Some generalized difference double sequence spaces ... - SciELO

CUBO A Mathematical Journal Vol.14, N¯o 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: [email protected], [email protected]

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 art´ıculo introducimos algunos espacios de sucesiones doble-diferencia generalizadas definidas por una sucesión de funciones de Orlicz. Estudiamos algunas propiedades topol´ ogicas y algunas relaciones de inclusi´ on 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, nnormed space 2010 AMS Mathematics Subject Classification: Primary 42B15; Secondary 40C05

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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 w2 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 C ¸ olak [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 m Z(∆m s ) = {x = (xk ) ∈ w : (∆s xk ) ∈ Z}, m m−1 where ∆m xk − ∆m−1 xk+1 ) and ∆0s xk = xk for all k ∈ N, which is equivalent s x = (∆s xk ) = (∆s s to the following binomial representation ! m X m ∆m xk+sv . (−1)v s xk = v v=0

Taking s = 1, we get the spaces which were studied by Et and C ¸ olak [5]. Taking m = s = 1, we get the spaces which were introduced and studied by Kızmaz [13].

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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: ∞

|x | X k lM = x ∈ w : M 0 : M ρ k=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 Zx M(x) = η(t)dt 0

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(λn xn − 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 seminormed space over the complex field C with the seminorm q. Now we define the following classes of sequences in the present paper: ∆m x − L ipk,l h

k,l 2 −s q n M c2 (∆m , M, u, p, q, s) = x = (x ) ∈ w : P − lim (kl) u = 0, k,l k,l k,l n k,l ρ for some ρ > 0, L and s ≥ 0 ,

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∆m x ipk,l h

k,l 2 −s = 0, c20 (∆m uk,l Mk,l q n n , M, u, p, q, s) = x = (xk,l ) ∈ w : P − lim(kl) k,l ρ for some ρ > 0 and s ≥ 0 and ∆m x ipk,l h

k,l 2 −s l2∞ (∆m uk,l Mk,l q n < ∞, n , M, u, p, q, s) = x = (xk,l ) ∈ w : sup(kl) ρ k,l for some ρ > 0 and s ≥ 0 . If we take M(x) = x, we get h ∆m x − L ipk,l

k,l 2 −s = 0, q n c2 (∆m , u, p, q, s) = x = (x ) ∈ w : P − lim (kl) u k,l k,l n k,l ρ for some ρ > 0, L and s ≥ 0 , h ∆m x ipk,l

k,l 2 −s = 0, c20 (∆m uk,l q n n , u, p, q, s) = x = (xk,l ) ∈ w : P − lim(kl) k,l ρ for some ρ > 0 and s ≥ 0

and

h ∆m x ipk,l

k,l 2 −s l2∞ (∆m uk,l q n < ∞, n , u, p, q, s) = x = (xk,l ) ∈ w : sup(kl) ρ k,l for some ρ > 0 and s ≥ 0 .

If we take p = (pk,l ) = 1, we get ∆m x − L i h

k,l 2 −s c2 (∆m uk,l Mk,l q n = 0, n , M, u, q, s) = x = (xk,l ) ∈ w : P − lim(kl) k,l ρ for some ρ > 0, L and s ≥ 0 , ∆m x i h

k,l 2 −s c20 (∆m uk,l Mk,l q n = 0, n , M, u, q, s) = x = (xk,l ) ∈ w : P − lim(kl) k,l ρ for some ρ > 0 and s ≥ 0

and ∆m x i h

k,l 2 −s < ∞, l2∞ (∆m uk,l Mk,l q n n , M, u, q, s) = x = (xk,l ) ∈ w : sup(kl) ρ k,l 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 :

h |x − L| ipk,l k,l c2 (M, u, p, s) = x = (xk,l ) ∈ w2 : P − lim(kl)−s uk,l Mk,l = 0, k,l ρ

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for some ρ > 0, L and s ≥ 0 , |x | ipk,l h

k,l = 0, c20 (M, u, p, s) = x = (xk,l ) ∈ w2 : P − lim(kl)−s uk,l Mk,l k,l ρ for some ρ > 0 and s ≥ 0 and h

|xk,l | ipk,l < ∞, l2∞ (M, u, p, s) = x = (xk,l ) ∈ w2 : sup(kl)−s uk,l Mk,l ρ k,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 : |∆x − L| ipk,l h

k,l 2 −s = 0, c2 (∆m uk,l Mk,l n , M, u, p, s) = x = (xk,l ) ∈ w : P − lim(kl) k,l ρ for some ρ > 0, L and s ≥ 0 , |∆x | ipk,l h

k,l 2 −s M = 0, c20 (∆m , M, u, p, s) = x = (x ) ∈ w : P − lim (kl) u k,l k,l k,l n k,l ρ for some ρ > 0 and s ≥ 0 and

h |∆m x | ipk,l n k,l 2 −s l2∞ (∆m uk,l Mk,l < ∞, n , M, u, p, s) = x = (xk,l ) ∈ w : sup(kl) ρ k,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 = H and let K = max{1, 2H−1 }. Then for the factorable k,l

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 difference 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.

