Some New Difference Sequence Spaces of Invariant Means Defined by Ideal and Modulus Function

Abstract

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 sequence spaces. Further, we introduce a new concept of 𝑆^Δ𝑚_𝜃𝜎(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=(𝑥_𝑛) has 𝑥_𝑛≥0, for all 𝑛;(ii)𝜑(𝑒)=1, where 𝑒=(1,1,1,…);(iii)𝜑(𝑥_𝜎(𝑛))=𝜑(𝑥), for all 𝑥∈ℓ_∞.

If 𝑥=(𝑥_𝑘), we write 𝑇𝑥=(𝑇𝑥_𝑘)=(𝑥_𝜎(𝑘)). It can be shown by Schaefer [1] that 𝑉_𝜎=𝑥∈ℓ_∞∶lim_𝑘→∞𝑡_𝑘𝑚(𝑥),=𝐿,uniformlyin𝑚,𝜎−lim𝑥=𝐿 (1) where 𝑡_𝑘𝑚(𝑥)=(𝑥_𝑚+𝑥_𝜎(𝑚)+⋯+𝑥_{𝜎𝑘(𝑚)})/(𝑘+1).

In case 𝜎 is the translation mapping 𝑛→𝑛+1, 𝜎-mean is often called a Banach limit and 𝑉_𝜎, 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]: 𝑤_𝜎=𝑥∶lim_𝑛→∞1𝑛+1^𝑛_𝑘=0𝑡_𝑘𝑚(),[𝑤]𝑥−ℓ⟶0,uniformlyin𝑚_𝜎=𝑥∶lim_𝑛→∞1𝑛+1^𝑛_𝑘=0𝑡_𝑘𝑚(,𝑤𝑥−ℓ)⟶0,uniformlyin𝑚_𝜎=𝑥∶lim_𝑛→∞1𝑛+1^𝑛_𝑘=0𝑡_𝑘𝑚(|𝑥−ℓ|)⟶0,uniformlyin𝑚 (2) 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 investigated by Connor [6], Fridy [7], Maddox [8], Šalát [9], and many others.

Definition 1 (see [5]). A number sequence 𝑥=(𝑥_𝑘) is said to be statistically convergent to a number 𝐿 (denoted by 𝑆−lim_𝑘→∞𝑥_𝑘=𝐿) provided that, for every 𝜖>0, lim_𝑛→∞1𝑛𝑘≤𝑛∶𝑥_𝑘−𝐿≥𝜖=0, (3) where the vertical bars denote the cardinality of the enclosed set.
By a lacunary sequence, we mean an increasing sequence 𝜃=(𝑘_𝑟) of positive integers such that 𝑘₀=0 and ℎ_𝑟=𝑘_𝑟−𝑘_𝑟−1→∞ as 𝑟→∞. The intervals determined by 𝜃 will be denoted by 𝐼_𝑟=(𝑘_𝑟−1,𝑘_𝑟], where the ratio 𝑘_𝑟/𝑘_𝑟−1 is denoted by 𝑞_𝑟. The space of lacunary strongly convergent sequence 𝑁_𝜃 was defined by Freedman et al. [10] as follows: 𝑁_𝜃=𝑥=𝑥_𝑘∶lim_𝑟→∞1ℎ_{𝑟𝑘∈𝐼𝑟}𝑥_𝑘−𝐿=0forsome𝐿. (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=(𝑘_𝑟) be a lacunary sequence. A sequence 𝑥=(𝑥_𝑘) is said to be lacunary strong 𝜎-lacunary statistically convergent if, for every 𝜖>0, lim_𝑟→∞1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑡_𝑘𝑚(𝑥−𝐿)≥𝜖=0,uniformlyin𝑚, (5) where 𝑆𝑡_𝜃𝜎 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.

