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Available online at www.worldscientificnews.com
( Received 10 September 2019; Accepted 27 September 2019; Date of Publication 28 September 2019 )
WSN 136 (2019) 52-65 EISSN 2392-2192
π-lacunary ππ¨ππππ -convergence defined by Musielak
Orlicz function
Ayten Esi1, Nagarajan Subramanian2 and Ayhan Esi1
1Department of Mathematics, Adiyaman University, 02040, Adiyaman, Turkey
2Department of Mathematics, SASTRA University, Thanjavur - 613 401, India
1-3E-mail address: [email protected] , [email protected] , [email protected]
ABSTRACT
We study some connections between π-lacunary strong ππ΄π’π£π€3 - convergence with respect to a
πππ sequence of Musielak Orlicz function and π-lacunary ππ΄π’π£π€3 - statistical convergence, where π΄ is
a sequence of four dimensional matrices π΄(π’π£π€) = (ππ1β¦ππβ1β¦βπ π1β¦πππ1β¦ππ (π’π£π€)) of complex numbers.
Keywords: Analytic sequence, x2 space, difference sequence space, Musielak-modulus function, p-
metric space, mn-sequences
2010 Mathematics Subject Classification: 40A05, 40C05, 40D05
1. INTRODUCTION
Throughout π€, π and Ξ denote the classes of all, gai and analytic scalar valued single
sequences, respectively. We write π€3 for the set of all complex triple sequences (π₯πππ), where
π, π, π β β, the set of positive integers. Then, π€3 is a linear space under the coordinate wise
addition and scalar multiplication.
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We can represent triple sequences by matrix. In case of double sequences we write in the
form of a square. In the case of a triple sequence it will be in the form of a box in three
dimensional case.
Some initial work on double series is found in Apostol [1] and double sequence spaces is
found in Hardy [9], Deepmala et al. [10, 11] and many others. Later on investigated by some
initial work on triple sequence spaces is found in Esi [2], Esi et al. [3-8], Εahiner et al. [12],
Subramanian et al. [13], Prakash et al. [14] and many others.
Let (π₯πππ) be a triple sequence of real or complex numbers. Then the series
ββπ,π,π=1 π₯πππ is called a triple series. The triple series ββπ,π,π=1 π₯πππ give one space is said
to be convergent if and only if the triple sequence (ππππ) is convergent, where
ππππ = β
π,π,π
π,π,π=1
π₯πππ (π, π, π = 1,2,3, . . . ).
A sequence π₯ = (π₯πππ) is said to be triple analytic if
supπ,π,π
|π₯πππ|1
π+π+π < β.
The vector space of all triple analytic sequences are usually denoted by Ξ3. A sequence
π₯ = (π₯πππ) is called triple entire sequence if
|π₯πππ|1
π+π+π β 0 as π, π, π β β.
A sequence π₯ = (π₯πππ) is called triple gai sequence if ((π + π + π)! |π₯πππ|)1
π+π+π β 0
as π, π, π β β. The triple gai sequences will be denoted by π3.
2. DEFINITIONS AND PRELIMINARIES
A triple sequence π₯ = (π₯πππ) has limit 0 (denoted by π β limπ₯ = 0) (i.e)
((π + π + π)! |π₯πππ|)1/π+π+π
β 0 as π, π, π β β. We shall write more briefly as π βππππ£ππππππ‘ π‘π 0.
Definition 2.1 An Orlicz function (see [15]) is a function M: [0,β) β [0,β) which is
continuous, non-decreasing and convex with M(0) = 0, M(x) > 0, for x > 0 and M(x) β β
as x β β. If convexity of Orlicz function M is replaced by M(x + y) β€ M(x) + M(y), then this
function is called modulus function.
Lindenstrauss and Tzafriri (see [16]) used the idea of Orlicz function to construct Orlicz
sequence space.
A sequence π = (πππ) defined by
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πππ(π£) = sup{|π£|π’ β (ππππ)(π’): π’ β₯ 0},π, π, π = 1,2, β¦
is called the complementary function of a Musielak-Orlicz function π. For a given Musielak-
Orlicz function π, (see [17]) the Musielak-Orlicz sequence space π‘π is defined as follows
π‘π = {π₯ β π€3: πΌπ(|π₯πππ|)
1/π+π+π β 0 as π, π, π β β},
where πΌπ is a convex modular defined by
πΌπ(π₯) = β
β
π=1
β
β
π=1
β
β
π=1
ππππ(|π₯πππ|)1/π+π+π, π₯ = (π₯πππ) β π‘π .