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Main Results

Theorem 2.1 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 2 m m the classes of sequences c20 (∆m n , M, u, p, q, s), c (∆n , M, u, p, q, s) and l∞ (∆n , M, u, p, q, s) are linear spaces over the field of complex numbers C. Proof. Let x = (xk,l ), y = (yk,l ) ∈ c20 (∆m n , M, u, p, q, s) and α, β ∈ C. Then there exist positive numbers ρ1 and ρ2 such that ∆m x ipk,l h k,l lim(kl)−s uk,l Mk,l q n = 0, for some ρ1 > 0 k,l ρ1 and

∆n y ipk,l h k,l = 0, for some ρ2 > 0. lim(kl)−s uk,l Mk,l q m k,l ρ2 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 h ∆m (αx + βy ) ipk,l k,l k,l lim(kl)−s uk,l Mk,l q n k,l ρ3 α∆m x h β∆m y ipk,l n k,l n k,l = lim(kl)−s uk,l Mk,l q +q k,l ρ3 ρ3 ∆m x ipk,l h 1 k,l ≤ K lim p (kl)−s uk,l Mk,l q n k,l 2 k,l ρ1 ∆m y ipk,l h 1 k,l + K lim p (kl)−s uk,l Mk,l q n k,l 2 k,l ρ2 h ∆m x ipk,l k,l ≤ K lim(kl)−s uk,l Mk,l q n k,l ρ1 ∆m y ipk,l h k,l + K lim(kl)−s uk,l Mk,l q n k,l ρ2 = 0. 2 n So, αx + βy ∈ c20 (∆n m , M, u, p, q, s). Hence c0 (∆m , M, u, p, q, s) is a linear space. Similarly, we 2 n can prove that c2 (∆n m , M, u, p, q, s) and l∞ (∆m , 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 Z2 (∆m n , M, u, p, q, s) are paranormed spaces, paranormed by g(x) =

nm X

k,l=1

pk,l ∆m x k,l q(xk,l ) + inf ρ H : sup(kl)−s uk,l Mk,l q n ≤1 ρ k,l

where H = max(1, sup pk,l ). k,l

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Proof. Clearly g(−x) = g(x), g(0) = 0. Let (xk,l ) and (yk,l ) be any two sequences belong 2 2 2 2 to any one of the spaces Z2 (∆n m , M, u, p, q, s), for Z = c0 , c and l∞ . Then, we get ρ1 , ρ2 > 0 such that ∆m x k,l ≤1 sup(kl)−s uk,l Mk,l q n ρ 1 k,l

and

∆m y ) k,l sup(kl)−s uk,l Mk,l q n ≤ 1. ρ2 k,l

Let ρ = ρ1 + ρ2 . Then by convexity of M = (Mk,l ), we have ∆m (x + y ) k,l k,l ≤ sup(kl)−s uk,l Mk,l q n ρ k,l + ≤

∆m x ρ1 k,l sup(kl)−s uk,l Mk,l q n ρ1 + ρ2 k,l ρ1 ∆m y ρ k,l 2 sup(kl)−s uk,l Mk,l q n ρ1 + ρ2 k,l ρ2 1.

Hence we have, g(x + y)

mn X

=

q(xk,l + yk,l )

k,l=1

∆m (x + y )

pk,l k,l k,l ≤1 inf ρ H : sup(kl)−s uk,l Mk,l q n ρ k,l mn

pk,l ∆m x X k,l q(xk,l ) + inf ρ1 H : sup(kl)−s uk,l Mk,l q n ≤1 ρ1 k,l

+ ≤

k,l=1 mn X

+

k,l=1

∆m y

pk,l k,l q(yk,l ) + inf ρ2 H : sup(kl)−s uk,l Mk,l q n ≤1 . ρ2 k,l

This implies that g(x + y) ≤ g(x) + g(y). The continuity of the scalar multiplication follows from the following inequality g(µx) = =

mn X

∆m µx

pk,l k,l q(µxk,l ) + inf ρ H : sup(kl)−s uk,l Mk,l q n ≤1 ρ k,l

k,l=1 mn X

|µ|

k,l=1

∆m x

pk,l k,l q(xk,l ) + inf (t|µ|) H : sup(kl)−s uk,l Mk,l q n ≤1 , t k,l

ρ 2 2 2 2 . Hence the space Z2 (∆n where t = |µ| m , M, u, p, q, s), for Z = c0 , c and l∞ 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.