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=(𝑥_𝑘) in 𝑋 is said to be ℐ-convergent to 𝜉 if, for each 𝜖>0, the set 𝐴(𝜖)={𝑘∈ℕ∶𝜌(𝑥_𝑘,𝜉)≥𝜖}∈ℐ. In this case, we write ℐ−lim_𝑘→∞𝑥_𝑘=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=(𝑘_𝑟) be a lacunary sequence. A sequence 𝑥=(𝑥_𝑘) of numbers is said to be ℐ-lacunay statistical convergent or 𝑆_𝜃(ℐ)-convergent to 𝐿, if, for every 𝜖>0 and 𝛿>0, 1𝑟∈ℕ∶ℎ_𝑟𝑘∈𝐼_𝑟∶𝑥_𝑘−𝐿≥𝜖≥𝛿∈ℐ. (6) In this case, we write 𝑥_𝑘→𝐿(𝑆_𝜃(ℐ)) or 𝑆_𝜃(ℐ)−lim_𝑘→∞𝑥_𝑘=𝐿. The set of all ℐ-lacunary statistically convergent sequences will be denoted by 𝑆_𝜃(ℐ).

Definition 5 (see [18]). Let 𝜃=(𝑘_𝑟) be a lacunary sequence. A sequence 𝑥=(𝑥_𝑘) of numbers is said to be 𝑁_𝜃(ℐ)-convergent to 𝐿 if, for every 𝜖>0, we have 1𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑥_𝑘−𝐿≥𝜖∈ℐ. (7) It is denoted by 𝑥_𝑘→𝐿(𝑁_𝜃(ℐ)).
In 1981, Kızmaz [19] introduced the notion of difference sequence space as follows: Δ(𝑋)=𝑥=𝑥_𝑘∶Δ𝑥_𝑘∈𝑋, (8) for 𝑋=ℓ_∞,𝑐 and 𝑐₀, where Δ𝑥_𝑘=𝑥_𝑘−𝑥_𝑘+1, for all 𝑘∈ℕ.
Continuing on this way, the notion was further generalized by Et and Çolak [20] by introducing the sequence spaces as follows: Δ^𝑚(𝑋)=𝑥=𝑥_𝑘∶Δ^𝑚𝑥_𝑘∈𝑋, (9) for 𝑋=ℓ_∞,𝑐, and 𝑐₀, where 𝑚∈ℕ and Δ^𝑚𝑥_𝑘=(Δ^𝑚−1𝑥_𝑘−Δ^𝑚−1𝑥_𝑘+1), so that Δ^𝑚𝑥_𝑘=∑^𝑚_𝑣=0(−1)^𝑣(𝑚/𝑣)𝑥_𝑘+𝑣. For extensive view in this area, we refer to the series of papers ([21–27]).

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.

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.

The following inequality will be used throughout the paper. Let 𝑝=(𝑝_𝑘) be a positive sequence of real numbers with 0<𝑝_𝑘≤sup_𝑘𝑝_𝑘=𝐻 and 𝐶=max(1,2^𝐻−1). Then, for all 𝑎_𝑘, 𝑏_𝑘 ∈ℂ, for all 𝑘∈ℕ, we have 𝑎_𝑘+𝑏_𝑘^𝑝𝑘≤𝐶𝑎_𝑘^𝑝𝑘+𝑏_𝑘^𝑝𝑘. (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 𝑆^Δ𝑚_𝜃𝜎(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=(𝑥_𝑘)∈𝑋.

Definition 7. Let ℐ be an admissible ideal, let 𝑓 be a modulus function, and let 𝑝=(𝑝_𝑘) be any sequence of strictly positive real numbers. Then, for each 𝜖>0, we define the following sequence spaces: 𝑤^𝑓_𝜎Δ^𝑚_𝑝,ℐ,𝜃₀=(𝑋)∶1𝑥∈𝑤𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚𝑥^𝑝𝑘,𝑤≥𝜖∈ℐ_𝑓𝜎Δ^𝑚_𝑝=(𝑋)1,ℐ,𝜃𝑥∈𝑤∶∃ℓ>0,𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚𝑥−ℓ^𝑝𝑘,𝑤≥𝜖∈ℐ_𝑓𝜎Δ^𝑚_𝑝,ℐ,𝜃_∞=(𝑋)1𝑥∈𝑤∶∃𝐾>0,𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚𝑥^𝑝𝑘≥𝐾∈ℐ (11) uniformly in 𝑛.

