Acta Mathematica Vietnamica

, Volume 38, Issue 3, pp 471–485

Lacunary statistical convergence of double sequences and some inclusion results in n-normed spaces

Authors

    • Department of MathematicsRajiv Gandhi University
  • Ekrem Savas
    • Department of MathematicsIstanbul Ticaret University
Article

DOI: 10.1007/s40306-013-0028-x

Cite this article as:
Hazarika, B. & Savas, E. Acta Math Vietnam. (2013) 38: 471. doi:10.1007/s40306-013-0028-x

Abstract

In this paper, we introduce the concept of lacunary statistical convergence of double sequences in n-normed spaces. Some inclusion relations between the sets of statistically convergent and lacunary statistically convergent cases of double sequences are established. In addition, we also define lacunary statistical Cauchy double sequences and prove that this notion is equivalent to the lacunary statistical convergence of double sequences.

Keywords

Statistical convergenceDouble lacunary sequenceP-convergentn-norm

Mathematics Subject Classification

40A0540B5046A1946A45

1 Introduction

The idea of statistical convergence was formerly given under the name “almost convergence” by Zygmund in the first edition of his celebrated monograph published in Warsaw in 1935 (see [43]). The concept was formally introduced by Steinhaus [42] and Fast [9] and later was reintroduced by Schoenberg [41], and also independently by Buck [1]. Although the notion of statistical convergence was introduced a long time ago, the related research has become active for 20 years. This concept has been applied in various areas, such as number theory [8], measure theory [25], trigonometric series [43], summability theory [10], locally convex spaces [23], the study of strong integral summability [6], metrizable topological groups [2], and topological spaces [24]. It should be mentioned that the notion of statistical convergence has been considered, by several people like Fridy [12], S̆alát [31], Gürdal and Pehlivan [20], Reddy [30], etc. In recent years, a generalization of statistical convergence has appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on Stone–C̆ech compactification of the natural numbers (see [6]). Moreover, statistical convergence is closely related to the concept of convergence in probability [5]. Mursaleen and Edely [27] extended the above idea from single to double sequences of scalars and established some relationships between statistical convergence and strongly Cesaro summable double sequences.

The notion of statistical convergence depends on the density of subsets of N. A subset of N is said to have density δ(E) if
$$\delta ( E ) =\lim_{n\rightarrow\infty}\frac{1}{n}\sum _{k=1}^{n}\chi_{E} ( k ) \quad \mbox{exists}. $$
A single sequence x=(xk) is said to be statistically convergent to L (see [12] and [3]) if, for every ε>0,
$$\delta \bigl( \bigl\{ k\in\mathbf{N} : \vert x_{k}-L\vert \geq \varepsilon \bigr\} \bigr) =0. $$
In this case, we write S-limkxk=L and denote the set of all statistical convergent sequences by S.

The concept of 2-normed space was initially introduced by Gähler [16], in the mid of 1960s, while that of n-normed spaces can be found in Misiak [26]. Since then, many other authors have studied this concept and obtained various results; see for instance Dutta [7], Gunawan [17], Gähler [15], Gunawan and Mashadi [18, 19], Lewandowska [21], Savas [34].

By a lacunary sequence θ=(kr), where k0=0, we mean an increasing sequence of non-negative integers with hr:krkr−1→∞ as r→∞. The intervals determined by θ will be denoted by Ir=(kr−1,kr] and the ratio \(\frac {k_{r}}{k_{r-1}}\) will be defined by qr. The space of lacunary strongly convergent sequences \(\mathcal{N}_{\theta}\) was defined by Freedman et al. [11] as follows:
$$\mathcal{N}_\theta= \biggl\{ x = (x_r) : \lim _{r} \frac{1}{h_r} \sum\limits _{k\in I_r} \vert x_k - L \vert=0 \mbox{ for some }L \biggr\}. $$
Let θ be a lacunary sequence and Ir={k:kr−1<kkr}. A set KN has lacunary density δθ(K) if
$$\lim_{r} \frac{1}{h_{r}} \bigl|\{i\in I_r : i\in K \}\bigr|=0. $$
A sequence x is said to be Sθ-convergent to if for each ε>0 we have
$$\lim_{r} \frac{1}{h_r} \bigl|\bigl\{k\in I_r : \vert x_k - l \vert\geq \varepsilon\bigr\}\bigr|=0. $$
In this case, we write Sθ−limx= or xk(Sθ) (for details see [13]).

A lacunary sequence \(\theta' = (k'_{r})\) is said to be a lacunary refinement of the lacunary sequence θ=(kr) if \((k_{r})\subseteq(k'_{r})\).

The notion of lacunary statistical convergence was studied by Çakalli [2], Fridy and Orhan [14], Li [22], Patterson [28], Savas and Patterson [35, 36, 39, 40], Savas [32, 33] and many others.

By the convergence of a double sequence we mean the convergence in Pringsheim’s sense [29]. A double sequence x=(xk,l) has a Pringsheim limit L (denoted by P−limx=L) provided that given an ε>0 there exists an NN such that |xk,lL|<ε whenever k,l>N. We shall describe such an x=(xk,l) more briefly as “P-convergent”. For details on double sequence spaces, see [37, 38].

Let KN×N and K(m,n) denotes the number of (i,j) in K such that im and jn, see [27]. Then the lower natural density of K is defined by \(\underline{\delta}_{2}(K) =\liminf_{{m,n}\rightarrow\infty, \infty} \frac{|K(m,n)|}{mn}\). In the case when the sequence \((\frac{K(m,n)}{ mn} )\) has a limit in Pringsheim’s sense, we say that K has a double natural density and it is defined by \(P-\lim_{{m,n}\rightarrow\infty, \infty} \frac{|K(m,n)|}{mn} = \delta_{2}(K)\).