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For Z2 = l2∞ , c2 and c20 , the spaces Z2 (∆m n , M, u, p, q, s) are complete paranormed spaces, paranormed by g(x) =

nm X

k,l=1

∆m x

pk,l k,l q(xk,l ) + inf ρ H : sup(kl)−s uk,l Mk,l q n ≤1 , ρ k,l

where H = max(1, sup pk,l ). k,l i Proof. We prove the result for the space l2∞ (∆m n , M, u, p, q, s). Let (xk,l ) be any Cauchy sem quence inl2∞ (∆ n , M, u, p, q, s). Let ǫ > 0 be given and for t > 0, choose x0 be fixed such that tx0 uk,l Mk,l 2 ≥ 1, then there exists a positive integer n0 ∈ N such that g(xik,l − xjk,l ) < xǫ0 t , for all i, j ≥ n0 . Using the definition of paranorm, we get mn X

k,l=1

∆m (xi − xj )

pk,l ǫ k,l < , q(xik,l − xjk,l ) + inf ρ H : sup(kl)−s uk,l Mk,l q n k,l ρ x0 t k,l for all i, j ≥ n0

Hence we have,

mn X

(2.1).

q(xik,l − xjk,l ) < ǫ, for all i, j ≥ n0 .

k,l=1

This implies that q(xik,l − xjk,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 (xik,l ) is convergent in C for k, l = 1, 2, ...., nm. Let lim xik,l = xk,l , say for k, l = 1, 2, ...., nm. (2.2) i→∞

Again from equation (2.1) we have, ∆m (xi − xj )

pk,l k,l ≤ 1 < ǫ, for all i, j ≥ n0 . inf ρ H : sup(kl)−s uk,l Mk,l q n k,l ρ k,l Hence we get ∆m (xi − xj ) k,l sup(kl)−s uk,l Mk,l q n k,l ≤ 1, for all i, j ≥ n0 . g(xi − xj ) k,l m i ∆ (x −xjk,l ) It follows that (kl)−s uk,l Mk,l q ng(xk,l ≤ 1, for each k, l ≥ 1 and for all i, j ≥ n0 . i −xj ) For t > 0 with (kl)−s uk,l Mk,l ( tx20 ) ≥ 1, we have

∆m (xi − xj ) tx0 k,l ≤ (kl)−s uk,l Mk,l ( ). (kl)−s uk,l Mk,l q n k,l g(xi − xj ) 2 This implies that i m j q(∆m n xk,l − ∆n xk,l )
0 and s ≥ 0 (2.4) k,l ρ Then from (2.4) we have ∆m x iPk,l+1 h k,l = 0, P − lim(kl)−s uk,l Mk,l q n k,l ρ

and

h ∆m x iPk+1,l k,l =0 P − lim(kl)−s uk,l Mk,l q n k,l ρ

h ∆m x iPk+1,l+1 k,l P − lim(kl)−s uk,l Mk,l q n = 0. k,l ρ

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Now for m m−1 m−1 ∆m xk,l − ∆n xk,l+1 − ∆m−1 xk+1,l + ∆m−1 xk+1,l+1 ), n x = (∆n xk,l ) = (∆n n n

we have h m iPk,l ∆ x (kl)−s uk,l Mk,l q n ρ k,l ≤ + ≤ + ≤ +

∆m−1 x ∆m−1 x h k,l+1 k,l +q n (kl)−s uk,l Mk,l q n ρ ρ ∆m−1 x iPk,l ∆m−1 x k+1,l+1 k+1,l +q n q n ρ ρ ∆m−1 x iPk,l

h ∆m−1 x iPk,l h k,l k+1,l K2 (kl)−s uk,l Mk,l q n + uk,l Mk,l q n ρ ρ ∆m−1 x iPk,l iPk,l h h ∆m−1 x k+1,l+1 k,l+1 + uk,l Mk,l q n uk,l M q n ρ ρ ∆m−1 x iPk,l h

h ∆m−1 x iPk+1,l k,l k+1,l K2 (kl)−s uk,l Mk,l q n + (kl)−s uk,l Mk,l q n ρ ρ iPk,l+1 iPk+1,l+1 h ∆m−1 x h ∆m−1 x k+1,l+1 k,l+1 + uk,l (kl)−s Mk,l q n (kl)−s uk,l Mk,l q n ρ ρ

