Generalized Statistical Convergence and Statistical Core of Double ...

Acta Mathematica Sinica, English Series Nov., 2010, Vol. 26, No. 11, pp. 2131–2144 Published online: October 15, 2010 DOI: 10.1007/s10114-010-9050-2 Http://www.ActaMath.com

Acta Mathematica Sinica, English Series Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2010

Generalized Statistical Convergence and Statistical Core of Double Sequences Mohammad MURSALEEN Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India Email : [email protected]

Celal C ¸ AKAN Faculty of Education, In¨ on¨ u University, 44280-Malatya, Turkey Email : [email protected]

Syed Abdul MOHIUDDINE Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India Email : [email protected]

Ekrem SAVAS ¸ Department of Mathematics, Istanbul Ticaret University, Uskudar, 34672 Istanbul, Turkey Email : [email protected] Abstract In this paper we extend the notion of λ-statistical convergence to the (λ, μ)statistical convergence for double sequences x = (xjk ). We also determine some matrix transformations and establish some core theorems related to our new space of double sequences Sλ,μ . Keywords

Double sequence, statistical convergence, matrix transformation, core

MR(2000) Subject Classification

1

40C05, 40H05

Introduction

The concept of statistical convergence for sequences of real numbers was introduced by Fast [1] and Steinhaus [2] independently in the same year 1951 and since then several generalizations and applications of this notion have been investigated by various authors, namely [3–10]. A subset E of the set N of natural numbers is said to have natural density δ(E) if 1 |{j ≤ n : j ∈ E}|, n where the vertical bars indicate the number of elements in the enclosed set. The number sequence x = (xj ) is said to be statistically convergent to the number  if for each  > 0, 1 lim |{j ≤ n : |xj − | ≥ }| = 0. n n δ(E) = lim n

Received February 6, 2009, accepted September 21, 2009 The first author is supported by the Department of Science and Technology, New Delhi (Grant No. SR/S4/MS:505/07)

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By the convergence of a double sequence we mean the convergence in the Pringsheim’s sense [11]. A double sequence x = (xjk ) is said to be Pringsheim’s convergent (or (P )convergent) if for given  > 0 there exists an integer N such that |xjk − | <  whenever j, k > N . In this case,  is called the Pringsheim limit of x = (xjk ) and it is written as (P ) lim x = . A double sequence x = (xjk ) is said to be bounded if there exists a positive number M such that |xjk | < M for all j, k, i.e., if x(∞,2) = sup |xjk | < ∞. j,k

Note that, in contrast to the case for single sequences, a convergent double sequence need ∞ not be bounded. Let us denote by c∞ 2 , 2 the space of all bounded convergent and bounded double sequences x = (xjk ) respectively. Let K ⊆ N × N be a two-dimensional set of positive integers and let Km,n = {(j, k) : j ≤ m, k ≤ n}. Then the two-dimensional analogue of natural density can be defined as follows. In case the sequence (K(m, n)/mn) has a limit in Pringsheim’s sense, then we say that K has a double natural density and is defined as (P ) lim

m,n

K(m, n) = δ2 (K). mn

For example, let K = {(i2 , j 2 ) : i, j ∈ N}. Then K(m, n) ≤ (P ) lim δ2 (K) = (P ) lim m,n m,n mn

√ √ m n = 0, mn

i.e. the set K has double natural density zero, while the set {(i, 2j) : i, j ∈ N} has double natural density 12 . Note that, if we set m = n, we have a two-dimensional natural density due to Christopher [12]. We can define the statistical convergence for double sequences analogues of single sequences as follows (see [13–15]): A real double sequence x = (xjk ) is said to be statistically convergent to the number  if for each  > 0, the set {(j, k), j ≤ m and k ≤ n :| xjk −  |≥ } has double natural density zero. In this case we write (st2 ) limj,k xjk =  and we denote the set of all statistically convergent double sequences by S2 and set of all bounded statistically convergent double sequences by S2∞ . The idea of (λ) statistical convergence was introduced in [9] for single sequences as follows: Let λ = (λn ) be a non-decreasing sequence of positive numbers tending to ∞ such that λn+1 ≤ λn + 1, λ1 = 0. The generalized de la Vall´ee–Pousin mean is defined by 1  xj , tn (x) =: λn j∈In

where In = [n − λn + 1, n].