Example 1.1

Let K={(i2,j2):i,jN}. Then
$$\delta_2(K) =P-\lim_{m,n\rightarrow\infty, \infty}\frac{|K(m,n)|}{mn} \leq P-\lim_{m,n\rightarrow\infty, \infty}\frac{\sqrt{m}\sqrt{n}}{mn}=0, $$
i.e., the set K has double natural density zero, while the set {(i,3j):i,jN} has double natural density \(\frac{1}{3}\).
Mursaleen and Edely [27] defined the statistical analogue for double sequences x=(xk,l) as follows. A real double sequence x=(xk,l) is said to be P-statistically convergent to if for each ε>0,
$$P-\lim_{r,s}\frac{1}{rs} \bigl\{(k,l) : k< r \mbox{~and~} l< s, |x_{k,l}-\ell|\geq\varepsilon \bigr\}=0. $$
In this case we write S2−limk,lxk,l= and denote the set of all statistical convergent double sequences by S2. It is clear that a convergent double sequence is also S2-convergent, but the converse is not true in general. Also, S2-convergent double sequence need not be bounded. For example, if x=(xk,l) is defined by
$$x_{k,l}=\left \{ \begin{array}{l@{\quad}l} kl , & \mbox{if } k, l= m^{2}, m = 1,2,3,\ldots,\\ 1, & \mbox{otherwise}, \end{array} \right . $$
then S2-limk,lxk,l=1. But x is neither convergent nor bounded.
Let θ1 and θ2 be arbitrary lacunary sequences and Ir={k:kr−1<kkr}, Is={l:ls−1<lls}. A set KN×N has a double lacunary density \(\delta_{2}^{\theta}(K)\) (see [4]) if
$$P-\lim_{r,s} \frac{1}{h_{r,s}} \bigl|\bigl\{i\in I_r, j\in I_s : (i,j)\in K \bigr\} \bigr|=0\ \mbox{exists}, $$
where \(h_{r,s}=h_{r}\overline{h}_{s}\), and
$$\begin{aligned} & k_0 = 0,\qquad h_r=k_r - k_{r-1} \rightarrow \infty\quad \mbox{as}\ r\rightarrow\infty;\\ & l_0 = 0,\qquad \overline{h}_s =l_s - l_{s-1}\rightarrow \infty \quad \mbox{as}\ s\rightarrow\infty. \end{aligned}$$

Example 1.2

Let θ1=(2r−1) and θ2=(3s−1) and K={(i2,j2):i,jN}. Then \(\delta_{2}(K)=0=\delta _{2}^{\theta}(K)\).

Example 1.3

Let θ1=(2r−1) and θ2=(3s−1) and K={(i,3j):i,jN}. Then \(\delta_{2}(K)=\frac{1}{3}\), but \(\delta_{2}^{\theta}(K)=0\).

A double sequence x=(xk,l) is said to be lacunary statistical convergent to a number if for every ε>0, the set {(k,l):|xk,l|≥ε} has double lacunary density zero. In this case, we write \(S_{2}^{\theta}-\lim x=\ell\) or \(x_{k,l}\rightarrow \ell(S^{\theta}_{2})\). Let \(S_{2}^{\theta}\) be the space of all \(S_{2}^{\theta}\)-convergent double sequences. It is clear that a convergent double sequence is \(S_{2}^{\theta}\)-convergent, but the converse is not true in general. Also, a \(S_{2}^{\theta}\)-convergent double sequence need not be bounded.

Example 1.4

Let θ1=(2r−1) and θ2=(3s−1) and a double sequence x=(xk,l) defined by
$$x_{k,l}=\left \{ \begin{array}{l@{\quad}l} kl , & \mbox{if } k,l= m^{2}, m = 1,2,3,\ldots;\\ 1, & \mbox{otherwise.} \end{array} \right . $$
Then \(S_{2}^{\theta}-\lim x=S_{2}-\lim x =1\).

Lemma 1.1

([4, Theorem 2.5])

For any setKN×N, \(\delta_{2}^{\theta }(K)\leq\delta_{2}(K)\)if and only ifP−liminfr,sqr,s>1, where\(q_{r,s}= \frac{k_{r}}{k_{r-1}}\frac{l_{s}}{l_{s-1}}\).

Lemma 1.2

([4, Theorem 2.4])

For any setKN×N, \(\delta_{2}(K)\leq \delta_{2}^{\theta}(K)\)if and only ifP−limsupr,sqr,s<∞.

2 Preliminaries

Let n be a non-negative integer and X be a real vector space of dimension dn (d may be infinite). A real-valued function ∥.,…,.∥ from Xn into R+ satisfying the conditions
  1. (1)

    x1,x2,…,xn∥=0 if and only if x1,x2,…,xn are linearly dependent,

     
  2. (2)

    x1,x2,…,xn∥ is invariant under permutation,

     
  3. (3)

    αx1,x2,…,xn∥=|α|∥x1,x2,…,xn∥, for any αR,

     
  4. (4)

    \(\|x+\overline{x},x_{2},\ldots,x_{n}\| \leq\|x,x_{2},\ldots,x_{n}\| + \|\overline{x},x_{2},\ldots,x_{n}\|\)

     
is called an n-norm on X and the pair (X,∥.,…,.∥) is called an n-normed space.
A trivial example of an n-normed space is X=Rn, equipped with the Euclidean n-norm ∥x1,x2,…,xnE= the volume of the n-dimensional parallelepiped spanned by the vectors x1,x2,…,xn, which may be given explicitly by the formula
$$\|x_1,x_2,\ldots,x_n\|_{E}=\bigl| \det(x_{ij})\bigr|= abs \bigl( \det\bigl(\langle x_i, x_j\rangle\bigr) \bigr), $$
where xi=(xi1,xi2,…,xin)∈Rn for each i=1,2,3,…,n.
Let (X,∥.,…,.∥) be an n-normed space of dimension dn≥2 and {a1,a2,…,an} be a linearly independent set in X. Then the function ∥.,…,.∥ from Xn−1 into R+ defined by
$$\|x_1,x_2,\ldots,x_n\|_{\infty} = \max_{1\leq i\leq n}\bigl\{\|x_1, x_2, \ldots,x_{n-1}, a_i\|\bigr\} $$
defines an (n−1)-norm on X with respect to {a1,a2,…,an} and this is known as the derived (n−1)-norm (for details see [18]).
The standard n-norm on X, a real inner product space of dimension dn, is
$$\|x_1, x_2,\ldots,x_n\|_S = \bigl[\det\bigl(\langle x_i, x_j\rangle\bigr)\bigr]^{\frac{1}{2}}, $$
where 〈 ,〉 denotes the inner product on X. For X=Rn, this n-norm is exactly the same as the Euclidean n-norm ∥x1,x2,…,xnE mentioned earlier. For n=1, this n-norm is the usual norm \(\|x_{1}\| = \sqrt{\langle x_{1},x_{1}\rangle }\) (for more details see [18]).

A double sequence (xk,l) in an n-normed space (X,∥.,…,.∥) is said to be statistical-P-convergent to X with respect to the n-norm if for each ε>0 the set {(k,l)∈N×N:∥xk,l,z1,z2,…,zn−1∥≥ε} has double natural density zero, for every z1,z2,…,zn−1X.

In other words, a double sequence (xk,l) statistical-P-converges to in an n-normed space X if
$$P-\lim_{r,s\rightarrow\infty} \frac{1}{rs}\bigl|\bigl\{k\leq r, ~l\leq s : \|x_{k,l} - \ell, z_1, z_2, \ldots, z_{n-1}\|\geq\varepsilon\bigr\}\bigr|=0, $$
for each z1,z2,…,zn−1X. Let S2(X) denote the set of all statistical-P-convergent double sequences in an n-normed space X.

A double sequence (xk,l) in an n-normed space (X,∥.,…,.∥) is said to be statistically P-Cauchy with respect to the n-norm if for each ε>0 there exist positive integers p,q such that the set {(k,l)∈N×N:∥xk,lxp,q,z1,z2,…,zn−1∥≥ε} has double natural density zero, for every z1,z2,…,zn−1X.