We consider π‘π equipped with the Luxemburg metric
π(π₯, π¦) = β
β
π=1
β
β
π=1
β
β
π=1
ππππ (|π₯πππ|
1/π+π+π
πππ)
is an extended real number.
Definition 2.2 Let mnk(β₯ 3) be an integer. A function x: (M Γ N Γ K) Γ (M Γ N Γ K) Γβ―Γ(M Γ N Γ K) Γ (M Γ N Γ K) [m Γ n Γ k β factors] β β(β) is called a real or complex mnk-
sequence, where β, β and β denote the sets of natural numbers and complex numbers
respectively. Let m1, m2, β¦mr, n1, n2, β¦ , ns, k1, k2, β¦ , kt β β and X be a real vector space of
dimension w, where m1, m2, β¦mr, n1, n2, β¦ , ns, k1, k2, β¦ , kt β€ w. A real valued function
ππ(π₯11, β¦ , π₯π1,π2,β¦ππ,π1,π2,β¦,ππ ,π1,π2,β¦,ππ‘)
=β₯ (π1(π₯11, 0), β¦ , ππ(π₯π1,π2,β¦ππ,π1,π2,β¦,ππ ,π1,π2,β¦,ππ‘ , 0)) β₯π
on π satisfying the following four conditions:
(i) β₯ (π1(π₯11, 0), β¦ , ππ(π₯π1,π2,β¦ππ,π1,π2,β¦,ππ ,π1,π2,β¦,ππ‘ , 0)) β₯π= 0 if and and only if
π1(π₯11, 0), β¦ , ππ1,π2,β¦ππ,π1,π2,β¦,ππ (π₯π1,π2,β¦ππ,π1,π2,β¦,ππ ,π1,π2,β¦,ππ‘ , 0) are linearly dependent,
(ii) β₯ (π1(π₯11, 0), β¦ , ππ1,π2,β¦ππ,π1,π2,β¦,ππ ,π1,π2,β¦,ππ‘(π₯π1,π2,β¦ππ,π1,π2,β¦,ππ ,π1,π2,β¦,ππ‘ , 0)) β₯π is
invariant under permutation,
(iii) For πΌ β β,
β₯ (πΌπ1(π₯11, 0), β¦ , ππ1,π2,β¦ππ,π1,π2,β¦,ππ,π1,π2,β¦,ππ‘(π₯π1,π2,β¦ππ,π1,π2,β¦,ππ ,π1,π2,β¦,ππ‘ , 0)) β₯π
= |πΌ| β₯ (π1(π₯11, 0), β¦ , ππ(π₯π1,π2,β¦ππ,π1,π2,β¦,ππ,π1,π2,β¦,ππ‘ , 0)) β₯π
(iv) For 1 β€ π < β,
ππ((π₯11, π¦11), (π₯12, π¦12)β¦ (π₯π1,π2,β¦ππ,π1,π2,β¦,ππ ,π1,π2,β¦,ππ‘ , π¦π1,π2,β¦ππ,π1,π2,β¦,ππ ,π1,π2,β¦,ππ‘))
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= (ππ(π₯11, π₯12, β¦ π₯π1,π2,β¦ππ,π1,π2,β¦,ππ ,π1,π2,β¦,ππ‘)π +
ππ(π¦11, π¦12, β¦ π¦π1,π2,β¦ππ,π1,π2,β¦,ππ ,π1,π2,β¦,ππ‘)π)1/π
(or)
(v) π((π₯11, π¦11), (π₯12, π¦12),β¦ (π₯π1,π2,β¦ππ,π1,π2,β¦,ππ , π¦π1,π2,β¦ππ,π1,π2,β¦,ππ ,π1,π2,β¦,ππ‘))
: = sup{ππ(π₯11, π₯12, β¦ π₯π1,π2,β¦ππ,π1,π2,β¦,ππ ,π1,π2,β¦,ππ‘),
ππ(π¦11, π¦12, β¦ π¦π1,π2,β¦ππ,π1,π2,β¦,ππ ,π1,π2,β¦,ππ‘)},
for π₯11, π₯12, β¦ π₯π1,π2,β¦ππ,π1,π2,β¦,ππ ,π1,π2,β¦,ππ‘ β π, π¦11, π¦12, β¦ π¦π1,π2,β¦ππ,π1,π2,β¦,ππ ,π1,π2,β¦,ππ‘ β π
is called the π-product metric of the Cartesian product of π1, π2, β¦ππ , π1, π2, β¦ ,ππ , π1, π2, β¦ , ππ‘ metric spaces is the π-norm of the πΓ π Γ π-vector of the norms of the
π1, π2, β¦ππ , π1, π2, β¦ , ππ , π1, π2, β¦ , ππ‘ subspaces.
