𝐿; that is, 𝑥𝑘 → 𝐿[𝑉, 𝜆]. We will write ̂ 𝜆]∞ = [𝑉, ̂ 𝜆] ∩ ℓ∞ . [𝑉, The 𝜆-statistical convergence was introduced by Mursaleen in [3] as follows. Let 𝜆 = (𝜆 𝑛 ) ∈ Λ. A sequence 𝑥 = (𝑥𝑘 ) is said to be 𝜆statistically convergent or 𝑆𝜆 -convergent to 𝐿 if for every 𝜀 > 0 𝑛→∞ 𝜆

󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨{𝑘 ∈ 𝐽𝑛 : 󵄨󵄨󵄨𝑥𝑘+𝑚 − 𝐿󵄨󵄨󵄨 ≥ 𝜀}󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 ≥ 󵄨󵄨󵄨{𝑘 ∈ 𝐼𝑛 : 󵄨󵄨󵄨𝑥𝑘+𝑚 − 𝐿󵄨󵄨󵄨 ≥ 𝜀}󵄨󵄨󵄨

=1

(11)

Theorem 4. Let 𝜆 = (𝜆 𝑛 ) and 𝜇 = (𝜇𝑛 ) ∈ Λ and suppose that 𝜆 𝑛 ≤ 𝜇𝑛 for all 𝑛 ∈ N𝑛𝑜 . Consider the following: ̂ 𝜇] ⊆ [𝑉, ̂ 𝜆]. (i) If (10) holds then [𝑉, ̂ 𝜆]∞ ⊆ [𝑉, ̂ 𝜇]. (ii) If (11) holds then [𝑉,

International Journal of Analysis

3

Proof. (i) Suppose that 𝜆 𝑛 ≤ 𝜇𝑛 for all 𝑛 ∈ N𝑛𝑜 . Then 𝐼𝑛 ⊆ 𝐽𝑛 so that we may write 1 󵄨 1 󵄨 󵄨 󵄨 − 𝐿󵄨󵄨󵄨 ≥ − 𝐿󵄨󵄨󵄨 ∑ 󵄨󵄨𝑥 ∑ 󵄨󵄨𝑥 𝜇𝑛 𝑘∈𝐽 󵄨 𝑘+𝑚 𝜇𝑛 𝑘∈𝐼 󵄨 𝑘+𝑚 𝑛

̂ 𝜇]. Now for Proof. (i) Let 𝜀 > 0 be given and let 𝑥𝑘 → 𝐿[𝑉, every 𝜀 > 0 we may write 󵄨 󵄨 󵄨 󵄨 ∑ 󵄨󵄨󵄨𝑥𝑘+𝑚 − 𝐿󵄨󵄨󵄨 ≥ ∑ 󵄨󵄨󵄨𝑥𝑘+𝑚 − 𝐿󵄨󵄨󵄨

(15)

𝑘∈𝐽𝑛

𝑛

𝑘∈𝐼𝑛

for all 𝑛 ∈ N𝑛𝑜 . This gives that



1 󵄨 𝜆 1 󵄨 󵄨 󵄨 − 𝐿󵄨󵄨󵄨 ≥ 𝑛 − 𝐿󵄨󵄨󵄨 . ∑ 󵄨󵄨𝑥 ∑ 󵄨󵄨𝑥 𝜇𝑛 𝑘∈𝐽 󵄨 𝑘+𝑚 𝜇𝑛 𝜆 𝑛 𝑘∈𝐼 󵄨 𝑘+𝑚 𝑛

𝑘∈𝐼𝑛 |𝑥𝑘+𝑚 −𝐿|≥𝜀

(16)

1 󵄨 󵄨 󵄨 1 󵄨 − 𝐿󵄨󵄨󵄨 ∑ 󵄨󵄨󵄨𝑥𝑘+𝑚 − 𝐿󵄨󵄨󵄨 = ∑ 󵄨󵄨𝑥 𝜇𝑛 𝑘∈𝐽 𝜇𝑛 𝑘∈𝐽 −𝐼 󵄨 𝑘+𝑚 𝑛

𝑛

+

𝑛

1 ∑ 𝜇𝑛 𝑘∈𝐼

󵄨󵄨 󵄨 󵄨󵄨𝑥𝑘+𝑚 − 𝐿󵄨󵄨󵄨

𝑛

𝜇 − 𝜆𝑛 1 󵄨 󵄨 ≤ 𝑛 𝑀+ − 𝐿󵄨󵄨󵄨 ∑ 󵄨󵄨𝑥 𝜇𝑛 𝜇𝑛 𝑘∈𝐼 󵄨 𝑘+𝑚

(17)