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A sequence x = (xj ) is said to be (V, λ)summable to a number  (see [16]) if tn (x) →  as n → ∞. The number sequence x = (xj ) is said to be (λ)statistically convergent to the number  if for each  > 0, 1 lim |{j ∈ In : |xj − | > }| = 0. n λn In this case we write (stλ ) limj xj =  and we denote the set of all (λ)statistically convergent sequences by Sλ . Let K ⊆ N be a set of positive integers. Then δλ (K) = lim n

1 |{n − λn + 1 ≤ j ≤ n : j ∈ K}| λn

is said to be (λ)density of K. In case λn = n, (λ)density reduces to the natural density. Also, since (λn /n) ≤ 1, δ(K) ≤ δλ (K) for every K ⊆ N. A real number sequence x = (xj ) is said to be (λ)statistical bounded if there exists a positive number M such that δλ ({(j ≤ n : |xj | > M }) = 0. As an analogue of (st)core (see [17]), (stλ )core of a sequence x = (xj ) can be defined as [(stλ ) lim inf x, (stλ ) lim sup x]. A number of papers have recently been written on the matrix summability of double sequences (cf. [18–29]). In this paper, we extend the concept of statistical convergence of double sequences by using two-dimensional analogue of the generalized de la Vall´ee–Pousin mean and study some important results, e.g., inclusion relations with the other related spaces of double sequences, matrix transformations and the results related to the core defined by this new method. 2

(λ, μ)Statistical Convergence

We define the following Definition 2.1 Let λ = (λm ) and μ = (μn ) be two non-decreasing sequences of positive real numbers such that each tending to ∞ and λm+1 ≤ λm + 1, λ1 = 0 and μn+1 ≤ μn + 1, μ1 = 0. Let K ⊆ N × N be a two-dimensional set of positive integers. Then the (λ, μ)density of K is defined as δλ,μ (K) = (P ) lim

m,n

1 |{m − λm + 1 ≤ j ≤ m, n − μn + 1 ≤ k ≤ n : (j, k) ∈ K}| λm μn

provided that the limit on the right-hand side exists. In case λm = m, μn = n, the (λ, μ)density reduces to the natural double density. Also, since (λm /m) ≤ 1, (μn /n) ≤ 1, δ2 (K) ≤ δλ,μ (K) for every K ⊆ N × N.

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Definition 2.2

We define the generalized double de la Val´ee–Pousin mean by   1 tm,n (x) = xjk , λm μn j∈Jm k∈In

where Jm = [m − λm + 1, m] and In = [n − μn + 1, n]. A double sequence x = (xjk ) is said to be strongly (V, λ, μ)summable to a number  if (P ) lim tm,n (|x − |) = 0. m,n

We denote the set of all double strongly (V, λ, μ)summable sequences by [V, λ, μ]. If λm = m for all m, and μn = n for all n, then strongly (V, λ, μ)summability is reduced to the strongly Ces` aro summability and [V, λ, μ] = [C, 1, 1] the space of strongly Ces` aro summable double sequences. Definition 2.3 A double sequence x = (xjk ) is said to be (λ, μ)statistically convergent to  if δλ,μ (E) = 0 where E = {j ∈ Jm , k ∈ In : |xjk − | ≥ }, i.e. if for every  > 0, (P ) lim

m,n

1 |{j ∈ Jm , k ∈ In : |xjk − | ≥ }| = 0. λm μn

In this case we write (stλ,μ ) limj,k xjk =  and we denote the set of all (λ, μ)statistically convergent double sequences by Sλ,μ . Here note that if λm = m for all m, and μn = n for all n, then the space Sλ,μ is reduced to the space S2 and since δ2 (K) ≤ δλ,μ (K), we have Sλ,μ ⊂ S2 . We write (Sλ,μ )0 to denote the space of all sequences which are (λ, μ)statistically convergent ∞ ∞ to zero and Sλ,μ for bounded (λ, μ)statistically convergent double sequences and we write (Sλ,μ )0 for double sequences which are bounded and (λ, μ)statistically convergent to zero. Definition 2.4

Let Bλ,μ (x) = {b ∈ R : δλ,μ {(j, k) : xjk > b} = 0}, Aλ,μ (x) = {a ∈ R : δλ,μ {(j, k) : xjk < a} = 0}.

