Some New Difference Sequence Spaces of Invariant Means Defined

Hindawi Publishing Corporation International Journal of Analysis Volume 2014, Article ID 631301, 7 pages http://dx.doi.org/10.1155/2014/631301

Research Article Some New Difference Sequence Spaces of Invariant Means Defined by Ideal and Modulus Function Sudhir Kumar,1 Vijay Kumar,2 and S. S. Bhatia1 1 2

School of Mathematics and Computer Applications, Thapar University, Patiala, Punjab 147004, India Department of Mathematics, Haryana College of Technology and Management, Kaithal, Haryana 136027, India

Correspondence should be addressed to Sudhir Kumar; [email protected] Received 15 February 2014; Accepted 9 May 2014; Published 28 May 2014 Academic Editor: Shamsul Qamar Copyright © 2014 Sudhir Kumar 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. The main objective of this paper is to introduce a new kind of sequence spaces by combining the concepts of modulus function, invariant means, difference sequences, and ideal convergence. We also examine some topological properties of the resulting 𝑚 sequence spaces. Further, we introduce a new concept of 𝑆𝜃Δ𝜎 (I)-convergence and obtain a condition under which this convergence coincides with above-mentioned sequence spaces.

1. Introduction and Background Let ℓ∞ and 𝐶 be the Banach spaces of real bounded and convergent sequences with the usual supremum norm. Let 𝜎 be the mapping of the set of all positive integers into itself. A continuous linear functional 𝜑 on ℓ∞ is said to be an invariant mean or 𝜎-mean if and only if (i) 𝜑(𝑥) ≥ 0, when the sequence 𝑥 = (𝑥𝑛 ) has 𝑥𝑛 ≥ 0, for all 𝑛; (ii) 𝜑(𝑒) = 1, where 𝑒 = (1, 1, 1, . . .); (iii) 𝜑(𝑥𝜎(𝑛) ) = 𝜑(𝑥), for all 𝑥 ∈ ℓ∞ . If 𝑥 = (𝑥𝑘 ), we write 𝑇𝑥 = (𝑇𝑥𝑘 ) = (𝑥𝜎(𝑘) ). It can be shown by Schaefer [1] that 𝑉𝜎 = {𝑥 ∈ ℓ∞ : lim 𝑡𝑘𝑚 (𝑥) = 𝐿, 𝑘→∞

(1)

uniformly in 𝑚, 𝜎 − lim 𝑥 = 𝐿} , where 𝑡𝑘𝑚 (𝑥) = (𝑥𝑚 + 𝑥𝜎(𝑚) + ⋅ ⋅ ⋅ + 𝑥𝜎𝑘 (𝑚) )/(𝑘 + 1). In case 𝜎 is the translation mapping 𝑛 → 𝑛 + 1, 𝜎-mean is often called a Banach limit and 𝑉𝜎 , the set of bounded sequences all whose invariant means are equal, is the set of almost convergent sequences (see Lorentz [2]). Using the

concept of invariant means Mursaleen et al. [3] introduced the following sequence spaces as a generalization of Das and Sahoo [4]: 𝑤𝜎 1 𝑛 ∑ 𝑡𝑘𝑚 (𝑥 − ℓ) 󳨀→ 0, uniformly in 𝑚} , 𝑛→∞𝑛 + 1 𝑘=0

= {𝑥 : lim [𝑤]𝜎

1 𝑛 󵄨󵄨 󵄨 ∑ 󵄨󵄨󵄨𝑡𝑘𝑚 (𝑥 − ℓ)󵄨󵄨󵄨󵄨 󳨀→ 0, uniformly in 𝑚} , 𝑛→∞𝑛 + 1 𝑘=0

= {𝑥 : lim [𝑤𝜎 ]

1 𝑛 ∑ 𝑡𝑘𝑚 (|𝑥 − ℓ|) 󳨀→ 0, uniformly in 𝑚} 𝑛→∞𝑛 + 1 𝑘=0 (2)

= {𝑥 : lim

and investigated some of its properties. The notion of statistical convergence for number sequences was studied at the initial stage by Fast [5] and later ˇ at [9], investigated by Connor [6], Fridy [7], Maddox [8], Sal´ and many others.

2

International Journal of Analysis

Definition 1 (see [5]). A number sequence 𝑥 = (𝑥𝑘 ) is said to be statistically convergent to a number 𝐿 (denoted by 𝑆 − lim𝑘 → ∞ 𝑥𝑘 = 𝐿) provided that, for every 𝜖 > 0, lim

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

𝑛→∞𝑛

(3)

where the vertical bars denote the cardinality of the enclosed set. By a lacunary sequence, we mean an increasing sequence 𝜃 = (𝑘𝑟 ) of positive integers such that 𝑘0 = 0 and ℎ𝑟 = 𝑘𝑟 − 𝑘𝑟−1 → ∞ as 𝑟 → ∞. The intervals determined by 𝜃 will be denoted by 𝐼𝑟 = (𝑘𝑟−1 , 𝑘𝑟 ], where the ratio 𝑘𝑟 /𝑘𝑟−1 is denoted by 𝑞𝑟 . The space of lacunary strongly convergent sequence 𝑁𝜃 was defined by Freedman et al. [10] as follows: } { 1 󵄨 󵄨 𝑁𝜃 = {𝑥 = (𝑥𝑘 ) : lim ∑ 󵄨󵄨󵄨𝑥𝑘 − 𝐿󵄨󵄨󵄨 = 0 for some 𝐿} . 𝑟→∞ℎ 𝑟 𝑘∈𝐼𝑟 } { (4) Fridy and Orhan [11] generalized the concept of statistical convergence by using lacunary sequence which is called lacunary statistical convergence. Further, lacunary sequences have been studied by Fridy and Orhan [12], Pehlivan and Fisher [13], Et and Gökhan [14], and Tripathy and Dutta [15]. Quite recently, Karakaya [16] combined the approach of lacunary sequence with invariant means and introduced the notion of strong 𝜎-lacunary statistically convergence as follows. Definition 2 (see [16]). Let 𝜃 = (𝑘𝑟 ) be a lacunary sequence. A sequence 𝑥 = (𝑥𝑘 ) is said to be lacunary strong 𝜎-lacunary statistically convergent if, for every 𝜖 > 0, lim

