/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 8, No. 2, 2015, 201-213 ISSN 1307-5543 – www.ejpam.com New Types of Generalized Difference Double A-sequence Spaces Defined by Orlicz Function Bipan Hazarika1, Ayhan Esi 2,∗ 1 Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh-791 112, Arunachal Pradesh, India 2 Adiyaman University, Science and Art Faculty, Department of Mathematics, 02040, Adiyaman, Turkey Abstract. In this paper we introduce some new generalized difference double sequence spaces defined by Orlicz function and study different topological properties of these spaces and also establish some inclusion results among them. 2010 Mathematics Subject Classifications: 40A05,40B05, 46A45. Key Words and Phrases: Orlicz function, difference space, double sequence, P-convergence. 1. Introduction In 1971 Lindenstrauss and Tzafriri [6] used the idea of Orlicz function to construct the sequence space for single sequences as follows: lM = ( x = � xk � : ∞ ∑ k=1 M �� �xk � � ρ � <∞, for some ρ > 0 ) , which is a Banach space normed by � xk � = inf ( ρ > 0 : ∞ ∑ k=1 M �� �xk � � ρ � ≤ 1 ) . Definition 1. An Orlicz function is a function M : [0,∞)→ [0,∞) which is continuous, non- decreasing and convex with M (0) = 0, M (x)> 0 for x > 0 and M (x)→∞ as x →∞. ∗Corresponding author. Email addresses: bh_rgu@yahoo.co.in (B. Hazarika), aesi23@adiyaman.edu.tr (A. Esi) http://www.ejpam.com 201 c© 2015 EJPAM All rights reserved. B. Hazarika, A. Esi / Eur. J. Pure Appl. Math, 8 (2015), 201-213 202 An Orlicz function M is said to satisfy the ∆2-condition for all values of u, if there exists a constant K > 0, such that M(2u)≤ KM(u), u≥ 0. Note that, if 0< λ < 1, then M (λx)≤ λM (x), for all x ≥ 0. In the later stage different Orlicz sequence spaces were introduced and studied by Parashar and Choudhary [8], Et and Colak [2] and many others. Kizmaz [5] introduced the notion of difference sequence spaces as follows: X (∆) = � x = � xk � : � ∆xk � ∈ X for X = l∞, c and co. Later on, the notion was generalized by Et and Colak [2] as follows: X (∆m) = � x = � xk � : � ∆m xk � ∈ X for X = l∞, c and co, where ∆m x = � ∆m xk � = � ∆m−1 xk −∆m−1 xk+1 � ,∆0 x = x and also this generalized difference notion has the following binomial representation: ∆m xk = m ∑ i=0 (−1)i � m i � xk+i for all k ∈ N. Definition 2 ([9]). A double sequence x = � xk,l � has a Pringsheim limit L (denoted by P − lim x = L) provided that given an ǫ > 0 there exists an N ∈ N such that � �xk,l − L � � < ǫ whenever k, l > N. We shall describe such an x = � xk,l � more briefly as "P-convergent". The four dimensional matrix A is said to be RH-regular if it maps every bounded P-convergent sequence into a P-convergent sequence with the same P-limit. The assumption of boundedness was made because a double