CUBO A Mathematical Journal Vol.14, No¯ 03, (167–190). October 2012 Some generalized difference double sequence spaces defined by a sequence of Orlicz-functions Kuldip Raj and Sunil K. Sharma School of Mathematics, Shri Mata Vaishno Devi University, Katra-182320, J&K, India email: kuldeepraj68@rediffmail.com, sunilksharma42@yahoo.co.in ABSTRACT In the present paper we introduce some generalized difference double sequence spaces defined by a sequence of Orlicz-functions. We study some topological properties and some inclusion relations between these spaces. We also make an effort to study these properties over n-normed spaces. RESUMEN En este art́ıculo introducimos algunos espacios de sucesiones doble-diferencia gen- eralizadas definidas por una sucesión de funciones de Orlicz. Estudiamos algunas propiedades topológicas y algunas relaciones de inclusión entre estos espacios. Además, hacemos un esfuerzo para estudiar estas propiedades en espacios n-normados. Keywords and Phrases: P-convergent, Orlicz function, sequence spaces, paranorm space, n- normed space 2010 AMS Mathematics Subject Classification: Primary 42B15; Secondary 40C05 168 K. Raj and S. K. Sharma CUBO 14, 3 (2012) 1 Introduction and Preliminaries The initial works on double sequences is found in Bromwich [4]. Later on, it was studied by Hardy [6], Moricz [17], Moricz and Rhoades [18], Tripathy ([33], [34]), Basarir and Sonalcan [2] and many others. Hardy[6] introduced the notion of regular convergence for double sequences. Quite recently, Zeltser [36] in her Ph.D thesis has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences. Mursaleen and Edely [21] have recently introduced the statistical convergence and Cauchy convergence for double sequences and given the relation between statistical convergent and strongly Cesaro summable double sequences. Nextly, Mursaleen [19] and Mursaleen and Edely [22] have defined the almost strong regularity of matrices for double sequences and applied these matrices to establish a core theorem and introduced the M-core for double sequences and determined those four dimensional matrices transforming every bounded double sequences x = (xmn) into one whose core is a subset of the M-core of x. More recently, Altay and Basar [1] have defined the spaces BS, BS(t), CSp, CSbp, CSr and BV of double sequences consisting of all double series whose sequence of partial sums are in the spaces Mu, Mu(t), Cp, Cbp, Cr and Lu, respectively and also examined some properties of these sequence spaces and determined the α-duals of the spaces BS, BV, CSbp and the β(v)-duals of the spaces CSbp and CSr of double series. Now, recently Basar and Sever [3] have introduced the Banach space Lq of double sequences corresponding to the well known space ℓq of single sequences and examined some properties of the space Lq. Let w 2 denote the set of all double sequences of complex numbers. By the convergence of a double sequence we mean the convergence of the Pringsheim sense i.e. a double sequence x = (xkl) has Pringsheim limit L (denoted by P − lim x = L) provided that given ǫ > 0 there exists n ∈ N such that |xkl − L| < ǫ whenever k, l > n see [26]. We shall write more briefly as P-convergent. We shall denote the space of all P-convergent sequences by c2. The double sequence x = (xkl) is bounded if there exists a positive number M such that |xkl| < M for all k and l. Let l2 ∞ the space of all bounded double sequence such that ||xkl||∞,2 = supkl |xkl| < ∞. For more details about double sequence spaces see ([30], [31],[32]) and references therein. The notion of difference sequence spaces was introduced by Kızmaz [13], who studied the difference sequence spaces l∞(∆), c(∆) and c0(∆). The notion was further generalized by Et. and Çolak [5] by introducing the spaces l∞(∆ n), c(∆n) and c0(∆ n). Let w be the space of all complex or real sequences x = (xk) and let m, s be non-negative integers, then for Z = l∞, c, c0 we have sequence spaces Z(∆ms ) = {x = (xk) ∈ w : (∆ m s xk) ∈ Z}, where ∆ms x = (∆ m s xk) = (∆ m−1 s xk − ∆ m−1 s xk+1) and ∆ 0 sxk = xk for all k ∈ N, which is equivalent to the following binomial representation ∆ms xk = m∑ v=0 (−1)v ( m v ) xk+sv. Taking s = 1, we get the spaces which were studied by Et and Çolak [5]. Taking m = s = 1, we get the spaces which were introduced and studied by Kızmaz [13]. CUBO 14, 3 (2012) Some generalized difference double sequence spaces ... 