1 󵄨󵄨 󵄨󵄨{𝑘 ∈ 𝐼𝑟 󵄨

𝑟→∞ℎ 𝑟

󵄨 󵄨 󵄨 : 󵄨󵄨󵄨󵄨𝑡𝑘𝑚 (𝑥 − 𝐿)󵄨󵄨󵄨󵄨 ≥ 𝜖}󵄨󵄨󵄨󵄨 = 0,

(5)

uniformly in 𝑚,

𝑋 is said to be I-convergent to 𝜉 if, for each 𝜖 > 0, the set 𝐴(𝜖) = {𝑘 ∈ N : 𝜌(𝑥𝑘 , 𝜉) ≥ 𝜖} ∈ I. In this case, we write I − lim𝑘 → ∞ 𝑥𝑘 = 𝜉. Recently, Das et al. [18] unified the idea of lacunary statistical convergence with ideal convergence and presented the following interesting generalization of statistical convergence. Definition 4 (see [18]). Let 𝜃 = (𝑘𝑟 ) be a lacunary sequence. A sequence 𝑥 = (𝑥𝑘 ) of numbers is said to be I-lacunay statistical convergent or 𝑆𝜃 (I)-convergent to 𝐿, if, for every 𝜖 > 0 and 𝛿 > 0, {𝑟 ∈ N :

1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨{𝑘 ∈ 𝐼𝑟 : 󵄨󵄨󵄨𝑥𝑘 − 𝐿󵄨󵄨󵄨 ≥ 𝜖}󵄨󵄨󵄨 ≥ 𝛿} ∈ I. ℎ𝑟 󵄨

(6)

In this case, we write 𝑥𝑘 → 𝐿(𝑆𝜃 (I)) or 𝑆𝜃 (I)−lim𝑘 → ∞ 𝑥𝑘 = 𝐿. The set of all I-lacunary statistically convergent sequences will be denoted by 𝑆𝜃 (I). Definition 5 (see [18]). Let 𝜃 = (𝑘𝑟 ) be a lacunary sequence. A sequence 𝑥 = (𝑥𝑘 ) of numbers is said to be 𝑁𝜃 (I)-convergent to 𝐿 if, for every 𝜖 > 0, we have } { 1 󵄨󵄨 󵄨󵄨 {𝑟 ∈ N : ℎ ∑ 󵄨󵄨𝑥𝑘 − 𝐿󵄨󵄨 ≥ 𝜖} ∈ I. 𝑟 𝑘∈𝐼𝑟 } {

(7)

It is denoted by 𝑥𝑘 → 𝐿(𝑁𝜃 (I)). In 1981, Kızmaz [19] introduced the notion of difference sequence space as follows: Δ (𝑋) = {𝑥 = (𝑥𝑘 ) : (Δ𝑥𝑘 ) ∈ 𝑋} ,

(8)

for 𝑋 = ℓ∞ , 𝑐 and 𝑐0 , where Δ𝑥𝑘 = 𝑥𝑘 − 𝑥𝑘+1 , for all 𝑘 ∈ N. Continuing on this way, the notion was further generalized by Et and C ¸ olak [20] by introducing the sequence spaces as follows: Δ𝑚 (𝑋) = {𝑥 = (𝑥𝑘 ) : Δ𝑚 𝑥𝑘 ∈ 𝑋} ,

(9)

for 𝑋 = ℓ∞ , 𝑐, and 𝑐0 , where 𝑚 ∈ N and Δ𝑚 𝑥𝑘 = (Δ𝑚−1 𝑥𝑘 − V Δ𝑚−1 𝑥𝑘+1 ), so that Δ𝑚 𝑥𝑘 = ∑𝑚 V=0 (−1) (𝑚/V)𝑥𝑘+V . For extensive view in this area, we refer to the series of papers ([21–27]).

where 𝑆𝑡𝜃𝜎 denotes the set of all lacunary strong 𝜎-lacunary statistically convergent sequences. Another interesting generalization of statistical convergence was introduced in [17] with the help of ideals of subsets of N. Let P(N) denote the power set of N. A family of sets I ⊂ P(N) is called an ideal in N if and only if (i) 0 ∈ I; (ii) 𝐴, 𝐵 ∈ I imply 𝐴 ∪ 𝐵 ∈ I; (iii) 𝐴 ∈ I and 𝐵 ⊂ 𝐴 imply 𝐵 ∈ I. A nonempty family of sets F ⊂ P(N) is called a filter on N if and only if (i) 0 ∉ F; (ii) 𝐴, 𝐵 ∈ F imply 𝐴 ∩ 𝐵 ∈ F; (iii) 𝐴 ∈ F and 𝐵 ⊃ 𝐴 imply 𝐵 ∈ F. An ideal I is called nontrivial if I ≠ 0 and N ∉ I. It immediately implies that I ⊂ P(N) is a nontrivial ideal if and only if the class F = F(𝐼) = {N − 𝐴 : 𝐴 ∈ I} is a filter on N. The filter F = F(𝐼) is called the filter associated with the ideal I. A nontrivial ideal I ⊂ P(N) is called an admissible ideal in N if and only if it contains all singletons, that is, if it contains {{𝑛} : 𝑛 ∈ N}. Throughout the present work, I denotes a nontrivial admissible ideal.

The notion of modulus function was introduced by Nakano [28] as follows: by a modulus function, we mean a function 𝑓 from [0, ∞) to [0, ∞) such that (i) 𝑓(𝑥) = 0 if and only if 𝑥 = 0, (ii) 𝑓(𝑥 + 𝑦) ≤ 𝑓(𝑥) + 𝑓(𝑦), for all 𝑥, 𝑦 ≥ 0, (iii) 𝑓 is increasing, and (iv) 𝑓 is continuous from right at 0. It follows that 𝑓 must be continuous everywhere on [0, ∞). A modulus function may be bounded or unbounded. In the recent past the notion of modulus function was investigated from different aspects and sequence spaces have been studied by Ruckle [29], Maddox [30], Et [31], Pehlivan and Fisher [13], Savas [32], Et and Gökhan [14], Kumar et al. [33], and many others. The following well-known lemma is required for establishing a very important result in our paper.