Then

⎧ ⎨ sup B (x), B (x) = ∅, λ,μ λ,μ (stλ,μ ) lim sup x = ⎩ −∞, Bλ,μ (x) = ∅,

and

⎧ ⎨ inf A (x), A (x) = ∅, λ,μ λ,μ (stλ,μ ) lim inf x = ⎩ ∞, Aλ,μ (x) = ∅.

The following theorem is a consequence of the definitions of (stλ,μ ) lim sup x and (stλ,μ )lim inf x. Theorem 2.5 (a) (stλ,μ ) lim sup x =  if and only if (i) δλ,μ ({(j, k) : xjk >  − }) = 0; (ii) δλ,μ ({(j, k) : xjk >  + }) = 0. (b) (stλ,μ ) lim inf x = s if and only if (i) δλ,μ ({(j, k) : xjk < s + }) = 0; (ii) δλ,μ ({(j, k) : xjk < s − }) = 0.

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Remark 2.6 It is easy to see that (i) For any sequence x = (xjk ), (stλ,μ ) lim inf x ≤ (stλ,μ ) lim sup x. (ii) (P ) lim inf x ≤ (stλ,μ ) lim inf x ≤ (stλ,μ ) lim sup x ≤ (P ) lim sup x, for every bounded sequence x = (xjk ). Definition 2.7 A real number sequence x = (xjk ) is said to be (λ, μ)statistical bounded if there exists a positive number M such that δλ,μ ({(j, k) : |xjk | > M }) = 0. It is clear that if x = (xjk ) is (λ, μ)statistically bounded, then it has both (stλ,μ ) lim inf and (stλ,μ ) lim sup. Hence, we have Definition 2.8 A (λ, μ)statistically bounded sequence x = (xjk ) is (λ, μ)statistically convergent if and only if (stλ,μ ) lim inf x = (stλ,μ ) lim sup x. Proof Let (stλ,μ ) lim inf x = (stλ,μ ) lim sup x = . For  > 0, Theorem 2.5 implies that δλ,μ ({(j, k) : xjk >  + /2}) = δλ,μ ({(j, k) : xjk <  − /2}) = 0. Hence (stλ,μ ) lim x = . Conversely, suppose that (stλ,μ ) lim x = . Then for any  > 0, δλ,μ ({(j, k) : |xjk − | ≥ }) = 0, and so δλ,μ ({(j, k) : xjk >  + }) = 0. Thus, (stλ,μ ) lim sup x ≤ . Also, δλ,μ ({(j, k) : xjk <  − }) = 0, which means that  ≤ (stλ,μ ) lim inf x. Therefore, (stλ,μ ) lim sup x ≤ (stλ,μ ) lim inf x and with Remark 2.6 (i) we conclude that (stλ,μ ) lim sup x = (stλ,μ ) lim inf x. Theorem 2.9 Let λ and μ be the sequences as defined above. Then (i) xjk →[V, λ, μ] implies xjk →(Sλ,μ ) but not conversely ; (ii) If x ∈ ∞ 2 and xjk →(Sλ,μ ), then xjk →[V, λ, μ] and hence xjk →[C, 1, 1]; ∞ (iii) Sλ,μ = [V, λ, μ] ∩ ∞ 2 . Proof (i) Let  > 0 and xjk →[V, λ, μ]. We have   |xjk − | ≥ |xjk − | ≥  |{j ∈ Jm , k ∈ In : |xjk − | ≥ }|. j∈Jm ,k∈In

j∈Jm ,k∈In

|xjk −L|≥

Hence xjk →(Sλ,μ ). For the converse, let x be defined by ⎧ ⎨ jk, if j and k are squares, xjk = ⎩ 0, otherwise.