2 m−1 from this it follows that x = (xk,l ) ∈ c20 (∆m , M, u, p, q, s) ⊂ n , M, u, p, q, s) and hence c0 (∆n 2 i c20 (∆m , M, u, p, q, s). On applying the principle of induction, it follows that c (∆ n n , M, u, p, q, s) ⊂ 0 2 m c0 (∆n , M, u, p, q, s) for i = 0, 1, 2, · · · , m − 1. Similarly, we can prove the other cases. 2 m Theorem 2.5 (a) If 0 < inf pk,l ≤ pk,l < 1, then Z2 (∆m n , M, u, p, q, s) ⊂ Z (∆n , M, u, q, s), k,l

2 m (b) If 1 < pk,l ≤ sup pk,l < ∞, then Z2 (∆m n , M, u, q, s) ⊂ Z (∆n , M, u, p, q, s), k,l

where Z2 = c2 , c20 and l2∞ .

Proof. (i) Let x = (xk,l ) ∈ l2∞ (∆m n , M, u, p, q, s). Since 0 < inf pk,l ≤ 1, we have ∆m x ipk,l h ∆m x i h k,l k,l ≤ sup(kl)−s uk,l Mk,l q n , sup(kl)−s uk,l Mk,l q n ρ ρ k,l k,l and hence x = (xk,l ) ∈ l2∞ (∆m n , M, u, p, q, s).

(ii) Let pk,l for each (k, l) and sup pk,l < ∞. Let x = (xk,l ) ∈ l2∞ (∆m n , M, u, q, s). Then, for k,l

each 0 < ǫ < 1, there exists a positive integer N such that ∆m x i h k,l ≤ ǫ < 1, sup(kl)−s uk,l Mk,l q n ρ k,l

for all m, n ∈ N. This implies that ∆m x ipk,l h ∆m x i h k,l k,l sup(kl)−s uk,l Mk,l q n . ≤ sup(kl)−s uk,l Mk,l q n ρ ρ k,l k,l

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Thus x = (xk,l ) ∈ l2∞ (∆m n , M, u, p, q, s) and this completes the proof. ′′ ′ ) be two sequences of Orlicz functions satisfying ) and M ′′ = (Mk,l Theorem 2.6 Let M ′ = (Mk,l ′′ Mk,l (t) 2 m ′′ ′ ≥ 1, then Z (∆n , M ′ , u, p, q, s) = Z2 (∆m ∆2 -condition. If β = lim n , M ◦M , u, p, q, s), t→∞ t where Z2 = c2 , c20 and l2∞ .

Proof. We prove it for Z2 = c2 and the other cases will follows on applying similar techniques. ′ Let x = (xk,l ) ∈ c2 (∆m n , M , u, p, q, s), then h ∆m x − L iPk,l k,l ′ = 0. q n P − lim(kl)−s Mk,l k,l ρ

′′ Let 0 < ǫ < 1 and δ with 0 < δ < 1 such that Mk,l (t) < ǫ for 0 ≤ t < δ. Let

∆m x − L k,l ′ q n yk,l = Mk,l ρ and consider ′′ ′′ ′′ [Mk,l (yk,l )]pk,l = [Mk,l (yk,l )]pk,l + [Mk,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 [Mk,l (yk,l )]pk,l < (kl)−s [Mk,l (2)]H [(yk,l )]pk,l

(2.6)

On the other hand, we use the fact that yk,l
0 and s ≥ 0 , c2 (M, ∆m n , p, u, s, ||·, · · · , ·||) =

