Strongly Almost Lacunary 𝐼-Convergent Sequences

Abstract

We study some new strongly almost lacunary 𝐼-convergent generalized difference sequence spaces defined by an Orlicz function. We give also some inclusion relations related to these sequence spaces.

1. Introduction

The notion of ideal convergence was first introduced by Kostyrko et al. [1] as a generalization of statistical convergence which was later studied by many other authors.

By a lacunary sequence, we mean an increasing integer sequence 𝜃=(𝑘_𝑟) such that 𝑘₀=0 and ℎ_𝑟 = 𝑘_𝑟−𝑘_𝑟−1→∞ as 𝑟→∞.

Throughout this paper, the intervals determined by 𝜃 will be denoted by 𝐽_𝑟=(𝑘_𝑟−1,𝑘_𝑟], and the ratio 𝑘_𝑟/𝑘_𝑟−1 will be defined by 𝜙_𝑟.

An Orlicz function is a function 𝑀∶[0,∞)→[0,∞), which is continuous, nondecreasing, and convex with 𝑀(0)=0, 𝑀(𝑥)>0, for 𝑥>0 and 𝑀(𝑥)→∞, as 𝑥→∞.

Let ℓ_∞, 𝑐, and 𝑐₀ be the Banach space of bounded, convergent, and null sequences 𝑥=(𝑥_𝑘), respectively, with the usual norm ‖𝑥‖=sup_𝑛|𝑥_𝑛|.

A sequence 𝑥∈ℓ_∞ is said to be almost convergent if all of its Banach limits coincide. Let 𝑐̂ denote the space of all almost convergent sequences.

Lorentz [2] introduced the following sequence space 𝑐̂=𝑥∈ℓ_∞∶lim_𝑚𝑡_𝑚,𝑛(𝑥)existsuniformlyin𝑛, (1) where 𝑡_𝑚,𝑛(𝑥) = (𝑥_𝑛+𝑥_𝑛+1+⋯+𝑥_𝑚+𝑛)/(𝑚+1).

The following space of strongly almost convergent sequence was introduced by Maddox [3]: [𝑐]̂=𝑥∈ℓ_∞∶lim_𝑚𝑡_𝑚,𝑛(|𝑥−𝐿𝑒|)existsuniformlyin𝑛forsome𝐿, (2) where 𝑒=(1,1,…).

Kızmaz [4] studied the difference sequence spaces ℓ_∞(Δ), 𝑐(Δ), and 𝑐₀(Δ) of crisp sets. The notion is defined as follows: 𝑍(Δ)=𝑥=𝑥_𝑘∶Δ𝑥_𝑘∈𝑍, (3) for 𝑍=ℓ_∞,𝑐, and 𝑐₀, where Δ𝑥=(Δ𝑥_𝑘) = (𝑥_𝑘−𝑥_𝑘+1), for all 𝑘∈𝑁.

The above spaces are Banach spaces, normed by ‖𝑥‖_Δ=𝑥₁+sup_𝑘Δ𝑥_𝑘. (4)

Tripathy et al. [5] introduced the generalized difference sequence spaces which are defined as, for 𝑚≥1 and 𝑛≥1, 𝑍Δ^𝑛_𝑚=𝑥=𝑥_𝑘∶Δ^𝑛_𝑚𝑥_𝑘∈𝑍,for𝑍=ℓ_∞,𝑐,𝑐₀. (5)

This generalized difference has the following binomial representation: Δ^𝑛_𝑚𝑥_𝑘=^𝑛_𝑟=0(−1)^𝑟𝑛𝑟𝑥_{𝑘+𝑟𝑚}. (6)

2. Definitions and Preliminaries

Kostyrko et al. [1] introduced the following three definitions.