169 An Orlicz function M : [0, ∞) → [0, ∞) is a continuous, non-decreasing and convex function such that M(0) = 0, M(x) > 0 for x > 0 and M(x) −→ ∞ as x −→ ∞. Lindenstrauss and Tzafriri [14] used the idea of Orlicz function to define the following sequence space: lM = { x ∈ w : ∞∑ k=1 M ( |xk| ρ ) < ∞ } which is called as an Orlicz sequence space. Also lM is a Banach space with the norm ||x|| = inf { ρ > 0 : ∞∑ k=1 M ( |xk| ρ ) ≤ 1 } . Also, it was shown in [14] that every Orlicz sequence space lM contains a subspace isomorphic to lp(p ≥ 1). The ∆2− condition is equivalent to M(Lx) ≤ LM(x), for all L with 0 < L < 1. An Orlicz function M can always be represented in the following integral form M(x) = ∫x 0 η(t)dt where η is known as the kernel of M, is right differentiable for t ≥ 0, η(0) = 0, η(t) > 0, η is non-decreasing and η(t) → ∞ as t → ∞. Let X be a linear metric space. A function p : X → R is called paranorm, if (1) p(x) ≥ 0, for all x ∈ X, (2) p(−x) = p(x), for all x ∈ X, (3) p(x + y) ≤ p(x) + p(y), for all x, y ∈ X, (4) if (λn) is a sequence of scalars with λn → λ as n → ∞ and (xn) is a sequence of vectors with p(xn − x) → 0 as n → ∞, then p(λnxn − λx) → 0 as n → ∞. A paranorm p for which p(x) = 0 implies x = 0 is called total paranorm and the pair (X, p) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [35], Theorem 10.4.2, P-183). For more details about sequence spaces see ([12], [15], [20], [23], [24], [25], [27]) and references therein. Let M = (Mk,l) be a sequence of orlicz functions,p = (pk,l) be a bounded sequence of positive real numbers and u = (uk,l) be a sequence of strictly positive real numbers. Let X be a semi- normed space over the complex field C with the seminorm q. Now we define the following classes of sequences in the present paper: c2(∆mn , M, u, p, q, s) = { x = (xk,l) ∈ w 2 : P − lim k,l (kl)−suk,l [ Mk,l ( q ( ∆mn xk,l − L ρ ))]pk,l = 0, for some ρ > 0, L and s ≥ 0 } , 170 K. Raj and S. K. Sharma CUBO 14, 3 (2012) c20(∆ m n , M, u, p, q, s) = { x = (xk,l) ∈ w 2 : P − lim k,l (kl)−suk,l [ Mk,l ( q ( ∆mn xk,l ρ ))]pk,l = 0, for some ρ > 0 and s ≥ 0 } and l2 ∞ (∆mn , M, u, p, q, s) = { x = (xk,l) ∈ w 2 : sup k,l (kl)−suk,l [ Mk,l ( q ( ∆mn xk,l ρ ))]pk,l < ∞, for some ρ > 0 and s ≥ 0 } . If we take M(x) = x, we get c2(∆mn , u, p, q, s) = { x = (xk,l) ∈ w 2 : P − lim k,l (kl)−suk,l [ q ( ∆mn xk,l − L ρ )]pk,l = 0, for some ρ > 0, L and s ≥ 0 } , c20(∆ m n , u, p, q, s) = { x = (xk,l) ∈ w 2 : P − lim k,l (kl)−suk,l [ q ( ∆mn xk,l ρ )]pk,l = 0, for some ρ > 0 and s ≥ 0 } and l2 ∞ (∆mn , u, p, q, s) = { x = (xk,l) ∈ w 2 : sup k,l (kl)−suk,l [ q ( ∆mn xk,l ρ )]pk,l < ∞, for some ρ > 0 and s ≥ 0 } . If we take p = (pk,l) = 1, we get c2(∆mn , M, u, q, s) = { x = (xk,l) ∈ w 2 : P − lim k,l (kl)−suk,l [ Mk,l ( q ( ∆mn xk,l − L ρ ))] = 0, for some ρ > 0, L and s ≥ 0 } , c20(∆ m n , M, u, q, s) = { x = (xk,l) ∈ w 2 : P − lim k,l (kl)−suk,l [ Mk,l ( q ( ∆mn xk,l ρ ))] = 0, for some ρ > 0 and s ≥ 0 } and l2 ∞ (∆mn , M, u, q, s) = { x = (xk,l) ∈ w 2 : sup k,l (kl)−suk,l [ Mk,l ( q ( ∆mn xk,l ρ ))] < ∞, for some ρ > 0 and s ≥ 0 } . If we take m = n = 0 and q(x) = |x|, then we get new double sequence spaces as follows : c2(M, u, p, s) = { x = (xk,l) ∈ w 2 : P − lim k,l (kl)−suk,l [ Mk,l ( |xk,l − L| ρ )]pk,l = 0, CUBO 14, 3 (2012) Some generalized difference double sequence spaces ... 171 for some ρ > 0, L and s ≥ 0 } , c20(M, u, p, s) = { x = (xk,l) ∈ w 2 : P − lim k,l (kl)−suk,l [ Mk,l ( |xk,l| ρ )]pk,l = 0, for some ρ > 0 and s ≥ 0 } and l2 ∞ (M, u, p, s) = { x = (xk,l) ∈ w 2 : sup k,l (kl)−suk,l [ Mk,l ( |xk,l| ρ )]pk,l < ∞, for some ρ > 0 and s ≥ 0 } . If we take m = n = 1 and q(x) = |x|, then we get new double sequence spaces as follows : c2(∆mn , M, u, p, s) = { x = (xk,l) ∈ w 2 : P − lim k,l (kl)−suk,l [ Mk,l ( |∆xk,l − L| ρ )]pk,l = 0, for some ρ > 0, L and s ≥ 0 } , c20(∆ m n , M, u, p, s) = { x = (xk,l) ∈ w 2 : P − lim k,l (kl)−suk,l [ Mk,l ( |∆xk,l| ρ )]pk,l = 0, for some ρ > 0 and s ≥ 0 } and l2 ∞ (∆mn , M, u, p, s) = { x = (xk,l) ∈ w 2 : sup k,l (kl)−suk,l [ Mk,l ( |∆mn xk,l| ρ )]pk,l < ∞, for some ρ > 0 and s ≥ 0 } . The following inequality will be used throughout the paper. Let p = (pk,l) be a double sequence of positive real numbers with 0 < pk,l ≤ sup k,l = H and let K = max{1, 2H−1}. Then for the factorable sequences {ak,l} and {bk,l} in the complex plane, we have |ak,l + bk,l| pk,l ≤ K(|ak,l| pk,l + |bk,l| pk,l). (1.1) The main goal of this paper is to extend a few known results in the literature from single differ- ence sequence spaces to double difference sequence spaces. We also make an effort to study some topological properties and inclusion relations between above defined sequence spaces. 