Remark 8. By taking some particular cases, we obtain the following. (i)If we take 𝑚=0, for all 𝑘∈ℕ, then the above spaces reduce to 𝑤^𝑓_𝜎[ℐ,𝑝,𝜃]₀, 𝑤^𝑓_𝜎[ℐ,𝑝,𝜃], and 𝑤^𝑓_𝜎[ℐ,𝑝,𝜃]_∞, respectively.(ii)If we choose 𝑚=0, for all 𝑘∈ℕ and 𝜎(𝑛)=𝑛+1, then we obtain 𝑤^𝑓_𝜎[ℐ,𝑝,𝜃]₀, 𝑤^𝑓_𝜎[ℐ,𝑝,𝜃], and 𝑤^𝑓_𝜎[ℐ,𝑝,𝜃]_∞ instead of 𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]₀, 𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃], and 𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]_∞.(iii)By taking ℐ=ℐ_𝑓={𝐸⊂ℕ∶if𝐸isafinitesubset}, 𝑚=0, 𝑓(𝑥)=𝑥, 𝜃=(2^𝑟), and 𝑝_𝑘=1, for all 𝑘∈ℕ, then we obtain [𝑤]_𝜎 defined by Mursaleen et al. [3] instead of 𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃] and 𝑤^𝑓_𝜎[ℐ,𝑝,𝜃] reduces to [𝑤] defined in Das and Sahoo [4].

Theorem 9. Let 𝑓 be a modulus function and 𝑝=(𝑝_𝑘) is a bounded sequence of strictly positive real numbers; then, 𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]₀, 𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃], and 𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]_∞ are linear spaces over ℂ.

Proof. We will prove the assertion only for 𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]₀ and others can be treated similarly. Suppose that 𝑥=(𝑥_𝑘), 𝑦=(𝑦_𝑘)∈𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]₀. Then, for every 𝜖>0 and uniformly in 𝑛, the sets 𝐴_𝜃𝜖2=1𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚𝑥^𝑝𝑘≥𝜖2,𝐵_𝜃𝜖2=1𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚𝑦^𝑝𝑘≥𝜖2 (12) belong to ℐ.
Let 𝛼, 𝛽∈ℂ and Δ^𝑚 is linear; then, 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚𝛼⋅𝑥+𝛽⋅𝑦^𝑝𝑘=1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛𝛼⋅Δ^𝑚𝑥+𝛽⋅Δ^𝑚𝑦^𝑝𝑘≤𝐶⋅𝐾_𝛼^𝐻⋅1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚𝑥^𝑝𝑘+𝐶⋅𝐾_𝛽^𝐻⋅1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚𝑦^𝑝𝑘,by(10), (13) where 𝐾_𝛼, 𝐾_𝛽 are two positive numbers such that |𝛼|≤𝐾_𝛼 and |𝛽|≤𝐾_𝛽.
Then, for given 𝜖>0, we have the following containment: 1𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚𝛼⋅𝑥+𝛽⋅𝑦^𝑝𝑘⊆1≥𝜖𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚𝑥^𝑝𝑘≥𝜖2𝐶⋅𝐾_𝛼^𝐻∪1𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚𝑦^𝑝𝑘≥𝜖2𝐶⋅𝐾_𝛽^𝐻 (14) uniformly in 𝑛.
Since 𝑥,𝑦∈𝑤^𝑓_𝜎[Δ^𝑚_𝑝,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 𝑤^𝑓_𝜎Δ_𝑝^𝑚−1,ℐ,𝜃_0,𝐿,∞⊂𝑤^𝑓_𝜎Δ^𝑚_𝑝,ℐ,𝜃_0,𝐿,∞ (15) is strict.