∆m x − L h ipk,l k,l x = (xk,l ) ∈ w2 : lim (kl)−s uk,l Mk,l || n , z1 , · · · , zn−1 || = 0, k,l→∞ ρ for some ρ > 0, L and s ≥ 0 , and l2∞ (M, ∆m n , p, u, s, ||·, · · · , ·||) = ∆m x h ipk,l

k,l , z1 , · · · , zn−1 || < ∞, x = (xk,l ) ∈ w2 : sup (kl)−s uk,l Mk,l || n ρ k,l≥1 for some ρ > 0 and s ≥ 0 . In this section of the present paper we shall study the topological properties and some interestm 2 ing inclusion relation between the spaces c2 (M, ∆m n , p, u, s, ||·, · · · , ·||), c0 (M, ∆n , p, u, s, ||·, · · · , ·||) and l2∞ (M, ∆m n , 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 m 2 2 m spaces c20 (M, ∆m n , p, u, s, ||·, · · · , ·||), c (M, ∆n , p, u, s, ||·, · · · , ·||) and l∞ (M, ∆n , p, u, s, ||·, · · · , ·||) are linear spaces. Proof. Let x = (xk,l ), y = (yk,l ) ∈ c20 (M, ∆m n , p, u, s, ||·, · · · , ·||) and α, β ∈ C. Then there exist positive number ρ1 and ρ2 such that ∆m x h ipk,l k,l lim (kl)−s uk,l Mk,l || n , z1 , · · · , zn−1 || = 0, for some ρ1 > 0 k,l→∞ ρ1 and ipk,l h ∆m y k,l = 0, for some ρ2 > 0. , z1 , · · · , zn−1 || lim (kl)−s uk,l Mk,l || n k,l→∞ ρ2 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

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ipk,l h ∆m (αx + βy )| k,l k,l , z1 , · · · , zn−1 || lim (kl)−s uk,l Mk,l || n k,l→∞ ρ3 α∆m x ipk,l h β∆m yk,l k,l = lim (kl)−s uk,l Mk,l || n , z1 , · · · , zn−1 || + || n , z1 , · · · , zn−1 || k,l→∞ ρ3 ρ3 ∆m x h ipk,l 1 k,l ≤ K lim p (kl)−s uk,l Mk,l || n , z1 , · · · , zn−1 || k,l→∞ 2 k,l ρ1 ∆m y h ipk,l 1 k,l + K lim p (kl)−s uk,l Mk,l || n , z1 , · · · , zn−1 || k,l→∞ 2 k,l ρ2 ∆m x h ipk,l k,l , z1 , · · · , zn−1 || ≤ K lim (kl)−s uk,l Mk,l || n k,l→∞ ρ1 ipk,l ∆m y h n k,l −s , z1 , · · · , zn−1 || + K lim (kl) uk,l Mk,l || k,l→∞ ρ2 = 0. n 2 So, αx + βy ∈ c20 (M, ∆n m , p, u, s, ||·, · · · , ·||). Hence c0 (M, ∆m , p, u, s, ||·, · · · , ·||) is a linear space. m 2 Similarly, we can prove that c2 (M, ∆m n , p, u, s, ||·, · · · , ·||) and l∞ (M, ∆n , p, u, s, ||·, · · · , ·||) are linear spaces.

Theorem 3.2 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 c2o , the spaces Z2 (M, ∆m n , p, u, s, ||·, · · · , ·||) are paranormed spaces, paranormed by nm

pk,l ∆m x X k,l , z1 , · · · , zn−1 || ≤ 1 g(x) = ||xk,l , z1 , · · · , zn−1 || + inf ρ H : sup(kl)−s uk,l Mk,l || n ρ k,l k,l=1


Proof. Clearly g(−x) = g(x), g(0) = 0. Let (xk ) and (yk ) be any two sequences belong 2 2 2 2 to any one of the spaces Z2 (M, ∆m n , p, u, s, ||·, · · · , ·||), for Z = c0 , c and l∞ . Then, we get ρ1 , ρ2 > 0 such that ∆m x h i k,l sup(kl)−s uk,l Mk,l || n , z1 , · · · , zn−1 || ≤ 1 ρ1 k,l and

∆m y h i k,l sup(kl)−s uk,l Mk,l || n , z1 , · · · , zn−1 || ≤ 1. ρ2 k,l Let ρ = ρ1 + ρ2 . Then by convexity of M = (Mk,l ), we have i h ∆m (x + y ) k,l k,l , z1 , · · · , zn−1 || sup(kl)−s uk,l Mk,l || n ρ k,l ∆m x h i ρ k,l 1 sup(kl)−s uk,l Mk,l || n , z1 , · · · , zn−1 || ≤ ρ1 + ρ2 k,l ρ1 i ρ h ∆m y 2 k,l , z1 , · · · , zn−1 || sup(kl)−s uk,l Mk,l || n + ρ1 + ρ2 k,l ρ2 ≤ 1.