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It is clear that x is an unbounded double sequence and for  > 0

√ 1 [ λm μn ] |{j ∈ Jm , k ∈ In : |xjk − 0| ≥ }| = (P ) lim = 0. (P ) lim m,n λm μn m,n λm μn

Therefore xjk →0 (Sλ,μ ). Also note that 1 m,n λm μn



(P ) lim

|xj,k − 0|

j∈Jm ,k∈In

does not exist, i.e., xjk →0 ([V, λ, μ]). (ii) Since x ∈ ∞ 2 , |xjk − | ≤ M for all j, k. Also for given  > 0 and enough large m, n, we obtain    1 1 1 |xjk − | = |xjk − | + |xjk − | λm μn λm μn j∈J ,k∈I λm μn j∈J ,k∈I j∈Jm ,k∈In

m

n

m

|xjk −|≥

≤

n

|xjk −|0 m

and

lim inf n

μn > 0. n

(2.1)

Proof For given  > 0 we have {j ≤ m and k ≤ n : |xjk − | ≥ } ⊃ {j ∈ Jm , k ∈ In : |xjk − | ≥ }. Therefore, 1 |{j ≤ m and k ≤ n : |xjk − | ≥ }| mn 1 ≥ |{j ∈ Jm , k ∈ In : |xjk − | ≥ }| mn λm μn 1 = |{j ∈ Jm , k ∈ In : |xjk − | ≥ }|. mn λm μn Taking the limit as m, n → ∞ and using hypothesis, we get xjk → (S2 ) =⇒ xjk → (Sλ,μ ). Conversely, suppose that x ∈ S2 , and either lim inf m λmm or lim inf n μnn or both are zero. λm(p) 1 ∞ Then as in [9] we can choose subsequences (m(p))∞ p=1 and (n(q))q=1 such that m(p) < p and

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< 1q . Define a sequence x = (xjk ) by ⎧ ⎨ 1, if j ∈ J m(p) and k ∈ In(q) (p, q = 1, 2, . . .), xjk = ⎩ 0, otherwise.

Then, clearly x is bounded and x ∈ [C, 1, 1] and hence, by Theorem 4.1 (a) of [14], x ∈ S2 . But ∞ on the other hand, x ∈ [V, λ, μ] and from Theorem 2.9, x ∈ Sλ,μ ; a contradiction and hence (2.1) must hold. 3

Characterizations of Some Matrix Classes

In this section we characterize some matrix transformations which will be used in the next section to establish some core theorems. Theorem 3.1 A ∈ (∞ , (S ∞ )0 ) if and only if 2  λ,μ (i) A = supm,n j k |amnjk | < ∞;   (ii) (stλ,μ ) limm,n j k |amnjk | = 0. ∞ ∞ Proof Let the conditions hold. (i) implies that A ∈ (∞ 2 , 2 ). Also, for any x ∈ 2       |Ax| =  amnjk xjk ≤ x(∞,2) |amnjk |, j

j

k

k

∞ 0. Hence A ∈ (∞ 2 , (Sλ,μ )0 ). Then, since (∞ 2 , (Sλ,μ )0 ) ⊂

and (ii) implies that (stλ,μ ) lim Ax = ∞ Conversely, let A ∈ (∞ 2 , (Sλ,μ )0 ). ∞ other hand, A ∈ (2 , (Sλ,μ )0 ) implies that

∞ (∞ 2 , 2 ), we get (i). On the

(stλ,μ ) lim Ax = 0, m,n

so, by choosing xjk = sgn amnjk , we get the condition (ii). Theorem 3.2 A ∈ (c∞ , S ∞ )reg if and only if 2  λ,μ (i) A = supm,n j k |amnjk | < ∞; (ii) (stλ,μ ) limm,n amnjk = 0, for each j, k;   (iii) (stλ,μ ) limm,n j k amnjk = 1;  (iv) (stλ,μ ) limm,n j |amnjk | = 0, for each k;  (v) (stλ,μ ) limm,n k |amnjk | = 0, for each j. ∞ ∞ ∞ Proof Let A ∈ (c∞ 2 , Sλ,μ )reg . Then A ∈ (c2 , 2 )reg , so that (i) holds. Now we define the (qr)

