Vol. 5, No. 1, 2013, pp. [1–10] www.cscanada.net DOI: 10.3968/j.pam.1925252820130501.3333 www.cscanada.org
Strongly Lacunary Summable Generalized
Difference Double Sequence Spaces in n−Normed
Spaces Defined by Ideal Convergence and an
Orlicz Function
Ayhan Esi
[a],*[a]Department of Mathematics, Science and Art Faculty, Adiyaman University, Turkey.
* Corresponding author.
Address: Department of Mathematics, Science and Art Faculty, Adiyaman Uni-versity, Adiyaman 02040, Turkey; E-Mail: [email protected]
Received: September 1, 2012/ Accepted: November 6, 2012/ Published: January 31, 2013
Abstract:
The notion of n-normed space was studied at the initial stage by Gahler (Gahler, 1965), Gunawan (Gunawan, 2001) and many others. In this paper, we introduce some certain new generalized difference double sequence spaces via ideal convergence, double lacunary sequence and an Orlicz function in n-normed spaces and examine some properties of the resulting these spaces.Key words:
P-convergent; Double lacunary sequence; n-normed space; Orlicz functionEsi, A. (2013). Strongly Lacunary Summable Generalized Difference Double Sequence S-paces in n−Normed SS-paces Defined by Ideal Convergence and an Orlicz Function. Progress in Applied Mathematics, 5 (1), 1–10. Available from http://www.cscanada.net/index.php/ pam/article/view/j.pam.1925252820130501.3333 DOI: 10.3968/j.pam.1925252820130501. 3333
1. INTRODUCTION
Let X be a non-empty set, then a family of sets I ⊂ 2X (the class of all subsets of X) is called an ideal if and only if for each A, B ∈ I, we have A ∪ B ∈ I and for each A ∈ I and each B ⊂ A, we have B ∈ I. A non-empty family of sets F ⊂ 2X is a filter on X if and only if ∅ /∈ F , for each A, B ∈ F , we have A ∩ B ∈ F and
n−Normed Spaces Defined by Ideal Convergence and an Orlicz Function each A ∈ F and each A ⊂ B, we have B ∈ F . An ideal I is called non-trivial ideal if I 6= ∅ and X /∈ I. Clearly I ⊂ 2X is a non-trivial ideal if and only if F = F (I) = {X/A : A ∈ I} is a filter on X. A non-trivial ideal I ⊂ 2X is called admissible if and only if {{x} .x ∈ X} ⊂ I. Further details on ideals of 2X can be found in Kostyrko, et al. [1]. The notion was further investigated by Salat, et al. [2] and others.
By the convergence of a double sequence we mean the convergence on the Pring-sheim sense that is, a double sequence x = (xk,l) has Pringsheim limit L (denoted by P − lim x = L) provided that given ε > 0 there exists n ∈ N such that |xk,l− L| < ε whenever k, l > n [3]. We shall write more briefly as “P −convergent ”.
The double sequence x = (xk,l) is bounded if there exists a positive number M such that |xk,l| < M for all k and l. Let l2∞ the space of all bounded double such that
kxk,lk(∞,2)= sup k,l
|xk,l| < ∞.
The double sequence θr,s = {(kr, ls)} is called double lacunary sequence [4] if there exist two increasing of integers such that
ko= 0, hr= kr− kr−1→ ∞ as r → ∞ and lo= 0, − hs= ls− ls−1→ ∞ as s → ∞. Notations: kr,s= krls, hr,s= hr − hs, θr,s is determined by Ir,s= {(k, l) : kr−1< k ≤ krand ls−1< l ≤ ls} , qr= kr kr−1 ,−qs= ls ls−1 and qr,s= qr − qs.
Recall in [5] that an Orlicz function M is continuous, convex, nondecreasing function define for x > 0 such that M (0) = 0 and M (x) > 0. If convexity of Orlicz function is replaced by M (x + y) ≤ M (x) + M (y) then this function is called the modulus function and characterized by Ruckle [6]. An Orlicz function M is said to satisfy ∆2−condition for all values u, if there exists K > 0 such that M (2u) ≤ KM (u), u ≥ 0.
Lemma 1. Let M be an Orlicz function which satisfies ∆2−condition and let 0 < δ < 1. Then for each t ≥ δ, we have M (t) < Ktδ−1M (2) for some constant K > 0. A double sequence space X is said to be solid or normal if (αk,lxk,l) ∈ X, and for all double sequences α = (αk,l) of scalars with |αk,l| ≤ 1 for all k, l ∈ N.