Definition 2.3 The triple sequence ΞΈi,β,j = {(mi, nβ, kj)} is called triple lacunary if there exist
three increasing sequences of integers such that
π0 = 0, βπ = ππ βππβ1 β β as π β β
and
π0 = 0, ββ = πβ β πββ1 β β as β β β.
π0 = 0, βπ = ππ β ππβ1 β β as π β β.
Let ππ,β,π = πππβππ, βπ,β,π = βπβββπ, and ππ,β,π is determine by
πΌπ,β,π = {(π, π, π):ππβ1 < π < ππ and πββ1 < π β€ πβ and ππβ1 < π β€ ππ},
ππ =ππππβ1
, πβ =πβπββ1
, ππ =ππ
ππβ1.
Let πΉ = (ππππ) be a πππ-sequence of Musielak Orlicz functions such that
limπ’β0+supπππππππ(π’) = 0. Throughout this paper ππ΄π’π£π€3 -convergence of π-metric of πππ-
sequence of Musielak Orlicz function determinated by πΉ will be denoted by ππππ β πΉ for every
π, π, π β β.
The purpose of this paper is to introduce and study a concept of triple lacunary strong
ππ΄π’π£π€3 -convergence of π-metric with respect to a πππ-sequence of Musielak Orlicz function.
We now introduce the generalizations of triple lacunary strongly ππ΄π’π£π€3 -convergence of
π-metric with respect a πππ-sequence of Musielak Orlicz function and investigate some
inclusion relations.
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Let π΄ denote a sequence of the matrices π΄π’π£π€ = (ππ1β¦ππβ1β¦βπ π1β¦πππ1β¦ππ ,π1,π2,β¦,ππ‘(π’π£π€)) of
complex numbers. We write for any sequence π₯ = (π₯πππ),
π¦ππ(π’π£) = π΄πππ’π£π€(π₯) = β
β
π1β¦ππ
β
β
π1β¦ππ
β
β
π1,π2,β¦,ππ‘
(ππ1β¦ππβ1β¦βπ π1β¦πππ1β¦ππ (π’π£π€)) β
((π1β¦ππ + π1β¦ππ + π1, π2, β¦ , ππ‘)! |π₯π1β¦πππ1β¦ππ π1,π2,β¦,ππ‘|)1/π1β¦ππ+π1β¦ππ +π1,π2,β¦,ππ‘
if it exits for each πππ and π’π£π€. We π΄π’π£π€(π₯) = (π΄ππππ’π£π€(π₯))
πππ, π΄π₯ = (π΄π’π£π€(π₯))
π’π£π€.
Definition 2.4 Let ΞΌ be a valued measure on β Γ β Γ β and F = (fm1β¦mrn1β¦nsk1,k2,β¦,ktijq
) be a
mnk-sequence of Musielak Orlicz function , A denote the sequence of four dimensional infinte
matrices of complex numbers and X be locally convex Hausdorff topological linear space whose
topology is determined by a set of continuous semi norms Ξ· and
(X, β(d(x111, 0), d(x122, 0), β¦ , d(xm1,m2,β¦mrβ1n1,n2,β¦nsβ1k1,k2,β¦,ktβ1 , 0))βp) be a p-metric
space, q = (qijq) be triple analytic sequence of strictly positive real numbers.
By w3(p β X) we denote the space of all sequences defined over
(X, β(d(x111, 0), d(x122, 0), β¦ , d(xm1,m2,β¦mrβ1n1,n2,β¦nsβ1k1,k2,β¦,ktβ1 , 0))βp)ΞΌ
.