In this paper we study lacunary statistical convergence of double sequences in n-normed spaces. We show that some properties of lacunary statistical convergence of double sequences of real numbers also hold for sequences in n-normed spaces. We also define the notion of statistical Cauchy double sequences in n-normed spaces.

3 Lacunary statistical convergence of double sequences in n-normed spaces

In this section we define lacunary statistical-P-convergent sequences and lacunary statistical-P-Cauchy sequences in an n-normed linear space X. Also we obtain some basic properties of these two notions and we prove that lacunary statistical-P-Cauchy sequence in n-normed space is equivalent to lacunary statistical-P-convergent sequence.

Definition 3.1

A double sequence (xk,l) in an n-normed space (X,∥.,…,.∥) is said to be lacunary statistical-P-convergent, or \(S_{2}^{\theta}\)-P-convergent to , with respect to the n-norm, denoted by \(S_{2}^{\theta}(X)-\lim_{k,l\rightarrow\infty}x_{k,l} = \ell\), if for each ε>0 we have
$$P-\lim_{ r,s \rightarrow\infty} \frac{1}{h_{r,s}} \bigl|\bigl\{(k,l)\in I_{r,s} : \|x_{k,l} - \ell, z_1, z_2, \ldots, z_{n-1}\| \geq\varepsilon\bigr\}\bigr| = 0, $$
where Ir,s={(k,l):kr−1<kkr and ls−1<lls}. Let \(S_{2}^{\theta}(X)\) denote the set of all lacunary statistical-P-convergent double sequences in X.

Definition 3.2

A double sequence (xk,l) in an n-normed space (X,∥.,…,.∥) is said to be lacunary statistical-P-Cauchy or \(S_{2}^{\theta}\)-P-Cauchy with respect to the n-norm if there is a subsequence \((x_{\overline{k}_{r},{\overline{l}_{s}}})\) of (xk,l) such that \((\overline{k}_{r},{\overline{l}_{s}})\in I_{r,s}\) for each r,s,
$$P-\lim_{r,s\rightarrow\infty, \infty} \|x_{\overline{k}_{r},{\overline {l}_{s}}} - \ell, z_1, z_2,\ldots,z_{n-1}\| =0 $$
and, for each ε>0,
$$P-\lim_{r,s\rightarrow\infty} \frac{1}{h_{r,s}} \bigl|\bigl\{(k,l)\in I_{r,s} : \|x_{k,l} - x_{\overline{k}_{r},{\overline {l}_{s}}}, z_1, z_2,\ldots, z_{n-1}\| \geq\varepsilon\bigr\}\bigr| = 0. $$

Theorem 3.1

LetXbe ann-normed space. If (xk,l) is a sequence in X such that\(S^{\theta}_{2}(X)-\lim x_{k,l} = \ell\)exists, then the limit is unique.

Proof

Suppose that there exist elements 1,2 (12) in X such that
$$S_2^{\theta}(X)- \lim_{k,l\rightarrow\infty}x_{k,l} = \ell _1;\qquad S_2^{\theta}(X)- \lim _{k,l\rightarrow\infty}x_{k,l} = \ell_2. $$
Since 12≠0, there exist z1,z2,…,zn−1X such that 12 and z1,z2,…,zn−1 are linearly independent. Then
$$\varepsilon:=\dfrac{1}{2}\|\ell_1 - \ell_2, z_1, z_2,\ldots, z_{n-1}\| $$
is positive. Note that
https://static-content.springer.com/image/art%3A10.1007%2Fs40306-013-0028-x/MediaObjects/40306_2013_28_Equu_HTML.gif
So
$$\begin{aligned} & \bigl\{(k,l)\in I_{r,s}: \|x_{k,l} -\ell_2, z_1, z_2, \ldots, z_{n-1}\|\geq \varepsilon \bigr\} \\ &\quad \subseteq\bigl\{(k,l)\in I_{r,s}: \|x_{k,l} - \ell_1, z_1, z_2,\ldots, z_{n-1}\| \geq\varepsilon\bigr\},\\ & \lim_{r,s \rightarrow\infty}\frac{1}{h_{r,s}}\bigl|\bigl\{(k,l)\in I_{r,s}: \|x_{k,l} -\ell_2, z_1, z_2, \ldots, z_{n-1}\|\geq\varepsilon \bigr\}\bigr|\\ &\quad \leq\lim_{r,s \rightarrow\infty}\frac{1}{h_{r,s}}\bigl|\bigl\{(k,l)\in I_{r,s}: \|x_{k,l} -\ell_1, z_1, z_2, \ldots, z_{n-1}\|\geq\varepsilon \bigr\}\bigr|. \end{aligned}$$
Since \(S_{2}^{\theta}(X)- \lim_{k,l\rightarrow\infty}x_{k,l} = \ell_{1}\), it follows that
$$P-\lim_{r,s \rightarrow\infty}\frac{1}{h_{r,s}}\bigl|\bigl\{(k,l)\in I_{r,s}: \|x_{k,l} -\ell_2, z_1, z_2, \ldots, z_{n-1}\|\geq\varepsilon \bigr\}\bigr|\leq0, $$
and consequently,
$$P-\lim_{r,s \rightarrow\infty}\frac{1}{h_{r,s}}\bigl|\bigl\{(k,l)\in I_{r,s}: \|x_{k,l} -\ell_2, z_1, z_2, \ldots, z_{n-1}\|\geq\varepsilon \bigr\}\bigr|=0, $$
as it cannot be negative. This contradicts the fact that \(S_{2}^{\theta}(X)- \lim_{k,l\rightarrow\infty}x_{k,l} = \ell_{2}\) and the proof is complete. □

The next theorem gives an algebraic characterization of lacunary statistical convergence on n-normed spaces.

Theorem 3.2

LetXbe ann-normed space andx=(xk,l) andy=(yk,l) be two sequences inX.
  1. (a)

    If\(S_{2}^{\theta}(X)- \lim_{k,l\rightarrow\infty }x_{k,l} = \ell\)andc (≠0)∈R, then\(S_{2}^{\theta}(X)- \lim_{k,l\rightarrow\infty,\infty} cx_{k,l} = c\ell\).

     
  2. (b)

    If\(S_{2}^{\theta}(X)- \lim_{k,l\rightarrow\infty }x_{k,l} = \ell_{1} \)and\(S_{2}^{\theta}(X)-\lim_{k,l\rightarrow\infty }y_{k,l} = \ell_{2}\), then\(S_{2}^{\theta}(X)- \lim_{k,l\rightarrow\infty, \infty}(x_{k,l}+ y_{k,l}) = \ell_{1} +\ell_{2}\).

     

The proof of the theorem is straightforward, thus omitted.