𝑛

(19)

so that 1 󵄨 󵄨 − 𝐿󵄨󵄨󵄨 ∑ 󵄨󵄨𝑥 𝜇𝑛 𝑘∈𝐽 󵄨 𝑘+𝑚 𝑛

𝜆 1 󵄨󵄨 󵄨 󵄨 󵄨 ≥ 𝑛 󵄨{𝑘 ∈ 𝐼𝑛 : 󵄨󵄨󵄨𝑥𝑘+𝑚 − 𝐿󵄨󵄨󵄨 ≥ 𝜀}󵄨󵄨󵄨 𝜀 𝜇𝑛 𝜆 𝑛 󵄨

(20)

for all 𝑛 ∈ N𝑛𝑜 . Then taking limit as 𝑛 → ∞, uniformly in 𝑚 in the last inequality, and using (10) we obtain that 𝑥𝑘 → ̂ 𝜇]. Since 𝑥 = (𝑥𝑘 ) ∈ [𝑉, ̂ 𝜇] is an 𝐿(𝑆̂𝜆 ) whenever 𝑥𝑘 → 𝐿[𝑉, ̂ ̂ arbitrary sequence we obtain that [𝑉, 𝜇] ⊂ 𝑆𝜆 . (ii) Suppose that 𝑥𝑘 → 𝐿(𝑆̂𝜆 ) and 𝑥 = (𝑥𝑘 ) ∈ ℓ∞ . Then there exists some 𝑀 > 0 such that |𝑥𝑘+𝑚 − 𝐿| ≤ 𝑀 for all 𝑘 and 𝑚. Since 1/𝜇𝑛 ≤ 1/𝜆 𝑛 , then for every 𝜀 > 0 we may write 1 󵄨 󵄨 󵄨 1 󵄨 − 𝐿󵄨󵄨󵄨 = − 𝐿󵄨󵄨󵄨 ∑ 󵄨󵄨𝑥 ∑ 󵄨󵄨𝑥 𝜇𝑛 𝑘∈𝐽 󵄨 𝑘+𝑚 𝜇𝑛 𝑘∈𝐽 −𝐼 󵄨 𝑘+𝑚

𝜆 1 󵄨 󵄨 ≤ (1 − 𝑛 ) 𝑀 + − 𝐿󵄨󵄨󵄨 ∑ 󵄨󵄨𝑥 𝜇𝑛 𝜆 𝑛 𝑘∈𝐼 󵄨 𝑘+𝑚

𝑛

𝑛

𝑛

for every 𝑛 ∈ N𝑛𝑜 . Since lim𝑛 (𝜆 𝑛 /𝜇𝑛 ) = 1 by (11) and since ̂ 𝜆] the first term and the second term of right hand 𝑥𝑘 → 𝐿[𝑉, side of above inequality tend to 0 as 𝑛 → ∞, uniformly in 𝑚 (note that 1 − 𝜆 𝑛 /𝜇𝑛 ≥ 0 for all 𝑛 ∈ N𝑛𝑜 ). Then we get ̂ 𝜆] ⇒ 𝑥𝑘 → 𝐿[𝑉, ̂ 𝜇]. Since 𝑥 = (𝑥𝑘 ) ∈ [𝑉, ̂ 𝜆]∞ is 𝑥𝑘 → 𝐿[𝑉, ̂ ̂ an arbitrary sequence we obtain [𝑉, 𝜆]∞ ⊆ [𝑉, 𝜇]. Since clearly (11) implies (10) from Theorem 4 we have the following result. Corollary 5. Let 𝜆, 𝜇 ∈ Λ such that 𝜆 𝑛 ≤ 𝜇𝑛 for all 𝑛 ∈ N𝑛𝑜 . If ̂ 𝜇]∞ . ̂ 𝜆]∞ = [𝑉, (11) holds then [𝑉, Theorem 6. Let 𝜆, 𝜇 ∈ Λ such that 𝜆 𝑛 ≤ 𝜇𝑛 for all 𝑛 ∈ N𝑛𝑜 . Consider the following:

+

1 󵄨 󵄨 − 𝐿󵄨󵄨󵄨 ∑ 󵄨󵄨𝑥 𝜇𝑛 𝑘∈𝐼 󵄨 𝑘+𝑚



𝜇𝑛 − 𝜆 𝑛 1 󵄨 󵄨 𝑀+ − 𝐿󵄨󵄨󵄨 ∑ 󵄨󵄨𝑥 𝜇𝑛 𝜇𝑛 𝑘∈𝐼 󵄨 𝑘+𝑚 𝑛

≤ (1 −

𝜆𝑛 1 󵄨 󵄨 )𝑀 + − 𝐿󵄨󵄨󵄨 ∑ 󵄨󵄨𝑥 𝜇𝑛 𝜆 𝑛 𝑘∈𝐼 󵄨 𝑘+𝑚 𝑛

≤ (1 − 1 + 𝜆𝑛 1 𝜆𝑛

(18)