Definition 3 (see [17]). Let I ⊂ P(N) be a nontrivial ideal in N and let (𝑋, 𝜌) be a metric space. A sequence 𝑥 = (𝑥𝑘 ) in

Lemma 6. Let 𝑓 be a modulus function and let 0 < 𝛿 < 1. Then, for each 𝑥 > 𝛿, we have 𝑓(𝑥) ≤ (2 ⋅ 𝑓(1)𝑥)/𝛿.


3

The following inequality will be used throughout the paper. Let 𝑝 = (𝑝𝑘 ) be a positive sequence of real numbers with 0 < 𝑝𝑘 ≤ sup𝑘 𝑝𝑘 = 𝐻 and 𝐶 = max(1, 2𝐻−1 ). Then, for all 𝑎𝑘 , 𝑏𝑘 ∈ C, for all 𝑘 ∈ N, we have

(i) If we take 𝑚 = 0, for all 𝑘 ∈ N, then the above spaces reduce to 𝑤𝜎𝑓 [I, 𝑝, 𝜃]0 , 𝑤𝜎𝑓 [I, 𝑝, 𝜃], and 𝑤𝜎𝑓 [I, 𝑝, 𝜃]∞ , respectively.

󵄨󵄨 󵄨𝑝 󵄨 󵄨𝑝 󵄨 󵄨𝑝 󵄨󵄨𝑎𝑘 + 𝑏𝑘 󵄨󵄨󵄨 𝑘 ≤ 𝐶 {󵄨󵄨󵄨𝑎𝑘 󵄨󵄨󵄨 𝑘 + 󵄨󵄨󵄨𝑏𝑘 󵄨󵄨󵄨 𝑘 } .

(ii) If we choose 𝑚 = 0, for all 𝑘 ∈ N and 𝜎(𝑛) = 𝑛 + 1, ̂𝜎𝑓 [I, 𝑝, 𝜃], ̂𝜎𝑓 [I, 𝑝, 𝜃]0 , 𝑤 then we obtain 𝑤 𝑓 𝑓 𝑚 ̂𝜎 [I, 𝑝, 𝜃]∞ instead of 𝑤𝜎 [Δ 𝑝 , I, 𝜃]0 , and 𝑤

(10)

Inspired by the above works, we presently introduce some new kind of sequence spaces by using ideal convergence, modulus function, and invariant mean. Further, we also obtain some relevant connections of these spaces with 𝑚 𝑆𝜃Δ𝜎 (I)-convergence.

2. Main Results Throughout the paper, I ⊂ P(N) is considered a nontrivial admissible ideal and 𝑤(𝑋) denotes the space of all sequences 𝑥 = (𝑥𝑘 ) ∈ 𝑋. Definition 7. Let I be an admissible ideal, let 𝑓 be a modulus function, and let 𝑝 = (𝑝𝑘 ) be any sequence of strictly positive real numbers. Then, for each 𝜖 > 0, we define the following sequence spaces: 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]

0

{ = {𝑥 ∈ 𝑤 (𝑋) : { } { } 1 󵄨󵄨 𝑚 󵄨󵄨 𝑝𝑘 {𝑟 ∈ N : ℎ ∑ [𝑓 (󵄨󵄨󵄨𝑡𝑘𝑛 (Δ 𝑥)󵄨󵄨󵄨)] ≥ 𝜖} ∈ I} , 𝑟 𝑘∈𝐼𝑟 } { } 𝑤𝜎𝑓

[Δ𝑚𝑝 , I, 𝜃]

{ = {𝑥 ∈ 𝑤 (𝑋) : ∃ℓ > 0, { } { } 1 󵄨󵄨 󵄨󵄨 𝑝𝑘 𝑚 {𝑟 ∈ N : ℎ ∑ [𝑓 (󵄨󵄨󵄨𝑡𝑘𝑛 (Δ 𝑥 − ℓ)󵄨󵄨󵄨)] ≥ 𝜖} ∈ I} , 𝑟 𝑘∈𝐼𝑟 } { } 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]

𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃], and 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]∞ .

(iii) By taking I = I𝑓 = {𝐸 ⊂ N : if 𝐸 is a finite subset}, 𝑚 = 0, 𝑓(𝑥) = 𝑥, 𝜃 = (2𝑟 ), and 𝑝𝑘 = 1, for all 𝑘 ∈ N, then we obtain [𝑤]𝜎 defined by Mursaleen ̂𝜎𝑓 [I, 𝑝, 𝜃] et al. [3] instead of 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃] and 𝑤 reduces to [̂ 𝑤] defined in Das and Sahoo [4]. Theorem 9. Let 𝑓 be a modulus function and 𝑝 = (𝑝𝑘 ) is a bounded sequence of strictly positive real numbers; then, 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]0 , 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃], and 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]∞ are linear spaces over C. Proof. We will prove the assertion only for 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]0 and others can be treated similarly. Suppose that 𝑥 = (𝑥𝑘 ), 𝑦 = (𝑦𝑘 ) ∈ 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]0 . Then, for every 𝜖 > 0 and uniformly in 𝑛, the sets { 𝜖 1 󵄨 󵄨 𝑝𝑘 𝜖 } 𝐴 𝜃 ( ) = {𝑟 ∈ N : ∑ [𝑓 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥)󵄨󵄨󵄨󵄨)] ≥ } , 2 ℎ𝑟 𝑘∈𝐼 2 𝑟 } { { 𝜖 1 󵄨 󵄨 𝑝𝑘 𝜖 } 𝐵𝜃 ( ) = {𝑟 ∈ N : ∑ [𝑓 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑦)󵄨󵄨󵄨󵄨)] ≥ } 2 ℎ𝑟 𝑘∈𝐼 2 𝑟 } {

belong to I. Let 𝛼, 𝛽 ∈ C and Δ𝑚 is linear; then, 1 󵄨 󵄨 𝑝𝑘 ∑ [𝑓 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 (𝛼 ⋅ 𝑥 + 𝛽 ⋅ 𝑦))󵄨󵄨󵄨󵄨)] ℎ𝑟 𝑘∈𝐼 𝑟

