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Thai Journal of Mathematics Volume 3 (2005) Number 2 : 209–218

Some Classes of Difference Paranormed Sequence Spaces Defined by Orlicz Functions B. C. Tripathy and B. Sarma Abstract : In this article we introduce the difference paranormed sequence spaces c0 {M, ∆, p, q}, c{M, ∆, p, q} and `∞ {M, ∆, p, q} defined by Orlicz functions. We study its different properties like solidness, symmetricity, completeness etc. and prove some inclusion results. Keywords : Orlicz sequence space, paranorm, solid space, symmetric space, completeness, difference sequence space. 2000 Mathematics Subject Classification : 40A05, 46A45, 46E30.

1

Introduction

Throughout the article w, c, c0 and `∞ denote the spaces of all convergent, null and bounded sequences, respectively. The zero sequence (0,0,0,. . . ) is denoted by θ and p = (pk ) is a sequence of strictly positive real numbers. Further the sequence (p−1 k ) will be represented by (tk ). The notion of paranormed sequences was introduced by Nakano [6] and Simons [8]. It was further investigated by Maddox [5], Lascarides [4] and many others. The notion of difference sequence space was introduced by Kizmaz [1] as follows: n o Z(∆) = x = (xk ) : (∆xk ) ∈ Z ,

for Z = c, c0 and `∞ , where (∆xk ) = (xk − xk+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 → ∞. If the convexity of the Orlicz function M is replaced by M (x + y) ≤ M (x) + M (y), then this function is called modulus function, introduced by Nakano [6] and studied by Ruckle [7] and further investigated by many others.

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Thai J. Math. 3(2005)/ B. C. Tripathy and B. Sarma

Lindenstrauss and Tzafriri [3] used the idea of Orlicz function to construct the sequence space   ∞ n o X |xk | `M = x ∈ w : M < ∞ for some ρ > 0 . ρ k=1

The space `M with the norm   ∞ n o X |xk | ||x|| = inf ρ > 0 : ≤1 . M ρ k=1

becomes a Banach space, called as an Orlicz sequence space. The space `M is closely related to the space `p which is an Orlicz sequence space with M (x) = |x|p for 1 ≤ p < ∞. An Orlicz function M is said to satisfy ∆2 -condition for all values of u, if there exists a constant K > 0, such that M (2u) ≤ KM (u), (u ≥ 0). The ∆2 -condition is equivalent to the inequality M (Lu) ≤ KLM (u) for all values of u and for L > 1 (see for instance Krasnosleskii and Rutitsky [2]).

2

Definitions and Preliminaries

A sequence space E is said to be solid (or normal ) if (αk xk ) ∈ E, whenever (xk ) ∈ E, for all sequence (αk ) of scalars with |αk | ≤ 1 for all k ∈ N. A sequence space E is said to be symmetric if (xπ(n) ) ∈ E, whenever (xn ) ∈ E, where π is a permutation on N . Let H = sup pk and D = max(1, 2H−1 ), then it is well known that n o |ak + bk |pk ≤ D |ak |pk + |bk |pk . Remark 1 Let M be an Orlicz function and 0 < λ < 1, then for all x > 0, M (λx) ≤ λM (x). Let M be an Orlicz function, then we have the following known Orlicz sequence spaces:   pk n o |∆xk | c0 (M, ∆, p) = (xk ) ∈ w : M → 0, as k → ∞, for some ρ > 0 . ρ   pk n |∆xk − L| c(M, ∆, p) = (xk ) ∈ w : M → 0, as k → ∞, for some ρ o L ∈ C and for some ρ > 0 .