172 K. Raj and S. K. Sharma CUBO 14, 3 (2012) 2 Main Results Theorem 2.1 Let M = (Mk,l) be a sequence of Orlicz functions, p = (pk,l) be a bounded se- quence of positive real numbers and u = (uk,l) be a sequence of strictly positive real numbers, then the classes of sequences c20(∆ m n , M, u, p, q, s), c 2(∆mn , M, u, p, q, s) and l∞(∆ m n , M, u, p, q, s) are linear spaces over the field of complex numbers C. Proof. Let x = (xk,l), y = (yk,l) ∈ c 2 0(∆ m n , M, u, p, q, s) and α, β ∈ C. Then there exist positive numbers ρ1 and ρ2 such that lim k,l (kl)−suk,l [ Mk,l ( q ( ∆mn xk,l ρ1 ))]pk,l = 0, for some ρ1 > 0 and lim k,l (kl)−suk,l [ Mk,l ( q ( ∆nmyk,l ρ2 ))]pk,l = 0, for some ρ2 > 0. Let ρ3 = max(2|α|ρ1, 2|β|ρ2). Since M = (Mk,l) is non-decreasing convex function and so by using inequality (1.1), we have lim k,l (kl)−suk,l [ Mk,l ( q ( ∆mn (αxk,l + βyk,l) ρ3 ))]pk,l = lim k,l (kl)−suk,l [ Mk,l ( q ( α∆mn xk,l ρ3 ) + q ( β∆mn yk,l ρ3 ))]pk,l ≤ K lim k,l 1 2pk,l (kl)−suk,l [ Mk,l ( q ( ∆mn xk,l ρ1 ))]pk,l + K lim k,l 1 2pk,l (kl)−suk,l [ Mk,l ( q ( ∆mn yk,l ρ2 ))]pk,l ≤ K lim k,l (kl)−suk,l [ Mk,l ( q ( ∆mn xk,l ρ1 ))]pk,l + K lim k,l (kl)−suk,l [ Mk,l ( q ( ∆mn yk,l ρ2 ))]pk,l = 0. So, αx + βy ∈ c20(∆ n m, M, u, p, q, s). Hence c 2 0(∆ n m, M, u, p, q, s) is a linear space. Similarly, we can prove that c2(∆nm, M, u, p, q, s) and l 2 ∞ (∆nm, M, u, p, q, s) are linear spaces. Theorem 2.2 Let M = (Mk,l) be a sequence of Orlicz functions, p = (pk,l) be a bounded sequence of positive real numbers and u = (uk,l) be a sequence of strictly positive real numbers. For Z2 = l2 ∞ , c2 and c20, the spaces Z 2(∆mn , M, u, p, q, s) are paranormed spaces, paranormed by g(x) = nm∑ k,l=1 q(xk,l) + inf { ρ pk,l H : sup k,l (kl)−suk,lMk,l ( q ( ∆mn xk,l ρ )) ≤ 1 } where H = max(1, sup k,l pk,l). CUBO 14, 3 (2012) Some generalized difference double sequence spaces ... 173 Proof. Clearly g(−x) = g(x), g(0) = 0. Let (xk,l) and (yk,l) be any two sequences belong to any one of the spaces Z2(∆nm, M, u, p, q, s), for Z 2 = c20, c 2 and l2 ∞ . Then, we get ρ1, ρ2 > 0 such that sup k,l (kl)−suk,lMk,l ( q ( ∆mn xk,l ρ1 )) ≤ 1 and sup k,l (kl)−suk,lMk,l ( q ( ∆mn yk,l) ρ2 ) ≤ 1. Let ρ = ρ1 + ρ2. Then by convexity of M = (Mk,l), we have sup k,l (kl)−suk,lMk,l ( q ( ∆mn (xk,l + yk,l) ρ )) ≤ ( ρ1 ρ1 + ρ2 ) sup k,l (kl)−suk,lMk,l ( q ( ∆mn xk,l ρ1 )) + ( ρ2 ρ1 + ρ2 ) sup k,l (kl)−suk,lMk,l ( q ( ∆mn yk,l ρ2 )) ≤ 1. Hence we have, g(x + y) = mn∑ k,l=1 q(xk,l + yk,l) + inf { ρ pk,l H : sup k,l (kl)−suk,lMk,l ( q ( ∆mn (xk,l + yk,l) ρ )) ≤ 1 } ≤ mn∑ k,l=1 q(xk,l) + inf { ρ pk,l H 1 : sup k,l (kl)−suk,lMk,l ( q ( ∆mn xk,l ρ1 )) ≤ 1 } + mn∑ k,l=1 q(yk,l) + inf { ρ pk,l H 2 : sup k,l (kl)−suk,lMk,l ( q ( ∆mn yk,l ρ2 )) ≤ 1 } . This implies that g(x + y) ≤ g(x) + g(y). The continuity of the scalar multiplication follows from the following inequality g(µx) = mn∑ k,l=1 q(µxk,l) + inf { ρ pk,l H : sup k,l (kl)−suk,lMk,l ( q ( ∆mn µxk,l ρ )) ≤ 1 } = |µ| mn∑ k,l=1 q(xk,l) + inf { (t|µ|) pk,l H : sup k,l (kl)−suk,lMk,l ( q ( ∆mn xk,l t )) ≤ 1 } , where t = ρ |µ| . Hence the space Z2(∆nm, M, u, p, q, s), for Z 2 = c20, c 2 and l2 ∞ is a paranormed space, paranormed by g. Theorem 2.3 Let M = (Mk,l) be a sequence of Orlicz functions, p = (pk,l) be a bounded sequence of positive real numbers and u = (uk,l) be a sequence of strictly positive real numbers. 174 K. Raj and S. K. Sharma CUBO 14, 3 (2012) For Z2 = l2 ∞ , c2 and c20, the spaces Z 2(∆mn , M, u, p, q, s) are complete paranormed spaces, para- normed by g(x) = nm∑ k,l=1 q(xk,l) + inf { ρ pk,l H : sup k,l (kl)−suk,lMk,l ( q ( ∆mn xk,l ρ )) ≤ 1 } , where H = max(1, sup k,l pk,l). Proof. We prove the result for the space l2 ∞ (∆mn , M, u, p, q, s). Let (x i k,l) be any Cauchy se- quence in l2 ∞ (∆mn , M, u, p, q, s). Let ǫ > 0 be given and for t > 0, choose x0 be fixed such that uk,lMk,l ( tx0 2 ) ≥ 1, then there exists a positive integer n0 ∈ N such that g(x i k,l − x j k,l ) < ǫ x0t , for all i, j ≥ n0. Using the definition of paranorm, we get mn∑ k,l=1 q(xik,l − x j k,l ) + inf { ρ pk,l H : sup k,l (kl)−suk,lMk,l ( q (∆mn (x i k,l − x j k,l ) ρ ))} < ǫ x0t , for all i, j ≥ n0 (2.1). Hence we have, mn∑ k,l=1 q(xik,l − x j k,l) < ǫ, for all i, j ≥ n0. This implies that q(xik,l − x j k,l ) < ǫ, for all i, j ≥ n0 and 1 ≤ k ≤ nm. Thus (xik,l) is a Cauchy sequence in C for k, l = 1, 2, ...., nm. Hence (x i k,l) is convergent in C for k, l = 1, 2, ...., nm. Let lim i→∞ xik,l = xk,l, say for k, l = 1, 2, ...., nm. (2.2) Again from equation (2.1) we have, inf { ρ pk,l H : sup k,l (kl)−suk,lMk,l ( q (∆mn (x i k,l − x j k,l ) ρ )) ≤ 1 } < ǫ, for all i, j ≥ n0. Hence we get sup k,l (kl)−suk,lMk,l ( q (∆mn (x i k,l − x j k,l ) g(xi − xj) )) ≤ 1, for all i, j ≥ n0. It follows that (kl)−suk,lMk,l ( q ( ∆ m n (x i k,l−x j k,l ) g(xi−xj) )) ≤ 1, for each k, l ≥ 1 and for all i, j ≥ n0. For t > 0 with (kl)−suk,lMk,l( tx0 2 ) ≥ 1, we have (kl)−suk,lMk,l ( q (∆mn (x i k,l − x j k,l ) g(xi − xj) )) ≤ (kl)−suk,lMk,l( tx0 2 ). This implies that q(∆mn x i k,l − ∆ m n x j k,l ) < tx0 2 ǫ tx0 = ǫ 2 . CUBO 14, 3 (2012) Some generalized difference double sequence spaces ... 