∞

{ = {𝑥 ∈ 𝑤 (𝑋) : ∃𝐾 > 0, { { } } 1 󵄨󵄨 𝑚 󵄨󵄨 𝑝𝑘 {𝑟 ∈ N : ℎ ∑ [𝑓 (󵄨󵄨󵄨𝑡𝑘𝑛 (Δ 𝑥)󵄨󵄨󵄨)] ≥ 𝐾} ∈ I} 𝑟 𝑘∈𝐼𝑟 { } } (11)

(12)

=

1 󵄨 󵄨 𝑝𝑘 ∑ [𝑓 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (𝛼 ⋅ Δ𝑚 𝑥 + 𝛽 ⋅ Δ𝑚 𝑦)󵄨󵄨󵄨󵄨)] ℎ𝑟 𝑘∈𝐼 𝑟

𝐻

≤ 𝐶 ⋅ (𝐾𝛼 ) ⋅

1 󵄨 󵄨 𝑝𝑘 ∑ [𝑓 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥)󵄨󵄨󵄨󵄨)] ℎ𝑟 𝑘∈𝐼 𝑟

𝐻

+ 𝐶 ⋅ (𝐾𝛽 ) ⋅

1 󵄨 󵄨 𝑝𝑘 ∑ [𝑓 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑦)󵄨󵄨󵄨󵄨)] , by (10) , ℎ𝑟 𝑘∈𝐼 𝑟

(13)

uniformly in 𝑛. Remark 8. By taking some particular cases, we obtain the following.

where 𝐾𝛼 , 𝐾𝛽 are two positive numbers such that |𝛼| ≤ 𝐾𝛼 and |𝛽| ≤ 𝐾𝛽 .

4

International Journal of Analysis Then, for given 𝜖 > 0, we have the following containment:

} { 1 󵄨󵄨 󵄨󵄨 𝑝𝑘 𝑚 {𝑟 ∈ N : ℎ ∑ [𝑓 (󵄨󵄨󵄨𝑡𝑘𝑛 (Δ 𝑥𝑘 )󵄨󵄨󵄨)] ≥ 𝜖} 𝑟 𝑘∈𝐼𝑟 } {

{ } 1 󵄨󵄨 󵄨󵄨 𝑝𝑘 𝑚 {𝑟 ∈ N : ℎ ∑ [𝑓 (󵄨󵄨󵄨𝑡𝑘𝑛 (Δ (𝛼 ⋅ 𝑥 + 𝛽 ⋅ 𝑦))󵄨󵄨󵄨)] ≥ 𝜖} 𝑟 𝑘∈𝐼𝑟 { } } { 𝜖 1 󵄨 󵄨 𝑝𝑘 ⊆ {𝑟 ∈ N : ∑ [𝑓 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥)󵄨󵄨󵄨󵄨)] ≥ } 𝐻 ℎ𝑟 𝑘∈𝐼 2𝐶 ⋅ (𝐾𝛼 ) 𝑟 } { } { 𝜖 1 󵄨 󵄨 𝑝𝑘 ∪ {𝑟 ∈ N : ∑ [𝑓 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑦)󵄨󵄨󵄨󵄨)] ≥ } 𝐻 ℎ𝑟 𝑘∈𝐼 2𝐶 ⋅ (𝐾𝛽 ) } 𝑟 { (14) uniformly in 𝑛. Since 𝑥, 𝑦 ∈ 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]0 , it follows that the later sets belong to I. By using the property of an ideal the set on the left hand side in the above expression also belongs to I. This completes the proof. Theorem 10. For 𝑚 ≥ 1, then the inclusion 𝑤𝜎𝑓 [Δ𝑚−1 𝑝 , I, 𝜃]

0,𝐿,∞

⊂ 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]

0,𝐿,∞

Now, for given 𝜖 > 0, we have the following containment:

(15)

{ 𝜖 } 1 󵄨 󵄨 𝑝𝑘 ⊆ {𝑟 ∈ N : ∑ [𝑓 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚−1 𝑥𝑘 )󵄨󵄨󵄨󵄨)] ≥ ℎ𝑟 𝑘∈𝐼 2𝐶 } 𝑟 } { { 𝜖 } 1 󵄨 󵄨 𝑝𝑘 ∪ {𝑟 ∈ N : , ∑ [𝑓 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚−1 𝑥𝑘+1 )󵄨󵄨󵄨󵄨)] ≥ ℎ𝑟 𝑘∈𝐼 2𝐶 } 𝑟 } { (18) uniformly in 𝑛. Both the sets on the right hand side in the above containment belong to I by (16). It follows that 𝑥 ∈ 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]0 . Since I is an admissible ideal and the inclusion is strict as the sequence 𝑥 = (𝑥𝑘 ) = (𝑘𝑚−1 ) belongs to 𝑥 ∈ 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]0 , it does not belong to 𝑤𝜎𝑓 [Δ𝑚−1 𝑝 , I, 𝜃]0 , for 𝑓(𝑥) = 𝑥, 𝑡0𝑛 (𝑥) = (𝑥𝑛 ), 𝜃 = (2𝑟 ), and 𝑝𝑘 = 1, for all 𝑘 ∈ N. Theorem 11. Let 𝑝 = (𝑝𝑘 ) be a bounded sequence of strictly positive real numbers and 𝑓󸀠 , 𝑓󸀠󸀠 are modulus functions. If

is strict.

lim sup

Proof. We will prove the result for Suppose that 𝑥 = (𝑥𝑘 ) ∈ each 𝜖 > 0,

𝑤𝜎𝑓 [Δ𝑚−1 𝑝 , I, 𝜃]0

𝑤𝜎𝑓 [Δ𝑚−1 𝑝 , I, 𝜃]0 ;

only.

by definition, for

{ } 1 󵄨󵄨 󵄨󵄨 𝑝𝑘 𝑚−1 {𝑟 ∈ N : ℎ ∑ [𝑓 (󵄨󵄨󵄨𝑡𝑘𝑛 (Δ 𝑥𝑘 )󵄨󵄨󵄨)] ≥ 𝜖} ∈ I 𝑟 𝑘∈𝐼𝑟 { }

𝑡→∞

(16)

uniformly in 𝑛. Since 𝑓 is a modulus function, therefore we have the following inequality:

󸀠

𝑓󸀠 (𝑡) = 𝑀 > 0, 𝑓󸀠󸀠 (𝑡)

(19)