Let n ∈ N and X be a real vector space of dimension d, where n ≤ d. A real-valued function k., ..., .k on X satisfying the following four conditions:
(i) kx1, x2, ..., xnk = 0 if and only if x1, x2, ..., xn are linearly dependent, (ii) kx1, x2, ..., xnk is invariant under permutation,
(iii) kαx1, x2, ..., xnk = |α| kx1, x2, ..., xnk , α ∈ R,
(iv) kx1+ xı1, x2, ..., xnk ≤ kx1, x2, ..., xnk + kxı1, x2, ..., xnk is called an n−norm on X, and the pair (X, k., ..., .k) is called an n−normed space [7,8]. Normed space was studied by Mursaleen and Mohiuddine [9,10], Mohiuddine and Lohani [11], Mohiuddine and Alghamdi [12] and many others from different aspects.
A trivial example of n−normed space is X = R equipped with the following Euclidean n−norm: kx1, x2, ..., xnkE= abs x11...x1n ... xn1...xnn where xi= (xi1, ..., xin) ∈ Rn for each i = 1, 2, . . . , n.
2. MAIN RESULTS
Let I2 be an ideal of 2N×N, θr,s be a double lacunary sequence, M be an Orlicz function, p = (pk,l) be a bounded double sequence of strictly positive real numbers and (X, k., ..., .k) be an n−normed space. Further w (n − X) denotes X−valued sequence space. Now, we define the following double generalized difference sequence spaces: wI2 θr,s[M, ∆ m, p, k., ..., .k] o= n x = (xk,l) ∈ w (n − X) : ∀ε > 0, (r, s) ∈ Ir,s: 1 hr,s X (k,l)∈Ir,s " M ∆mxk,l ρ , z1, z2, ..., zn−1 !#pk,l ≥ ε ∈ I2
for some ρ > 0 and for every z1, z2, ..., zn−1∈ X o , wI2 θr,s[M, ∆ m, p, k., ..., .k] =nx = (x k,l) ∈ w (n − X) : ∀ε > 0, (r, s) ∈ Ir,s: 1 hr,s X (k,l)∈Ir,s " M ∆mx k,l− L ρ , z1, z2, ..., zn−1 !#pk,l ≥ ε ∈ I2
for some ρ > 0, L ∈ X and for every z1, z2, ..., zn−1∈ X o , wI2 θr,s[M, ∆ m, p, k., ..., .k] ∞= n x = (xk,l) ∈ w (n − X) : ∃K > 0, (r, s) ∈ Ir,s: 1 hr,s X (k,l)∈Ir,s " M ∆mx k,l ρ , z1, z2, ..., zn−1 !#pk,l ≥ K ∈ I2
for some ρ > 0 and for every z1, z2, ..., zn−1∈ X o , and wθr,s[M, ∆m, p, k., ..., .k]∞= n x = (xk,l) ∈ w (n − X) : ∃K > 0, 1 hr,s X (k,l)∈Ir,s " M ∆mx k,l ρ , z1, z2, ..., zn−1 !#pk,l ≤ K
for some ρ > 0 and for every z1, z2, ..., zn−1∈ X o
n−Normed Spaces Defined by Ideal Convergence and an Orlicz Function
where ∆mx = (∆mx
k,l) = ∆m−1xk,l− ∆m−1xk,l+1− ∆m−1xk+1,l+ ∆m−1xk+1,l+1, ∆1x
k,l = (∆xk,l) = (xk,l− xk,l+1− xk+1,l+ xk+1,l+1), ∆0x = (xk,l) and also this generalized difference double notion has the following binomial representation:
∆mxk,l= m X i=0 m X j=0 (−1)i+jm i m j xk+i,l+j. If m = 0 and θr,s= {(2r, 2s)} , we obtain wI2 θr,s[M, ∆ m, p, k., ..., .k] o= w I2[M, p, k., ..., .k] o, wθr,sI2 [M, ∆m, p, k., ..., .k] = wI2[M, p, k., ..., .k] , wθI2 r,s[M, ∆ m, p, k., ..., .k] ∞= w I2[M, p, k., ..., .k] ∞ and wθr,s[M, ∆m, p, k., ..., .k]∞= w [M, p, k., ..., .k]∞ which were defined and studied by Esi [13].