Theorem 3.3

\(S_{2}^{\theta}(X)\cap\ell^{2}_{\infty}(X)\)is a closed subset of\(\ell^{2}_{\infty}(X)\), where\(\ell^{2}_{\infty}(X)\)denotes the space of all bounded double sequences of real numbers in ann-normed spaceX.

Proof

Suppose that (xm)mN is a convergent sequence in \(S_{2}^{\theta}(X)\cap\ell^{2}_{\infty}(X)\) and it converges to \(x\in\ell^{2}_{\infty}(X)\). We need to prove that \(x\in S_{2}^{\theta}(X)\cap\ell^{2}_{\infty}(X)\). Assume that \(x^{m} \rightarrow\ell_{m}(S_{2}^{\theta}(X))\), for all mN. Take a positive decreasing convergent sequence (εm)mN, where \(\varepsilon_{m} = \frac{\varepsilon}{2^{m}}\), for a given ε>0. Clearly (εm)mN converges to 0. Choose a positive integer m such that \(\|x-x^{m}, z_{1}, z_{2},\ldots,z_{n-1}\|_{\infty} < \frac {\varepsilon_{m}}{4}\), for every z1,z2,…,zn−1X. Then we have
$$\frac{1}{h_{r,s}}\biggl|\biggl\{(k,l)\in I_{r,s}: \bigl\|x_{k,l}^{m}- \ell_m, z_1,z_2,\ldots,z_{n-1}\bigr\| \geq\frac{\varepsilon_m}{4}\biggr\}\biggr|< \frac{1}{4} $$
and
$$\frac{1}{h_{r,s}}\biggl|\biggl\{(k,l)\in I_{r,s}: \bigl\|x_{k,l}^{m+1}- \ell_{m+1}, z_1,z_2,\ldots,z_{n-1}\bigr\| \geq\frac{\varepsilon_{m+1}}{4}\biggr\}\biggr|< \frac{1}{4}. $$
Therefore
$$\begin{aligned} & \biggl\{(k,l)\in I_{r,s}: \bigl\|x_{k,l}^{m}- \ell_m, z_1,z_2,\ldots,z_{n-1}\bigr\|< \frac{\varepsilon_m}{4}\biggr\}\\ &\quad {}\cap\biggl\{(k.l)\in I_{r,s}: \bigl\|x_{k,l}^{m+1} - \ell_{m+1}, z_1,z_2,\ldots,z_{n-1}\bigr\|< \frac{\varepsilon_{m+1}}{4}\biggr\} \neq\phi. \end{aligned}$$
Choosing a member k in this intersection, we can write
$$\begin{aligned} & \|\ell_m - \ell_{m+1}, z_1, z_2, \ldots,z_{n-1}\| \\ &\quad \leq\bigl\|\ell_m - x_{k,l}^{m}, z_1,z_2,\ldots,z_{n-1}\bigr\| + \bigl\|x_{k,l}^{m} - x_{k,l}^{m+1}, z_1,z_2, \ldots,z_{n-1}\bigr\| \\ &\qquad{} + \bigl\|x_{k,l}^{m+1} -\ell_{m+1}, z_1,z_2,\ldots,z_{n-1}\bigr\| \\ &\quad \leq \bigl\|x_{k,l}^{m}-\ell_{m}, z_1,z_2,\ldots,z_{n-1}\bigr\| + \bigl\|x_{k,l}^{m+1} -\ell_{m+1}, z_1,z_2,\ldots,z_{n-1}\bigr\| \\ &\qquad {}+\bigl\|x-x^{m}, z_1,z_2, \ldots,z_{n-1}\bigr\|_{\infty} + \bigl\|x -x^{m+1}, z_1,z_2,\ldots,z_{n-1}\bigr\|_{\infty}\\ &\quad \leq \frac{\varepsilon_m}{4} + \frac{\varepsilon _{m+1}}{4} +\frac{\varepsilon_m}{4} + \frac{\varepsilon_{m+1}}{4}\leq \varepsilon_m. \end{aligned}$$
This implies that (m) is a Cauchy sequence in R and there is a real number such that m as m→∞. We need to prove that \(x\rightarrow\ell(S_{2}^{\theta}(X))\). For any ε>0, choose mN such that \(\varepsilon _{m} <\frac{\varepsilon}{4}\),
$$ \bigl\|x-x^{m}, z_1, z_2,\ldots,z_{n-1}\bigr\|_{\infty}< \frac{\varepsilon }{4},\qquad \|\ell_m-\ell, z_1,z_2, \ldots,z_{n-1}\|< \frac{\varepsilon}{4}. $$
Then
$$\begin{aligned} & \frac{1}{h_{r,s}}\bigl|\bigl\{(k,l)\in I_{r,s}: \|x_{k,l} -\ell, z_1,z_2,\ldots ,z_{n-1}\|\geq\varepsilon\bigr\}\bigr|\\ &\quad \leq\frac{1}{h_{r,s}}\bigl|\bigl\{(k,l)\in I_{r,s}: \bigl\|x_{k,l}^{m} -\ell_m, z_1,z_2,\ldots,z_{n-1}\bigr\|\\ &\qquad {} + \bigl\|x_{k,l} -x_{k,l}^{m}, z_1,z_2,\ldots,z_{n-1}\bigr\|_{\infty} + \bigl\| \ell_m -\ell, z_1,z_2,\ldots,z_{n-1}\bigr\| \geq\varepsilon\bigr\}\bigr|\\ &\quad \leq \frac{1}{h_{r,s}}\biggl|\biggl\{(k,l)\in I_{r,s}: \bigl\|x_{k,l}^{m} -\ell_m, z_1,z_2,\ldots,z_{n-1}\bigr\| + \frac{\varepsilon}{4} + \frac{\varepsilon}{4} \geq\varepsilon\biggr\}\biggr| \\ &\quad \leq\frac{1}{h_{r,s}}\biggl|\biggl\{(k,l)\in I_{r,s}: \bigl\|x_{k,l}^{m} -\ell_m, z_1,z_2,\ldots,z_{n-1}\bigr\| \geq\frac{\varepsilon}{2}\biggr\}\biggr| \rightarrow0 \quad {\rm as}~r,s\rightarrow \infty. \end{aligned}$$
Therefore \(x\rightarrow\ell(S_{2}^{\theta}(X))\). This completes the proof of the theorem. □

Theorem 3.4

LetXbe ann-normed space. For any lacunary sequenceθ, \(S_{2}(X)\subset S_{2}^{\theta}(X)\)if and only ifP−liminfr,sqr,s>1, where\(q_{r,s}= \frac{k_{r}}{k_{r-1}}\frac {l_{s}}{l_{s-1}}\).

Proof

The proof of the theorem follows from Lemma 1.1. □

Theorem 3.5

LetXbe ann-normed space. For any lacunary sequenceθ, ifP−limsupr,sqr,s<∞, then\(S^{\theta}_{2}(X)\subset S_{2}(X)\).