󸀠󸀠

then 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]0 ⊂ 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]0 . Proof. Assume that lim sup𝑡 → ∞ (𝑓󸀠 (𝑡)/𝑓󸀠󸀠 (𝑡)) = 𝑀; there exists a positive number 𝐾 > 0 such that 𝑓󸀠 (𝑡) ≥ 𝐾 ⋅ 𝑓󸀠󸀠 (𝑡), for all 𝑡 ≥ 0, which implies that 1 󵄨 󵄨 𝑝𝑘 ∑ [𝑓󸀠 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥)󵄨󵄨󵄨󵄨)] ℎ𝑟 𝑘∈𝐼 𝑟

1 󵄨 󵄨 𝑝𝑘 ≥ (𝐾) ⋅ ∑ [𝑓󸀠󸀠 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥)󵄨󵄨󵄨󵄨)] ℎ𝑟 𝑘∈𝐼

(20)

𝐻

𝑟

1 󵄨 󵄨 𝑝𝑘 ∑ [𝑓 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥𝑘 )󵄨󵄨󵄨󵄨)] ℎ𝑟 𝑘∈𝐼

uniformly in 𝑛. Thus, for any 𝜖 > 0,

𝑟

≤

1 󵄨 󵄨 󵄨 󵄨 𝑝𝑘 ∑ [𝑓 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚−1 𝑥𝑘 )󵄨󵄨󵄨󵄨) + 𝑓 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚−1 𝑥𝑘+1 )󵄨󵄨󵄨󵄨)] ℎ𝑟 𝑘∈𝐼 𝑟

≤𝐶⋅

1 󵄨 󵄨 𝑝𝑘 ∑ [𝑓 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚−1 𝑥𝑘 )󵄨󵄨󵄨󵄨)] ℎ𝑟 𝑘∈𝐼 𝑟

+𝐶⋅

1 󵄨 󵄨 𝑝𝑘 ∑ [𝑓 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚−1 𝑥𝑘+1 )󵄨󵄨󵄨󵄨)] by (10) ℎ𝑟 𝑘∈𝐼 𝑟

uniformly in 𝑛.

(17)

{ } 1 󸀠󸀠 󵄨󵄨 𝑚 󵄨󵄨 𝑝𝑘 {𝑟 ∈ N : ℎ ∑ [𝑓 (󵄨󵄨󵄨𝑡𝑘𝑛 (Δ 𝑥)󵄨󵄨󵄨)] ≥ 𝜖} 𝑟 𝑘∈𝐼𝑟 { } } { 1 󵄨 󵄨 𝑝𝑘 ⊆ {𝑟 ∈ N : ∑ [𝑓󸀠 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥)󵄨󵄨󵄨󵄨)] ≥ 𝜖 ⋅ (𝐾)𝐻} ℎ𝑟 𝑘∈𝐼 𝑟 } { (21) uniformly in 𝑛. Therefore, the above containment gives the result.


5

Theorem 12. If 𝑓, 𝑓󸀠 , and 𝑓󸀠󸀠 are modulus functions, then 󸀠

󸀠

(i) 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃] ⊂ 𝑤𝜎𝑓∘𝑓 [Δ𝑚𝑝 , I, 𝜃], (ii)

󸀠 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]

∩

󸀠󸀠 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]

⊂

󸀠 󸀠󸀠 𝑤𝜎𝑓 +𝑓 [Δ𝑚𝑝 , I, 𝜃].

󸀠 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃] and let 𝜖

Proof. (i) Let 𝑥 = (𝑥𝑘 ) ∈ > 0 be given. Choose 𝛿 ∈ (0, 1) such that 𝑓(𝑡) < 𝜖, for all 0 < 𝑡 < 𝛿, since 󸀠 𝑥 ∈ 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃] such that { } 1 󵄨 󵄨 𝑝𝑘 𝐴 𝛿 = {𝑛 ∈ N : ∑ [𝑓󸀠 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥 − ℓ)󵄨󵄨󵄨󵄨)] ≥ 𝛿} ∈ I ℎ𝑟 𝑘∈𝐼 𝑟 { } (22) uniformly in 𝑛. On the other hand, we have

Theorem 13. If 𝑓 is a modulus function and 𝑝 = (𝑝𝑘 ) is a sequence of positive real numbers, then 𝑤𝜎 [Δ𝑚𝑝 , I, 𝜃] ⊆ 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃].

Proof. This can be proved similarly as in Theorem 12(i). Theorem 14. Let 𝑓 be a modulus lim sup𝑡 → ∞ (𝑓(𝑡)/𝑡) = 𝑀 > 0, then 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃] ⊆ 𝑤𝜎 [Δ𝑚𝑝 , I, 𝜃] .

+

∑ 𝑝

𝑘∈𝐼𝑟 &[𝑓󸀠 (|𝑡𝑘𝑛 (Δ𝑚 𝑥−ℓ)|)] 𝑘 0, { } 1 󵄨󵄨 𝑝𝑘 󸀠 󵄨󵄨 𝑚 {𝑛 ∈ N : ℎ ∑ [𝑓 ∘ 𝑓 (󵄨󵄨󵄨𝑡𝑘𝑛 (Δ 𝑥 − ℓ)󵄨󵄨󵄨)] ≥ 𝜂} 𝑟 𝑘∈𝐼𝑟 { }

≥ (𝐾)𝐻 ⋅

𝐻

where 𝐾 = max(1, (2, 𝑓(1)/𝛿) ). 󸀠 Since 𝑥 = (𝑥𝑘 ) ∈ 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃], so by (22) the latter set belongs to I and therefore the theorem is proved. (ii) This result can be proved by the following inequality: 1 󵄨 󵄨 𝑝𝑘 ∑ [(𝑓󸀠 + 𝑓󸀠󸀠 ) (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥 − ℓ)󵄨󵄨󵄨󵄨)] ℎ𝑟 𝑘∈𝐼 𝑟

(25)

𝑟

+

1 󵄨 󵄨𝑝𝑘 ∑ 󵄨󵄨󵄨𝑡 (Δ𝑚 𝑥 − ℓ)󵄨󵄨󵄨󵄨 ℎ𝑟 𝑘∈𝐼 󵄨 𝑘𝑛

(27)