(q)

(r)

sequences B (qr) = {bjk : j, k = 1, 2, 3, . . .}, B (q) = {bjk }, C (r) = {cjk } and C = {cjk } as follows: For all j, k, q, r, we set ⎧ ⎨ 1, if (j, k) = (q, r), (qr) bjk = ⎩ 0, otherwise; ⎧ ⎨ 1, if j = q, (q) bjk = ⎩ 0, otherwise; ⎧ ⎨ 1, if k = r, (r) cjk = ⎩ 0, otherwise;

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and cjk = 1 for all j, k. (qr)

Then, the necessity of (ii) follows from (stλ,μ ) lim A(bjk ), (iii) from (stλ,μ ) lim A(cjk ), (iv) from (q)

(r)

(stλ,μ ) lim A(bjk ) and (v) from (stλ,μ ) lim A(cjk ). Conversely, let the conditions hold and x = (xjk ) ∈ c∞ 2 with (P ) limj,k→∞ xjk = , say. Then, for  > 0 there exists integers j0 , k0 > 0 such that |xjk − | ≤ /A for j > j0 , k > k0 , ∞ so that by (i) Ax exists. Now, to show that Ax ∈ Sλ,μ , write    amnjk xjk = amnjk (xjk − ) +  amnjk , (3.1) j

and  j

Since

j

k

amnjk (xjk − ) =

 

+

j≤j0 k≤k0

k

j

k

 

+

j≤j0 k>k0

 

+

j>j0 k≤k0

k

  amnjk (xjk − ). j>j0 k>k0

           ≤ x a (x − ) |a | + x |amnjk | mnjk jk mnjk   j j≤j0 k≤k0 j≤j0 k>k0 k          + x |amnjk | + amnjk ,  A j>j j>j 0

k≤k0

0

(3.2)

k>k0

by taking (stλ,μ ) limm,n in (3.1) and (3.2) and using the conditions, we get  amnjk xjk = . (stλ,μ ) lim m,n

Hence A ∈

j

k

∞ (c∞ 2 , Sλ,μ )reg .

∞ ∞ Theorem 3.3 A ∈ (Sλ,μ , Sλ,μ )reg if and only if ∞ ∞ (i) A ∈ (c2 , Sλ,μ )reg ;  (ii) (stλ,μ ) limm,n (j,k)∈E |amnjk | = 0; for every E ⊆ N × N with δλ,μ (E) = 0. ∞ ∞ ∞ ∞ ∞ ∞ , Sλ,μ )reg . Then, since c∞ Proof Let A ∈ (Sλ,μ 2 ⊂ Sλ,μ , A ∈ (c2 , Sλ,μ )reg . For any x ∈ 2 and E ⊆ N × N with δλ,μ (E) = 0, define the sequence z = (zjk ) by ⎧ ⎨ x , if (j, k) ∈ E, jk (3.3) zjk = ⎩ 0, if (j, k) ∈ E. ∞ Then, it is clear that z ∈ (Sλ,μ )0 which implies Az ∈ (Sλ,μ )0 . Since   amnjk zjk = amnjk xjk , j

k

the matrix B = (bmnjk ) defined by bmnjk

(j,k)∈E

⎧ ⎨ a mnjk , if (j, k) ∈ E, = ⎩ 0, if (j, k) ∈ E,

(3.4)

∞ for all m, n, belongs to the class (∞ 2 , (Sλ,μ )0 ). Hence (ii) follows from the condition (ii) of Theorem 3.1.