sequence which is P-convergent is not necessarily bounded. Using this definition Robison and Hamilton, independently, both presented the following Silverman- Toeplitz type characterization of RH-regularity. Lemma 1 ([4, 10]). The four dimensional matrix A is RH-regular if and only if RH1: P − limm,n am,n,k,l = 0 for each k and l; RH2: P − limm,n ∑∞,∞ k,l=1,1 am,n,k,l = 1; RH3: P − limm,n ∑∞,∞ k,l=1,1 � �am,n,k,l � �= 0 for each l; RH4: P − limm,n ∑∞,∞ k,l=1,1 � �am,n,k,l � �= 0, for each k; RH5: ∑∞,∞ k,l=1,1 � �am,n,k,l � � is P-convergent; RH6: There exist finite positive integers E and F such that ∑ k,l>F � �am,n,k,l � �< E. B. Hazarika, A. Esi / Eur. J. Pure Appl. Math, 8 (2015), 201-213 203 2. New Generalized Difference Double Sequence Spaces Let M be an Orlicz function, p = � pk,l � be a factorable double sequence of strictly positive real numbers and A= � am,n,k,l � be a nonnegative RH-regular summability matrix method. We now define the following new difference double sequence spaces (for some ρ > 0 and L): w2 o � A, M , p � (∆r) = ( x = � xk,l � : P − lim m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �∆r xk,l � � ρ ��pk,l = 0, ) , w2 � A, M , p � (∆r) = ( x = � xk,l � : P − lim m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �∆r xk,l − L � � ρ ��pk,l = 0, ) , w2 ∞ � A, M , p � (∆r) = ( x = � xk,l � : sup m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �∆r xk,l � � ρ ��pk,l <∞, ) . When M(x) = x , we have the following difference sequence spaces: w2 o � A, p � (∆r) = ( x = � xk,l � : P − lim m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � �∆r xk,l � � pk,l = 0 ) , w2 � A, p � (∆r) = ( x = � xk,l � : P − lim m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � �∆r xk,l − L � � pk,l = 0, for some L ) , w2 ∞ � A, p � (∆r) = ( x = � xk,l � : sup m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � �∆r xk,l � � pk,l <∞ ) . Some spaces are defined by specializing A, M , r and p = � pk,l � . For example, if A= (C , 1, 1) the difference sequence spaces defined above become w2 o � M , p � (∆r), w2 � M , p � (∆r) and w2 ∞ � M , p � (∆r) which are as follows (for some ρ > 0 and L): w2 o � M , p � (∆r) = ( x = � xk,l � ∈ w2 : P − lim m,n 1 mn m−1,n−1 ∑ k,l=0,0 � M �� �∆r xk,l � � ρ ��pk,l = 0, ) , w2 � M , p � (∆r) = ( x = � xk,l � : P − lim m,n 1 mn m−1,n−1 ∑ k,l=0,0 � M �� �∆r xk,l − L � � ρ ��pk,l = 0, ) , w2 ∞ � M , p � (∆r) = ( x = � xk,l � : sup m,n 1 mn m−1,n−1 ∑ k,l=0,0 � M �� �∆r xk,l � � ρ ��pk,l <∞, ) . Let A= (C , 1, 1), pk,l = 1, for all k, l ∈ N and M(x) = x , we obtain the following difference sequence spaces: w2 o(∆ r) = ( x = � xk,l � : P − lim m,n 1 mn m−1,n−1 ∑ k,l=0,0 � �∆r xk,l � �= 0 ) , B. Hazarika, A. Esi / Eur. J. Pure Appl. Math, 8 (2015), 201-213 204 w2(∆r) = ( x = � xk,l � : P − lim m,n 1 mn m−1,n−1 ∑ k,l=0,0 � �∆r xk,l − L � �= 0, for some L ) , w2 ∞(∆ r) = ( x = � xk,l � : sup m,n 1 mn m−1,n−1 ∑ k,l=0,0 � �∆r xk,l � �<∞ ) . If r = 1 the we obtain the following difference sequence spaces (for some ρ > 0 and L): w2 o � A, M , p � (∆) = ( x = � xk,l � : P − lim m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �∆xk,l � � ρ ��pk,l = 0, ) , w2 � A, M , p � (∆) = ( x = � xk,l � : P − lim m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �∆xk,l − L � � ρ ��pk,l = 0, ) , w2 ∞ � A, M , p � (∆) = ( x = � xk,l � : sup m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �∆xk,l � � ρ ��pk,l <∞, ) which were defined and studied by Esi [1]. 3. Main Results In this section we shall establish some basic properties for the difference sequence spaces defined above. Theorem 1. Let p = � pk,l � be bounded. The classes of sequences w2 o � A, M , p � (∆r), w2 � A, M , p � (∆r) and w2 ∞ � A, M , p � (∆r) are linear spaces. Proof. The proof of the theorem is easy, so omitted. Theorem 2. If 0 < h = inf pk,l ≤ sup pk,l = H <∞, then for any Orlicz function M and a nonnegative RH-regular summability matrix method A, then w2 � A, p � (∆r) ⊂ w2 � A, M , p � (∆r). Proof. Let 0 < h = inf pk,l ≤ sup pk,l = H <∞ and x = � xk,l � ∈ w2 � A, p � (∆r) and let 0 < ǫ < 1 and δ with 0 < δ < 1 such that M (t) < ǫ for 0 ≤ t < δ. We can write for each m and n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �∆r xk,l − L � � ρ ��pk,l = ∞,∞ ∑ k,l=0,0 & |∆r xk,l−L|≤δ am,n,k,l � M �� �∆r xk,l − L � � ρ ��pk,l + ∞,∞ ∑ k,l=0,0 & |∆r xk,l−L|>δ am,n,k,l � M �� �∆r xk,l − L � � ρ ��pk,l . Then ∞,∞ ∑ k,l=0,0 & |∆r xk,l−L|≤δ am,n,k,l � M �� �∆r xk,l − L � � ρ ��pk,l ≤ ǫh ∞,∞ ∑ k,l=0,0 am,n,k,l . (1) B. Hazarika, A. Esi / Eur. J. Pure Appl. Math, 8 (2015), 201-213 205 On the other hand, we use the fact that � �∆r xk,l − L � �< 1+   � � � � � � �∆r xk,l − L � � ρ � � � � �   where [|t|] denotes the integer part of t. Since M is Orlicz function we have M �� �∆r xk,l − L � � ρ � ≥ M (1) . Now, let us consider the second part where the sum is taken over � �∆r xk,l − L � �> δ. Thus ∑ k,l=0,0 & |∆r xk,l−L|>δ ∞,∞am,n,k,l � M �� �∆r xk,l − L � � ρ ��pk,l ≤ ∞,∞ ∑ k,l=0,0 & |∆r xk,l−L|>δ am,n,k,l  M  1+   � � � � � � �∆r xk,l − L � � ρ � � � � �       pk,l ≤ �2M (1)δ−1 �H ∞,∞ ∑ k,l=0,0 am,n,k,l �� �∆r xk,l − L � � ρ �pk,l This inequality and from (1) and RH-regularity of A, we are granted that x = � xk,l � ∈ w2 � A, M , p � (∆r) and this completes the proof. Theorem 3. w2 o � A, M , p � (∆r), w2 � A, M , p � (∆r) and w2 ∞ � A, M , p � (∆r) are complete linear topological spaces with the paranorm g �� xk,l �� = r ∑ k=1 |xk,1|+ r ∑ l=1 |x1,l | + inf    ρ pk,l T > 0 : sup m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M � |∆r xk,l | ρ ��pk,l ! 