175 Hence q(∆mn x i k,l) is a Cauchy sequence in C for all k, l ∈ N. This implies that q(∆ m n x i k,l) is convergent in C for all k, l ∈ N. Let lim i→∞ q(∆mn x i k,l) = yk,l for each k, l ∈ N. Let k, l = 1, then we have lim i→∞ q(∆mn x i 1,1) = lim i→∞ m∑ v=0 (−1)v ( m v ) xi1+nv,1+mv = y1,1. (2.3) We have by equation (2.2) and equation (2.3) lim i→∞ ximn+1 = xmn+1, exists. Proceeding in this way inductively, we have lim i→∞ xik,l = xk,l exists for each k, l ∈ N. Now we have for all i, j ≥ n0, mn∑ k,l=1 q(xik,l − x j k,l ) + inf { ρ pk,l H : sup k,l (kl)−suk,lMk,l ( q (∆mn (x i k,l − x j k,l ) ρ )) ≤ 1 } < ǫ. This implies that lim j→∞ { mn∑ k,l=1 q(xik,l − x j k,l ) + inf { ρ pk,l H : sup k,l (kl)−suk,lMk,l ( q (∆mn (x i k,l − x j k,l) ρ )) ≤ 1 }} < ǫ, for all i ≥ n0. Using the continuity of Mk,l, we have mn∑ k,l=1 q(xik,l − xk,l) + inf { ρ pk,l H : sup k,l (kl)−suk,lMk,l ( q ( ∆mn x i k,l − ∆ m n xk,l ρ )) ≤ 1 } < ǫ, for all i ≥ n0. It follows that (x i −x) ∈ l2 ∞ (∆mn , M, u, p, q, s). Since x i ∈ l2 ∞ (∆nm, M, u, p, q, s) and l2 ∞ (∆mn , M, u, p, q, s) is a linear space, so we have x = x i − (xi − x) ∈ l2 ∞ (∆nm, M, u, p, q, s). This completes the proof. Similarly, we can prove that c2(∆nm, M, u, p, q, s) and c 2 0(∆ n m, M, u, p, q, s) are complete paranormed spaces in view of the above proof. Theorem 2.4 Let m ≥ 1, then for all 0 < i ≤ m, Z2(∆in, M, u, p, q, s) ⊂ Z 2(∆mn , M, u, p, q, s), where Z2 = c2, c20 and l 2 ∞ . Proof. We will prove it for only c20(∆ m−1 n , M, u, p, q, s). Let x = (xk,l) ∈ c 2 0(∆ m−1 n , M, u, p, q, s). Then P − lim k,l (kl)−suk,l [ Mk,l ( q(∆m−1n xk,l) ρ )]Pk,l = 0, for some ρ > 0 and s ≥ 0 (2.4) Then from (2.4) we have P − lim k,l (kl)−suk,l [ Mk,l ( q ( ∆mn xk,l ρ ))]Pk,l+1 = 0, P − lim k,l (kl)−suk,l [ Mk,l ( q ( ∆mn xk,l ρ ))]Pk+1,l = 0 and P − lim k,l (kl)−suk,l [ Mk,l ( q ( ∆mn xk,l ρ ))]Pk+1,l+1 = 0. 176 K. Raj and S. K. Sharma CUBO 14, 3 (2012) Now for ∆mn x = (∆ m n xk,l) = (∆ m−1 n xk,l − ∆ m−1 n xk,l+1 − ∆ m−1 n xk+1,l + ∆ m−1 n xk+1,l+1), we have (kl)−suk,l [ Mk,l ( q ( ∆ m n xk,l ρ ))]Pk,l ≤ (kl)−suk,l [ Mk,l ( q ( ∆m−1n xk,l ρ ) + q ( ∆m−1n xk,l+1 ρ ) + q ( ∆m−1n xk+1,l ρ ) + q ( ∆m−1n xk+1,l+1 ρ ))]Pk,l ≤ K2(kl)−suk,l {[ Mk,l ( q ( ∆m−1n xk,l ρ ))]Pk,l + uk,l [ Mk,l ( q ( ∆m−1n xk+1,l ρ ))]Pk,l + uk,l [ M ( q ( ∆m−1n xk,l+1 ρ ))]Pk,l + uk,l [ Mk,l ( q ( ∆m−1n xk+1,l+1 ρ ))]Pk,l } ≤ K2 {[ (kl)−suk,lMk,l ( q ( ∆m−1n xk,l ρ ))]Pk,l + [ (kl)−suk,lMk,l ( q ( ∆m−1n xk+1,l ρ ))]Pk+1,l + [ (kl)−suk,lMk,l ( q ( ∆m−1n xk,l+1 ρ ))]Pk,l+1 + uk,l [ (kl)−sMk,l ( q ( ∆m−1n xk+1,l+1 ρ ))]Pk+1,l+1 } from this it follows that x = (xk,l) ∈ c 2 0(∆ m n , M, u, p, q, s) and hence c 2 0(∆ m−1 n , M, u, p, q, s) ⊂ c20(∆ m n , M, u, p, q, s). On applying the principle of induction, it follows that c 2 0(∆ i n, M, u, p, q, s) ⊂ c20(∆ m n , M, u, p, q, s) for i = 0, 1, 2, · · · , m − 1. Similarly, we can prove the other cases. Theorem 2.5 (a) If 0 < inf k,l pk,l ≤ pk,l < 1, then Z 2(∆mn , M, u, p, q, s) ⊂ Z 2(∆mn , M, u, q, s), (b) If 1 < pk,l ≤ sup k,l pk,l < ∞, then Z 2(∆mn , M, u, q, s) ⊂ Z 2(∆mn , M, u, p, q, s), where Z2 = c2, c20 and l 2 ∞ . Proof. (i) Let x = (xk,l) ∈ l 2 ∞ (∆mn , M, u, p, q, s). Since 0 < inf pk,l ≤ 1, we have sup k,l (kl)−suk,l [ Mk,l ( q ( ∆mn xk,l ρ ))] ≤ sup k,l (kl)−suk,l [ Mk,l ( q ( ∆mn xk,l ρ ))]pk,l , and hence x = (xk,l) ∈ l 2 ∞ (∆mn , M, u, p, q, s). (ii) Let pk,l for each (k, l) and sup k,l pk,l < ∞. Let x = (xk,l) ∈ l 2 ∞ (∆mn , M, u, q, s). Then, for each 0 < ǫ < 1, there exists a positive integer N such that sup k,l (kl)−suk,l [ Mk,l ( q ( ∆mn xk,l ρ ))] ≤ ǫ < 1, for all m, n ∈ N. This implies that sup k,l (kl)−suk,l [ Mk,l ( q (∆mn xk,l ρ ))]pk,l ≤ sup k,l (kl)−suk,l [ Mk,l ( q (∆mn xk,l ρ ))] . CUBO 14, 3 (2012) Some generalized difference double sequence spaces ... 177 Thus x = (xk,l) ∈ l 2 ∞ (∆mn , M, u, p, q, s) and this completes the proof. Theorem 2.6 Let M′ = (M′k,l) and M ′′ = (M′′k,l) be two sequences of Orlicz functions satisfying ∆2-condition. If β = lim t→∞ M′′k,l(t) t ≥ 1, then Z2(∆mn , M ′, u, p, q, s) = Z2(∆mn , M ′′◦M′, u, p, q, s), where Z2 = c2, c20 and l 2 ∞ . Proof. We prove it for Z2 = c2 and the other cases will follows on applying similar techniques. Let x = (xk,l) ∈ c 2(∆mn , M ′, u, p, q, s), then P − lim k,l (kl)−s [ M′k,l ( q ( ∆mn xk,l − L ρ ))]Pk,l = 0. Let 0 < ǫ < 1 and δ with 0 < δ < 1 such that M′′k,l(t) < ǫ for 0 ≤ t < δ. Let yk,l = M ′ k,l ( q ( ∆mn xk,l − L ρ )) and consider [M′′k,l(yk,l)] pk,l = [M′′k,l(yk,l)] pk,l + [M′′k,l(yk,l)] pk,l (2.5) where the first term is over yk,l ≤ δ and the second is over yk,l > δ. From the first term in (2.5), we have (kl)−s[M′′k,l(yk,l)] pk,l < (kl)−s[M′′k,l(2)] H [(yk,l)] pk,l (2.6) On the other hand, we use the fact that yk,l < yk,l δ < 1 + yk,l δ . Since (M′′k,l) for each k, l is non-decreasing and convex, it follows that M′′k,l(yk,l) < M ′′ k,l ( 1 + yk,l δ ) < 1 2 M′′k,l(2) + 1 2 M′′k,l( 2yk,l δ ). Since (M′′k,l) for each k, l satisfies ∆2-condition, we have M′′k,l(yk,l) < 1 2 K yk,l δ M′′k,l(2) + 1 2 K yk,l δ M′′k,l(2) = K yk,l δ M′′k,l(2). Hence, from the second term in (2.5), it follows that (kl)−s[M′′(yk,l)] pk,l ≤ max ( 1, (KM′′(2)δ−1)H ) (kl)−s[(yk,l)] pk,l (2.7) By the inequalities (2.6) and (2.7), taking limit in the Pringsheim sense, we have x = (xk,l) ∈ c2(∆mn , M ′′ ◦ M′, u, p, q, s). Observe that in this part of the proof we did not need β ≥ 1. Now, let β ≥ 1 and x = (xk,l) ∈ c 2(M′, ∆rn, u, q, p). Then, we have M ′′ k,l(t) ≥ β(t) for all t ≥ 0. It follows that x = (xk,l) ∈ c 2(∆mn , M ′′ ◦ M′, u, p, q, s) implies x = (xk,l) ∈ c 2(∆mn , M ′, u, p, q, s). 