Proof

The proof of the theorem follows from Lemma 1.2. □

Combining Theorem 3.4 and Theorem 3.5 we get the following corollary.

Corollary 3.1

LetXbe ann-normed space. For any lacunary sequenceθ, if 1<P−liminfr,sqr,sP−limsupr,sqr,s<∞, then\(S^{\theta}_{2}(X)=S_{2}(X)\).

Theorem 3.6

LetXbe ann-normed space. Then
  1. (i)

    \(x_{k,l}\rightarrow\ell(\mathcal{N}_{2}^{\theta}(X)) \Rightarrow x_{k,l} \rightarrow\ell(S_{2}^{\theta}(X))\),

     
  2. (ii)

    \(\mathcal{N}_{2}^{\theta}(X)\)is a proper subset of\(S_{2}^{\theta}(X)\),

     
  3. (iii)

    \(x\in\ell^{2}_{\infty}(X)\)and\(x_{k,l} \rightarrow \ell(S_{2}^{\theta}(X))\Rightarrow x_{k,l}\rightarrow\ell(\mathcal {N}_{2}^{\theta}(X))\),

     
  4. (iv)
    \(S_{2}^{\theta}(X)\cap\ell^{2}_{\infty}(X) =\mathcal {N}_{2}^{\theta}(X)\cap\ell^{2}_{\infty}(X)\), where
    $$\begin{aligned} \mathcal{N}_{2}^{\theta}(X) =& \biggl\{ x = (x_{k,l}): P-\lim_{r,s} \dfrac {1}{h_{r,s}} \sum\limits _{(k,l)\in I_{r,s}} \| x_{k,l} - \ell,z_1,z_2,\ldots,z_{n-1}\| =0, \\ &\quad \mbox{for~some~}\ell\in\mathbf{R} \biggr\}. \end{aligned}$$
     

Proof

(i) If ε>0 and \(x_{k,l} \rightarrow\ell(\mathcal {N}_{2}^{\theta}(X))\), we can write
$$\begin{aligned} & \sum_{(k,l)\in I_{r,s}} \|x_{k,l}-\ell, z_1, z_2,\ldots,z_{n-1}\|\\ &\quad \geq\sum_{{(k,l)\in I_{r,s}}, \|x_{k,l}-\ell, z_1, z_2,\ldots ,z_{n-1}\|\geq\varepsilon} \|x_{k,l}-\ell, z_1,z_2,\ldots,z_{n-1}\|\\ &\quad \geq\varepsilon\bigl|\bigl\{(k,l)\in I_{r,s} : \|x_{k,l}-\ell, z_1, z_2, \ldots ,z_{n-1}\|\geq\varepsilon\bigr \}\bigr| \end{aligned}$$
and so
$$\begin{aligned} & \frac{1}{\varepsilon h_{r,s}} \sum_{(k,l)\in I_{r,s}} \|x_{k,l}- \ell, z_1, z_2,\ldots,z_{n-1}\| \\ &\quad \geq\frac{1}{h_{r,s}} \bigl|\bigl\{(k,l)\in I_{r,s} : \|x_{k,l}- \ell, z_1, z_2, \ldots,z_{n-1}\|\geq\varepsilon \bigr\}\bigr|. \end{aligned}$$
This proves the result.
(ii) In order to establish that the inclusion \(\mathcal{N}_{2}^{\theta }(X)\subseteq S_{2}^{\theta}(X)\) is proper, we define a double sequence x=(xk,l) as follows:
https://static-content.springer.com/image/art%3A10.1007%2Fs40306-013-0028-x/MediaObjects/40306_2013_28_Equah_HTML.gif
Then for every ε>0,
$$ P-\lim_{r,s}\frac{1}{h_{r,s}}\bigl \vert \bigl\{ ( k,l ) \in I_{r,s}:\| x_{k,l}-L, z_1, z_2,\ldots,z_{n-1}\| \geq\varepsilon \bigr\} \bigr \vert =P- \lim_{r,s}\frac{ [ \sqrt[3]{h_{r,s}} ] }{h_{r,s}}=0. $$
Therefore \(S_{2}^{\theta}(X)-\lim x=0\), i.e. \(x_{k,l}\rightarrow 0(S_{2}^{\theta}(X))\). But
$$\begin{aligned} & P-\lim_{r,s}\frac{1}{h_{r,s}}\sum _{ ( k,l ) \in I_{r,s}}\| x_{k,l}-0,z_1,z_2, \ldots,z_{n-1}\| \\ &\quad =P-\lim_{r,s}\text{ }\frac{ [ \sqrt [3]{h_{r,s}} ] ( [ \sqrt[3]{h_{r,s}} ] ( [ \sqrt[3]{h_{r,s}} ] +1 ) ) }{2h_{r,s}}=\frac{1}{2}. \end{aligned}$$
Hence (xk,l) does not converge to 0 in \(\mathcal{N}_{2}^{\theta}(X)\).
(iii) Suppose that \(x_{k} \rightarrow\ell(S_{2}^{\theta}(X))\) and \(x\in \ell^{2}_{\infty}(X)\). Then there exists M>0 such that ∥xk,l,z1,z2,…,zn−1∥≤M for all k,lN. Given ε>0, we have
$$\begin{aligned} & \frac{1}{ h_{r,s}} \sum_{(k,l)\in I_{r,s}} \|x_{k,l}- \ell, z_1, z_2,\ldots,z_{n-1}\| \\ &\quad =\frac{1}{h_{r,s}}\sum_{{(k,l)\in I_{r,s}},\|x_{k,l}-\ell , z_1, z_2,\ldots,z_{n-1}\|\geq\frac{\varepsilon}{2}} \|x_{k,l}-\ell, z_1,z_2,\ldots,z_{n-1}\| \\ &\qquad {}+\frac{1}{h_{r,s}}\sum_{{(k,l)\in I_{r,s}},\|x_{k,l}-\ell, z_1, z_2,\ldots,z_{n-1}\|< \frac{\varepsilon}{2}} \|x_{k,l}- \ell, z_1,z_2,\ldots,z_{n-1}\| \\ &\quad \leq\frac{M}{h_{r,s}}\biggl|\biggl\{(k,l)\in I_{r,s}: \|x_{k,l}-\ell , z_1, z_2, \ldots,z_{n-1}\|\geq\frac{\varepsilon}{2}\biggr\}\biggr|+ \frac {\varepsilon}{2}. \end{aligned}$$

This proves the result.

(iv) This is an immediate consequence of (i), (ii) and (iii). □

Theorem 3.7

LetXbe ann-normed space andθbe any lacunary sequence. If\(x=(x_{k,l})\in S^{\theta}_{2}(X)\cap S_{2}(X)\), then\(S^{\theta}_{2}(X)-\lim_{k,l\rightarrow\infty} x_{k,l} =S_{2}(X)-\lim_{k,l\rightarrow\infty} x_{k,l}\).