In the present paper we define the following sequence spaces:
[ππ΄πππ
πΌ3ππ
, β(π(π₯111), π(π₯122),β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1))βπ]π
= πlimππ π‘[ππππ(βππ
πΌ(π₯), , (π(π₯111, 0), π(π₯122, 0), β¦ ,
π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ)]ππππ
β₯ π = 0,
where
πππΌ(π₯) =
1
βππ π‘πΌ β
πβπΌππ π‘
β
πβπΌππ π‘
β
πβπΌππ π‘
ππ΄πππ’π£π€ (((π1β¦ππ + π1β¦ππ + π1, π2, β¦ , ππ‘)!
|π₯π1β¦πππ1β¦ππ π1,π2,β¦,ππ‘|)1/π1β¦ππ+π1β¦ππ +π1,π2,β¦,ππ‘
),
uniformly in π’, π£, π€
[Ξπ΄πππ
πΌ3ππ
, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
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= πsupππ π‘[ππ’π£π€(βππ
πΌ(π₯), (π(π₯111, 0), π(π₯122, 0), β¦ ,
π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ)]ππππ
β₯ π = 0,
where π =
(
1 1 β―11 1 β―1...1 1 β―1
)
.
The main aim of this paper is to introduce the idea of summability of triple lacunary
sequence spaces in π-metric spaces using a three valued measure. We also make an effort to
study π-of lacunary triple sequences with respect to a sequence of Musielak Orlicz function in
π-metric spaces and three valued measure π. We also plan to study some topological properties
and inclusion relation between these spaces.
3. MAIN RESULTS
Proposition 3.1 Let ΞΌ be a three valued measure,
[ππ΄πππ
πΌ3ππ
, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
and
[Ξπ΄πππ
πΌ3ππ
, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
are linear spaces.
Proof. It is routine verification. Therefore the proof is omitted.
The inclusion relation between
[πππππ
πππͺπ
, β(π(π±πππ, π), π(π±πππ, π), β¦ , π(π±π¦π,π¦π,β¦π¦π«βππ§π,π§π,β¦π§π¬βππ€π,π€π,β¦,π€πβπ , π))βπ©]π
and
[π²ππππ
πππͺπ
, β(π(π±πππ, π), π(π±πππ, π),β¦ , π(π±π¦π,π¦π,β¦π¦π«βππ§π,π§π,β¦π§π¬βππ€π,π€π,β¦,π€πβπ , π))βπ©]π
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Theorem 3.1 Let ΞΌ be a three valued measure and A be a mnk-sequence the four dimensional
infinite matrices Auv = (ak1β¦krβ1β¦βsm1β¦mrn1β¦nsk1,k2,β¦,kt(uvw)) of complex numbers and F = (fmnk
ijq) be
a mn-sequence of Musielak Orlicz function. If x = (xmnk) triple lacunary strong Auvw-
convergent of orer Ξ± to zero then x = (xmnk) triple lacunary strong Auvw-convergent of order
Ξ± to zero with respect to mnk-sequence of Musielak Orlicz function, (i.e)
[ππ΄ππ
πΌ3ππ, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))β
π]π
β [ππ΄πππ
πΌ3ππ
, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
.
Proof. Let F = (fmnkijq) be a mnk-sequence of Musielak Orlicz function and put supfmnk
ijq (1) =T. Let
π₯ = (π₯πππ) β [ππ΄πππΌ
2ππ, β(π(π₯11, 0), π(π₯12, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))β
π]π
and π > 0. We choose 0 < πΏ < 1 such that πππππππ (π’) < π for every π’ with 0 β€ π’ β€
πΏ (π, π, π β β). We can write
[ππ΄πππ
πΌ3ππ
, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
= [ππ΄πππ
πΌ3ππ
, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
+
[ππ΄πππ
πΌ3ππ
, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
where the first part is over β€ πΏ and second part is over > πΏ. By definition of Musielak Orlicz
function of πππππππ
for every πππ, we have
[ππ΄πππ
πΌ3ππ
, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
β€ ππ»2 + (3ππΏβ1)π»2 β
[ππ΄πππ
πΌ3ππ
, β(π(π₯111, 0), π(π₯122, 0), β¦ ,
π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
.