Let 𝑋 be a nonempty set. Then a family of sets 𝐼⊆2^𝑋 (power sets of 𝑋) is said to be 𝑖𝑑𝑒𝑎𝑙 if 𝐼 is additive, that is, 𝐴,𝐵∈𝐼⇒𝐴∪𝐵∈𝐼, and hereditary, that is, 𝐴∈𝐼,𝐵⊆𝐴⇒𝐵∈𝐼.

A sequence (𝑥_𝑘) in a normed space (𝑋,‖⋅‖) is said to be I-convergent to 𝑥₀∈𝑋 if for each 𝜀>0, the set 𝐸(𝜀)=𝑘∈𝑁∶𝑥_𝑘−𝑥₀≥𝜀belongsto𝐼. (7)

A sequence (𝑥_𝑘) in a normed space (𝑋,‖⋅‖) is said to be I-bounded if there exists 𝑀>0 such that the set {𝑘∈𝑁∶‖𝑥_𝑘‖>𝑀} belongs to 𝐼.

Freedman et al. [6] defined the space 𝑁_𝜃. For any lacunary sequence 𝜃=(𝑘_𝑟), 𝑁_𝜃=𝑥_𝑘∶lim_𝑟→∞ℎ_𝑟⁻¹_{𝑘∈𝐽𝑟}𝑥_𝑘−𝐿=0,forsome𝐿. (8)

The space 𝑁_𝜃 is a 𝐵𝐾 space with the norm (𝑥_𝑘)_𝜃=sup_𝑟ℎ_𝑟⁻¹_{𝑘∈𝐽𝑟}𝑥_𝑘. (9)

The notion of lacunary ideal convergence of real sequences introduced by Tripathy et al. in [7, 8] and Hazarika [9, 10] introduced the lacunary ideal convergent sequences of fuzzy real numbers and studied some properties. In [5, 7], the lacunary ideal convergence is defined as follows.

Let 𝜃=(𝑘_𝑟) be a lacunary sequence. Then a sequence (𝑥_𝑘) is said to be lacunary 𝐼-convergent if for every 𝜀>0, such that 𝑟∈𝑁∶ℎ_𝑟⁻¹_{𝑘∈𝐽𝑟}𝑥_𝑘−𝑥≥𝜀∈𝐼, (10)

we write 𝐼_𝜃−lim𝑥_𝑘=𝑥.

Lindenstrauss and Tzafriri [11] used the idea of Orlicz function to construct the sequence space: ℓ_𝑀=𝑥_𝑘∈𝑤∶^∞_𝑘=1𝑀|𝑥|𝜌<∞,forsome𝜌>0. (11)

The space ℓ_𝑀 with the norm ‖𝑥‖=inf𝜌>0∶^∞_𝑘=1𝑀𝑥_𝑘𝜌≤1 (12) becomes a Banach space which is called an Orlicz sequence space.

In this paper, we defined some new generalized difference lacunary 𝐼-convergent sequence spaces defined by Orlicz function. We also introduce and examine some new sequence spaces and study their different properties.

3. Main Results

Esi [12] introduced the strongly almost ideal convergent sequence spaces in 2-normed spaces. In this paper we introduced the strongly almost lacunary ideal convergent sequence spaces using generalized difference operator and Orlicz function.

Let 𝐼 be an admissible ideal of 𝑁,  𝑀 an Orlicz function, and 𝜃=(𝑘_𝑟) a lacunary sequence. Further, let 𝑠=(𝑠_𝑘) be a bounded sequence of positive real numbers and Δ^𝑞_𝑝 a generalized difference operator.