1 T ≤ 1    . where T =max (1, H), H = supk,l pk,l . Proof. Clearly g (0) = 0, g (−x) = g (x). Let x = � xk,l � , y = � yk,l � ∈ w2 ∞ � A, M , p � (∆r). Then there exist some ρ1 and ρ2 such that sup m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �∆r xk,l � � ρ1 ��pk,l ! 1 T ≤ 1 B. Hazarika, A. Esi / Eur. J. Pure Appl. Math, 8 (2015), 201-213 206 and sup m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �∆r yk,l � � ρ2 ��pk,l ! 1 T ≤ 1. Let ρ = ρ1 +ρ2. Then we have sup m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �∆r � xk,l + yk,l �� � ρ ��pk,l ! 1 T sup m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �∆r xk,l +∆ r yk,l � � ρ1 +ρ2 ��pk,l ! 1 T ≤ sup m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � ρ1 ρ1 +ρ2 M �� �∆r xk,l � � ρ1 � + ρ2 ρ1 +ρ2 M �� �∆r yk,l � � ρ2 ��pk,l ! 1 T By Minkowsky’s inequality ≤ � ρ1 ρ1 +ρ2 � sup m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �∆r xk,l � � ρ1 ��pk,l ! 1 T + � ρ2 ρ1 +ρ2 � sup m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �∆r yk,l � � ρ2 ��pk,l ! 1 T ≤ 1. Now g �� xk,l � + � yk,l �� = r ∑ k=1 �|xk,1|+ |yk,1| � + r ∑ l=1 �|x1,l |+ |y1,l | � + inf    ρ pk,l T > 0 : sup m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �∆r � xk,l + yk,l �� � ρ ��pk,l ! 1 T ≤ 1    ≤ r ∑ k=1 |xk,1|+ r ∑ k=1 |yk,1|+ r ∑ l=1 |x1,l |+ r ∑ l=1 |y1,l | + inf    ρ pk,l T 1 > 0 : sup m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �∆r xk,l � � ρ1 ��pk,l ! 1 T    + inf    ρ pk,l T 2 > 0 : sup m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �∆r yk,l � � ρ2 ��pk,l ! 1 T    =g �� xk,l �� + g �� yk,l �� . B. Hazarika, A. Esi / Eur. J. Pure Appl. Math, 8 (2015), 201-213 207 Let λ ∈ C, then the continuity of the product follows from the following equality: g � λ � xk,l �� = r ∑ k=1 |λxk,1|+ r ∑ l=1 |λx1,l | + inf    ρ pk,l T > 0 : sup m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �λ∆r xk,l � � ρ ��pk,l ! 1 T ≤ 1,ρ > 0    =|λ| r ∑ k=1 |xk,1|+ |λ| r ∑ l=1 |x1,l | + inf    (|λ| r) pk,l T > 0 : sup m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �λ∆r xk,l � � ρ ��pk,l ! 1 T ≤ 1, r > 0    = |λ| g ��xk,l �� where 1 r = |λ| ρ . Now let � x s k,l � be a Cauchy sequence in w2 ∞ � A, M , p � (∆r). Then g �� x s k,l − x t k,l �� → 0 as s, t →∞. For given ǫ > 0, choose b > 0 and xo > 0 be such that ǫ bxo > 0 and M � bxo 2 � ≥ 1. Now g �� x s k,l − x t k,l �� → 0 as s, t →∞ implies that there exists no ∈ N such that g �� x s k,l − x t k,l �� < ǫ bxo for all s, t ≥ no. ⇒ r ∑ k=1 � � �x s k,1 − x t k,1 � � �+ r ∑ l=1 � � �x s 1,l − x t 1,l � � � + inf        ρ pk,l T > 0 : sup m,n    ∞,∞ ∑ k,l=0,0 am,n,k,l   M    � � �∆r x s k,l −∆r x t k,l � � � ρ       pk,l    1 T leq1        < ǫ bxo (2) This implies that r ∑ k=1 � � �x s k,1 − x t k,1 � � �+ r ∑ l=1 � � �x s 1,l − x t 1,l � � �< ǫ, f or al l s, t ≥ n0. This shows that � x s k,1 � , � x t 1,l � are Cauchy