The following well-known inequality will be used in this study:
If 0 ≤ infk,lpk,l= Ho≤ pk,l≤ supk,l= H < ∞, D = max 1, 2H−1, then |xk,l+ yk,l| pk,l≤ D {|x k,l| pk,l+ |y k,l| pk,l} for all k, l ∈ N and xk,l, yk,l∈ C. Also |xk,l|
pk,l≤ max1, |x k,l|
H
for all xk,l∈ C. Theorem 1. The sets wI2
θr,s[M, ∆ m, p, k, ., k] o, w I2 θr,s[M, ∆ m, p, k, ., k] and wI2 θr,s[M, ∆ m, p, k, ., k]
∞ are linear spaces over the complex field C. Proof. We will prove only for wI2
θr,s[M, ∆
m, p, k, ., k]
oand the others can be proved similarly. Let x, y ∈ wI2θr,s[M, ∆m, p, k, ., k] o and α, β ∈ C. Then (r, s) ∈ Ir,s: 1 hr,s X (k,l)∈Ir,s M ∆mx k,l ρ1 , z1, z2, ..., zn−1 pk,l ≥ε 2 ∈ I2, for some ρ1> 0 and (r, s) ∈ Ir,s: 1 hr,s X (k,l)∈Ir,s M ∆my k,l ρ2 , z1, z2, ..., zn−1 pk,l ≥ε 2 ∈ I2, for some ρ2> 0.
Since k., ..., .k is a n−norm and M is an Orlicz function, the following inequality holds: 1 hr,s X (k,l)∈Ir,s M ∆m(αx k,l+ βyk,l) |α| ρ1+ |β| ρ2 , z1, z2, ..., zn−1 pk,l ≤ D hr,s X (k,l)∈Ir,s |α| |α| ρ1+ |β| ρ2 M ∆mx k,l ρ1 , z1, z2, ..., zn−1 pk,l
+ D hr,s X (k,l)∈Ir,s |β| |α| ρ1+ |β| ρ2 M ∆my k,l ρ2 , z1, z2, ..., zn−1 pk,l ≤ D hr,s X (k,l)∈Ir,s M ∆mxk,l ρ1 , z1, z2, ..., zn−1 pk,l + D hr,s X (k,l)∈Ir,s M ∆my k,l ρ2 , z1, z2, ..., zn−1 pk,l
From the above inequality, we get (r, s) ∈ Ir,s : 1 hr,s X (k,l)∈Ir,s M ∆m(αx k,l+ βyk,l) |α| ρ1+ |β| ρ2 , z1, z2, ..., zn−1 pk,l ≥ ε ⊂ (r, s) ∈ Ir,s: D hr,s X (k,l)∈Ir,s M ∆mxk,l ρ1 , z1, z2, ..., zn−1 pk,l ≥ ε 2 ∪ (r, s) ∈ Ir,s: D hr,s X (k,l)∈Ir,s M ∆my k,l ρ2 , z1, z2, ..., zn−1 pk,l ≥ ε 2 .
Two sets on the right hand side belong to I2 and this completes the proof. It is also easy verify that the space wθr,s[M, ∆m, p, k., ..., .k]∞ is also a linear space.
Theorem 2. For fixed (n, m) ∈ N × N, wθr,s[M, p, k., ..., .k]∞ paranormed space with respect to the paranorm defined by
h(n,m)(x) = m,m X k,l=1,1 kxk,l, z1, z2, ..., zn−1k + infρpn,mH > 0 : sup r,s 1 hr,s X (k,l)∈Ir,s M ∆mxk,l ρ , z1, z2, ..., zn−1 pk,l 1 H ≤ 1, for some ρ > 0 and for every z1, z2, ..., zn−1∈ X.
Proof. h(n,m)(θ) = 0 and h(n,m)(−x) = h(n,m)(x) are easy to prove, so we omit them. Let us take x, y ∈ wθr,s[M, ∆
m, p, k., ..., .k] ∞. Let A (x) = {ρ > 0 : sup r,s 1 hr,s X (k,l)∈Ir,s M ∆mx k,l ρ , z1, z2, ..., zn−1 pk,l ≤ 1, ∀z1, z2, ..., zn−1∈ X} and A (y) = {ρ > 0 : sup r,s 1 hr,s X (k,l)∈Ir,s M ∆myk,l ρ , z1, z2, ..., zn−1 pk,l ≤ 1, ∀z1, z2, ..., zn−1∈ X} .