Therefore
π₯ = (π₯πππ) β [ππ΄ππππΌ
3ππ, β(π(π₯111, 0), π(π₯122, 0), β¦ ,
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π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
.
Theorem 3.2 Let ΞΌ be a three valued measure and A be a mnk-sequence of the four dimensional
infinite matrices Auvw = (ak1β¦krβ1β¦βsm1β¦mrn1β¦ns(uvw)) of complex numbers, q = (qijq) be a mnk-
sequence of positive real numbers with 0 < infqijq = H1 β€ supqijq = H2 > β and F = (fmnkijq)
be a mnk-sequence of Musielak Orlicz function. If limmu,v,wββinfijqfijq(uvw)
uvw> 0, then
[ππ΄πππ
πΌ3ππ
, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
= [ππ΄ππ
πΌ3ππ, β(π(π₯111, 0), π(π₯122, 0),β― , π(π₯π1,π2,β―ππβ1π1,π2,β―ππ β1π1,π2,β―,ππ‘β1 , 0))β
π]π
.
Proof. If limπ’,π£,π€ββinfππππππππ(π’π£π€)
π’π£π€> 0, then there exists a number π½ > 0 such that
ππππ(π’π£π€) β₯ π½π’ for all π’ β₯ 0 and π, π, π β β. Let
π₯ = (π₯π1 β¦πππ1β¦ππ π1, π2, β¦ , ππ‘) β [ππ΄ππππΌ
3ππ, β(π(π₯111, 0), π(π₯122, 0), β¦ ,
π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
.
Clearly
[ππ΄πππ
πΌ3ππ
, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
β₯ π½ [ππ΄ππ
πΌ3ππ, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))β
π]π
.
Therefore
π₯ = (π₯π1 β¦πππ1β¦ππ π1, π2, β¦ , ππ‘) β [ππ΄πππΌ
3ππ, β(π(π₯111, 0), π(π₯122, 0), β¦ ,
π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
.
By using Theorem 3.1, the proof is complete.
We now give an example to show that
[ππ΄πππ
πΌ3ππ
, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
β [ππ΄ππ
πΌ3ππ, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))β
π]π
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in the case when π½ = 0. Consider π΄ = πΌ, unit matrix,
π(π₯) = ((π1β―ππ + π1β―ππ + π1, π2, β¦ , ππ‘)!
|π₯π1β¦πππ1β¦ππ π1,π2,β¦,ππ‘|)1/π1β―ππ+π1β―ππ +π1,π2,β¦,ππ‘
, ππππ = 1
for every π, π, π β β and
πππππππ (π₯) =
|π₯π1β¦πππ1β¦ππ π1,π2,β¦,ππ‘|1/((π1β¦ππ+π1β¦ππ +π1,π2,β¦,ππ‘)(π+1)(π+1)(π+1))
((π1β¦ππ + π1β¦ππ + π1, π2, β¦ , ππ‘)!)1/π1β¦ππ+π1β¦ππ +π1,π2,β¦,ππ‘
(π, π, π β₯ 1, π₯ > 0)
in the case π½ > 0. Now we define π₯πππ = βππ π‘πΌ if π, π, π = ππππ ππ‘ for some π, π , π‘ β₯ 1 and π₯πππ =
0 otherwise. Then we have,
[ππ΄πππ
πΌ3ππ
, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
β 1
as π, π , π‘ β β
and so
π₯ = (π₯π1β¦πππ1β¦ππ π1,π2,β¦,ππ‘) β [ππ΄πππΌ
3ππ, β(π(π₯111, 0), π(π₯122, 0), β¦ ,
π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
.
In this section we introduce natural relationship between π be a three valued measure of
triple lacunary π΄π’π£π€- statistical convergence of order πΌ and π be a three valued measure of
triple lacunary strong π΄π’π£π€-convergence of order πΌ with respect to πππ-sequence of Musielak
Orlicz function.