Proof. We will prove the result for 𝑤^𝑓_𝜎[Δ_𝑝^𝑚−1,ℐ,𝜃]₀ only. Suppose that 𝑥=(𝑥_𝑘)∈𝑤^𝑓_𝜎[Δ_𝑝^𝑚−1,ℐ,𝜃]₀; by definition, for each 𝜖>0, 1𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚−1𝑥_𝑘^𝑝𝑘≥𝜖∈ℐ (16) uniformly in 𝑛.
Since 𝑓 is a modulus function, therefore we have the following inequality: 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚𝑥_𝑘^𝑝𝑘≤1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚−1𝑥_𝑘+𝑓𝑡_𝑘𝑛Δ^𝑚−1𝑥_𝑘+1^𝑝𝑘1≤𝐶⋅ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚−1𝑥_𝑘^𝑝𝑘1+𝐶⋅ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚−1𝑥_𝑘+1^𝑝𝑘by(10) (17) uniformly in 𝑛.
Now, for given 𝜖>0, we have the following containment: 1𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚𝑥_𝑘^𝑝𝑘⊆1≥𝜖𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚−1𝑥_𝑘^𝑝𝑘≥𝜖∪12𝐶𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚−1𝑥_𝑘+1^𝑝𝑘≥𝜖,2𝐶 (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∈𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]₀.
Since ℐ is an admissible ideal and the inclusion is strict as the sequence 𝑥=(𝑥_𝑘)=(𝑘^𝑚−1) belongs to 𝑥∈𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]₀, it does not belong to 𝑤^𝑓_𝜎[Δ_𝑝^𝑚−1,ℐ,𝜃]₀, for 𝑓(𝑥)=𝑥, 𝑡_0𝑛(𝑥)=(𝑥_𝑛), 𝜃=(2^𝑟), and 𝑝_𝑘=1, for all 𝑘∈ℕ.

Theorem 11. Let 𝑝=(𝑝_𝑘) be a bounded sequence of strictly positive real numbers and 𝑓^′, 𝑓^′′ are modulus functions. If limsup_𝑡→∞𝑓^′(𝑡)𝑓^′′(𝑡)=𝑀>0, (19) then 𝑤^𝑓′_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]₀⊂𝑤^𝑓′′_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]₀.

Proof. Assume that limsup_𝑡→∞(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 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓^′𝑡_𝑘𝑛Δ^𝑚𝑥^𝑝𝑘≥(𝐾)^𝐻⋅1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓^′′𝑡_𝑘𝑛Δ^𝑚𝑥^𝑝𝑘 (20) uniformly in 𝑛.
Thus, for any 𝜖>0, 1𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑓^′′𝑡_𝑘𝑛Δ^𝑚𝑥^𝑝𝑘⊆1≥𝜖𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑓^′𝑡_𝑘𝑛Δ^𝑚𝑥^𝑝𝑘≥𝜖⋅(𝐾)^𝐻 (21) uniformly in 𝑛. Therefore, the above containment gives the result.

Theorem 12. If 𝑓,𝑓^′, and 𝑓^′′ are modulus functions, then (i)𝑤^𝑓′_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]⊂𝑤^{𝑓∘𝑓′}_𝜎[Δ^𝑚_𝑝,ℐ,𝜃], (ii)𝑤^𝑓′_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]∩𝑤^𝑓′′_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]⊂𝑤^{𝑓′+𝑓′′}_𝜎[Δ^𝑚_𝑝,ℐ,𝜃].