n−Normed Spaces Defined by Ideal Convergence and an Orlicz Function
Let ρ1∈ A (x) and ρ2∈ A (y). If ρ = ρ1+ ρ2, then we have
sup r,s 1 hr,s X (k,l)∈Ir,s M ∆m(xk,l+ yk,l) ρ , z1, z2, ..., zn−1 ≤ ρ1 ρ1+ ρ2 sup r,s 1 hr,s X (k,l)∈Ir,s M ∆mx k,l ρ1 , z1, z2, ..., zn−1 + ρ2 ρ1+ ρ2 sup r,s 1 hr,s X (k,l)∈Ir,s M ∆my k,l ρ1 , z1, z2, ..., zn−1 . Thus sup r,s 1 hr,s X (k,l)∈Ir,s M ∆m(x k,l+ yk,l) ρ1+ ρ2 , z1, z2, ..., zn−1 pk,l ≤ 1 and h(n,m)(x + y) = m,m X k,l=1,1 kxk,l+ yk,l, z1, z2, ..., zn−1k + infn(ρ1+ ρ2) pn,m H : ρ 1∈ A (x) and ρ2∈ A (y) o ≤ m,m X k,l=1,1 kxk,l, z1, z2, ..., zn−1k + inf n (ρ1) pn,m H : ρ 1∈ A (x) o + m,m X k,l=1,1 kyk,l, z1, z2, ..., zn−1k + inf n (ρ2) pn,m H : ρ 2∈ A (y) o = h(n,m)(x) + h(n,m)(y) . Now, let λu k,l → λ, where λuk,l, λ ∈ C and h(n,m) xu k,l− xk,l → 0 as u → ∞. We have to show that h(n,m)
λu k,lx u k,l− λxk,l → 0 as u → ∞. Let λk,l→ α, where λk,l, λ ∈ C and h(n,m) xuk,l− xk,l → 0 as u → ∞. Let A (xu) ={ρu> 0 : sup r,s 1 hr,s X (k,l)∈Ir,s M ∆mxu k,l ρu , z1, z2, ..., zn−1 pk,l ≤ 1, ∀z1, z2, ..., zn−1∈ X o . and A (xu− x) ={ρıu> 0 : sup r,s 1 hr,s X (k,l)∈Ir,s M ∆mxu k,l− xk,l ρı u , z1, z2, ..., zn−1 pk,l ≤ 1, ∀z1, z2, ..., zn−1∈ X} .
If ρu∈ A (xu) and ρıu∈ A (xu− x) then we observe that M ∆mλu k,lxuk,l− λxk,l ρu λ u k,l− λ + ρ ı u|λ| , z1, z2, ..., zn−1 ≤M ∆mλu k,lxuk,l− λxuk,l ρu λ u k,l− λ +ρ ı u|λ| , z1, z2, ..., zn−1 + ∆mλxu k,l− λxk,l ρu λ u k,l− λ +ρ ı u|λ| , z1, z2, ..., zn−1 ≤ ρu λ u k,l− λ ρu λ u k,l− λ + ρ ı u|λ| M ∆mxu k,l ρu , z1, z2, ..., zn−1 + ρ ı u|λ| ρu λ u k,l− λ + ρ ı u|λ| M ∆mxuk,l− xk,l ρı u , z1, z2, ..., zn−1 .
From this inequality, it follows that M ∆mλu k,lxuk,l− λxk,l ρu λ u k,l− λ + ρ ı u|λ| , z1, z2, ..., zn−1 pk,l ≤ 1 and consequently h(n,m) λuk,lxuk,l− λxk,l = m,m X k,l=1,1 λuk,lxk,l− λxk,l, z1, z2, ..., zn−1 + infn ρu λuk,l− λ + ρıu|λ| pn,mH : ρ u∈ A (xu) and ρıu∈ A (x u− x)o ≤ λuk,l− λ m,m X k,l=1,1 xuk,l, z1, z2, ..., zn−1 + |λ| m,m X k,l=1,1 xuk,l− xk,l, z1, z2, ..., zn−1 + λuk,l− λ pn,mH infn(ρ u) pn,m H : ρ u∈ A (xu) o + (|λ|) pn,m H inf n (ρıu) pn,m H : ρı u∈ A (xu− x) o ≤ maxn λuk,l− λ , λuk,l− λ pn,mH o h(n,m) xuk,l + maxn|λ| , (|λ|)pn,mH o h(n,m) xuk,l− xk,l .