Definition 3.1 Let ΞΌ be a three valued measure and ΞΈ be a triple lacunary mnk-sequence. Then
a mnk-sequence x = (xm1β¦mrn1β¦nsk1,k2,β¦,kt) is said to be ΞΌ-lacunary statistically convergent of
order Ξ± to a number zero if for every Ο΅ > 0, ΞΌ(limmrstββhrstβΞ±|KΞΈ(Ο΅)|) = 0, where |KΞΈ(Ο΅)|
denotes the number of elements in
πΎπ(π) = π {(π, π, π) β πΌππ π‘: ((π1β―ππ + π1β―ππ + π1, π2, β¦ , ππ‘)! β
|π₯π1β¦πππ1β¦ππ π1,π2,β¦,ππ‘|)1/π1β―ππ+π1β―ππ +π1,π2,β―,ππ‘
β₯ π = 0}.
The set of all triple lacunary statistical convergent of order πΌ of πππ β sequences is
denoted by (πππΌ)π.
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Let π be a three valued measure and π΄π’π£π€ = (ππ1β―ππβ1β―βπ π1β―πππ1β―ππ π1,π2,β¦,ππ‘(π’π£π€)) be an four
dimensional infinite matrix of complex numbers. Then a πππ-sequence π₯ =
(π₯π1β―πππ1β―ππ π1,π2,β¦,ππ‘) is said to be π-triple lacunary π΄-statistically convergent of order πΌ to
a number zero if for every π > 0, π(limππ π‘βββππ π‘βπΌ|πΎπ΄π(π)|) = 0, where |πΎπ΄π(π)| denotes the
number of elements in
πΎπ΄π(π) = π {(π, π, π) β πΌππ π‘: ((π1β―ππ + π1β―ππ + π1, π2, β¦ , ππ‘)! β
|π₯π1β―πππ1β―ππ π1,π2,β¦,ππ‘|)1/π1β―ππ+π1β―ππ +π1,π2,β¦,ππ‘
β₯ π = 0}.
The set of all triple lacunary π΄-statistical convergent of order πΌ of πππ-sequences is
denoted by (πππΌ(π΄))
π.
Definition 3.2 Let ΞΌ be a three valued measure and A be a mnk-sequence of the four
dimensional infinite matrices Auvw = (ak1β―krβ1β―βsm1β―mrn1β―nsk1,k2,β¦,kt(uvw)) of complex numbers and
let q = (qijq) be a mnk-sequence of positive real numbers with 0 < infqijq = H1 β€ supqijq =
H2 < β. Then a mnk-sequence x = (xm1β―mrn1β―nsk1,k2,β¦,kt) is said to be ΞΌ-lacunary Auvw-
statistically convergent of order Ξ± to a number zero if for every Ο΅ > 0,
ΞΌ(limrstββhrstβΞ±|KAΞΈΞ·(Ο΅)|) = 0, where |KAΞΈΞ·(Ο΅)| denotes the number of elements in
πΎπ΄ππ(π) = π {(π, π, π) β πΌππ π‘: ((π1β―ππ + π1β―ππ + π1, π2, β¦ , ππ‘)! β
|π₯π1β―πππ1β―ππ π1,π2,β¦,ππ‘|)1/π1β―ππ+π1β―ππ +π1,π2,β¦,ππ‘
β₯ π = 0}.
The set of all π-lacunary π΄π-statistical convergent of order πΌ of πππ-sequences is
denoted by (πππΌ(π΄, π))
π.
The following theorems give the relations between π-lacunary π΄π’π£π€-statistical
convergence of order πΌ and π-lacunary strong π΄π’π£π€-convergence of order πΌ with respect to a
πππ-sequence of Musielak Orlicz function.
Theorem 3.3 Let ΞΌ be a three valued measure and F = (fijq) be a mnk-sequence of Musielak
Orlicz function. Then
[ππ΄πππ
πΌ3ππ
, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
β [ππ΄ππ
πΌ3π, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β―ππβ1π1,π2,β―ππ β1π1,π2,β¦,ππ‘β1 , 0))β
π]π
if and only if π (limπππββππππ(π’)) > 0, (π’ > 0).
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Proof. Let π > 0 and π₯ = (π₯π1β―πππ1β―ππ π1,π2,β¦,ππ‘) β [ππ΄ππππΌ
3ππ, β(π(π₯111, 0), π(π₯122, 0), β¦ ,
π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
. If π (limπππββππππ(π’)) > 0, (π’ > 0), then
there exists a number π > 0 such that ππππ(π) > π for π’ > π and π, π, π β β. Let
[ππ΄πππ
πΌ3ππ
, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
β₯ βππ π‘βπΌππ»1πΎπ΄ππ(π).