178 K. Raj and S. K. Sharma CUBO 14, 3 (2012) This implies c2(∆mn , M ′, u, p, q, s) = c2(∆mn , M ′′ ◦ M′, u, p, q, s). Theorem 2.7 Let M′ = (M′k,l) and M ′′ = (M′′k,l) be two sequences of Orlicz functions, q, q1 and q2 be seminorms and s, s1 and s2 be positive real numbers. Then (1) Z2(∆mn , M ′, u, p, q, s) ∩ Z2(∆mn , M ′′, u, p, q, s) ⊂ Z2(∆mn , M ′ + M′′, u, p, q, s), (2) Z2(∆mn , M, u, p, q1, s) ∩ Z 2(∆mn , M, u, p, q2, s) ⊂ Z 2(∆mn , M, u, p, q1 + q2, s), (3) If q1 is stronger than q2, then Z 2(∆mn , M, u, p, q1, s) ⊂ Z 2(∆mn , M, u, p, q2, s) (4) If s1 ≤ s2, then Z 2(∆mn , M, u, p, q, s1) ⊂ Z 2(∆mn , M, u, p, q, s2), where Z2 = c2, c20 and l 2 ∞ . Proof. (1) Let x = (xk,l) ∈ c 2(∆mn , M ′, u, p, q, s) ∩ c2(∆m, M′′, u, p, q, s). Then P − lim k,l (kl)−suk,l [ M′k,l ( q ( ∆mn xk,l − L ρ1 ))]Pk,l = 0, for some ρ1 > 0, P − lim k,l (kl)−suk,l [ M′′k,l ( q ( ∆mn xk,l − L ρ2 ))]Pk,l = 0, for some ρ2 > 0. Let ρ = max(ρ1, ρ2). The result follows from the following inequality (kl)−s [ (M′ + M′′) ( q ( ∆ m n xk,l−L ρ ))]Pk,l ≤ K { (kl)−suk,l [ M ′ ( q ( ∆mn xk,l − L ρ1 ))]Pk,l + (kl)−suk,l [ M ′′ ( q ( ∆mn xk,l − L ρ2 ))]Pk,l } . The proofs of (2), (3) and (4) follows by same pattern. Theorem 2.8 For any sequence of orlicz functions, if q1 ≡ (equivalent to) q2, then Z2(∆mn , M, u, p, q1, s) = Z 2(∆mn , M, u, p, q2, s), where Z 2 = c2, c20 and l 2 ∞ . Proof.It is easy to prove so we omit the details. 3 Some generalized difference double sequence spaces over n-normed spaces The concept of 2-normed spaces was initially developed by Gähler[8] in the mid of 1960’s, while that of n-normed spaces one can see in Misiak[16]. Since then, many others have studied this concept and obtained various results, see Gunawan ([9],[10]), Gunawan and Mashadi [11] and many others. CUBO 14, 3 (2012) Some generalized difference double sequence spaces ... 179 Let n ∈ N and X be a linear space over the field K, where K is the field of real or complex numbers of dimension d, where d ≥ n ≥ 2. A real valued function ||·, · · · , ·|| on Xn satisfying the following four conditions: (1) ||x1, x2, · · · , xn|| = 0 if and only if x1, x2, · · · , xn are linearly dependent in X; (2) ||x1, x2, · · · , xn|| is invariant under permutation; (3) ||αx1, x2, · · · , xn|| = |α| ||x1, x2, · · · , xn|| for any α ∈ K, and (4) ||x + x′, x2, · · · , xn|| ≤ ||x, x2, · · · , xn|| + ||x ′, x2, · · · , xn|| is called an n-norm on X and the pair (X, ||·, · · · , ·||) is called a n-normed space over the field K. For example, we may take X = Rn being equipped with the n-norm ||x1, x2, · · · , xn||E = the volume of the n-dimensional parallelopiped spanned by the vectors x1, x2, · · · , xn which may be given explicitly by the formula ||x1, x2, · · · , xn||E = | det(xij)|, where xi = (xi1, xi2, · · · , xin) ∈ R n for each i = 1, 2, · · · , n and ||.||E denotes the Euclidean norm. Let (X, ||·, · · · , ·||) be an n-normed space of dimension d ≥ n ≥ 2 and {a1, a2, · · · , an} be linearly independent set in X. Then the following function ||·, · · · , ·||∞ on X n−1 defined by ||x1, x2, · · · , xn−1||∞ = max{||x1, x2, · · · , xn−1, ai|| : i = 1, 2, · · · , n} defines an (n − 1)-norm on X with respect to {a1, a2, · · · , an}. A sequence (xk) in a n-normed space (X, ||·, · · · , ·||) is said to converge to some L ∈ X if lim k→∞ ||xk − L, z1, · · · , zn−1|| = 0 for every z1, · · · , zn−1 ∈ X. A sequence (xk) in a n-normed space (X, ||·, · · · , ·||) is said to be Cauchy if lim k,p→∞ ||xk − xp, z1, · · · , zn−1|| = 0 for every z1, · · · , zn−1 ∈ X. If every Cauchy sequence in X converges to some L ∈ X, then X is said to be complete with respect to the n-norm. Any complete n-normed space is said to be n-Banach space. For more details about sequence spaces see ([28], [29]) and references therein. Let M = (Mk,l) be a sequence of Orlicz functions, p = (pk,l) be a bounded sequence of positive real numbers and u = (uk,l) be a sequence of positive reals such that uk,l 6= 0 for all k, then we define the following sequences spaces in the present paper: c20(M, ∆ m n , p, u, s, ||·, · · · , ·||) = { x = (xk,l) ∈ w 2 : lim k,l→∞ (kl)−suk,l [ Mk,l ( || ∆mn xk,l ρ , z1, · · · , zn−1|| )]pk,l = 0, 180 K. Raj and S. K. Sharma CUBO 14, 3 (2012) for some ρ > 0 and s ≥ 0 } , c2(M, ∆mn , p, u, s, ||·, · · · , ·||) = { x = (xk,l) ∈ w 2 : lim k,l→∞ (kl)−suk,l [ Mk,l ( || ∆mn xk,l − L ρ , z1, · · · , zn−1|| )]pk,l = 0, for some ρ > 0, L and s ≥ 0 } , and l2 ∞ (M, ∆mn , p, u, s, ||·, · · · , ·||) = { x = (xk,l) ∈ w 2 : sup k,l≥1 (kl)−suk,l [ Mk,l ( || ∆mn xk,l ρ , z1, · · · , zn−1|| )]pk,l < ∞, for some ρ > 0 and s ≥ 0 } . In this section of the present paper we shall study the topological properties and some interest- ing inclusion relation between the spaces c2(M, ∆mn , p, u, s, ||·, · · · , ·||), c 2 0(M, ∆ m n , p, u, s, ||·, · · · , ·||) and l2 ∞ (M, ∆mn , p, u, s, ||·, · · · , ·||). Theorem 3.1 Let M = (Mk,l) be a sequence of Orlicz functions, p = (pk,l) be a bounded sequence of positive real numbers and u = (uk,l) be a sequence of strictly positive real numbers, then the spaces c20(M, ∆ m n , p, u, s, ||·, · · · , ·||), c 2(M, ∆mn , p, u, s, ||·, · · · , ·||) and l 2 ∞ (M, ∆mn , p, u, s, ||·, · · · , ·||) are linear spaces. Proof. Let x = (xk,l), y = (yk,l) ∈ c 2 0(M, ∆ m n , p, u, s, ||·, · · · , ·||) and α, β ∈ C. Then there exist positive number ρ1 and ρ2 such that lim k,l→∞ (kl)−suk,l [ Mk,l ( || ∆mn xk,l ρ1 , z1, · · · , zn−1|| )]pk,l = 0, for some ρ1 > 0 and lim k,l→∞ (kl)−suk,l [ Mk,l ( || ∆mn yk,l ρ2 , z1, · · · , zn−1|| )]pk,l = 0, for some ρ2 > 0. Let ρ3 = max(2|α|ρ1, 2|β|ρ2). Since M = (Mk,l) is non-decreasing convex function and so by using inequality (1.1), we have CUBO 14, 3 (2012) Some generalized difference double sequence spaces ... 181 lim k,l→∞ (kl)−suk,l [ Mk,l ( || ∆mn (αxk,l + βyk,l)| ρ3 , z1, · · · , zn−1|| )]pk,l = lim k,l→∞ (kl)−suk,l [ Mk,l ( || α∆mn xk,l ρ3 , z1, · · · , zn−1|| + || β∆mn yk,l ρ3 , z1, · · · , zn−1|| )]pk,l ≤ K lim k,l→∞ 1 2pk,l (kl)−suk,l [ Mk,l ( || ∆mn xk,l ρ1 , z1, · · · , zn−1|| )]pk,l + K lim k,l→∞ 1 2pk,l (kl)−suk,l [ Mk,l ( || ∆mn yk,l ρ2 , z1, · · · , zn−1|| )]pk,l ≤ K lim k,l→∞ (kl)−suk,l [ Mk,l ( || ∆mn xk,l ρ1 , z1, · · · , zn−1|| )]pk,l + K lim k,l→∞ (kl)−suk,l [ Mk,l ( || ∆mn yk,l ρ2 , z1, · · · , zn−1|| )]pk,l = 0. So, αx + βy ∈ c20(M, ∆ n m, p, u, s, ||·, · · · , ·||). Hence c 2 0(M, ∆ n m, p, u, s, ||·, · · · , ·||) is a linear space. Similarly, we can prove that c2(M, ∆mn , p, u, s, ||·, · · · , ·||) and l 2 ∞ (M, ∆mn , p, u, s, ||·, · · · , ·||) are lin- ear spaces. Theorem 3.2 Let M = (Mk,l) be a sequence of Orlicz functions, p = (pk,l) be a bounded se- quence of positive real numbers and u = (uk,l) be a sequence of strictly positive real numbers. For Z2 = l2 ∞ , c2 and c2o, the spaces Z 2(M, ∆mn , p, u, s, ||·, · · · , ·||) are paranormed spaces, paranormed by g(x) = nm∑ k,l=1 ||xk,l, z1, · · · , zn−1|| + inf { ρ pk,l H : sup k,l (kl)−suk,lMk,l ( || ∆mn xk,l ρ , z1, · · · , zn−1|| ) ≤ 1 } where H = max(1, sup k,l pk,l). Proof. Clearly g(−x) = g(x), g(0) = 0. Let (xk) and (yk) be any two sequences belong to any one of the spaces Z2(M, ∆mn , p, u, s, ||·, · · · , ·||), for Z 2 = c20, c 2 and l2 ∞ . Then, we get ρ1, ρ2 > 0 such that sup k,l (kl)−suk,l [ Mk,l ( || ∆mn xk,l ρ1 , z1, · · · , zn−1|| )] ≤ 1 and sup k,l (kl)−suk,l [ Mk,l ( || ∆mn yk,l ρ2 , z1, · · · , zn−1|| )] ≤ 1. Let ρ = ρ1 + ρ2. Then by convexity of M = (Mk,l), we have sup k,l (kl)−suk,l [ Mk,l ( || ∆mn (xk,l + yk,l) ρ , z1, · · · , zn−1|| )] ≤ ( ρ1 ρ1 + ρ2 ) sup k,l (kl)−suk,l [ Mk,l ( || ∆mn xk,l ρ1 , z1, · · · , zn−1|| )] + ( ρ2 ρ1 + ρ2 ) sup k,l (kl)−suk,l [ Mk,l ( || ∆mn yk,l ρ2 , z1, · · · , zn−1|| )] ≤ 1. 182 K. Raj and S. K. Sharma CUBO 14, 3 (2012) Hence we have, g(x + y) = mn∑ k,l=1 ||(xk,l + yk,l), z1, · · · , zn−1|| + inf { ρ pk,l H : sup k,l (kl)−suk,l [ Mk,l ( || ∆mn (xk,l + yk,l) ρ , z1, · · · , zn−1|| )] ≤ 1 } ≤ mn∑ k,l=1 ||xk,l, z1, · · · , zn−1|| + inf { ρ pk,l H 1 : sup k,l (kl)−suk,l [ Mk,l ( || ∆mn xk,l ρ1 , z1, · · · , zn−1|| )] ≤ 1 } + mn∑ k,l=1 ||yk,l, z1, · · · , zn−1|| + inf { ρ pk,l H 2 : sup k,l (kl)−suk,l [ Mk,l ( || ∆mn yk,l ρ2 , z1, · · · , zn−1|| )] ≤ 1 } . This implies that g(x + y) ≤ g(x) + g(y). The continuity of the scalar multiplication follows from the following inequality g(µx) = mn∑ k,l=1 ||µxk,l, z1, · · · , zn−1|| + inf { ρ pk,l H : sup k,l (kl)−suk,l [ Mk,l ( || ∆mn µxk,l ρ , z1, · · · , zn−1|| )] ≤ 1 } = |µ| mn∑ k,l=1 ||xk,l, z1, · · · , zn−1|| + inf { (t|µ|) pk,l H : sup k,l (kl)−suk,l [ Mk,l ( || ∆mn xk,l t , z1, · · · , zn−1|| )] ≤ 1 } , where t = ρ |µ| . Hence the space Z2(M, ∆mn , p, u, s, ||·, · · · , ·||), for Z 2 = c20, c 2 and l2 ∞ is a para- normed space, paranormed by g. Theorem 3.3 Let M = (Mk,l) be a sequence of Orlicz functions, p = (pk,l) be a bounded sequence of positive real numbers and u = (uk,l) be a sequence of strictly positive real numbers. For Z2 = l2 ∞ , c2 and c20, the spaces Z 2(M, ∆mn , p, u, s, ||·, · · · , ·||) are complete paranormed spaces, paranormed by g(x) = nm∑ k,l=1 ||xk,l, z1, · · · , zn−1|| + inf { ρ pk,l H : sup k,l (kl)−suk,lMk,l ( || ∆mn xk,l ρ , z1, · · · , zn−1|| ) ≤ 1 } , where H = max(1, sup k,l pk,l). Proof. We prove the result for the space l2 ∞ (M, ∆mn , p, u, s, ||·, · · · , ·||). Let (x i) be any Cauchy CUBO 14, 3 (2012) Some generalized difference double sequence spaces ... 