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∞ Conversely, let the conditions hold and x = (xjk ) ∈ Sλ,μ with (stλ,μ ) lim x = . Let E = {(j, k), j ≤ m and k ≤ n : |xjk − | ≥ }. Then δλ,μ (E) = 0 and we can write the equality (3.1). Since          amnjk (xjk − ), amnjk (xjk − ) = + + + j

j≤n k≤m

k

we have

j>n k≤m

j≤n k>m

j>n k>m

        amnjk (xjk − ) ≤ x |amnjk | + x |amnjk |  j

k

j≤n k>m

(j,k)∈E

+ x



|amnjk | +

j>n k≤m

       a mnjk .  A j>n

(3.5)

k>m

Now taking (stλ,μ ) limm,n in (3.1) and (3.5) and using the conditions, we get  amnjk xjk = . (stλ,μ ) lim m,n

Hence A ∈

j

k

∞ ∞ , Sλ,μ )reg . (Sλ,μ

From Theorem 3.3, we have the following results. Theorem 3.4 A ∈ (S2∞ , S2∞ )reg if and only if ∞ (i) A ∈ (c∞ 2 , S2 )reg ,  (ii) (st2 ) limmn (j,k)∈E |amnjk | = 0, for every E ⊆ N × N with δ2 (E) = 0. ∞ Theorem 3.5 A ∈ (Sλ,μ , S2∞ )reg if and only if ∞ (i) A ∈ (c∞ 2 , S2 )reg ,  (ii) (st2 ) limm,n (j,k)∈E |amnjk | = 0, for every E ⊆ N × N with δλ,μ (E) = 0. ∞ Theorem 3.6 A ∈ (S2∞ , Sλ,μ )reg if and only if ∞ ∞ (i) A ∈ (c2 , Sλ,μ )reg ,  (ii) (stλ,μ ) limm,n (j,k)∈E |amnjk | = 0, for every E ⊆ N × N with δ2 (E) = 0. ∞ Theorem 3.7 A ∈ (Sλ,μ , c∞ 2 )reg if and only if ∞ (i) A is bounded regular, i.e. A ∈ (c∞ 2 , c2 )reg (see [30–31]),  (ii) (P ) limm,n (j,k)∈E |amnjk | = 0, for every E ⊆ N × N with δλ,μ (E) = 0. ∞ , c∞ Proof Let A ∈ (Sλ,μ 2 )reg . Then the bounded regularity of A follows from the fact that ∞,0 ∞ ∞ c2 ⊂ Sλ,μ . Also for z as in (3.3) and B as in (3.4), we have that B ∈ (∞ ) (see [20]). 2 , c2 Hence (ii) holds. The converse follows directly from (3.1) and (3.5).

4

(λ, μ)Statistical Core Theorems

The Knopp core (or (K)core) of a real number single sequence x = (xk ) is defined to be the closed interval [lim inf x, lim sup x].

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The well-known Knopp core theorem states that (cf. Cooke [32, p.137]) in order that lim inf Ax ≤ lim sup Ax for every bounded real sequence x, it is necessary and sufficient that ∞ A = (ajk ) should be regular and limn→∞ k=0 |ank | = 1. Patterson [28] extended this idea for double sequences by defining the Pringsheim core (or P -core or K2 -core) of a real bounded double sequence x = (xjk ) as the closed interval [(P ) lim inf x, (P ) lim sup x]. Statistical core (or st2 -core) of a real double sequence was defined in [20] as the closed interval (st2 )core(x) = [(st2 ) lim inf x, (st2 ) lim sup x]. Analogously, we can define the following: Definition 4.1 Let x = (xjk ) be a (λ, μ)statistically bounded sequence. Then (stλ,μ )core(x) is defined as [(stλ,μ ) lim inf x, (stλ,μ ) lim sup x]. For core of real bounded sequences, we refer to [4, 19–20, 22–23, 25–26] and [28]. Remark 4.2 i) From Remark 2.6 (ii), we have (stλ,μ )core(x) ⊆ (K2 )core(x) for all x. ii) Since Sλ,μ ⊆ S2 for all λ, μ, (st2 )core(x) ⊆ (stλ,μ )core(x). iii) In the case λm = m, μn = n, (stλ,μ )core(x) = (st2 )core(x) for every x. Now, we prove a relation between (stλ,μ )core and (st2 )core. Theorem 4.3