Proof

Suppose that S2(X)−limk,l→∞xk,l=1 and \(S^{\theta}_{2}(X)-\lim_{k,l\rightarrow\infty} x_{k,l} =\ell _{2}\), where 12.

Let ε>0 and m,n be two positive integers. Since 12 we have
$$\begin{aligned} & \bigl\{k\leq m; l\leq n: \|x_{k,l} -\ell_1, z_1, z_2,\ldots,z_{n-1}\|\geq \varepsilon\bigr\} \\ &\quad{} \cap\bigl\{k\leq m; l\leq n: \|x_{k,l} -\ell_2, z_1, z_2,\ldots ,z_{n-1}\|\geq\varepsilon\bigr\}\neq \phi. \end{aligned}$$
It follows that
$$\begin{aligned} & \bigl\{k\leq m; l\leq n: \|x_{k,l} -\ell_2, z_1, z_2,\ldots,z_{n-1}\|\geq \varepsilon\bigr\} \\ &\quad{} \subseteq\bigl\{k\leq m; l\leq n: \|x_{k,l} -\ell_1, z_1, z_2,\ldots ,z_{n-1}\|\geq\varepsilon\bigr\} \end{aligned}$$
and therefore we have
$$\begin{aligned} & \lim_{m,n\rightarrow\infty}\frac{1}{mn}\bigl|\bigl\{k\leq m; l\leq n: \|x_{k,l} -\ell_2, z_1, z_2, \ldots,z_{n-1}\|\geq\varepsilon\bigr\}\bigr| \\ &\quad \leq \lim_{m,n\rightarrow\infty} \frac{1}{mn}\bigl|\bigl\{k\leq m; l\leq n: \|x_{k,l} -\ell_1, z_1, z_2, \ldots,z_{n-1}\|\geq\varepsilon\bigr\}\bigr|. \end{aligned}$$
Since S2(X)−limk,l→∞xk,l=1, it follows that
$$ P-\lim_{m,n\rightarrow\infty}\frac{1}{mn}\bigl|\bigl\{k\leq m; l\leq n: \|x_{k,l} -\ell_2, z_1, z_2, \ldots,z_{n-1}\|\geq\varepsilon\bigr\}\bigr|=1. $$
(3.1)
Consider the kp,qth term of the statistical limit expression
$$\begin{aligned} & \frac{1}{mn}\bigl|\bigl\{k\leq m; l\leq n: \|x_{k,l} - \ell_2, z_1, z_2,\ldots ,z_{n-1}\| \geq\varepsilon\bigr\}\bigr| \\ &\quad \leq\frac{1}{k_{p,q}}\Biggl \vert \Biggl\{(k,l)\in\bigcup _{r,s=1,1}^{p,q}I_{r,s}: \|x_{k,l} -\ell_2, z_1, z_2,\ldots,z_{n-1}\|\geq\varepsilon \Biggr \} \Biggr \vert \\ &\quad =\frac{1}{k_{p,q}} \sum_{r,s=1,1}^{p,q}\bigl|\bigl\{ (k,l)\in I_{r,s}: \|x_{k,l} -\ell_2, z_1, z_2,\ldots,z_{n-1}\|\geq\varepsilon\bigr \}\bigr| \\ &\quad = \frac{1}{\sum_{r,s=1,1}^{p,q}h_{r,s}} \sum_{r,s=1,1}^{p,q}h_{r,s}.t_{r,s}, \end{aligned}$$
(3.2)
where \(t_{r,s}= \frac{1}{h_{r,s}} |\{ (k,l)\in I_{r,s}: \|x_{k,l} -\ell _{2}, z_{1}, z_{2},\ldots,z_{n-1}\|\geq\varepsilon\}|\rightarrow0\), because \(x_{k,l}\rightarrow\ell_{2}(S^{\theta}_{2}(X))\). Since θ is an arbitrary lacunary sequence and (3.2) is a regular weighted mean transform of tr,s, (3.2) tends to zero as m,n→∞. Since this is a subsequence of
$$\biggl(\frac{1}{m,n}\bigl| \bigl\{k\leq m, l\leq n: \|x_{k,l} - \ell_2, z_1, z_2,\ldots,z_{n-1}\| \geq\varepsilon \bigr\}\bigr| \biggr)_{m,n=1,1}^{\infty ,\infty}, $$
it follows that
$$P-\lim_{m,n} \frac{1}{m,n}\bigl| \bigl\{k\leq m; l\leq n: \|x_{k,l} -\ell_2, z_1, z_2, \ldots,z_{n-1}\|\geq\varepsilon \bigr\}\bigr|\neq1; $$
thus we obtain a contradiction to (3.1). This completes the proof. □
Savas and Patterson [35] defined the notion of double lacunary sequence as follows. A double index sequence θ=(kr,ls) is called double lacunary if there exist two increasing sequences of integers such that
$$k_0 = 0,\qquad h_r=k_r - k_{r-1} \rightarrow\infty\quad{\rm as}\ r\rightarrow\infty $$
and
$$l_0 = 0,\qquad \overline{h}_s =l_s - l_{s-1}\rightarrow\infty\quad{\rm as}\ s\rightarrow\infty. $$

Savas and Patterson [35] also presented the definition of double lacunary refinement as follows.

The double index sequence \(\rho=(\overline{k}_{r},\overline{l}_{s})\) is called a double lacunary refinement of the double lacunary sequence θ=(kr,ls) if \((k_{r},l_{s}) \subseteq(\overline {k}_{r},\overline{l}_{s})\).

Theorem 3.8

LetXbe ann-normed space. Ifρis a double lacunary refinement ofθand\(x_{k,l}\rightarrow\ell(S^{\rho}_{2}(X))\), then\(x_{k,l}\rightarrow\ell(S^{\theta}_{2}(X))\).