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Hence we have, g(x + y)

=

mn X

||(xk,l + yk,l ), z1 , · · · , zn−1 ||

k,l=1

i ∆m (x + y )

pk,l h k,l k,l , z1 , · · · , zn−1 || ≤ 1 inf ρ H : sup(kl)−s uk,l Mk,l || n ρ k,l mn X ≤ ||xk,l , z1 , · · · , zn−1 || +

k,l=1

+ +

∆m x h i

pk,l k,l inf ρ1 H : sup(kl)−s uk,l Mk,l || n , z1 , · · · , zn−1 || ≤ 1 ρ1 k,l mn X ||yk,l , z1 , · · · , zn−1 || k,l=1

+

∆m y h i

pk,l k,l , z1 , · · · , zn−1 || ≤ 1 . inf ρ2 H : sup(kl)−s uk,l Mk,l || n ρ2 k,l

This implies that g(x + y) ≤ g(x) + g(y). The continuity of the scalar multiplication follows from the following inequality g(µx)

=

mn X

||µxk,l , z1 , · · · , zn−1 ||

k,l=1

i

pk,l ∆m µx h k,l , z1 , · · · , zn−1 || ≤ 1 + inf ρ H : sup(kl)−s uk,l Mk,l || n ρ k,l mn X = |µ| ||xk,l , z1 , · · · , zn−1 || k,l=1

i ∆m x h pk,l k,l + inf (t|µ|) H : sup(kl)−s uk,l Mk,l || n , z1 , · · · , zn−1 || ≤ 1 , t k,l

ρ 2 2 2 2 where t = |µ| . Hence the space Z2 (M, ∆m n , p, u, s, ||·, · · · , ·||), for Z = c0 , c and l∞ is a paranormed 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 Z2 (M, ∆m n , p, u, s, ||·, · · · , ·||) are complete paranormed spaces, paranormed by g(x) =

nm X

k,l=1

∆m x

pk,l k,l ||xk,l , z1 , · · · , zn−1 || + inf ρ H : sup(kl)−s uk,l Mk,l || n , z1 , · · · , zn−1 || ≤ 1 , ρ k,l


Proof.

i We prove the result for the space l2∞ (M, ∆m n , p, u, s, ||·, · · · , ·||). Let (x ) be any Cauchy

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sequence in l2∞(M,∆m n , p, u, s, ||·, · · · , ·||). Let ǫ > 0 be given and for t > 0, choose x0 be fixed such that uk,l Mk,l tx20 ≥ 1, then there exists a positive integer n0 ∈ N such that g(xik,l − xjk,l ) < xǫ0 t , for all i, j ≥ n0 . Using the definition of paranorm, we get mn X

k,l=1

pk,l ∆m (xi − xj ) k,l , z1 , · · · , zn−1 || ||(xik,l −xjk,l ), z1 , · · · , zn−1 ||+inf ρ H : sup(kl)−s uk,l Mk,l || n k,l ρ k,l
0 with (kl)−s uk,l Mk,l ( tx20 ) ≥ 1, we have ∆m (xi − xj ) tx0 k,l , z , · · · , z || ≤ (kl)−s uk,l Mk,l ( ). (kl)−s uk,l Mk,l || n k,l 1 n−1 i j g(x − x ) 2 This implies that i m j ||(∆m n xk,l − ∆n xk,l ), z1 , · · · , zn−1 ||
0 and L ∈ X such that ∆m x − L pk,l k,l lim (kl)−s uk,l Mk,l || n , z1 , · · · , zn−1 || = 0. k,l→∞ ρ ∆m x − L k,l , z1 , · · · , zn−1 || < ǫ (0 < ǫ < 1) for sufficiently This implies that (kl)−s uk,l Mk,l || n ρ large k, l. Hence we get ∆m x − L qk,l k,l lim (kl)−s uk,l Mk,l || n , z1 , · · · , zn−1 || k,l→∞ ρ pk,l ∆m x − L k,l , z1 , · · · , zn−1 || = 0. ≤ lim (kl)−s uk,l Mk,l || n k,l→∞ ρ This implies that x = (xk,l ) ∈ c2 (M, ∆m n , q, u, s, ||·, · · · , ·||). This completes the proof. Similarly, we can prove for the case Z2 = c20 .