It follows that
[ππ΄πππ
πΌ3π
, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
.
Conversely, suppose that π (limπππββππππ(π’)) > 0 does not hold, then there is a number
π‘ > 0 such that π (limπππββπππ(π‘)) = 0. We can select a lacunary ππ-sequence π =
(π1β―πππ1β―ππ π1, π2, β¦ , ππ‘) such that ππππ(π‘) < 3βππ π‘ for any π > π1β―ππ , π >
π1β―ππ , π > π1, π2, β¦ , ππ‘. Let π΄ = πΌ, unit matrix, define the πππ-sequence π₯ by putting π₯πππ =
π‘ if
π1, π2, β―ππβ1π1, π2, β―ππ β1π1, π2, β¦ , ππ‘β1 < π, π, π <
π1, π2, β¦πππ1, π2, β¦ ππ π1, π2, β¦ , ππ‘ +π1,π2, β¦ππβ1π1, π2, β¦ ππ β1π1, π2, β¦ , ππ‘β12
and π₯πππ = 0 if
π1, π2, β¦πππ1, π2, β¦ ππ π1, π2, β¦ , ππ‘ +π1,π2, β¦ππβ1π1, π2, β¦ ππ β1π1, π2, β¦ , ππ‘β1
2
β€ π, π, π β€ π1,π2, β―πππ1, π2, β―ππ π1, π2, β¦ , ππ‘.
We have
π₯ = (π₯π1β―πππ1β―ππ π1,π2,β¦,ππ‘) β [ππ΄ππππΌ
3ππ, β(π(π₯111, 0), π(π₯122, 0), β¦ ,
π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
.
but π₯ β [ππ΄ππ
πΌ3π, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))β
π]π
.
Theorem 3.4 Let ΞΌ be a three valued measure and F = (fijq) be a mnk-sequence of Musielak
Orlicz function. Then
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[ππ΄πππ
πΌ3ππ
, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
β [ππ΄ππ
πΌ3π, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))β
π]π
if and only if π (supπ’supπππππππ(π’)) < β.
Proof. Let
π₯ β [ππ΄ππ
πΌ3π, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))β
π]π
.
Suppose that β(π’) = supπππππππ(π’) and β = supππ’β(π’). Since ππππ(π’) β€ β for all π, π, π
and π’ > 0, we have for all π’, π£, π€
[ππ΄ππ
πΌ3π, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))β
π]π
β€ βπ»2βππ π‘βπΌ|πΎπ΄ππ(π)| + |β(π)|
π»2 .
It follows from π β 0 that
π₯ β [ππ΄πππ
πΌ3ππ
, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
.
Conversely, suppose that π (supπ’supπππππππ(π’)) = β. Then we have
0 < π’111 < β― < π’πβ1π β1π‘β1 < π’ππ π‘ < β―, such that πππππ ππ‘(π’ππ π‘) β₯ βππ π‘πΌ for π, π , π‘ β₯ 1. Let
π΄ = πΌ, unit matrix, define the πππ-sequence π₯ by putting π₯πππ = π’ππ π‘ if π, π, π =
π1π2β―πππ1π2β―ππ for some π, π , π‘ = 1,2, β¦ and π₯πππ = 0 otherwise. Then we have
π₯ β [ππ΄ππ
πΌ3π, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))β
π]π
but
π₯ β [ππ΄πππ
πΌ3ππ
, β(π(π₯111, 0), π(π₯122, 0), β¦ , π(π₯π1,π2,β¦ππβ1π1,π2,β¦ππ β1π1,π2,β¦,ππ‘β1 , 0))βπ]π
.
4. CONCLUSION
In this paper we have studied study some connections between π-lacunary strong ππ΄π’π£π€3 -
convergence with respect to a πππ sequence of Musielak Orlicz function and π-lacunary
ππ΄π’π£π€3 -statistical convergence, where π΄ is a sequence of four dimensional matrices π΄(π’π£π€) =
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(ππ1β¦ππβ1β¦βπ π1β¦πππ1β¦ππ (π’π£π€)) of complex numbers. The results of this of this paper are more general
than earlier results.
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