Hence by our assumption the right hand side tends to zero as u → ∞. This completes the proof.
Corollary 1. It can be noted that h = infn,m∈Nh(n,m) also gives a paranorm on the above sequence spaces. However if one consider the sequence space
n−Normed Spaces Defined by Ideal Convergence and an Orlicz Function
which is larger space than the space wI2
θr,s[M, ∆
m, p, k., . . . , .k]
∞the construction of the paranorm is not clear and we leave it as an open problem. However it should be noted that for a fixed F ∈ I2, the space
wFθr,s[M, ∆ m, p, k., ..., .k] ∞ =nx = (xk,l) ∈ w(n − X) : ∃K > 0, n (n, m) ∈ N × N : sup (r,s)∈N×N/F 1 hr,s X (k,l)∈Ir,s M ∆mx k,l ρ , z1, z2, ..., zn−1 pk,l ≥ K ∈ I2,
for some ρ > 0 and for every z1, z2, ..., zn−1∈ X o which is subspace of the space wθr,sI2 [M, ∆m, p, k., ..., .k]
∞is a paranormed space with the paranorms h(n,m) for (n, m) /∈ F and hF = inf(n,m)∈N×N/Fh(n,m).
Theorem 3. Let M, M1 and M2 be Orlicz functions. Then we have (i) wθI2 r,s[M1, ∆ m, p, k., ..., .k] o⊂ w I2 θr,s[M oM1, ∆ m, p, k., ..., .k] oprovided that p = (pk,l) is such that Ho> 0. (ii) wθr,sI2 [M1, ∆m, p, k., ..., .k]o∩ wI2θr,s[M2, ∆m, p, k., ..., .k]o ⊂ wI2 θr,s[M1+ M2, ∆ m, p, k., ..., .k] o.
Proof. (i). For given ε > 0, we first choose εo > 0 such that maxεHo, ε Ho o < ε. Now using the continuity of M , choose 0 < δ < 1 such that 0 < t < δ implies M (t) < εo. Let x ∈ wIθ2r,s[M1, ∆
m, p, k., ..., .k]
o. Now from the definition of the space wI2[M 1, ∆m, p, k., ..., .k]o, for some ρ > 0 A (δ) = (r, s) ∈ Ir,s : 1 hr,s X (k,l)∈Ir,s M1 ∆mxk,l ρ , z1, z2, ..., zn−1 pk,l ≥ δH ∈ I2. Thus if (n, m) /∈ A (δ) then 1 hr,s X (k,l)∈Ir,s M1 ∆mx k,l ρ , z1, z2, ..., zn−1 pk,l < δH ⇒ X (k,l)∈Ir,s M1 ∆mxk,l ρ , z1, z2, ..., zn−1 pk,l < hr,sδH, ⇒ M1 ∆mx k,l ρ , z1, z2, ..., zn−1 pk,l < δH for all (k, l) ∈ Ir,s, ⇒M1 ∆mx k,l ρ , z1, z2, ..., zn−1 < δ for all (k, l) ∈ Ir,s. Hence from above inequality and using continuity of M , we must have
M M1 ∆mx k,l ρ , z1, z2, ..., zn−1 < εo for all (k, l) ∈ Ir,s which consequently implies that
X (k,l)∈Ir,s " M M1 ∆mx k,l ρ , z1, z2, ..., zn−1 !!#pk,l < hr,smaxεHo, εHoo < hr,sε,
⇒ 1 hr,s X (k,l)∈Ir,s " M M1 ∆mxk,l ρ , z1, z2, ..., zn−1 !!#pk,l < ε.
This shows that {(r, s) ∈ Ir,s : 1 hr,s X (k,l)∈Ir,s M M1 ∆mxk,l ρ , z1, z2, ..., zn−1 pk,l ≥ ε} ⊂ A (δ)
and so belongs to I2. This completes the proof. (ii) Let x ∈ wI2 θr,s[M1, ∆ m, p, k., ..., .k] o∩ w I2 θr,s[M2, ∆ m, p, k., ..., .k] o. Then the fact that 1 hr,s " (M1+ M2) ∆mx k,l ρ , z1, z2, ..., zn−1 !#pk,l ≤ D hr,s " M1 ∆mxk,l ρ , z1, z2, ..., zn−1 !#pk,l + D hr,s " M2 ∆mx k,l ρ , z1, z2, ..., zn−1 !#pk,l
gives us the result.