Proof. (i) Let 𝑥=(𝑥_𝑘)∈𝑤^𝑓′_𝜎[Δ^𝑚_𝑝,ℐ,𝜃] and let 𝜖>0 be given. Choose 𝛿∈(0,1) such that 𝑓(𝑡)<𝜖, for all 0<𝑡<𝛿, since 𝑥∈𝑤^𝑓′_𝜎[Δ^𝑚_𝑝,ℐ,𝜃] such that 𝐴_𝛿=1𝑛∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑓^′𝑡_𝑘𝑛Δ^𝑚𝑥−ℓ^𝑝𝑘≥𝛿∈ℐ (22) uniformly in 𝑛.
On the other hand, we have 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓∘𝑓^′𝑡_𝑘𝑛Δ^𝑚𝑥−ℓ^𝑝𝑘=1ℎ_{𝑟𝑘∈𝐼𝑟&𝑓′𝑡𝑘𝑛(Δ𝑚𝑥−ℓ)𝑝𝑘<𝛿}𝑓∘𝑓^′𝑡_𝑘𝑛Δ^𝑚𝑥−ℓ^𝑝𝑘+1ℎ_{𝑟𝑘∈𝐼𝑟&𝑓′𝑡𝑘𝑛(Δ𝑚𝑥−ℓ)𝑝𝑘≥𝛿}𝑓∘𝑓^′𝑡_𝑘𝑛Δ^𝑚𝑥−ℓ^𝑝𝑘≤(𝜖)^𝐻𝑓(1)+max1,2⋅𝛿^𝐻⋅1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓^′𝑡_𝑘𝑛Δ^𝑚𝑥−ℓ^𝑝𝑘byLemma6, (23) uniformly in 𝑛.
Then, for any 𝜂>0, 1𝑛∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑓∘𝑓^′𝑡_𝑘𝑛Δ^𝑚𝑥−ℓ^𝑝𝑘⊆1≥𝜂𝑛∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑓^′𝑡_𝑘𝑛Δ^𝑚𝑥−ℓ^𝑝𝑘≥𝜂−𝜖𝐾, (24) where 𝐾=max(1,(2,𝑓(1)/𝛿)^𝐻).
Since 𝑥=(𝑥_𝑘)∈𝑤^𝑓′_𝜎[Δ^𝑚_𝑝,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: 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓^′+𝑓^′′𝑡_𝑘𝑛Δ^𝑚𝑥−ℓ^𝑝𝑘≤𝐶ℎ_{𝑟𝑘∈𝐼𝑟}𝑓^′𝑡_𝑘𝑛Δ^𝑚𝑥−ℓ^𝑝𝑘+𝐶ℎ_{𝑟𝑘∈𝐼𝑟}𝑓^′′𝑡_𝑘𝑛Δ^𝑚𝑥−ℓ^𝑝𝑘, (25) uniformly in 𝑛, where sup_𝑘𝑝_𝑘=𝐻 and 𝐶=max(1,2^𝐻−1).

Theorem 13. If 𝑓 is a modulus function and 𝑝=(𝑝_𝑘) is a sequence of positive real numbers, then 𝑤_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]⊆𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃].

Proof. This can be proved similarly as in Theorem 12(i).

Theorem 14. Let 𝑓 be a modulus function. If  limsup_𝑡→∞(𝑓(𝑡)/𝑡)=𝑀>0, then 𝑤^𝑓_𝜎Δ^𝑚_𝑝,ℐ,𝜃⊆𝑤_𝜎Δ^𝑚_𝑝,ℐ,𝜃. (26)

