Volume 2013, Issue 1 642535

Research Article

Open Access

Strongly Almost Lacunary I-Convergent Sequences

Corresponding Author

Adem Kılıçman

[email protected]

Department of Mathematics and Institute for Mathematical Research, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia upm.edu.my

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Stuti Borgohain,

Stuti Borgohain

orcid.org/0000-0002-5951-2735

Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai, Maharashtra 400076, India iitb.ac.in

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Adem Kılıçman,

Corresponding Author

Adem Kılıçman

[email protected]

Department of Mathematics and Institute for Mathematical Research, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia upm.edu.my

Search for more papers by this author

Stuti Borgohain,

Stuti Borgohain

orcid.org/0000-0002-5951-2735

Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai, Maharashtra 400076, India iitb.ac.in

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First published: 06 November 2013

https://doi.org/10.1155/2013/642535

Academic Editor: S. A. Mohiuddine

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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 denote the space of all almost convergent sequences.

Lorentz [2] introduced the following sequence space

(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]:

(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:

(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

(4)

Tripathy et al. [5] introduced the generalized difference sequence spaces which are defined as, for m ≥ 1 and n ≥ 1,

(5)

This generalized difference has the following binomial representation:

(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

(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),

(8)

The space N_θ is a BK space with the norm

(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

(10)

we write I_θ − lim x_k = x.

Lindenstrauss and Tzafriri [11] used the idea of Orlicz function to construct the sequence space:

(11)

The space ℓ_M with the norm

(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 a generalized difference operator.

For every ε > 0 and for some ρ > 0, we have introduced the following sequence spaces:

(13)

Particular Cases. Consider the following.

(1)
If θ = (2^r), we have , , and .
(2)
If M(x) = x, then , , and .
(3)
If s_k = 1 for all k ∈ N, M(x) = x, and θ = (2^r), then , , and .

Theorem 1. Let the sequence (s_k) be bounded; then .

Proof. Let . Then, for some ρ > 0, we have

(14)

where sup _ks_k = H and D = max (1, 2^H−1).

Hence, .

The inclusion is obvious.

Theorem 2. Let the sequence (s_k) be bounded; then , , and are closed under the operations of addition and scalar multiplication.

Theorem 3. Let M₁, M₂ be Orlicz functions; then we have

(1)
,
(2)
,
(3)
.

Theorem 4. Let 0 < s_k ≤ u_k for all k ∈ N, and let (u_k/s_k) be bounded; then we have .

Theorem 5. Let θ = (k_r) be a lacunary sequence with 1 < lim inf _ru_r ≤ sup _ru_r < ∞. Then, for any Orlicz function M, .

Proof. Suppose lim inf _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 , we have

(15)

Let

(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,

(17)

Hence, .

Next, suppose that lim sup _rq_r < ∞. Then, there exists β > 0, such that, q_r < β for all r ≥ 1.

Let and ε > 0. There exists R > 0 such that for every j ≥ R,

(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,

(19)

Since k_r−1 → ∞ as r → ∞, it follows that

(20)

Hence, .

Theorem 6. If lim s_k > 0 and x is strongly almost lacunary convergent to x₀, with respect to the Orlicz function M, that is, , then x₀ is unique.

Proof. Let lim s_k = s > 0 and suppose that , .

Then there exist ρ₁ and ρ₂ such that

(21)

Let ρ = max (2ρ₁, 2ρ₂). Then we have

(22)

where sup _ks_k = H and D = max (1, 2^H−1).

Thus, from (21), we get

(23)

Further, as k → ∞, and, therefore,

(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.

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Strongly Almost Lacunary I-Convergent Sequences

Abstract

1. Introduction

2. Definitions and Preliminaries

3. Main Results

4. Conclusion

Acknowledgments

References

References

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Strongly Almost Lacunary I-Convergent Sequences

Abstract

1. Introduction

2. Definitions and Preliminaries

3. Main Results

4. Conclusion

Acknowledgments

References

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