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′′ ′ ) be a sequence of Orlicz functions. Then ) and M ′′ = (Mk,l Theorem 3.5 If M ′ = (Mk,l 2 ′ m 2 ′′ ′ (i) Z (M , ∆n , p, u, s, ||·, · · · , ·||) ⊆ Z (M ◦ M , ∆m n , p, u, s, ||·, · · · , ·||), (ii) 2 ′′ m Z2 (M ′ , ∆m n , p, u, s, ||·, · · · , ·||) ∩ Z (M , ∆n , p, u, s, ||·, · · · , ·||)

⊆ Z2 (M ′ + M ′′ , ∆m n , 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 ) ∈ l2∞ (M ′ , ∆m n , p, u, s, ||·, · · · , ·||), then there exists 0 < U < ∞ such that ∆m x pk,l k,l ′ || n (kl)−s uk,l Mk,l , z1 , · · · , zn−1 || ≤ U, for all k, l ∈ N. ρ m 1 ∆ x ′ || n ρ k,l , z1 , · · · , zn−1 || . Then yk,l ≤ U pk,l ≤ V, say for all k, l ∈ N. Let yk,l = (kl)−s uk,l Mk,l Hence we have ∆m x pk,l k,l ′′ ′ ′′ ′′ (Mk,l ◦ Mk,l ) || n , z1 , · · · , zn−1 || = (Mk,l (yk,l ))pk,l ≤ (Mk,l (V))pk,l < ∞, ρ for all k, l ∈ N.

pk,l ∆m x k,l ′′ ′ , z1 , · · · , zn−1 || < ∞. Thus x = (xk,l ) ∈ l2∞ (M ′′ ◦ ◦Mk,l ) || n Hence sup uk,l (Mk,l ρ k,l M ′ , ∆m n , p, u, s, ||·, · · · , ·||). (ii) We prove the result for the case Z2 = c2 and the rest of the cases will follow similarly. Let 2 ′′ m x = (xk,l ) ∈ c2 (M ′ , ∆m n , p, u, s, ||·, · · · , ·||) ∩ c (M , ∆n , p, u, s, ||·, · · · , ·||), then there exist some ρ1 , ρ2 > 0 and L ∈ X such that ∆m x − L pk,l k,l ′ || n lim (kl)−s uk,l Mk,l , z1 , · · · , zn−1 || =0 k→∞ ρ1

and

pk,l ∆m x − L k,l ′′ , z1 , · · · , zn−1 || = 0. || n lim (kl)−s uk,l Mk,l k→∞ ρ2 Let ρ = ρ1 + ρ2 . Then we have

pk,l m ∆ x −L ′′ ′ + Mk,l ) || n ρk,l , z1 , · · · , zn−1 || (kl)−s uk,l (Mk,l ≤ + This implies that

ipk,l ∆m x − L ρ1 k,l ′ , z1 , · · · , zn−1 || (kl)−s uk,l Mk,l || n ρ1 + ρ2 ρ1 ∆m x − L ipk,l h ρ k,l 2 ′′ || n (kl)−s uk,l Mk,l , z1 , · · · , zn−1 || . K ρ1 + ρ2 ρ2 K

h

∆m x − L pk,l k,l ′′ ′ lim (kl)−s uk,l (Mk,l + Mk,l ) || n = 0. , z1 , · · · , zn−1 || k→∞ ρ

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Thus x = (xk,l ) ∈ c2 (M ′ + M ′′ , ∆m m , 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, 2 2 2 2 , p, u, s, ||·, · · · , ·||) ⊂ Z2 (M, ∆m then Z2 (M, ∆m−1 n , p, u, s, ||·, · · · , ·||) , for Z = l∞ , c and co . n Proof. We prove the result for the case Z2 = l2∞ and the rest of the cases will follow simim−1 larly. Let x = (xk,l ) ∈ l2∞ (M, ∆n , p, u, s, ||·, · · · , ·||). Then we can have ρ > 0 such that ∆m−1 x pk,l k,l (kl)−s uk,l Mk,l || n , z1 , · · · , zn−1 || < ∞, ρ

for all k ∈ N.

(3.4)

On considering 2ρ and using the convexity of (Mk,l ), we have ∆m x k,l , z1 , · · · , zn−1 || ≤ (kl)−s uk,l Mk,l || n 2ρ +

∆m−1 x 1 k,l (kl)−s uk,l Mk,l || n , z1 , · · · , zn−1 || 2 ρ m−1 1 ∆ xk+n,l+m (kl)−s uk,l Mk,l || n , z1 , · · · , zn−1 || . 2 ρ

Hence we have m pk,l ∆ x (kl)−s uk,l Mk,l || n2ρk,l , z1 , · · · , zn−1 || ≤ +

1

∆m−1 x pk,l k,l K (kl)−s uk,l Mk,l || n , z1 , · · · , zn−1 || 2 ρ ∆m−1 x pk,l 1 k+n,l+m , z1 , · · · , zn−1 || . (kl)−s uk,l Mk,l || n 2 ρ

Then using equation (3.4), we have pk,l ∆m x k,l < ∞, for all k, l ∈ N. (kl)−s uk,l Mk,l || n , z1 , · · · , zn−1 || ρ

Thus l2∞ (M, ∆m−1 , p, u, s, ||·, · · · , ·||) ⊂ l2∞ (M, ∆m n n , p, u, s, ||·, · · · , ·||).