𝑟

uniformly in 𝑛. Then, for each 𝜖 > 0, { } 1 󵄨󵄨 󵄨󵄨 𝑝𝑘 𝑚 {𝑛 ∈ N : ℎ ∑ [󵄨󵄨󵄨𝑡𝑘𝑛 (Δ 𝑥 − ℓ)󵄨󵄨󵄨] ≥ 𝜖} 𝑟 𝑘∈𝐼𝑟 { } } { 1 󵄨 󵄨 𝑝𝑘 ⊆ {𝑛 ∈ N : ∑ [𝑓 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥 − ℓ)󵄨󵄨󵄨󵄨)] ≥ 𝜖 ⋅ (𝐾)𝐻} , ℎ𝑟 𝑘∈𝐼 𝑟 } { (28) uniformly in 𝑛. Since 𝑥 ∈ 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃], therefore the latter set belongs to I. It follows that 𝑥 ∈ 𝑤𝜎 [Δ𝑚𝑝 , I, 𝜃]. Theorem 15. If 0 < 𝑝𝑘 ≤ 𝑞𝑘 and (𝑞𝑘 /𝑝𝑘 ) are bounded, then 𝑤𝜎𝑓 [Δ𝑚𝑞 , I, 𝜃] ⊂ 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃].

{ 1 󵄨 󵄨 𝑝𝑘 (𝜂 − 𝜖) } ⊆ {𝑛 ∈ N : , ∑ [𝑓󸀠 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥 − ℓ)󵄨󵄨󵄨󵄨)] ≥ ℎ𝑟 𝑘∈𝐼 𝐾 } 𝑟 } { (24)

𝐶 󵄨 󵄨 𝑝𝑘 ≤ ∑ [𝑓󸀠 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥 − ℓ)󵄨󵄨󵄨󵄨)] ℎ𝑟 𝑘∈𝐼

(26)

1 󵄨 󵄨 𝑝𝑘 ∑ [𝑓 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥 − ℓ)󵄨󵄨󵄨󵄨)] ℎ𝑟 𝑘∈𝐼

𝑟

1 ℎ𝑟

If

Proof. Suppose that 𝑥 = (𝑥𝑘 ) ∈ 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]. It is given that lim sup𝑡 → ∞ (𝑓(𝑡)/𝑡) = 𝑀 > 0; there exists a constant 𝐾 > 0 such that 𝑓(𝑡) ≥ 𝐾 ⋅ 𝑡, for all 𝑡 ≥ 0, which implies that

1 󵄨 󵄨 𝑝𝑘 ∑ [𝑓 ∘ 𝑓󸀠 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥 − ℓ)󵄨󵄨󵄨󵄨)] ℎ𝑟 𝑘∈𝐼 =

function.

𝐶 󵄨 󵄨 𝑝𝑘 ∑ [𝑓󸀠󸀠 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥 − ℓ)󵄨󵄨󵄨󵄨)] , ℎ𝑟 𝑘∈𝐼 𝑟

uniformly in 𝑛, where sup𝑘 𝑝𝑘 = 𝐻 and 𝐶 = max(1, 2𝐻−1 ).

Proof. The proof of this theorem is easy and so it is omitted. 𝑚

3. 𝑆𝜃Δ𝜎 (I)-Convergence 𝑚

In this section, we define the notion of 𝑆𝜃Δ𝜎 (I)-convergence with the help of ideal and invariant means and difference sequences. Further, we also establish some relations between 𝑚 𝑆𝜃Δ𝜎 (I)-convergence and 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]0 . Definition 16. Let I ⊆ P(N) be a nontrivial ideal. A sequence 𝑥 = (𝑥𝑘 ) is said to be Δ𝑚 (I)-strong lacunary 𝜎𝑚 statistically convergent or 𝑆𝜃Δ𝜎 (I)-convergent to a number ℓ, provided that, for every 𝜖 > 0 and 𝛿 > 0, {𝑟 ∈ N :

1 ℎ𝑟

󵄨󵄨 󵄨󵄨{𝑘 ∈ 𝐼𝑟 󵄨

󵄨 󵄨 󵄨 : 󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥 − ℓ)󵄨󵄨󵄨󵄨 ≥ 𝜖}󵄨󵄨󵄨󵄨 ≥ 𝛿} ∈ I, uniformly in 𝑛.

(29)

6

International Journal of Analysis 𝑚

𝑚

ℓ(𝑆𝜃Δ𝜎 (I)) or 𝑆𝜃Δ𝜎 (I) −

In this case, we write 𝑥𝑘

→

lim𝑘 → ∞ 𝑥𝑘 = ℓ. Let convergent sequences.

denote the set of all

𝑚 𝑆𝜃Δ𝜎 (I)

𝑚 𝑆𝜃Δ𝜎 (I)-

Theorem 17. Let 𝑓 be a modulus function and let 𝑝 = (𝑝𝑘 ) be a sequence of strictly positive real numbers. If 0 < inf 𝑘 𝑝𝑘 = 𝑚 ℎ ≤ 𝑝𝑘 ≤ sup 𝑘 𝑝𝑘 = 𝐻 < ∞, then 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]0 ⊂ 𝑆𝜃Δ𝜎 (I). Proof. Assume that 𝑥 ∈ Then, we have

𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]0

and 𝜖 > 0 is given.

1 󵄨 󵄨 𝑝𝑘 ∑ [𝑓 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥)󵄨󵄨󵄨󵄨)] ℎ𝑟 𝑘∈𝐼 1 󵄨 󵄨 𝑝𝑘 [𝑓 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥)󵄨󵄨󵄨󵄨)] ∑ ℎ𝑟 𝑘∈𝐼 & 𝑡 𝑚 𝑥) ≥𝜖 | 𝑟 | 𝑘𝑛 (Δ

1 󵄨 󵄨 𝑝𝑘 ≥ [𝑓 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥)󵄨󵄨󵄨󵄨)] ∑ ℎ𝑟 𝑘∈𝐼 & 𝑡 𝑚 𝑥) ≥𝜖 | 𝑟 | 𝑘𝑛 (Δ

lim

≤

𝐻

{𝑟 ∈ N :

uniformly in 𝑛. Then, for every 𝛿 > 0, we have the following containment: 1 ℎ𝑟

󵄨󵄨 󵄨󵄨{𝑘 ∈ 𝐼𝑟 󵄨

(33)

󵄨 󵄨 󵄨 : 󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥)󵄨󵄨󵄨󵄨 ≥ 𝜖}󵄨󵄨󵄨󵄨 ≥ 𝛿}

{ } 1 󵄨 󵄨 𝑝𝑘 ⊆ {𝑟 ∈ N : ∑ [𝑓 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥)󵄨󵄨󵄨󵄨)] ≥ 𝐾 ⋅ 𝛿} ℎ𝑟 𝑘∈𝐼 𝑟 { }

(34)

𝑟 ≥ 𝑟0 .