For every 𝜀>0 and for some 𝜌>0, we have introduced the following sequence spaces: 𝑤^𝐼𝑀,Δ^𝑞_𝑝=𝑥,𝑠,𝜃_𝑘∶1𝑟∈𝑁∶ℎ_{𝑟𝑘∈𝐽𝑟}𝑀Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)−𝑙𝜌^𝑠𝑘≥𝜀∈𝐼,𝑚∈𝑁,forsome,𝑤𝑙∈𝑅_𝐼0𝑀,Δ^𝑞_𝑝=𝑥,𝑠,𝜃_𝑘∶1𝑟∈𝑁∶ℎ_{𝑟𝑘∈𝐽𝑟}𝑀Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)𝜌^𝑠𝑘,𝑤≥𝜀∈𝐼,𝑚∈𝑁_𝐼∞𝑀,Δ^𝑞_𝑝=𝑥,𝑠,𝜃_𝑘∶𝑟∈𝑁∶∃𝐾>0s.t.1ℎ_{𝑟𝑘∈𝐽𝑟}𝑀Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)𝜌^𝑠𝑘.≥𝐾∈𝐼,𝑚∈𝑁 (13)

Particular Cases. Consider the following.(1)If 𝜃=(2^𝑟), we have 𝑤^𝐼(𝑀,Δ^𝑞_𝑝,𝑠,𝜃)=𝑤^𝐼(𝑀,Δ^𝑞_𝑝,𝑠), 𝑤^𝐼₀(𝑀,Δ^𝑞_𝑝,𝑠,𝜃)=𝑤^𝐼₀(𝑀,Δ^𝑞_𝑝,𝑠), and 𝑤^𝐼_∞(𝑀,Δ^𝑞_𝑝,𝑠,𝜃)=𝑤^𝐼_∞(𝑀,Δ^𝑞_𝑝,𝑠).(2)If 𝑀(𝑥)=𝑥, then 𝑤^𝐼(𝑀,Δ^𝑞_𝑝,𝑠,𝜃)=𝑤^𝐼(Δ^𝑞_𝑝,𝑠,𝜃), 𝑤^𝐼₀(𝑀,Δ^𝑞_𝑝,𝑠,𝜃)=𝑤^𝐼₀(Δ^𝑞_𝑝,𝑠,𝜃), and 𝑤^𝐼_∞(𝑀,Δ^𝑞_𝑝,𝑠,𝜃)=𝑤^𝐼_∞(Δ^𝑞_𝑝,𝑠,𝜃).(3)If 𝑠_𝑘=1 for all 𝑘∈𝑁, 𝑀(𝑥)=𝑥, and 𝜃=(2^𝑟), then 𝑤^𝐼(𝑀,Δ^𝑞_𝑝,𝑠,𝜃)=𝑤^𝐼(Δ^𝑞_𝑝), 𝑤^𝐼₀(𝑀,Δ^𝑞_𝑝,𝑠,𝜃)=𝑤^𝐼(Δ^𝑞_𝑝), and 𝑤^𝐼_∞(𝑀,Δ^𝑞_𝑝,𝑠,𝜃)=𝑤^𝐼(Δ^𝑞_𝑝).

Theorem 1. Let the sequence (𝑠_𝑘) be bounded; then 𝑤^𝐼₀(𝑀,Δ^𝑞_𝑝,𝑠,𝜃)⊂𝑤^𝐼(𝑀,Δ^𝑞_𝑝,𝑠,𝜃)⊂𝑤^𝐼_∞(𝑀,Δ^𝑞_𝑝,𝑠,𝜃).

Proof. Let 𝑥∈𝑤^𝐼(𝑀,Δ^𝑞_𝑝,𝑠,𝜃). Then, for some 𝜌>0, we have 1ℎ_{𝑟𝑘∈𝐽𝑟}𝑀Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)2𝜌^𝑠𝑘≤𝐷ℎ_{𝑟𝑘∈𝐽𝑟}12^𝑝𝑘𝑀Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)−𝑙𝜌^𝑠𝑘+𝐷ℎ_{𝑟𝑘∈𝐽𝑟}12^𝑝𝑘𝑀|𝑙|𝜌^𝑠𝑘≤𝐷ℎ_{𝑟𝑘∈𝐽𝑟}𝑀Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)−𝑙𝜌^𝑠𝑘𝑀|𝑙|+𝐷max1,sup𝜌^𝐻, (14) where sup_𝑘𝑠_𝑘=𝐻 and 𝐷=max(1,2^𝐻−1).
Hence, 𝑥∈𝑤^𝐼_∞(𝑀,Δ^𝑞_𝑝,𝑠,𝜃).
The inclusion 𝑤^𝐼₀(𝑀,Δ^𝑞_𝑝,𝑠,𝜃)⊂𝑤^𝐼(𝑀,Δ^𝑞_𝑝,𝑠,𝜃) is obvious.