sequences of real numbers. As the set of real numbers is complete so there exists real numbers xk,1, x1,l such that lim s→∞ x s k,1 = xk,1 and lim t→∞ x t 1,l = x1,l . B. Hazarika, A. Esi / Eur. J. Pure Appl. Math, 8 (2015), 201-213 208 Now from 0(2) we have, M    � � �∆r x s k,l −∆r x t k,l � � � ρ    ≤ 1≤ M � bxo 2 � ⇒ � � �∆r x s k,l −∆r x t k,l � � � g �� x s k,l − x t k,l �� ≤ bxo 2 ⇒ � � �∆r x s k,l −∆r x t k,l � � �< bxo 2 . ǫ bxo = ǫ 2 . This implies that � ∆r x s k,l � is a Cauchy sequence of real numbers. Let lims→∞∆r x s k,l = zk,l for all k, l ∈ N . Let k, l = 1, we have lims→∞∆r x s 1,1 = lims→∞ r ∑ i=0 r ∑ j=0 (−1)i+ j � r i �� r j � x1+i,1+ j = z1,1. Similarly we have lims→∞∆r x s k,l = lims→∞ x s k,l = zk,l for k, l = 1,2, . . . , r. Thus we have lims→∞ x s 1+r,1+r exists. Let lims→∞ x s 1+r,1+r = x1+r,1+r . Proceeding in this way inductively we conclude that lims→∞ x s k,l = xk,l exists for each k, l ∈ N. Using continuity of M , we have lim t→∞M    � � �∆r x s k,l −∆r x t k,l � � � ρ   ≤ 1 ⇒M    � � �∆r x s k,l −∆xk,l � � � ρ    ≤ 1. Let s ≥ no, then taking the infimum of such ρ′s we have g �� x s k,l − xk,l �� < ǫ. Thus (x s k,l − xk,l) ∈ w2 ∞ � A, M , p � (∆r). By linearity of the space w2 ∞ � A, M , p � (∆r) we have � xk,l � ∈ w2 ∞ � A, M , p � (∆r). Hence w2 ∞ � A, M , p � (∆r) is complete space. Proposition 1. (a) w2 � A, M , p � (∆r) ⊂ w2 ∞ � A, M , p � (∆r), (b) w2 o � A, M , p � (∆r) ⊂ w2 ∞ � A, M , p � (∆r). Proof. The proof is easy. Theorem 4. The spaces w2 o � A, M , p � (∆r) and w2 � A, M , p � (∆r) are nowhere dense subsets of w2 ∞ � A, M , p � (∆r). B. Hazarika, A. Esi / Eur. J. Pure Appl. Math, 8 (2015), 201-213 209 Proof. The proof is clear in view of Theorem 3 and Proposition 1. Theorem 5. (a) If 0< h= inf pk,l < pk,l ≤ 1, then w2 � A, M , p � (∆r) ⊂ w2 [A, M] (∆r). (b) If 1≤ pk,l ≤ sup pk,l <∞, then w2 [A, M] (∆r) ⊂ w2 � A, M , p � (∆r). Proof. (a) Let x = � xk,l � ∈ w2 � A, M , p � (∆r), since 0 < h = inf pk,l < pk,l ≤ 1, we obtain the following: ∞,∞ ∑ k,l=0,0 am,n,k,l M �� �∆r xk,l − L � � ρ � ≤ ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �∆r xk,l − L � � ρ ��pk,l thus x = � xk,l � ∈ w2 [A, M] (∆r). (b) Let pk,l ≥ 1 for each k , l and sup pk,l <∞ and let x = � xk,l � ∈ w2 [A, M] (∆r). Then for each 0< ǫ < 1 there exists a positive integer K such that ∞,∞ ∑ k,l=0,0 am,n,k,l M �� �∆r xk,l − L � � ρ � ≤ ǫ < 1 for all n, m≥ K . This implies that ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �∆r xk,l − L � � ρ ��pk,l ≤ ∞,∞ ∑ k,l=0,0 am,n,k,l M �� �∆r xk,l − L � � ρ � . Thus x = � xk,l � ∈ w2 � A, M , p � (∆r). This completes the proof. Corollary 1. Let A= (C , 1, 1). Then (a) If 0< h= inf pk,l < pk,l ≤ 1, then w2 � M , p � (∆r) ⊂ w2 [M] (∆r). (b) If 1≤ pk,l ≤ sup pk,l <∞, then w2 [M] (∆r) ⊂ w2 � M , p � (∆r). B. Hazarika, A. Esi / Eur. J. Pure Appl. Math, 8 (2015), 201-213 210 Theorem 6. If sup pk,l pi, j <∞ for all k ≥ i , l ≥ j, then w2 � A, M , p � ⊂ w2 � A, M , p � (∆r) and the inclusion is strict, where w2 � A, M , p � = ( x = � xk,l � ∈ w2 : P − lim m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �xk,l − L � � ρ ��pk,l = 0, for some ρ > 0 and L . Proof. Let x = � xk,l � ∈ w2 � A, M , p � . Then P − lim m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �xk,l − L � � ρ ��pk,l = 0, for some ρ > 0 and L. (3) Since sup pk,l pi, j <∞, so there exists C > 0 such that pk,l < C pi, j for all k ≥ i , l ≥ j. Thus from (3) we have, P − lim m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �xk,l − L � � ρ ��pk,l+1 = 0, P − lim m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �xk,l − L � � ρ ��pk+1,l = 0, P − lim m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �xk,l − L � � ρ ��pk+1,l+1 = 0. Now for � �∆r xk,l � � = � �∆r−1 xk,l −∆r−1 xk,l+1 −∆r−1 xk+1,l +∆ r−1 xk+1,l+1 + L − L + L − L � � we have, P − lim m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �∆r xk,l − L � � ρ ��pk,l ≤P − lim m,n ∞,∞ ∑ k,l=0,0 am,n,k,l � M �� �∆r−1 xk,l − L � � ρ + � �∆r−1 xk+1,l − L � � ρ + � �∆r−1 xk,l+1 − L � � ρ + � �∆r−1 xk+1,l+1 − L � � ρ ��pk,l ≤D2P − lim m,n ∞,∞ ∑ k,l=0,0 am,n,k,l   � M �� �∆r−1 xk,l − L � � ρ ��pk,l + � M �� �∆r−1 xk+1,l − L � � ρ ��pk,l + � M �� �∆r−1 xk,l+1 − L � � ρ ��pk,l + � M �� �∆r−1 xk+1,l+1 − L � � ρ ��pk,l   B. Hazarika, A. Esi / Eur. J. Pure Appl. Math, 8 (2015), 201-213 211 ≤D2P − lim m,n ∞,∞ ∑ k,l=0,0 am,n,k,l   � M �� �∆r−1 xk,l − L � � ρ ��pk,l + � M �� �∆r−1 xk+1,l − L � � ρ ��pk+1,l + � M �� �∆r−1 xk,l+1 − L � � ρ ��pk,l+1 + � M �� �∆r−1 xk+1,l+1 − L � � ρ ��pk+1,l+1  = 0 where D =max � 1,2H−1 � . Thus x = � xk,l � ∈ w2 � A, M , p � (∆r). This completes the proof. The inclusion is strict follows from the following example. Example 1. Let A= (C , 1, 1), M (x) = x p, pk,l = 1 for all k odd and for all l ∈ N and pk,l = 2 otherwise. Consider the sequence x = � xk,l � defined by xk,l = (k + l)r for all k, l ∈ N. We have ∆r xk,l = 0 for all k, l ∈ N. Hence x = � xk,l � ∈ w2 � A, M , p � (∆r) but x = � xk,l � /∈ w2 � A, M , p � . Let E be a sequence space. Then E is called (a) solid (or normal) if (αk xk) ∈ E whenever (xk) ∈ E for all sequences (αk) of scalars with |αk| ≤ 1 for all k ∈ N; (b) monotone provided E contains the canonical preimages of all its step spaces. It is a well known result that if E is normal then it is monotone. Theorem 7. The spaces w2 o � A, M , p � (∆r) and w2 ∞ � A, M , p � (∆r) are normal as well as mono- tone. Proof. Let (αk,l) be a double sequences of scalars such that |αk,l | ≤ 1 for all k, l ∈ N. Since M is monotone, we get for some ρ > 0 ∞,∞ ∑ k,l=0,0 am,n,k,l � M � |∆r(αk,l xk,l)| ρ ��pk,l ≤ ∞,∞ ∑ k,l=0,0 am,n,k,l � M � sup |αk,l | |∆r xk,l | ρ ��pk,l ≤ ∞,∞ ∑ k,l=0,0 am,n,k,l � M � |∆r xk,l | ρ ��pk,l which leads us to the desired results. 