Theorem 3.7 Let M = (Mk,l ) be a sequence of Orlicz functions. Then 2 m 2 m c20 (M, ∆m n , p, u, s, ||·, · · · , ·||) ⊂ c (M, ∆n , p, u, s, ||·, · · · , ·||) ⊂ l∞ (M, ∆n , p, u, s, ||·, · · · , ·||).

2 m Proof. It is obvious that c20 (M, ∆m n , p, u, s, ||·, · · · , ·||) ⊂ c (M, ∆n , p, u, s, ||·, · · · , ·||). We shall 2 m 2 m prove that c (M, ∆n , p, u, s, ||·, · · · , ·||) ⊂ l∞ (M, ∆n , p, u, s, ||·, · · · , ·||). Let x = (xk,l ) ∈ c2 (M, ∆m n , p, u, s, ||·, · · · , ·||). Then there exists some ρ > 0 and L ∈ X such that

∆m x − L pk,l k,l lim (kl)−s uk,l Mk,l || n = 0. , z1 , · · · , zn−1 || k,l→∞ ρ

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On taking ρ = 2ρ1 , we have ∆m x pk,l k,l , z1 , · · · , zn−1 sup(kl)−s uk,l Mk,l || n ρ k,l ≤ + ≤ +

ipk,l ∆m x − L k,l (kl)−s uk,l Mk,l || n , z1 , · · · , zn−1 || 2 ρ1 k,l L h1 ipk,l sup K (kl)−s uk,l Mk,l || , z1 , · · · , zn−1 || 2 ρ1 k,l ∆m x − L h ipk,l 1 pk,l k,l (kl)−s uk,l Mk,l || n , z1 , · · · , zn−1 || sup K( ) 2 ρ1 k,l L H 1 pk,l sup K( ) (kl)−s max(1, uk,l Mk,l || , z1 , · · · , zn−1 || , 2 ρ1 k,l sup K

h1

where H = max(1, sup pk,l ). Thus we get x = (xk,l ) ∈ l2∞ (M, ∆m n , p, u, s, ||·, · · · , ·||). Hence 2 m 2 m c20 (M, ∆m n , p, u, s, ||·, · · · , ·||) ⊂ c (M, ∆n , p, u, s, ||·, · · · , ·||) ⊂ l∞ (M, ∆n , p, u, s, ||·, · · · , ·||).

Theorem 3.8 The sequence space l2∞ (M, ∆m n , p, u, s, ||·, · · · , ·||) is solid. Proof. Let x = (xk,l ) ∈ l2∞ (M, ∆m n , p, u, s, ||·, · · · , ·||), that is ∆m x h ipk,l k,l sup (kl)−s uk,l Mk,l || n , z1 , · · · , zn−1 || < ∞. ρ k,l→∞ Let (αk,l ) be a sequence of scalars such that |αk,l | ≤ 1 for all k, l ∈ N. Thus we have ipk,l h α ∆m x supk,l→∞ (kl)−s uk,l Mk,l || k,l ρn k,l , z1 , · · · , zn−1 || ∆m x h ipk,l k,l ≤ sup (kl)−s uk,l Mk,l || n , z1 , · · · , zn−1 || < ∞. ρ k,l→∞ This shows that (αk,l xk,l ) ∈ l2∞ (M, ∆m n , p, u, s, ||·, · · · , ·||) for all sequences of scalars (αk,l ) with |αk,l | ≤ 1 for all k, l ∈ N, whenever (xk,l ) ∈ l2∞ (M, ∆m n , p, u, s, ||·, · · · , ·||). Hence the space 2 m l∞ (M, ∆n , p, u, s, ||·, · · · , ·||) is a solid sequence space. Theorem 3.9 The sequence space l2∞ (M, ∆m n , p, u, s, ||·, · · · , ·||) is monotone. Proof. The proof of the Theorem is obvious and so we omit it.

Received:

October 2011.

Revised:

August 2012.

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