Some new lacunary statistical convergence with ideals

Journal of Inequalities and Applications, Jan 2017

In this paper, the idea of lacunary I λ -statistical convergent sequence spaces is discussed which is defined by a Musielak-Orlicz function. We study relations between lacunary I λ -statistical convergence with lacunary I λ -summable sequences. Moreover, we study the I λ -lacunary statistical convergence in probabilistic normed space and discuss some topological properties.

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Some new lacunary statistical convergence with ideals

Kilicman and Borgohain Journal of Inequalities and Applications Some new lacunary statistical convergence with ideals Adem Kilicman 0 Stuti Borgohain 0 1 0 Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia , Serdang, 43400 Selangor , Malaysia 1 Department of Mathematics, Indian Institute of Technology , Bombay, Powai, 400076 Mumbai , India In this paper, the idea of lacunary Iλ-statistical convergent sequence spaces is discussed which is defined by a Musielak-Orlicz function. We study relations between lacunary Iλ-statistical convergence with lacunary Iλ-summable sequences. Moreover, we study the Iλ-lacunary statistical convergence in probabilistic normed space and discuss some topological properties. - i ∈ N : |xi – M| ≥ ξ ∈ I. where χK(ξ) denotes the characteristic function of K (ξ ). A sequence (xi) of elements of R is I-convergent to M ∈ R if, for each ξ > , © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. i∈Kj |xi – M| = , for some M . Ni(a) = sup |a|b – Mj(b) : b ≥  , i = , , . . . , which is named the complementary function of a Musielak-Orlicz function M (see []) (throughout the paper M is a Musielak-Orlicz function). If λ = (λi) is a non-decreasing sequence of positive integers such that denotes the set of all non-decreasing sequences of positive integers. We call a sequence {xi}i∈N lacunary Iλ-statistically convergent of order α to M, if, for each γ >  and ξ > ,  i ∈ N : λiα  j ≤ i : hi j∈Ii ∈ I. 3 Main results Theorem . Let λ = (λi) and μ = (μi) be two sequencesλiαn such that λi ≤ μi for all i ∈ N and  < α ≤ β ≤  for fixed reals α and β. If lim infi→∞ μiβ > , then SIβμ (M, θ ) ⊆ SIαλ (M, θ ). i λα Proof Suppose that λi ≤ μi for all i ∈ N and lim infi→∞ μiβ > . Since Ii ⊂ Ji, where Ji = i [i – μi + , i], so for γ > , we can write Since I is admissible and (xj) is a lacunary Iμ-statistically convergent sequence of order β defined by M, by using the continuity of M, we see with the lacunary sequence θ = (hi), the right hand side belongs to I, which completes the proof. Theorem . If limi→∞ λμβi = , for λ = (λi) and μ = (μi) two sequences of i λi ≤ μi, ∀i ∈ N and  < α ≤ β ≤  for fixed α, β reals, then SIαλ (M, θ ) ⊆ SIβμ (M, θ ). Proof Let (xj) be lacunary Iλ-statistically convergent to M of order α defined by M. Also assume that limi→∞ λμβi = . Choose m ∈ N such that | λμβi – | <  , ∀i ≥ m. ξ i i Since Ii ⊂ Ji, for γ > , we may write = μiβ i – μi +  ≤ j ≤ i – λi : |xj – M| ≥ γ ≤ μiλ–iβλiβ + μiβ j ∈ Ii : |xj – M| ≥ γ ≤ λμiβi –  + λiα j ∈ Ii : |xj – M| ≥ γ = ξ + λiα j ∈ Ii : |xj – M| ≥ γ . Since (xj) is lacunary Iλ-statistically convergent sequence of order α defined by M and since I is admissible, by using the continuity of M, it follows that the set on the right hand wIαλ (M, θ ) = i ∈ N : λiα j ≤ i : hi j∈Ii ∈ I. λα Theorem . Let λi ≤ μi for all i ∈ N, where λ, μ ∈ . Then, if lim infi→∞ μiβ > , and if i (xj) is lacunary Iμ-summable of order β defined by M, then it is lacunary Iλ-statistically convergent of order α defined by M. Here  < α ≤ β ≤ , for fixed reals α and β. j∈Ji |xj – M| = |xj – M| + |xj – M| + ≥ j ∈ Ii : |xj – M| ≥ γ .