For every x, (stλ,μ )core(x) ⊆ (st2 )core(x) if and only if (P ) lim inf m,n

λm μn > 0. mn

Proof For any b ∈ R, since {(j, k), j ≤ m and k ≤ n : xjk > b} ⊃ {m − λm + 1 ≤ j ≤ m, n − μn + 1 ≤ k ≤ n : xjk > b}, we have 1 |{(j, k), j ≤ m and k ≤ n : xjk > b}| mn 1 |{m − λm + 1 ≤ j ≤ m, n − μn + 1 ≤ k ≤ n : xjk > b}| ≥ mn λm μn 1 = |{m − λm + 1 ≤ j ≤ m, n − μn + 1 ≤ k ≤ n : xjk > b}|. mn λm μn Hence (stλ,μ )core(x) ⊆ (st2 )core(x) if and only if (P ) lim inf m,n

λm μn > 0. mn

In this section we will apply the result of Section 3 to establishing some core theorems. First we state the following useful lemma which can be proved exactly in the same manner as Lemma 3.1 of [28] and by Lemma 2.4 of [5] just by replacing (st) lim by (stλ,μ ) lim. Lemma 4.4 Let A < ∞ and (stλ,μ ) limm,n |amnjk | = 0. Then, there exists a sequence y = (yjk ) ∈ ∞ 2 such that y ≤ 1 and   amnjk yjk = (stλ,μ ) lim sup |amnjk |. (4.1) (stλ,μ ) lim sup m,n

Lemma 4.5

For every x ∈

j

k

m,n

j

k

∞ 2 , (stλ,μ )core(Ax) ⊆ (K2 )core(x)

(4.2)

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if and only if ∞ (i) A ∈ (c∞ 2 , Sλ,μ )reg ;   (ii) (stλ,μ ) lim supm,n j k |amnjk | ≤ 1. Proof Let (4.2) hold for all x ∈ ∞ 2 and let (x) = (P ) lim inf x and L(x) = (P ) lim sup x. Then (stλ,μ ) lim sup(Ax) ≤ L(x) for all x ∈ ∞ 2 . Now, replacing x by −x, we have (x) ≤ (stλ,μ ) lim inf(Ax) ≤ (stλ,μ ) lim sup(Ax) ≤ L(x) ∞ for all x ∈ ∞ 2 . If x ∈ c2 then (x) = L(x) = (P ) lim x, so that

(stλ,μ ) lim inf(Ax) = (stλ,μ ) lim sup(Ax) = (stλ,μ ) lim(Ax) = (P ) lim x. ∞ Hence A ∈ (c∞ 2 , Sλ,μ )reg . Now, by Lemma 4.4, there is y = (yjk ) ∈ ∞ 2 such that y ≤ 1 and   |amnjk | = (stλ,μ ) lim sup amnjk yjk (stλ,μ ) lim sup m,n

j

m,n

k

j

k

≤ L(y) ≤ y ≤ 1. Conversely, let the conditions hold. If x ∈ ∞ 2 , for given  > 0, we can find j0 , k0 ∈ N such that xjk ≤ L(x) +  for j ≥ j0 , k ≥ k0 . So, we can write  amnjk xjk (Ax)mn = j

=

k

  |amnjk | + amnjk

−

|amnjk | − amnjk xjk 2

2 j k       ≤ |amnjk |xjk  + (|amnjk | − amnjk )|xjk | j

≤ x

j

k

 

|amnjk | + x

 

|amnjk |xjk + x

j>j0 k>k0

≤ x

 

j≤j0 k≤k0

+ (L(x) + )

|amnjk | + x

j>j0 k≤k0

j≤j0 k≤k0

+

k

   j

|amnjk | + x  

(|amnjk | − amnjk )

  j>j0 k≤k0

j>j0 k>k0

|amnjk |

j≤j0 k>k0

k

|amnjk | + x

 

|amnjk | + x  j

 

|amnjk |

j≤j0 k>k0

(|amnjk | − amnjk ).

k

Now taking (stλ,μ ) lim supm,n and using conditions (i) and (ii), we get (stλ,μ ) lim sup(Ax)mn ≤ L(x) + . m,n

Since  was arbitrary, we get the result. Theorem 4.6

For every x ∈ ∞ 2 , (stλ,μ )core(Ax) ⊆ (stλ,μ )core(x)

if and only if

(4.3)

Mursaleen M., et al.