Theorem 2. Let the sequence (𝑠_𝑘) be bounded; then 𝑤^𝐼(𝑀,Δ^𝑞_𝑝,𝑠,𝜃), 𝑤^𝐼₀(𝑀,Δ^𝑞_𝑝,𝑠,𝜃), and 𝑤^𝐼_∞(𝑀,Δ^𝑞_𝑝,𝑠,𝜃) are closed under the operations of addition and scalar multiplication.

Theorem 3. Let 𝑀₁,𝑀₂ be Orlicz functions; then we have(1)𝑤^𝐼₀(𝑀₁,Δ^𝑞_𝑝,𝑠,𝜃)∩𝑤^𝐼₀(𝑀₂,Δ^𝑞_𝑝,𝑠,𝜃)⊂𝑤^𝐼₀(𝑀₁+𝑀₂,Δ^𝑞_𝑝,𝑠,𝜃),(2)𝑤^𝐼(𝑀₁,Δ^𝑞_𝑝,𝑠,𝜃)∩𝑤^𝐼(𝑀₂,Δ^𝑞_𝑝,𝑠,𝜃)⊂𝑤^𝐼(𝑀₁+𝑀₂,Δ^𝑞_𝑝,𝑠,𝜃), (3)𝑤^𝐼_∞(𝑀₁,Δ^𝑞_𝑝,𝑠,𝜃)∩𝑤^𝐼_∞(𝑀₂,Δ^𝑞_𝑝,𝑠,𝜃)⊂𝑤^𝐼_∞(𝑀₁+𝑀₂,Δ^𝑞_𝑝,𝑠,𝜃).

Theorem 4. Let 0<𝑠_𝑘≤𝑢_𝑘 for all 𝑘∈𝑁, and let (𝑢_𝑘/𝑠_𝑘) be bounded; then we have 𝑤^𝐼(𝑀₁,Δ^𝑞_𝑝,𝑢,𝜃)⊆𝑤^𝐼(𝑀₁,Δ^𝑞_𝑝,𝑠,𝜃).

Theorem 5. Let 𝜃=(𝑘_𝑟) be a lacunary sequence with 1<liminf_𝑟𝑢_𝑟≤sup_𝑟𝑢_𝑟<∞. Then, for any Orlicz function 𝑀, 𝑤^𝐼(𝑀,Δ^𝑞_𝑝,𝑠)=𝑤^𝐼(𝑀,Δ^𝑞_𝑝,𝑠,𝜃).