Proof

Suppose each Ir,s of θ contains the points \((\overline{k}_{r,i},\overline{l}_{s,j} )_{i,j=1}^{v(r),u(s)}\) of ρ so that
$$k_{r-1}<\overline{ k}_{r,1}< \overline{k}_{r,2}< \cdots<\overline{k}_{r, v(r)}=k_r, $$
where
$$\overline{I}_{r,i}=(\overline{k}_{r,i-1}, \overline{k}_{r,i}], $$
$$l_{s-1}<\overline{l}_{s,1}< \overline{l}_{s,2}< \cdots<\overline{l}_{s, u(s)}=l_s, $$
with
$$\overline{I}_{s,j}=(\overline{l}_{s,j-1}, \overline{l}_{s,j}] $$
and
$$\overline{I}_{r,s,i,j}=\bigl\{ (k,l): \overline{k}_{r,i-1}<k\leq \overline {k}_r ;\overline{l}_{s,j-1}<l\leq \overline{l}_s\bigr\}, $$
for all r,s and let v(r)≥1,u(s)≥1. This implies that \((k_{r}, l_{s})\subseteq(\overline{k}_{r}, \overline{l}_{s})\). Let \((\overline{I}_{i,j} )_{i,j=1,1}^{\infty,\infty}\) be the sequence of abutting blocks of \((\overline{I}_{r,s,i,j})\) ordered by increasing a lower right index points. Since \(x_{k,l}\rightarrow\ell(S^{\rho}_{2}(X))\), for each ε>0, we have the following:
$$ P-\lim_{i,j\rightarrow\infty,\infty} \sum_{\overline{I}_{i,j}\subset I_{r,s}} \frac{1}{\overline{h}_{r,s}}\bigl|\bigl\{(k,l)\in I_{i,j}: \|x_{k,l} -\ell, z_1, z_2,\ldots,z_{n-1}\|\geq\varepsilon\bigr \}\bigr|=0. $$
(3.3)
As before, we write \(h_{r,s}=h_{r} \overline{h}_{s}; \overline{h}_{r,i} = \overline{k}_{r,i} - \overline{k}_{r,i-1}, \overline{h}_{s,j} = \overline{l}_{s,j} - \overline{l}_{s,j-1}\).
For each ε>0 we have
$$\begin{aligned} & \frac{1}{h_{r,s}}\bigl|\bigl\{(k,l)\in I_{r,s}: \|x_{k,l} -\ell, z_1, z_2,\ldots ,z_{n-1}\|\geq\varepsilon\bigr \}\bigr| \\ &\quad =\frac{1}{h_{r,s}} \sum_{\overline{I}_{i,j}\subset I_{r,s}} \overline {h}_{i,j} \biggl(\frac{1}{\overline{h}_{i,j}}\bigl|\bigl\{(k,l)\in \overline{I}_{i,j}: \|x_{k,l} -\ell, z_1, z_2,\ldots,z_{n-1}\|\geq \varepsilon\bigr\}\bigr| \biggr). \end{aligned}$$
(3.4)
By (3.3), \(t_{i,j}= (\frac{1}{\overline{h}_{i,j}}|\{(k,l)\in \overline{I}_{i,j}: \|x_{k,l} -\ell, z_{1}, z_{2},\ldots,z_{n-1}\|\geq \varepsilon\}| )_{j=1}^{\infty}\) is a Pringsheim null sequence. The transformation
$$(At)_{r,s}=\frac{1}{h_{r,s}} \sum_{\overline{I}_{i,j}\subset I_{r,s}} \overline{h}_{i,j} \biggl(\frac{1}{\overline{h}_{i,j}}\bigl|\bigl\{(k,l)\in \overline{I}_{i,j}: \|x_{k,l} -\ell, z_1, z_2,\ldots,z_{n-1}\|\geq\varepsilon\bigr\}\bigr| \biggr) $$
satisfies all the conditions for a matrix transformation to map a Pringsheim null sequence into a Pringsheim null sequence. Therefore \(x_{k,l}\rightarrow\ell (S^{\theta}_{2}(X))\). This completes the proof of the theorem. □

Theorem 3.9

If\(\rho=(\overline{k}_{r},\overline{l}_{s})\)is a double lacunary refinement ofθ=(kr,ls). LetIr,sand\(\overline{I}_{r,s}\)forr,s=1,2,3,… be defined as above. If there existsδ>0 such that
$$\frac{|\overline{I}_{\alpha,\beta}|}{|I_{r,s}|}\geq\delta\quad \mathit{for~every}~\overline{I}_{\alpha,\beta}\subseteq I_{r,s} $$
then\(S^{\theta}_{2}(X)\subseteq S^{\rho}_{2}(X)\).

Proof

Let \(x=(x_{k,l})\rightarrow\ell(S^{\theta}_{2}(X))\). We shall show that \(x=(x_{k,l})\rightarrow\ell(S^{\rho}_{2}(X))\). For any given ε>0 and for every \(\overline{I}_{\alpha,\beta }\) we can find Ir,s such that \(\overline{I}_{\alpha,\beta }\subseteq I_{r,s}\). Then we have
$$\begin{aligned} & \frac{1}{|I_{r,s}|}\bigl|\bigl\{(k,l)\in I_{r,s}: \|x_{k,l} -\ell, z_1, z_2,\ldots,z_{n-1}\|\geq\varepsilon\bigr \}\bigr| \\ &\quad =\frac{|\overline{I}_{\alpha,\beta}|}{|I_{r,s}|} \frac{1}{|\overline {I}_{\alpha,\beta}|}\bigl|\bigl\{(k,l)\in I_{r,s}: \|x_{k,l} -\ell, z_1, z_2, \ldots,z_{n-1}\|\geq\varepsilon\bigr\}\bigr| \\ &\quad \leq\frac{|\overline{I}_{\alpha,\beta}|}{|I_{r,s}|} \frac {1}{|\overline{I}_{\alpha,\beta}|}\bigl| \bigl\{(k,l)\in\overline{I}_{\alpha,\beta}: \|x_{k,l} -\ell, z_1, z_2,\ldots ,z_{n-1}\|\geq\varepsilon\bigr\}\bigr| \\ &\quad \leq\frac{1}{\delta} \frac{1}{|\overline{I}_{\alpha,\beta}|} \bigl|\bigl\{ (k,l)\in\overline{I}_{\alpha,\beta}: \|x_{k,l} -\ell, z_1, z_2,\ldots ,z_{n-1}\|\geq\varepsilon\bigr\}\bigr|. \end{aligned}$$

This completes the proof of the theorem. □

Theorem 3.10

If\(\rho=(\overline{k}_{r},\overline{l}_{s})\)andθ=(kr,ls) are two double lacunary sequences. LetIr,sand\(\overline{I}_{r,s}\)forr,s=1,2,3,… be defined as above and\(I_{r,s,\alpha,\beta} =I_{r,s} \cap\overline {I}_{\alpha,\beta}\)forr,s,α,β=1,2,3,… If there existsδ>0 such that
$$\frac{|I_{r,s,\alpha,\beta}|}{|I_{r,s}|}\geq\delta $$
for everyr,s,α,β=1,2,3,… withIr,s,α,βϕ, then\(S^{\theta}_{2}(X)\subseteq S^{\rho}_{2}(X)\).