Proof. Suppose that 𝑥=(𝑥_𝑘)∈𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]. It is given that limsup_𝑡→∞(𝑓(𝑡)/𝑡)=𝑀>0; there exists a constant 𝐾>0 such that 𝑓(𝑡)≥𝐾⋅𝑡, for all 𝑡≥0,
which implies that 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚𝑥−ℓ^𝑝𝑘≥(𝐾)^𝐻⋅1ℎ_{𝑟𝑘∈𝐼𝑟}𝑡_𝑘𝑛Δ^𝑚𝑥−ℓ^𝑝𝑘 (27) uniformly in 𝑛.
Then, for each 𝜖>0, 1𝑛∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑡_𝑘𝑛Δ^𝑚𝑥−ℓ^𝑝𝑘⊆1≥𝜖𝑛∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚𝑥−ℓ^𝑝𝑘≥𝜖⋅(𝐾)^𝐻, (28) uniformly in 𝑛.
Since 𝑥∈𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃], therefore the latter set belongs to ℐ. It follows that 𝑥∈𝑤_𝜎[Δ^𝑚_𝑝,ℐ,𝜃].

Theorem 15. If 0<𝑝_𝑘≤𝑞_𝑘 and (𝑞_𝑘/𝑝_𝑘) are bounded, then 𝑤^𝑓_𝜎[Δ^𝑚_𝑞,ℐ,𝜃]⊂𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃].

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

3. 𝑆^Δ𝑚_𝜃𝜎(ℐ)-Convergence

In this section, we define the notion of 𝑆^Δ𝑚_𝜃𝜎(ℐ)-convergence with the help of ideal and invariant means and difference sequences. Further, we also establish some relations between 𝑆^Δ𝑚_𝜃𝜎(ℐ)-convergence and 𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]₀.

Definition 16. Let ℐ⊆𝒫(ℕ) be a nontrivial ideal. A sequence 𝑥=(𝑥_𝑘) is said to be Δ^𝑚(ℐ)-strong lacunary 𝜎-statistically convergent or 𝑆^Δ𝑚_𝜃𝜎(ℐ)-convergent to a number ℓ, provided that, for every 𝜖>0 and 𝛿>0, 1𝑟∈ℕ∶ℎ_𝑟𝑘∈𝐼_𝑟∶𝑡_𝑘𝑛Δ^𝑚𝑥−ℓ≥𝜖≥𝛿∈ℐ,uniformlyin𝑛. (29) In this case, we write 𝑥_𝑘→ℓ(𝑆^Δ𝑚_𝜃𝜎(ℐ)) or 𝑆^Δ𝑚_𝜃𝜎(ℐ)−lim_𝑘→∞𝑥_𝑘=ℓ. Let 𝑆^Δ𝑚_𝜃𝜎(ℐ) denote the set of all 𝑆^Δ𝑚_𝜃𝜎(ℐ)-convergent sequences.

Theorem 17. Let 𝑓 be a modulus function and let 𝑝=(𝑝_𝑘) be a sequence of strictly positive real numbers. If 0<inf_𝑘𝑝_𝑘=ℎ≤𝑝_𝑘≤sup_𝑘𝑝_𝑘=𝐻<∞, then 𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]₀⊂𝑆^Δ𝑚_𝜃𝜎(ℐ).

Proof. Assume that 𝑥∈𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]₀ and 𝜖>0 is given. Then, we have 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚𝑥^𝑝𝑘=1ℎ_{𝑟𝑘∈𝐼𝑟&𝑡𝑘𝑛(Δ𝑚𝑥)≥𝜖}𝑓𝑡_𝑘𝑛Δ^𝑚𝑥^𝑝𝑘+1ℎ_{𝑟𝑘∈𝐼𝑟&𝑡𝑘𝑛(Δ𝑚𝑥)<𝜖}𝑓𝑡_𝑘𝑛Δ^𝑚𝑥^𝑝𝑘≥1ℎ_{𝑟𝑘∈𝐼𝑟&𝑡𝑘𝑛(Δ𝑚𝑥)≥𝜖}𝑓𝑡_𝑘𝑛Δ^𝑚𝑥^𝑝𝑘≥1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓(𝜖)^𝑝𝑘≥1ℎ_{𝑟𝑘∈𝐼𝑟}min𝑓(𝜖)^ℎ,𝑓(𝜖)^𝐻≥1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑡_𝑘𝑛Δ^𝑚𝑥𝑓(𝜖≥𝜖⋅min)^ℎ,𝑓(𝜖)^𝐻 (30) uniformly in 𝑛.
Then, for every 𝛿>0, we have the following containment: 1𝑟∈ℕ∶ℎ_𝑟𝑘∈𝐼_𝑟∶𝑡_𝑘𝑛Δ^𝑚𝑥⊆1≥𝜖≥𝛿𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝑡_𝑘𝑛Δ^𝑚𝑥^𝑝𝑘≥𝐾⋅𝛿 (31) uniformly in 𝑛, where 𝐾=min([𝑓(𝜖)]^ℎ,[𝑓(𝜖)]^𝐻), since 𝑥∈𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]₀, which implies that 𝑥∈𝑆^Δ𝑚_𝜃𝜎(ℐ).