Proof. Suppose liminf_𝑟𝑢_𝑟>1 then there exists 𝛿>0 such that 𝑢_𝑟=𝑘_𝑟/𝑘_𝑟−1≥1+𝛿 for all 𝑟≥1.
Then, for 𝑥∈𝑤^𝐼(𝑀,Δ^𝑞_𝑝,𝑠), we have 𝑚∈𝑁∶_{𝑘∈𝐽𝑟}𝑀|Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)−𝑙|𝜌^𝑠𝑘≥𝜀∈𝐼. (15)
Let 𝐴_𝑟=1ℎ_{𝑟𝑘∈𝐽𝑟}𝑀Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)−𝑙𝜌^𝑠𝑘=1ℎ_𝑟^𝑘𝑟_𝑘=1𝑀|Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)−𝑙|𝜌^𝑠𝑘−1ℎ_𝑟^𝑘𝑟−1_𝑘=1𝑀Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)−𝑙𝜌^𝑠𝑘=𝑘_𝑟ℎ_𝑟1𝑘_𝑟^𝑘𝑟_𝑘=1𝑀Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)−𝑙𝜌^𝑠𝑘−𝑘_𝑟−1ℎ_𝑟1𝑘^𝑘_𝑟−1^𝑟−1_𝑘=1𝑀Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)−𝑙𝜌^𝑠𝑘. (16)
Since ℎ_𝑟=𝑘_𝑟−𝑘_𝑟−1, we have 𝑘_𝑟/ℎ_𝑟≤(1+𝛿)/𝛿 and 𝑘_𝑟−1/ℎ_𝑟≤1/𝛿.
So, for 𝜀>0 and for some 𝜌>0, 𝑘𝑟∈𝑁∶^𝑟ℎ_𝑟1𝑘_𝑟^𝑘𝑟_𝑘=1𝑀|Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)−𝑙|𝜌^𝑠𝑘≥𝜀∈𝐼,𝑚∈𝑁,forsome𝑘𝑙∈𝑅,𝑟∈𝑁∶_𝑟−1ℎ_𝑟1𝑘^𝑘_𝑟−1^𝑟−1_𝑘=1𝑀Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)−𝑙𝜌^𝑠𝑘≥𝜀∈𝐼,𝑚∈𝑁,forsome𝑙∈𝑅. (17)
Hence, 𝑤^𝐼(𝑀,Δ^𝑞_𝑝,𝑠)⊂𝑤^𝐼(𝑀,Δ^𝑞_𝑝,𝑠,𝜃).
Next, suppose that limsup_𝑟𝑞_𝑟<∞. Then, there exists 𝛽>0, such that, 𝑞_𝑟<𝛽 for all 𝑟≥1.
Let 𝑥∈𝑤^𝐼(𝑀,Δ^𝑞_𝑝,𝑠,𝜃) and 𝜀>0. There exists 𝑅>0 such that for every 𝑗≥𝑅, 𝐴_𝑗=1𝑟∈𝑁∶ℎ_{𝑟𝑘∈𝐽𝑗}𝑀|Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)−𝑙|𝜌^𝑝𝑘≥𝜀∈𝐼. (18)
Let 𝐾>0 such that 𝐴_𝑗≤𝐾 for all 𝑗=1,2,…. Now let 𝑛 be any integer with 𝑘_𝑟−1<𝑛≤𝑘_𝑟, where 𝑟>𝑅. Then, 1𝑛^𝑛_𝑘=1𝑀Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)−𝑙𝜌^𝑠𝑘≤1𝑘^𝑘_𝑟−1^𝑟_𝑘=1𝑀Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)−𝑙𝜌^𝑝𝑘=1𝑘_{𝑟−1𝑘∈𝐽1}𝑀Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)−𝑙𝜌^𝑝𝑘+_𝑘∈𝐽2𝑀Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)−𝑙𝜌^𝑝𝑘+⋯+_{𝑘∈𝐽𝑟}𝑀Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)−𝑙𝜌^𝑝𝑘=𝑘₁𝑘_𝑟−1𝐴₁+𝑘₂−𝑘₁𝑘_𝑟−1𝐴₂𝑘+⋯+^𝑅−𝑘_𝑅−1𝑘_𝑟−1𝐴_𝑅𝑘+⋯+^𝑟−𝑘_𝑟−1𝑘_𝑟−1𝐴_𝑟=sup_𝑗≥1𝐴_𝑗𝑘_𝑅𝑘_𝑟−1+sup_𝑗≥𝑅𝐴_𝑗𝑘_𝑟−𝑘_𝑅𝑘_𝑟−1𝑘<𝐾^𝑅𝑘_𝑟−1+𝜀𝛽. (19)
Since 𝑘_𝑟−1→∞ as 𝑟→∞, it follows that 1𝑚∈𝑁∶𝑛^𝑛_𝑘=1𝑀Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)−𝑙𝜌^𝑠𝑘≥𝜀∈𝐼. (20)
Hence, 𝑤^𝐼(𝑀,Δ^𝑞_𝑝,𝑠,𝜃)⊂𝑤^𝐼(𝑀,Δ^𝑞_𝑝,𝑠).