Now, we have

⋅ min ([𝑓 (𝜖)] , [𝑓 (𝜖)] )

{𝑟 ∈ N :

󳨀→ 0,

for 𝑟 ∉ 𝐴,

󵄨 󵄨 󵄨 : 󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥)󵄨󵄨󵄨󵄨 ≥ 𝜖}󵄨󵄨󵄨󵄨 ℎ

ℎ𝑟

󵄨󵄨 𝑚 󵄨󵄨 󵄨󵄨 : 󵄨󵄨󵄨󵄨𝑡𝑘𝑛 ((−1)𝑚+2 (𝑚 + 1)! (𝑘 + ) − 0)󵄨󵄨󵄨󵄨 ≥ 𝜖}󵄨󵄨󵄨󵄨 2 󵄨 󵄨 󵄨

1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨{𝑘 ∈ 𝐼𝑟 : 󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥 − 0)󵄨󵄨󵄨 ≥ 𝜖}󵄨󵄨󵄨 < 𝛿 󵄨 󵄨 󵄨 ℎ𝑟 󵄨

1 ℎ 𝐻 ∑ min ([𝑓 (𝜖)] , [𝑓 (𝜖)] ) ℎ𝑟 𝑘∈𝐼 󵄨󵄨 󵄨󵄨{𝑘 ∈ 𝐼𝑟 󵄨

[√ℎ𝑟 ]

󵄨󵄨 󵄨󵄨{𝑘 ∈ 𝐼 󵄨󵄨 𝑟 󵄨

for all 𝑟 ∉ 𝐴, and uniformly in 𝑛. Hence, for 𝛿 > 0, there exists a positive integer 𝑟0 such that

1 𝑝 ∑ [𝑓 (𝜖)] 𝑘 ℎ𝑟 𝑘∈𝐼

1 ℎ𝑟

1 ℎ𝑟

󵄨 󵄨 󵄨 : 󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥 − 0)󵄨󵄨󵄨󵄨 ≥ 𝜖}󵄨󵄨󵄨󵄨

(30)

𝑟

≥

1 󵄨󵄨 󵄨󵄨{𝑘 ∈ 𝐼𝑟 󵄨

= 𝑟lim →∞

𝑟

≥

𝑘𝑚+1 , if 𝑟 ∉ 𝐴, 2𝑟−1 + 1 ≤ 𝑘 ≤ 2𝑟−1 + [√ℎ𝑟 ] , { { 𝑚+1 (32) 𝑥𝑘 = {𝑘 , if 𝑟 ∈ 𝐴, 2𝑟−1 < 𝑘 ≤ 2𝑟−1 + ℎ𝑟 , { otherwise, {0,

𝑟→∞ℎ 𝑟

1 󵄨 󵄨 𝑝𝑘 + [𝑓 (󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥)󵄨󵄨󵄨󵄨)] ∑ ℎ𝑟 𝑘∈𝐼 & 𝑡 𝑚 𝑥) 0, we have

𝑟

=

Proof. This part is the direct consequence of Theorems 17 and 18.

(31)

uniformly in 𝑛, where 𝐾 = min([𝑓(𝜖)]ℎ , [𝑓(𝜖)]𝐻), since 𝑥 ∈ 𝑚 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]0 , which implies that 𝑥 ∈ 𝑆𝜃Δ𝜎 (I). Theorem 18. Let 𝑓 be a bounded modulus function and 0 < 𝑚 inf 𝑘 𝑝𝑘 = ℎ ≤ 𝑝𝑘 ≤ sup 𝑘 𝑝𝑘 = 𝐻 < ∞; then, 𝑆𝜃Δ𝜎 (I) ⊂ 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]0 . Proof. Using the same technique of [26, Theorem 3.3], it is easy to prove. Theorem 19. If 0 < inf 𝑘 𝑝k = ℎ ≤ 𝑝𝑘 ≤ sup 𝑘 𝑝𝑘 = 𝐻 < ∞, 𝑚 then 𝑆𝜃Δ𝜎 (I) = 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]0 if and only if 𝑓 is bounded.

1 ℎ𝑟

󵄨󵄨 󵄨󵄨{𝑘 ∈ 𝐼𝑟 󵄨

󵄨 󵄨 󵄨 : 󵄨󵄨󵄨󵄨𝑡𝑘𝑛 (Δ𝑚 𝑥 − 0)󵄨󵄨󵄨󵄨 ≥ 𝜖}󵄨󵄨󵄨󵄨 ≥ 𝛿}

(35)

⊂ (𝐴 ∪ (1, 2, . . . , 𝑟0 − 1)) . 𝑚

Since I is an admissible ideal, it follows that 𝑆𝜃Δ𝜎 (I) − lim𝑘 → ∞ 𝑥𝑘 = 0. If we take 𝑝𝑘 = 1, for all 𝑘 = 1, 2, . . ., then 𝑥𝑘 ∉ 𝑚 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]0 . This contradicts the fact that 𝑆𝜃Δ𝜎 (I) = 𝑤𝜎𝑓 [Δ𝑚𝑝 , I, 𝜃]0 .

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

References [1] P. Schaefer, “Infinite matrices and invariant means,” Proceedings of the American Mathematical Society, vol. 36, pp. 104–110, 1972. [2] G. G. Lorentz, “A contribution to the theory of divergent sequences,” Acta Mathematica, vol. 80, pp. 167–190, 1948. [3] Mursaleen, A. K. Gaur, and T. A. Chishti, “On some new sequence spaces of invariant means,” Acta Mathematica Hungarica, vol. 75, no. 3, pp. 209–214, 1997.

