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 Iλstatistical convergent sequence spaces is discussed which is defined by a MusielakOrlicz function. We study relations between lacunary Iλstatistical convergence with lacunary Iλsummable sequences. Moreover, we study the Iλlacunary 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 Iconvergent to M ∈ R if, for each ξ > ,
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i∈Kj
xi – M = , for some M .
Ni(a) = sup ab – Mj(b) : b ≥ , i = , , . . . ,
which is named the complementary function of a MusielakOrlicz function M (see [])
(throughout the paper M is a MusielakOrlicz function).
If λ = (λi) is a nondecreasing sequence of positive integers such that denotes the set
of all nondecreasing sequences of positive integers. We call a sequence {xi}i∈N lacunary
Iλstatistically 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 Iμstatistically 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 Iλstatistically 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
= μiβ i – μi + ≤ j ≤ i – λi : xj – M ≥ γ
≤ μiλ–iβλiβ + μiβ j ∈ Ii : xj – M ≥ γ
≤ λμiβi – + λiα j ∈ Ii : xj – M ≥ γ
= ξ + λiα j ∈ Ii : xj – M ≥ γ .
Since (xj) is lacunary Iλstatistically 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 : λiα 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 Iμsummable of order β defined by M, then it is lacunary Iλstatistically
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 Iλstatistically convergent sequence of order α defined by M,
then it is also a lacunary Iμsummable 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 Iλstatistically 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 MusielakOrlicz 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 MusielakOrlicz 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 Iλstatistical 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 : λi j ∈ Ii : hi j∈Ii Mj
j ≤ i :
hi j∈Ii
≤ – r
= .
Theorem . Let (X, ν, τ ) be a PNS. If x is lacunary λstatistical convergent to M, then
Iλν (θ ) – lim x = M.
Proof Let x = (xi) be lacunary λstatistically 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 λstatistical convergent, then it has a
unique limit.
Theorem . Let (X, ν, τ ) be a PNS. If x is lacunary λstatistically convergent, then there
exists a subsequence (xmk ) of x such that it is also lacunary λstatistically convergent to M.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both of the authors jointly worked on deriving the results and approved the final manuscript.
⊆ {, , , . . . i – }.
Acknowledgements
The authors would like to extend their sincere appreciation to the referees for very useful comments and remarks for the
earlier version of the manuscript.
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