Theorem 6. If lim𝑠_𝑘>0 and 𝑥 is strongly almost lacunary convergent to 𝑥₀, with respect to the Orlicz function 𝑀, that is, (𝑥_𝑘)→𝑙(𝑤^𝐼(𝑀,Δ^𝑞_𝑝,𝑠,𝜃)), then 𝑥₀ is unique.

Proof. Let lim𝑠_𝑘=𝑠>0 and suppose that 𝑥_𝑘→𝑥₁(𝑤^𝐼(𝑀,Δ^𝑞_𝑝,𝑠,𝜃)), 𝑥_𝑘→𝑥₀(𝑤^𝐼(𝑀,Δ^𝑞_𝑝,𝑠,𝜃)).
Then there exist 𝜌₁ and 𝜌₂ such that 1𝑟∈𝑁∶ℎ_{𝑟𝑘∈𝐽𝑟}𝑀|Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)−𝑥₀|𝜌₁^𝑠𝑘1≥𝜀∈𝐼,𝑚∈𝑁,𝑟∈𝑁∶ℎ_{𝑟𝑘∈𝐽𝑟}𝑀|Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)−𝑥₁|𝜌₁^𝑠𝑘≥𝜀∈𝐼,𝑚∈𝑁. (21)
Let 𝜌=max(2𝜌₁,2𝜌₂). Then we have 1ℎ_{𝑟𝑘∈𝐽𝑟}𝑀𝑥₀−𝑥₁𝜌^𝑠𝑘≤𝐷ℎ_{𝑟𝑘∈𝐽𝑟}𝑀Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)−𝑥₀𝜌₁^𝑠𝑘+𝐷ℎ_{𝑟𝑘∈𝐽𝑟}𝑀Δ^𝑞_𝑝𝑡_𝑘𝑚(𝑥)−𝑥₁𝜌₂^𝑠𝑘, (22) where sup_𝑘𝑠_𝑘=𝐻 and 𝐷=max(1,2^𝐻−1).
Thus, from (21), we get 1𝑟∈𝑁∶ℎ_{𝑟𝑘∈𝐽𝑟}𝑀|𝑥₀−𝑥₁|𝜌₁^𝑠𝑘≥𝜀∈𝐼. (23)
Further, 𝑀(|𝑥₀−𝑥₁|/𝜌)^𝑠𝑘→𝑀(|𝑥₀−𝑥₁|/𝜌)^𝑠 as 𝑘→∞, and, therefore, 𝑀|𝑥₀−𝑥₁|𝜌^𝑠𝑘⟶𝑀|𝑥₀−𝑥₁|𝜌^𝑠=0. (24)
Hence, 𝑥₀=𝑥₁.

4. Conclusion

The concept of lacunary 𝐼-convergence has been studied by various mathematicians. In this paper, we have introduced some fairly wide classes of strongly almost lacunary 𝐼-convergent sequences of real numbers using Orlicz function with the generalized difference operator. Giving particular values to the sequence 𝜃=(𝑘_𝑟) and 𝑀, we obtain some new sequence spaces which are the special cases of the sequence spaces we have defined. There are lots more to be investigated in the future.

Acknowledgments

First of all, the authors sincerely thank the referees for the valuable comments. The first author gratefully acknowledges that part of this research was partially supported by the University Putra Malaysia under the ERGS Grant Scheme having Project no. 5527068. The work of the second author was carried under the Postdoctoral Fellow under National Board of Higher Mathematics, DAE (Government of India), Project no. NBHM/PDF.50/2011/64.