183 sequence in l2 ∞ (M, ∆mn , p, u, s, ||·, · · · , ·||). Let ǫ > 0 be given and for t > 0, choose x0 be fixed such that uk,lMk,l ( tx0 2 ) ≥ 1, then there exists a positive integer n0 ∈ N such that g(x i k,l −x j k,l ) < ǫ x0t , for all i, j ≥ n0. Using the definition of paranorm, we get mn∑ k,l=1 ||(xik,l−x j k,l ), z1, · · · , zn−1||+inf { ρ pk,l H : sup k,l (kl)−suk,lMk,l ( || ∆mn (x i k,l − x j k,l ) ρ , z1, · · · , zn−1|| )} < ǫ x0t , for all i, j ≥ n0 (3.1). Hence we have, mn∑ k,l=1 ||(xik,l) − x j k,l ), z1, · · · , zn−1|| < ǫ, for all i, j ≥ no. This implies that ||(xik,l − x j k,l ), z1, · · · , zn−1|| < ǫ, for all i, j ≥ n0 and 1 ≤ k ≤ nm. Thus (xik,l) is a Cauchy sequence in C for k, l = 1, 2, ...., nm. Hence (x i k,l) is convergent in C for k, l = 1, 2, ...., nm. Let lim i→∞ xik,l = xk,l, say for k, l = 1, 2, ...., nm. (3.2) Again from equation (3.1) we have, inf { ρ pk,l H : sup k,l (kl)−suk,lMk,l ( || ∆mn (x i k,l − x j k,l) ρ , z1, · · · , zn−1|| ) ≤ 1 } < ǫ, for all i, j ≥ n0. Hence we get sup k,l (kl)−suk,lMk,l ( || ∆mn (x i k,l − x j k,l ) g(xi − xj) , z1, · · · , zn−1|| ) ≤ 1, for all i, j ≥ n0. It follows that (kl)−suk,lMk,l ( || ∆ m n (x i k,l−x j k,l ) g(xi−xj) , z1, · · · , zn−1|| ) ≤ 1, for each k, l ≥ 1 and for all i, j ≥ n0. For t > 0 with (kl)−suk,lMk,l( tx0 2 ) ≥ 1, we have (kl)−suk,lMk,l ( || ∆mn (x i k,l − x j k,l ) g(xi − xj) , z1, · · · , zn−1|| ) ≤ (kl)−suk,lMk,l( tx0 2 ). This implies that ||(∆mn x i k,l − ∆ m n x j k,l ), z1, · · · , zn−1|| < tx0 2 ǫ tx0 = ǫ 2 . Hence (∆mn x i k,l) is a Cauchy sequence in C for all k, l ∈ N. This implies that (∆ m n x i k,l) is convergent in C for all k, l ∈ N. Let lim i→∞ ∆mn x i k,l = yk,l for each k, l ∈ N. Let k, l = 1, then we have lim i→∞ ∆mn x i 1,1 = lim i→∞ n∑ v=0 (−1)v ( m v ) xi1+nv,mv+1 = y1. (3.3) 184 K. Raj and S. K. Sharma CUBO 14, 3 (2012) We have by equation (3.2) and equation (3.3) lim i→∞ ximn+1 = xmn+1, exists. Proceeding in this way inductively, we have lim i→∞ xik,l = xk,l exists for each k, l ∈ N. Now we have for all i, j ≥ n0, mn∑ k,l=1 ||(xik,l − x j k,l ), z1, · · · , zn−1|| + inf { ρ pk,l H : sup k,l (kl)−suk,lMk,l ( || ∆mn (x i k,l − x j k,l ) ρ , z1, · · · , zn−1|| ) ≤ 1 } < ǫ. This implies that lim j→∞ { mn∑ k,l=1 ||(xik,l − x j k,l ), z1, · · · , zn−1|| + inf { ρ pk,l H : sup k,l (kl)−suk,lMk,l ( || ∆mn (x i k,l − x j k,l ) ρ , z1, · · · , zn−1|| ) ≤ 1 }} < ǫ, for all i ≥ n0. Using the continuity of (Mk,l), we have mn∑ k,l=1 ||(xik,l − xk,l), z1, · · · , zn−1|| + inf { ρ pk,l H : sup k,l (kl)−suk,lMk,l ( || (∆mn x i k,l − ∆ m n xk,l) ρ , z1, · · · , zn−1|| ) ≤ 1 } < ǫ, for all i ≥ n0. It follows that (x i − x) ∈ l2 ∞ (M, ∆mn , p, u, s, ||·, · · · , ·||). Since x i ∈ l2 ∞ (M, ∆nm, p, u, s, ||·, · · · , ·||) and l 2 ∞ (M, ∆nm, p, u, s, ||·, · · · , ·||) is a linear space, so we have x = xi − (xi − x) ∈ l2 ∞ (M, ∆nm, p, u, s, ||·, · · · , ·||). This completes the proof. Similarly, we can prove that c2(M, ∆nm, p, u, ||·, · · · , ·||) and c 2 0(M, ∆ n m, p, u, ||·, · · · , ·||) are complete paranormed spaces in view of the above proof. Theorem 3.4 If 0 < pk,l ≤ qk,l < ∞ for each k, l, then Z 2(M, ∆mn , p, u, s, ||·, · · · , ·||) ⊆ Z2(M, ∆mn , q, u, s, ||·, · · · , ·||), for Z 2 = c20 and c 2. Proof. Let x = (xk,l) ∈ c 2(M, ∆mn , p, u, s, ||·, · · · , ·||). Then there exists some ρ > 0 and L ∈ X such that lim k,l→∞ (kl)−suk,l ( Mk,l ( || ∆mn xk,l − L ρ , z1, · · · , zn−1|| ))pk,l = 0. This implies that (kl)−suk,lMk,l ( || ∆mn xk,l − L ρ , z1, · · · , zn−1|| ) < ǫ (0 < ǫ < 1) for sufficiently large k, l. Hence we get lim k,l→∞ (kl)−suk,l ( Mk,l ( || ∆mn xk,l − L ρ , z1, · · · , zn−1|| ))qk,l ≤ lim k,l→∞ (kl)−suk,l ( Mk,l ( || ∆mn xk,l − L ρ , z1, · · · , zn−1|| ))pk,l = 0. This implies that x = (xk,l) ∈ c 2(M, ∆mn , q, u, s, ||·, · · · , ·||). This completes the proof. Similarly, we can prove for the case Z2 = c20. CUBO 14, 3 (2012) Some generalized difference double sequence spaces ... 185 Theorem 3.5 If M′ = (M′k,l) and M ′′ = (M′′k,l) be a sequence of Orlicz functions. Then (i) Z2(M′, ∆mn , p, u, s, ||·, · · · , ·||) ⊆ Z 2(M′′ ◦ M′, ∆mn , p, u, s, ||·, · · · , ·||), (ii) Z2(M′, ∆mn , p, u, s, ||·, · · · , ·||) ∩ Z 2(M′′, ∆mn , p, u, s, ||·, · · · , ·||) ⊆ Z2(M′ + M′′, ∆mn , p, u, s, ||·, · · · , ·||), for Z2 = l2 ∞ , c2 and c20. Proof. (i) We prove this part for Z2 = l2 ∞ and the rest of the cases will follow similarly. Let (xk,l) ∈ l 2 ∞ (M′, ∆mn , p, u, s, ||·, · · · , ·||), then there exists 0 < U < ∞ such that (kl)−suk,l ( M′k,l ( || ∆mn xk,l ρ , z1, · · · , zn−1|| ))pk,l ≤ U, for all k, l ∈ N. Let yk,l = (kl) −suk,lM ′ k,l ( || ∆ m n xk,l ρ , z1, · · · , zn−1|| ) . Then yk,l ≤ U 1 pk,l ≤ V, say for all k, l ∈ N. Hence we have ( (M′′k,l ◦ M ′ k,l) ( || ∆mn xk,l ρ , z1, · · · , zn−1|| ))pk,l = (M′′k,l(yk,l)) pk,l ≤ (M′′k,l(V)) pk,l < ∞, for all k, l ∈ N. Hence sup k,l uk,l ( (M′′k,l◦M ′ k,l) ( || ∆mn xk,l ρ , z1, · · · , zn−1|| ))pk,l < ∞. Thus x = (xk,l) ∈ l 2 ∞ (M′′◦ M′, ∆mn , p, u, s, ||·, · · · , ·||). (ii) We prove the result for the case Z2 = c2 and the rest of the cases will follow similarly. Let x = (xk,l) ∈ c 2(M′, ∆mn , p, u, s, ||·, · · · , ·||) ∩ c 2(M′′, ∆mn , p, u, s, ||·, · · · , ·||), then there exist some ρ1, ρ2 > 0 and L ∈ X such that lim k→∞ (kl)−suk,l ( M′k,l ( || ∆mn xk,l − L ρ1 , z1, · · · , zn−1|| ))pk,l = 0 and lim k→∞ (kl)−suk,l ( M′′k,l ( || ∆mn xk,l − L ρ2 , z1, · · · , zn−1|| ))pk,l = 0. Let ρ = ρ1 + ρ2. Then we have (kl)−suk,l ( (M′k,l + M ′′ k,l) ( || ∆ m n xk,l−L ρ , z1, · · · , zn−1|| ))pk,l ≤ K [( ρ1 ρ1 + ρ2 ) (kl)−suk,lM ′ k,l ( || ∆mn xk,l − L ρ1 , z1, · · · , zn−1|| )]pk,l + K [( ρ2 ρ1 + ρ2 ) (kl)−suk,lM ′′ k,l ( || ∆mn xk,l − L ρ2 , z1, · · · , zn−1|| )]pk,l . This implies that lim k→∞ (kl)−suk,l ( (M′k,l + M ′′ k,l) ( || ∆mn xk,l − L ρ , z1, · · · , zn−1|| ))pk,l = 0. 186 K. Raj and S. K. Sharma CUBO 14, 3 (2012) Thus x = (xk,l) ∈ c 2(M′ + M′′, ∆mm, p, u, s, ||·, · · · , ·||). This completes the proof. Theorem 3.6 Let M = (Mk,l) be a sequence of Orlicz functions, p = (pk,l) be a bounded sequence of positive real numbers and u = (uk,l) be a sequence of strictly positive real numbers, then Z2(M, ∆m−1n , p, u, s, ||·, · · · , ·||) ⊂ Z 2(M, ∆mn , p, u, s, ||·, · · · , ·||) , for Z 2 = l2 ∞ , c2 and c2o. Proof. We prove the result for the case Z2 = l2 ∞ and the rest of the cases will follow simi- larly. Let x = (xk,l) ∈ l 2 ∞ (M, ∆m−1n , p, u, s, ||·, · · · , ·||). Then we can have ρ > 0 such that (kl)−suk,l ( Mk,l ( || ∆m−1n xk,l ρ , z1, · · · , zn−1|| ))pk,l < ∞, for all k ∈ N. (3.4) On considering 2ρ and using the convexity of (Mk,l), we have (kl)−suk,lMk,l ( || ∆mn xk,l 2ρ , z1, · · · , zn−1|| ) ≤ 1 2 (kl)−suk,lMk,l ( || ∆m−1n xk,l ρ , z1, · · · , zn−1|| ) + 1 2 (kl)−suk,lMk,l ( || ∆m−1n xk+n,l+m ρ , z1, · · · , zn−1|| ) . Hence we have (kl)−suk,l ( Mk,l ( || ∆ m n xk,l 2ρ , z1, · · · , zn−1|| ))pk,l ≤ K { (kl)−suk,l ( 1 2 Mk,l ( || ∆m−1n xk,l ρ , z1, · · · , zn−1|| ))pk,l + (kl)−suk,l ( 1 2 Mk,l ( || ∆m−1n xk+n,l+m ρ , z1, · · · , zn−1|| ))pk,l } . Then using equation (3.4), we have (kl)−suk,l ( Mk,l ( || ∆mn xk,l ρ , z1, · · · , zn−1|| ))pk,l < ∞, for all k, l ∈ N. Thus l2 ∞ (M, ∆m−1n , p, u, s, ||·, · · · , ·||) ⊂ l 2 ∞ (M, ∆mn , p, u, s, ||·, · · · , ·||). Theorem 3.7 Let M = (Mk,l) be a sequence of Orlicz functions. Then c20(M, ∆ m n , p, u, s, ||·, · · · , ·||) ⊂ c 2 (M, ∆mn , p, u, s, ||·, · · · , ·||) ⊂ l 2 ∞ (M, ∆mn , p, u, s, ||·, · · · , ·||). Proof. It is obvious that c20(M, ∆ m n , p, u, s, ||·, · · · , ·||) ⊂ c 2(M, ∆mn , p, u, s, ||·, · · · , ·||). We shall prove that c2(M, ∆mn , p, u, s, ||·, · · · , ·||) ⊂ l 2 ∞ (M, ∆mn , p, u, s, ||·, · · · , ·||). Let x = (xk,l) ∈ c 2(M, ∆mn , p, u, s, ||·, · · · , ·||). Then there exists some ρ > 0 and L ∈ X such that lim k,l→∞ (kl)−suk,l ( Mk,l ( || ∆mn xk,l − L ρ , z1, · · · , zn−1|| ))pk,l = 0. CUBO 14, 3 (2012) Some generalized difference double sequence spaces ... 187 On taking ρ = 2ρ1, we have sup k,l (kl)−suk,l ( Mk,l ( || ∆mn xk,l ρ , z1, · · · , zn−1 ))pk,l ≤ sup k,l K [ 1 2 (kl)−suk,l ( Mk,l ( || ∆mn xk,l − L ρ1 , z1, · · · , zn−1|| ))]pk,l + sup k,l K [ 1 2 (kl)−suk,lMk,l ( || L ρ1 , z1, · · · , zn−1|| )]pk,l ≤ sup k,l K( 1 2 )pk,l(kl)−suk,l [ Mk,l ( || ∆mn xk,l − L ρ1 , z1, · · · , zn−1|| )]pk,l + sup k,l K( 1 2 )pk,l(kl)−s max(1, uk,l ( Mk,l ( || L ρ1 , z1, · · · , zn−1|| ))H) , where H = max(1, sup pk,l). Thus we get x = (xk,l) ∈ l 2 ∞ (M, ∆mn , p, u, s, ||·, · · · , ·||). Hence c20(M, ∆ m n , p, u, s, ||·, · · · , ·||) ⊂ c 2 (M, ∆mn , p, u, s, ||·, · · · , ·||) ⊂ l 2 ∞ (M, ∆mn , p, u, s, ||·, · · · , ·||). Theorem 3.8 The sequence space l2 ∞ (M, ∆mn , p, u, s, ||·, · · · , ·||) is solid. Proof. Let x = (xk,l) ∈ l 2 ∞ (M, ∆mn , p, u, s, ||·, · · · , ·||), that is sup k,l→∞ (kl)−suk,l [ Mk,l ( || ∆mn xk,l ρ , z1, · · · , zn−1|| )]pk,l < ∞. Let (αk,l) be a sequence of scalars such that |αk,l| ≤ 1 for all k, l ∈ N. Thus we have supk,l→∞(kl) −suk,l [ Mk,l ( || αk,l∆ m n xk,l ρ , z1, · · · , zn−1|| )]pk,l ≤ sup k,l→∞ (kl)−suk,l [ Mk,l ( || ∆mn xk,l ρ , z1, · · · , zn−1|| )]pk,l < ∞. This shows that (αk,lxk,l) ∈ l 2 ∞ (M, ∆mn , p, u, s, ||·, · · · , ·||) for all sequences of scalars (αk,l) with |αk,l| ≤ 1 for all k, l ∈ N, whenever (xk,l) ∈ l 2 ∞ (M, ∆mn , p, u, s, ||·, · · · , ·||). Hence the space l2 ∞ (M, ∆mn , p, u, s, ||·, · · · , ·||) is a solid sequence space. Theorem 3.9 The sequence space l2 ∞ (M, ∆mn , p, u, s, ||·, · · · , ·||) is monotone. Proof. The proof of the Theorem is obvious and so we omit it. Received: October 2011. Revised: August 2012. 188 K. Raj and S. K. Sharma CUBO 14, 3 (2012) References [1] B. Altay and F. Başar, Some new spaces of double sequencs, J. 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