Theorem 18. Let 𝑓 be a bounded modulus function and 0<inf_𝑘𝑝_𝑘=ℎ≤𝑝_𝑘≤sup_𝑘𝑝_𝑘=𝐻<∞; then, 𝑆^Δ𝑚_𝜃𝜎(ℐ)⊂𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]₀.

Proof. Using the same technique of [26, Theorem 3.3], it is easy to prove.

Theorem 19. If 0<inf_𝑘𝑝_k=ℎ≤𝑝_𝑘≤sup_𝑘𝑝_𝑘=𝐻<∞, then 𝑆^Δ𝑚_𝜃𝜎(ℐ)=𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]₀ if and only if 𝑓 is bounded.

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

Conversely. Suppose that 𝑓 is unbounded defined by 𝑓(𝑘)=𝑘, for all 𝑘∈ℕ, and let 𝜃=(2^𝑟) be a lacunary sequence. We take a fixed set 𝐴∈ℐ and define 𝑥=(𝑥_𝑘) as follows: 𝑥_𝑘=𝑘^𝑚+1,if𝑟∉𝐴,2^𝑟−1+1≤𝑘≤2^𝑟−1+ℎ_𝑟,𝑘^𝑚+1,if𝑟∈𝐴,2^𝑟−1<𝑘≤2^𝑟−1+ℎ_𝑟,0,otherwise, (32) where 𝐼_𝑟=(2^𝑟−1,2^𝑟] and ℎ_𝑟=2^𝑟−2^𝑟−1.

For given 𝜖>0, we have lim_𝑟→∞1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑡_𝑘𝑛Δ^𝑚𝑥−0≥𝜖=lim_𝑟→∞1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑡_𝑘𝑛(−1)^𝑚+2()!𝑚𝑚+1𝑘+2≤−0≥𝜖ℎ_𝑟ℎ_𝑟⟶0, (33) for all 𝑟∉𝐴, and uniformly in 𝑛.

Hence, for 𝛿>0, there exists a positive integer 𝑟₀ such that 1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑡_𝑘𝑛Δ^𝑚𝑥−0≥𝜖<𝛿for𝑟∉𝐴,𝑟≥𝑟₀. (34) Now, we have 1𝑟∈ℕ∶ℎ_𝑟𝑘∈𝐼_𝑟∶𝑡_𝑘𝑛Δ^𝑚⊂𝑥−0≥𝜖≥𝛿𝐴∪1,2,…,𝑟₀−1. (35) Since ℐ is an admissible ideal, it follows that 𝑆^Δ𝑚_𝜃𝜎(ℐ)−lim_𝑘→∞𝑥_𝑘=0.

If we take 𝑝_𝑘=1, for all 𝑘=1,2,…, then 𝑥_𝑘∉𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]₀. This contradicts the fact that 𝑆^Δ𝑚_𝜃𝜎(ℐ)=𝑤^𝑓_𝜎[Δ^𝑚_𝑝,ℐ,𝜃]₀.

Conflict of Interests

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