Theorem 4. (i) If 0 < Ho≤ pk,l< 1, then wI2θ r,s[M, ∆ m, p, k.,...,.k] o⊂ w I2 θr,s[M, ∆ m, k.,...,.k] o . (ii) If 1 ≤ pk,l≤ H < ∞, then wI2 θr,s[M, ∆ m, k., ..., .k] o⊂ w I2 θr,s[M, ∆ m, p, k., ..., .k] o . (iii) If 0 < pk,l< qk,l < ∞ and qk,l pk,l is bounded, then wθr,sI2 [M, ∆m, p, k., ..., .k]o⊂ wI2 θr,s[M, ∆ m, q, k., ..., .k] o .
Proof. The proof is standard, so we omit it.
Theorem 5. The sequence spaces wθr,sI2 [M, ∆m, p, k., ..., .k] o, w I2 θr,s[M, ∆ m, p, k., ..., .k], wI2 θr,s[M, ∆ m, p, k., ..., .k] ∞ and wθr,s[M, ∆ m, p, k., ..., .k] ∞ are solid. Proof. We give the proof for only wI2
θr,s[M, ∆
m, p, k., ..., .k]
o. The others can be proved similarly. Let x ∈ wI2
θr,s[M1, ∆
m, p, k., ..., .k]
o and α = (αk,l) be a double sequence of scalars such that |αk,l| ≤ 1 for all k, l ∈ N. Then we have
(r, s) ∈ Ir,s: 1 hr,s X (k,l)∈Ir,s M ∆m(α k,lxk,l) ρ , z1, z2, ..., zn−1 pk,l ≤ ε
n−Normed Spaces Defined by Ideal Convergence and an Orlicz Function ⊂ (r, s) ∈ Ir,s : T hr,s X (k,l)∈Ir,s M ∆mx k,l ρ , z1, z2, ..., zn−1 pk,l ≤ ε ∈ I2, where T = maxk,l n 1, |αk,l| Ho . Hence αx ∈ wI2θ r,s[M1, ∆ m, p, k., ..., .k]
o for all double sequences α = (αk,l) with |αk,l| ≤ 1 for all k, l ∈ N whenever x ∈ wI2θr,s[M1, ∆m, p, k., ..., .k]o.
REFERENCES
[1] Kostyrko, P., Salat, T., & Wilczynski, W. (2000/2001). I-convergence. Real Analysis Exchange, 26 (2), 669-686.
[2] Salat, T., Tripathy, B. C., & Ziman, M. (2005). On I-convergence field. Italian J. Pure and Appl. Math., 17, 45-54.
[3] Pringsheim, A. (1900). Zur theori der zweifach unendlichen zahlenfolgen. Math. Ann., 53, 289-321.
[4] Sava¸s, E., & Patterson, R. F. (2011). Some double lacunary sequence spaces defined by orlicz functions. Southeast Asian Bulletin of Mathematics, 35 (1), 103-110.
[5] Krasnoselski, M. A., & Rutickii, Y. B. (1961). Convex function and orlicz spaces. Nederland: Groningen.
[6] Ruckle, W. H. (1973). FK-spaces in which the sequence of coordinate vectors is bounded. Canad. J. Math., 25, 973-978.
[7] Gunawan, H. (2001). On n-inner product, n-norms and the Cauchy-Schwarz inequality. Scientiae Mathematicae Japonicae Online, 5, 47-54.
[8] Gahler, S. (1965). Lineare 2-normierte. Rume, Math. Nachr., 28 (164), 1-43. [9] Mursaleen, M., & Mohiuddine, S. A. (2009). On lacunary statistical
conver-gence with respect to the intuitionistic fuzzy normed space. J. Comput. Appl. Math., 233, 142-149.
[10] Mursaleen, M., & Mohiuddine, S. A. (2012). On ideal convergence in proba-bilistic normed spaces. Math. Slovaca, 62 (1), 49-62.
[11] Mohiuddine, S. A., & Danish Lohani, Q. M. (2009). On generalized statistical convergence in intuitionistic fuzzy normed space. Chaos, Solitons Fractals, 42, 1731-1737.
[12] Mohiuddine, S. A., & Alghamdi, M. A. (2012). Statistical summability through a lacunary sequence in locally solid Riesz spaces. J. Inequal. Appl., 2012, 225. [13] Esi, A. (2012). Strongly summable double sequence spaces in n-normed space defined by ideal convergence and an orlicz function. Kyungpook Math. J., 52, 137-147.