International Journal of Analysis [4] G. Das and S. K. Sahoo, “On some sequence spaces,” Journal of Mathematical Analysis and Applications, vol. 164, no. 2, pp. 381– 398, 1992. [5] H. Fast, “Sur la convergence statistique,” Colloquium Mathematicum, vol. 2, pp. 241–244, 1951. [6] J. S. Connor, “The statistical and strong 𝑝-Cesàro convergence of sequences,” Analysis. International Mathematical Journal of Analysis and its Applications, vol. 8, no. 1-2, pp. 47–63, 1988. [7] J. A. Fridy, “On statistical convergence,” Analysis, vol. 5, no. 4, pp. 301–313, 1985. [8] I. J. Maddox, “Statistical convergence in a locally convex space,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 104, no. 1, pp. 141–145, 1988. ˇ at, “On statistically convergent sequences of real numbers,” [9] T. Sal´ Mathematica Slovaca, vol. 30, no. 2, pp. 139–150, 1980. [10] A. R. Freedman, J. J. Sember, and R. Raphael, “Some 𝑝Cesàro type summability spaces,” Proceedings of the London Mathematical Society, vol. 37, no. 2, pp. 508–529, 1978. [11] J. A. Fridy and C. Orhan, “Lacunary statistical convergence,” Pacific Journal of Mathematics, vol. 160, no. 1, pp. 43–51, 1993. [12] J. A. Fridy and C. Orhan, “Lacunary statistical summability,” Journal of Mathematical Analysis and Applications, vol. 173, no. 2, pp. 497–504, 1993. [13] S. Pehlivan and B. Fisher, “Lacunary strong convergence with respect to a sequence of modulus functions,” Commentationes Mathematicae Universitatis Carolinae, vol. 36, no. 1, pp. 69–76, 1995. [14] M. Et and A. Gökhan, “Lacunary strongly almost summable sequences,” Studia. Universitatis Babes¸-Bolyai. Mathematica, vol. 53, no. 4, pp. 29–38, 2008. [15] B. C. Tripathy and H. Dutta, “On some lacunary difference sequence spaces defined by a sequence of Orlicz functions and 𝑞-lacunary Δ𝑛𝑚 -statistical convergence,” Analele Stiintifice ale Universitatii Ovidius Constanta, vol. 20, no. 1, pp. 417–430, 2012. [16] V. Karakaya, “Some new sequence spaces defined by a sequence of Orlicz functions,” Taiwanese Journal of Mathematics, vol. 9, no. 4, pp. 617–627, 2005. ˇ at, and W. Wilczyński, “I−convergence,” Real [17] P. Kostyrko, T. Sal´ Analysis Exchange, vol. 26, no. 2, pp. 669–686, 2000. [18] P. Das, E. Savas, and S. Kr. Ghosal, “On generalizations of certain summability methods using ideals,” Applied Mathematics Letters, vol. 24, no. 9, pp. 1509–1514, 2011. [19] H. Kızmaz, “On certain sequence spaces,” Canadian Mathematical Bulletin, vol. 24, no. 2, pp. 169–176, 1981. [20] M. Et and R. C ¸ olak, “On some generalized difference sequence spaces,” Soochow Journal of Mathematics, vol. 21, no. 4, pp. 377– 386, 1995. [21] A. A. Bakery, E. A. E. Mohamed, and M. A. Ahmed, “Some generalized difference sequence spaces defined by ideal convergence and Musielak-Orlicz function,” Abstract and Applied Analysis, vol. 2013, Article ID 972363, 9 pages, 2013. [22] M. Bas¸arir, S¸. Konca, and E. E. Kara, “Some generalized difference statistically convergent sequence spaces in 2-normed space,” Journal of Inequalities and Applications, vol. 2013, article 177, pp. 1–10, 2013. [23] V. K. Bhardwaj and S. Gupta, “Cesàro summable difference sequence space,” Journal of Inequalities and Applications, vol. 2013, article 315, pp. 1–9, 2013. [24] M. Et and M. Basarir, “On some new generalized difference sequence spaces,” Periodica Mathematica Hungarica, vol. 35, no. 3, pp. 169–175, 1997.

7 [25] M. Et and A. Bektas¸, “Generalized strongly (𝑉, 𝜆)-summable sequences defined by Orlicz functions,” Mathematica Slovaca, vol. 54, no. 4, pp. 411–422, 2004. [26] M. Et, Y. Altin, and H. Altinok, “On some generalized difference sequence spaces defined by a modulus function,” Filomat, no. 17, pp. 23–33, 2003. [27] B. Hazarika and A. Esi, “On some I-convergent generalized difference lacunary double sequence spaces defined by orlicz functions,” Acta Scientiarum: Technology, vol. 35, no. 3, pp. 527– 537, 2013. [28] H. Nakano, “Concave modulars,” Journal of the Mathematical Society of Japan, vol. 5, pp. 29–49, 1953. [29] W. H. Ruckle, “𝐹𝐾 spaces in which the sequence of coordinate vectors is bounded,” Canadian Journal of Mathematics. Journal Canadien de Mathématiques, vol. 25, pp. 973–978, 1973. [30] I. J. Maddox, “Sequence spaces defined by a modulus,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 100, no. 1, pp. 161–166, 1986. [31] M. Et, “Spaces of Cesaro difference sequences of order 𝑟 defined by a modulus function in a locally convex space,” Taiwanese Journal of Mathematics, vol. 10, no. 4, pp. 865–879, 2006. [32] E. Savas, “On some generalized sequence spaces defined by a modulus,” Indian Journal of Pure and Applied Mathematics, vol. 30, no. 5, pp. 459–464, 1999. [33] S. Kumar, V. Kumar, and S. S. Bhatia, “Generalized sequence spaces in 2-normed spaces defined by ideal and a modulus function,” Analele Stiintifice ale Universitatii Alexandruioan Cuza din Iasi-Serie Noua- Mathematica. In press.

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


Volume 2014

Journal of

Probability and Statistics Volume 2014

The Scientific World Journal 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


Volume 2014

Volume 2014


Volume 2014

Discrete Mathematics

Journal of

Volume 2014


Discrete Dynamics in Nature and Society

Journal of

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

Abstract and Applied Analysis

Volume 2014


Volume 2014


Volume 2014

International Journal of

Journal of

Stochastic Analysis

Optimization



Volume 2014

Volume 2014