4. Double ∆r− Statistical Convergence The concept of statistical convergence for single sequences was introduced by Fast [3] in 1951. Later, Mursaleen and Edely [7] defined the statistical analogue for double sequence x = � xk,l � as follows: A real double sequence x = � xk,l � is said to be P-statistically convergent to L provided that for each ǫ > 0 P − lim m,n 1 mn � the number of (k, l) : k < m, l < n; � �xk,l − L � �≥ ǫ = 0. B. Hazarika, A. Esi / Eur. J. Pure Appl. Math, 8 (2015), 201-213 212 In this case, we write st2− limk,l xk,l = L and we denote the set of all P-statistically convergent double sequences by st2. Definition 3. A real double sequence x = � xk,l � is said to be P-statistically ∆r -convergent to L provided that for each ǫ > 0 P − lim m,n 1 mn � the number of (k, l) : k < m, l < n; � �∆r xk,l − L � �≥ ǫ = 0. In this case, we write st2(∆ r) − limk,l xk,l = L and we denote the set of all P-statistically ∆r - convergent double sequences by st2(∆ r). Theorem 8. If M be an Orlicz function, then w2 [M] (∆r) ⊂ st2(∆ r). Proof. Suppose that x = � xk,l � ∈ w2 [M] (∆r) and ǫ > 0, then we obtain the following for every n and m 1 mn m−1,n−1 ∑ k,l=0,0 M �� �∆r xk,l − L � � ρ � ≥ 1 mn m−1,n−1 ∑ k,l=0,0 & |∆r xk,l−L|≥ǫ M �� �∆r xk,l − L � � ρ � ≥M (ǫ) mn � the number of (k, l) : k < m, l < n; � �∆r xk,l − L � �≥ ǫ . Hence x = � xk,l � ∈ st2(∆ r). Theorem 9. st2(∆ r) = w2 o [M] (∆ r) if and only if the Orlicz function M is bounded. Proof. Suppose that M is bounded and x = � xk,l � ∈ st2(∆ r). Since M is bounded then there exists an integer K such that M(x)≤ K , for all x ≥ 0. Then for each m and n, we have 1 mn m−1,n−1 ∑ k,l=0,0 M �� �∆r xk,l � � ρ � = 1 mn m−1,n−1 ∑ k,l=0,0 & |∆r xk,l−L|≥ǫ M �� �∆r xk,l � � ρ � + 1 mn m−1,n−1 ∑ k,l=0,0 & |∆r xk,l−L|<ǫ M �� �∆r xk,l � � ρ � ≤ K mn � the number of (k, l) : k < m, l < n; � �∆r xk,l � �≥ ǫ +M (ǫ) and thus the Pringsheim’s limit on m and n grant us the result. Conversely, suppose that M is unbounded so that there is a positive double sequence � zmn � with M � zmn � = (mn)2 for m, n= 1,2, . . .. Now the sequence x = � xk,l � defined by ∆r xk,l = zmn if k, l = (mn)2 for m, n= 1,2, . . . and ∆r xk,l = 0, otherwise. Then we have 1 mn � the number of (k, l) : k < m, l < n; � �∆r xk,l � �≥ ǫ ≤ p mn mn → 0, as m, n→∞. Hence xk,l → 0 � st2(∆ r) � . But x = � xk,l � /∈ w2 o [M] (∆ r), contradicting st2(∆ r) = w2 o [M] (∆ r). This completes the proof. REFERENCES 213 References [1] A. Esi. On some new difference double sequence spaces via Orlicz function. Journal of Advanced Studies in Topology, 2(2):16–25, 2011. [2] M. 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