2142 ∞ ∞ (i) A ∈ (Sλ,μ , Sλ,μ )reg ;   (ii) (stλ,μ ) lim supm,n j k |amnjk | ≤ 1.

Proof Let (4.3) hold for all x ∈ ∞ 2 . Then (stλ,μ ) lim sup(Ax) ≤ (stλ,μ ) lim sup x, that is, (stλ,μ ) lim inf x ≤ (stλ,μ ) lim inf(Ax) ≤ (stλ,μ ) lim sup(Ax) ≤ (stλ,μ ) lim sup x ∞ ∞ ∞ for all x ∈ ∞ 2 . For x ∈ Sλ,μ , we have A ∈ (Sλ,μ , Sλ,μ )reg . Since

(stλ,μ )core(Ax) ⊆ (K2 )core(x). the necessity of (ii) follows from Theorem 4.5. ∞ Conversely, let the conditions hold. If x = (xjk ) ∈ Sλ,μ , then (stλ,μ ) lim sup x = C(x) is finite. Hence from Theorem 2.5, for any  > 0, δλ,μ (E) = 0 where E = {(j, k), j ≤ m and k ≤ n : xjk > C(x) + } and xjk ≤ C(x) +  whenever (j, k) ∈ E. Write   amnjk xjk + amnjk xjk . (Ax)mn = (j,k)∈E

Now (Ax)mn =

(j,k)∈E

  |amnjk | + amnjk

−

|amnjk | − amnjk xjk 2

2 j k      ≤  |amnjk |xjk  + (|amnjk | − amnjk )|xjk | j

≤ x

j

k



|amnjk | + x



|amnjk |xjk + x

j>n k>m

Thus (Ax)mn ≤ x



|amnjk | + x

j>n k≤m

j≤n k≤m

+

k



(j,k)∈E

+ (C(x) + )



|amnjk |

j≤n k>m

 (|amnjk | − amnjk ). j

|amnjk | + x



k



|amnjk | + x

j>n k≤m

|amnjk | + x

j>n k>m

 j



|amnjk |

j≤n k>m

(|amnjk | − amnjk ).

k

Now taking (stλ,μ ) lim supm,n and using the conditions, we get (stλ,μ ) lim sup(Ax)mn ≤ C(x) + . m,n

Since  was arbitrary, (4.3) is proved. From Theorem 4.6, we have the following results: Theorem 4.7

For every x ∈ ∞ 2 , (st2 )core(Ax) ⊆ (st2 )core(x)

if and only if

(4.4)

Generalized Statistical Convergence and Statistical Core of Double Sequences

2143

(i) A ∈ (S2∞ , S2∞ )reg ;   (ii) (st2 ) lim supm,n j k |amnjk | ≤ 1. Theorem 4.8

For every x ∈ ∞ 2 , (st2 )core(Ax) ⊆ (stλ,μ )core(x)

(4.5)

if and only if ∞ , S2∞ )reg ; (i) A ∈ (Sλ,μ (ii) the same as in Theorem 4.7. Theorem 4.9

For every x ∈ ∞ 2 , (stλ,μ )core(Ax) ⊆ (st2 )core(x)

(4.6)

if and only if ∞ )reg ; (i) A ∈ (S2∞ , Sλ,μ (ii) the same as in Theorem 4.6. By the same methods used in the above theorems and Theorem 3.4 in [20], we can prove the following Theorem 4.10

For every x ∈ ∞ 2 , (K2 )core(Ax) ⊆ (stλ,μ )core(x)

(4.7)

if and only if ∞ , c∞ (i) A ∈ (Sλ,μ 2 )reg ;   (ii) (P ) lim supm,n j k |amnjk | ≤ 1. Acknowledgements

We are grateful to the referees for their valuable suggestions.

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