̂ 𝜇] ⊂ 𝑆̂𝜆 holds for some 𝜆, 𝜇 ∈ Λ. and the inclusion [𝑉, ̂ 𝜇], (ii) If (𝑥𝑘 ) ∈ ℓ∞ and 𝑥𝑘 → 𝐿(𝑆̂𝜆 ) then 𝑥𝑘 → 𝐿[𝑉, whenever (11) holds. ̂ 𝜇]∞ . (iii) If (11) holds then ℓ∞ ∩ 𝑆̂𝜆 = [𝑉,

𝑛

𝑛

+

(i) If (10) holds then ̂ 𝜇] 󳨐⇒ 𝑥𝑘 󳨀→ 𝐿 (𝑆̂𝜆 ) 𝑥𝑘 󳨀→ 𝐿 [𝑉,

󵄨󵄨 󵄨 󵄨󵄨𝑥𝑘+𝑚 − 𝐿󵄨󵄨󵄨

󵄨 󵄨 󵄨 󵄨 ≥ 𝜀 󵄨󵄨󵄨{𝑘 ∈ 𝐼𝑛 : 󵄨󵄨󵄨𝑥𝑘+𝑚 − 𝐿󵄨󵄨󵄨 ≥ 𝜀}󵄨󵄨󵄨

𝑛

Then taking limit as 𝑛 → ∞, uniformly in 𝑚 in the last ̂ 𝜇] ⇒ 𝑥𝑘 → inequality, and using (10) we obtain 𝑥𝑘 → 𝐿[𝑉, ̂ 𝜆]. 𝐿[𝑉, ̂ 𝜆]∞ be any sequence. Suppose that (ii) Let 𝑥 = (𝑥𝑘 ) ∈ [𝑉, ̂ 𝜇] and that (11) holds. Since 𝑥 = (𝑥𝑘 ) ∈ 𝑙∞ then 𝑥𝑘 → 𝐿[𝑉, there exists some 𝑀 > 0 such that |𝑥𝑘+𝑚 − 𝐿| ≤ 𝑀 for all 𝑘 and 𝑚. Since 𝜆 𝑛 ≤ 𝜇𝑛 so that 1/𝜇𝑛 ≤ 1/𝜆 𝑛 , and 𝐼𝑛 ⊂ 𝐽𝑛 for all 𝑛 ∈ N𝑛𝑜 , we may write



≤ (1 − +

𝜆𝑛 )𝑀 𝜇𝑛 ∑ 𝑘∈𝐼𝑛 |𝑥𝑘+𝑚 −𝐿|≥𝜀

∑ 𝑘∈𝐼𝑛 |𝑥𝑘+𝑚 −𝐿| 0 then 𝑆̂𝜇 ∩ [𝑉, If we take 𝜇𝑛 = 𝑛 for all 𝑛 in Theorem 6 then we have the following results, because lim𝑛→∞ (𝜆 𝑛 /𝜇𝑛 ) = 1 implies that lim inf 𝑛→∞ (𝜆 𝑛 /𝜇𝑛 ) = 1 > 0; that is, (11) ⇒ (10). Corollary 8. If lim𝑛→∞ (𝜆 𝑛 /𝑛) = 1 then (i) if (𝑥𝑘 ) ∈ ℓ∞ and 𝑥𝑘 → 𝐿(𝑆̂𝜆 ) then 𝑥𝑘 → 𝐿[𝐶, 1], (ii) if 𝑥𝑘 → 𝐿[𝐶, 1] then 𝑥𝑘 → 𝐿(𝑆̂𝜆 ).

Competing Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

References [1] E. Savaˇs, “Strong almost convergence and almost 𝜆-statistical convergence,” Hokkaido Mathematical Journal, vol. 29, no. 3, pp. 531–536, 2000. ¨ [2] L. Leindler, “Uber die de la vall´ee-pousinsche summierbarkeit allgemeiner Orthogonalreihen,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 16, pp. 375–387, 1965. [3] M. Mursaleen, “𝜆-statistical convergence,” Mathematica Slovaca, vol. 50, no. 1, pp. 111–115, 2000.

Advances in

Operations Research Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Advances in

Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Applied Mathematics

Algebra

Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Probability and Statistics Volume 2014

The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of

Differential Equations Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014

Submit your manuscripts at http://www.hindawi.com International Journal of

Advances in

Combinatorics Hindawi Publishing Corporation http://www.hindawi.com

Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of Mathematics and Mathematical Sciences

Mathematical Problems in Engineering

Journal of

Mathematics Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Discrete Mathematics

Journal of

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Discrete Dynamics in Nature and Society

Journal of

Function Spaces Hindawi Publishing Corporation http://www.hindawi.com

Abstract and Applied Analysis

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of

Journal of

Stochastic Analysis

Optimization

Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014