Proof

Let \(x=(x_{k,l})\rightarrow\ell(S^{\theta}_{2}(X))\). We shall show that \(x=(x_{k,l})\rightarrow\ell(S^{\rho}_{2}(X))\). Let ψ=ρθ. Then ψ is a lacunary refinement of the lacunary double sequences ρ and θ. By Theorem 3.9, \(x_{k,l}\rightarrow\ell(S^{\theta}_{2}(X))\) implies \(x_{k,l}\rightarrow\ell(S^{\psi}_{2}(X))\). Since ψ is a lacunary refinement of ρ, by Theorem 3.8 we see that \(x_{k,l}\rightarrow\ell(S^{\psi}_{2}(X))\) implies \(x_{k,l}\rightarrow \ell(S^{\rho}_{2}(X))\). This completes the proof of the theorem. □

Theorem 3.11

If\(\rho=(\overline{k}_{r},\overline{l}_{s})\)andθ=(kr,ls) are two lacunary sequences. LetIr,sand\(\overline{I}_{r,s}\)forr,s=1,2,3,… be defined as above and\(I_{r,s,\alpha,\beta} =I_{r,s} \cap\overline{I}_{\alpha,\beta}\)forr,s,α,β=1,2,3,… If there existsδ>0 such that
$$\frac{|I_{r,s,\alpha,\beta}|}{|I_{r,s}|+|I_{\alpha,\beta}|}\geq\delta $$
for everyr,s,α,β=1,2,3,… withIr,s,α,βϕ, then\(S^{\theta}_{2}(X)= S^{\rho}_{2}(X)\).

Proof

The proof of the theorem follows from Theorem 3.9 and Theorem 3.10. □

Finally, we prove the Cauchy convergence criterion for lacunary statistical convergence of double sequences in n-normed spaces.

Theorem 3.12

LetXbe ann-normed space andθbe a double lacunary sequence. A double sequence (xk,l) inXis lacunary statistically P-convergent if and only if it is lacunary statistically P-Cauchy.

Proof

Let (xk,l) be a lacunary statistically P-convergent sequence in X. We assume that \(S^{\theta}_{2}-\lim_{k,l\rightarrow\infty} x_{k,l} =\ell\). For (i,j)∈N×N, let
$$K(i,j)= \biggl\{(k,l)\in\mathbf{N}\times\mathbf{N}: \|x_{k,l} -\ell, z_1, z_2,\ldots,z_{n-1}\|< \frac{1}{ij} \biggr\}. $$
Then we have the following:
  1. (i)

    K(i+1,j+1)⊂K(i,j),

     
  2. (ii)

    \(\frac{|K(i,j)\cap I_{r,s}|}{h_{r,s}}\rightarrow1 \ {\rm as} \ r,s\rightarrow\infty\).

     
This implies that we can choose positive integers m(1) and n(1) such that for rm(1) and sn(1) we have \(\frac{|K(1,1)\cap I_{r,s}|}{h_{r,s}}> 0\), i.e. K(1,1)∩Ir,sϕ. Next we can choose m(2)>m(1) and n(2)>n(1) so that rm(2) and sn(2) imply that K(2,2)∩Ir,sϕ. Thus for each pair (r,s) satisfying m(1)≤rm(2) and n(1)≤sn(2), we can choose \((\overline{k}_{r}, \overline{l}_{s})\in I_{r,s}\) such that \((\overline{k}_{r}, \overline{l}_{s})\in K(1,1)\cap I_{r,s}\), i.e.
$$\|x_{\overline{k}_{r}, \overline{l}_s} -\ell, z_1, z_2,\ldots ,z_{n-1}\|< 1. $$
In general, we can choose m(i+1)>m(i) and n(j+1)>n(j) such that r>m(i+1) and s>n(j+1) imply that K(i+1,j+1)∩Ir,sϕ. Then for each pair (r,s) satisfying m(i)≤rm(i+1) and n(j)≤sn(j+1) we can choose \((\overline{k}_{r}, \overline {l}_{s})\in I_{r,s}\), i.e.
$$ \|x_{\overline{k}_{r}, \overline{l}_s} -\ell, z_1, z_2,\ldots ,z_{n-1}\|< \frac{1}{ij}. $$
(3.5)
Thus \((\overline{k}_{r}, \overline{l}_{s})\in I_{r,s}\) for each r,s. Together with (3.5), this implies that
$$P-\lim_{r,s \rightarrow\infty,\infty} \|x_{\overline{k}_{r}, \overline {l}_s} -\ell, z_1, z_2,\ldots,z_{n-1}\| =0. $$
For ε>0, setting
https://static-content.springer.com/image/art%3A10.1007%2Fs40306-013-0028-x/MediaObjects/40306_2013_28_Equbf_HTML.gif
we find that BCA and therefore AcBcCc. Then we have
$$\begin{aligned} & \frac{1}{h_{r,s}}\bigl|\bigl\{(k,s)\in I_{r,s}; \|x_{k,l} - x_{\overline{k}_{r}, \overline{l}_s}, z_1, z_2,\ldots,z_{n-1}\| \geq\varepsilon\bigr\}\bigr| \\ &\quad \leq \frac{1}{h_{r,s}}\biggl|\biggl\{(k,s)\in I_{r,s}; \|x_{k,l} - \ell, z_1, z_2, \ldots,z_{n-1}\| \geq\frac{\varepsilon}{2}\biggr\}\biggr|\\ &\qquad {}+\frac{1}{h_{r,s}}\biggl|\biggl\{(k,s)\in I_{r,s}; \| x_{\overline{k}_{r}, \overline{l}_s}- \ell, z_1, z_2,\ldots,z_{n-1}\| \geq \frac{\varepsilon }{2}\biggr\}\biggr|. \end{aligned}$$
Since \(S^{\theta}_{2}-\lim_{k,l\rightarrow\infty} x_{k,l} =\ell\) and \(P-\lim_{r,s \rightarrow\infty,\infty} \|x_{\overline{k}_{r}, \overline {l}_{s}} -\ell, z_{1}, z_{2},\ldots,z_{n-1}\| =0\), it follows that (xk,l) is lacunary statistically P-Cauchy in X.
Next, suppose that x is lacunary statistically P-Cauchy in X. Then we have
$$\begin{aligned} & \bigl|\bigl\{(k,s)\in I_{r,s}; \|x_{k,l} - \ell, z_1, z_2,\ldots,z_{n-1}\| \geq \varepsilon\bigr\}\bigr| \\ &\quad \leq\biggl|\biggl\{(k,s)\in I_{r,s}; \|x_{k,l} - x_{\overline {k}_{r}, \overline{l}_s}, z_1, z_2,\ldots,z_{n-1}\| \geq \frac{\varepsilon }{2}\biggr\}\biggr| \\ &\qquad {}+ \biggl|\biggl\{(k,s)\in I_{r,s}; \| x_{\overline{k}_{r}, \overline{l}_s}-\ell, z_1, z_2,\ldots,z_{n-1}\| \geq \frac{\varepsilon}{2}\biggr\}\biggr|. \end{aligned}$$

Therefore \(x=(x_{k,l})\rightarrow\ell(S^{\theta}_{2}(X))\). This completes the proof of the theorem. □

Corollary 3.2

LetXbe ann-normed space. Then any lacunary statistical-P-convergent sequence inXhas aP-convergent subsequence.

Acknowledgements

The authors express their gratitude to the anonymous referee for excellent comments and suggestions, which have enormously enhanced the quality and presentation of this paper.

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© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2013