γ . λα α If lim infi→∞ μiβ = a, then {i ∈ N : λiβ < a } is finite. So, for δ > , we get i μi  i ∈ N : λiα j ≤ i : j∈Ji Theorem . Let limi→∞ λμβi = , where  < α ≤ β ≤  for fixed reals α and β and λi ≤ μi, for all i ∈ N, where λ, μ ∈ i . Also let θ ! be a refinement of θ . Let (xj) to be a bounded sequence. If (xj) is lacunary Iλ-statistically convergent sequence of order α defined by M, then it is also a lacunary Iμ-summable sequence of order β defined by M. i.e. SIαλ (M, θ ) ⊆ wIβμ (M, θ !). ∀i ∈ N L hi–hi hi –L + λiα j ∈ Ii : hi–hi hi –|xj – M| ≥ ε = δ hi–hi hi –L + λLiα j ∈ Ii : hi–hi hi –|xj – M| ≥ ε j∈Ji j∈Ii j∈Ii j∈Ii j∈Ii ∪ {, , , . . . , s}. Since (xj) is lacunary Iλ-statistically convergent sequence of order α defined by M and since I is admissible, by using the continuity of M, we can say that λα Corollary . Let λ ≤ μi for all i ∈ N and  < α ≤ β ≤ . Let lim infi→∞ μiβ > , θ ! be the i refinement of θ . Also let M = (Mi) be a Musielak-Orlicz function where (Mi) is pointwise β γ convergent. Then wIμ (M, θ !) ⊂ SIαλ (M, θ ) iff limi Mi( ρ(i) ) > , for some γ > , ρ(i) > . Corollary . Let M = (Mi) be a Musielak-Orlicz function and limi→∞ λμβi = , for fixed i numbers α and β such that  < α ≤ β ≤ . Then SIαλ (M, θ ) ⊂ wIβμ (M, θ ) iff supν supi( ρν(i) ). 4 Lacunary Iλ-statistical convergence in probabilistic normed spaces hi j∈Ii  j ≤ i : hi j∈Ii which implies K ∈ I and Iλν (θ ) – lim = . Theorem . Let (X, ν, τ ) be a PNS. If x = (xi) is lacunary Iλν -statistical convergent, then it has a unique limit. Proof Suppose x = (xi) to be lacunary Iλν -statistical convergent in X which has two limits, M and M. For β >  and t > , let us choose ξ >  such that τ (( – ξ ), ( – ξ )) ≥  – β. Take the following sets:  j ≤ i :  j ≤ i : hi j∈Ii hi j∈Ii Since τ (( – ξ ), ( – ξ )) ≥  – β, it follows that νM–M (t) >  – β. For arbitrary β > , we get νM–M (t) =  for all t > , which proves M = M. Theorem . Let (X, ν, τ ) be a PNS. If x is lacunary Iν -statistical convergent, then it is lacunary Iλν -statistical convergent if limi λii > .  j ≤ i : hi j∈Ii hi j∈Ii ≥ λii i ∈ N : λi j ∈ Ii : hi j∈Ii Mj  j ≤ i : hi j∈Ii ≤  – r = . Theorem . Let (X, ν, τ ) be a PNS. If x is lacunary λ-statistical convergent to M, then Iλν (θ ) – lim x = M. Proof Let x = (xi) be lacunary λ-statistically convergent to M, then, for every t > , ξ >  and μ > , there exists i ∈ N such that hi j∈Ii = , for all i ≥ i. Therefore the set hi j∈Ii Theorem . Let (X, ν, τ ) be a PNS. If x is lacunary λ-statistical convergent, then it has a unique limit. Theorem . Let (X, ν, τ ) be a PNS. If x is lacunary λ-statistically convergent, then there exists a subsequence (xmk ) of x such that it is also lacunary λ-statistically convergent to M. Competing interests The authors declare that they have no competing interests. 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Adem Kilicman, Stuti Borgohain. Some new lacunary statistical convergence with ideals, Journal of Inequalities and Applications, 2017, 15, DOI: 10.1186/s13660-016-1284-9