Some Classes of Difference Paranormed Sequence Spaces

211

  pk n o |∆xk | `∞ (M, ∆, p) = (xk ) ∈ w : sup M < ∞, for some ρ > 0 . ρ k Let (X, q) be a seminormed space seminormed by q. Now we introduce the following sequence spaces.   pk n o q(∆xk ) c0 {M, ∆, p, q} = (xk ) ∈ w : M tk → 0, as k → ∞, for some ρ > 0 . ρ   pk n q(∆xk − L) c{M, ∆, p, q} = (xk ) ∈ w : M tk → 0, as k → ∞, for some ρ o L ∈ X and for some ρ > 0 . o n   q(∆x ) pk o n k tk < ∞, for some ρ > 0 . `∞ {M, ∆, p, q} = (xk ) ∈ w : sup M ρ k Lascarides [4] has shown that the sequence spaces c0 {p} and `∞ {p} are linear spaces for any positive sequence p = (pk ). Two sets of sequences E and F are said to be equivalent if there exists a sequence u = (uk ) of strictly positive numbers such that the mapping u : E → F defined by (uk xk ) ∈ F, whenever (xk ) ∈ E is a one to one correspondence. We write E ∼ = F (u). Clearly E ∼ = F (u) implies F ∼ = E(u−1 ), where u−1 = (u−1 k ). The following results will be used for establishing some results of this article. It were proved in Lascarides [4] : Lemma 2.1 Let h = inf pk and H = sup pk , then the following are equivalent : (i) H < ∞ and h > 0. (ii) c0 (p) = c0 or `∞ (p) = `∞ . (iii) `∞ {p} = `∞ (p). (iv) c0 {p} = c0 (p). (v) `{p} = `(p). Lemma 2.2 Let p, q be two sequences of strictly positive numbers. Then c0 {p} ∼ = c0 {q} if and only if there exists a sequence u = (uk ) of strictly positive numbers such that p−1 −(1+ p1 ) qk k ) (uk pkk N lim lim sup =0 (1) N qk k and

q −1

−(1+ q1 ) pk k

(uk qkk N lim lim sup N pk k

)

= 0.

(2)

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Thai J. Math. 3(2005)/ B. C. Tripathy and B. Sarma q −1 −p−1 k

Lemma 2.3 Let the sequence a = (ak ) = (qkk pk and only if the following conditions hold −1

lim lim sup N qk (1+pk N

)

). Then c0 {p} ∼ = c0 {q} if

=0

(3)

= 0.

(4)

k

and

−1

lim lim sup N −pk (1+qk N

)

k q −1 −p−1

Lemma 2.4 Let the sequence a = (ak ) = (qkk pk k ). Then   1 1 − = 0 implies c0 {p} ∼ lim = c0 {q}. k→∞ pk qk Lemma 2.5 Let fk = pqkk for every k ∈ N. Let (fk ) and (fk−1 ) be both in `∞ . Then `∞ {p} ∼ = `∞ {q}(f ). Lemma 2.6 Let q ∈ `∞ . Then `∞ {p} ⊆ `∞ {q} if and only if −1

lim inf qk (N pk )−qk pk > 0, k

(5)

for every integer N > 1. Lemma 2.7 Let q ∈ `∞ and c0 {p} ∼ = c0 {q}, then c0 (p) ∼ = c0 (q).

3

Main Results In this section we prove the results of this article.

Theorem 3.1 The classes c0 {M, ∆, p, q}, c{M, ∆, p, q} and `∞ {M, ∆, p, q} are linear spaces, for any sequence p = (pk ) of strictly positive numbers. Proof. We establish it for the case c0 {M, ∆, p, q} and rest of the cases will follow similarly. Let (xk ), (yk ) ∈ c0 {M, ∆, p, q} and α, β ∈ C. Then there exists ρ1 > 0 and ρ2 > 0 such that   pk q(∆xk ) M tk → 0, as k → ∞ (6) ρ1 and

pk   q(∆yk ) tk → 0, as k → ∞. M ρ2

(7)

Let ρ = max{2|α|ρ1 , 2|β|ρ2 }. By (6) and (7), we then have   pk   pk   pk q(α∆xk + β∆yk ) q(∆xk ) q(∆yk ) M tk ≤ D M tk + D M tk ρ ρ1 ρ2 → 0, as k → ∞. Hence (αxk + βyk ) ∈ c0 {M, ∆, p, q}. Therefore c0 {M, ∆, p, q} is a linear space. 

Some Classes of Difference Paranormed Sequence Spaces

213

Theorem 3.2 The space `∞ {M, ∆, p, q} is paranormed by  1    n pk o q(∆xk ) p g(x) = q(x1 ) + inf ρ J : sup M tk k ≤ 1, ρ > 0 ρ k≥1 where J = max(1, H). Proof. Clearly g(θ) = 0, g(−x) = g(x). Next let x = (xk ), y = (yk ) ∈ `∞ {M, ∆, p, q}. Then there exists some ρ1 > 0 and ρ2 > 0 such that M(

q(∆yk ) p1k q(∆xk ) p1k )tk ≤ 1 and M ( )tk ≤ 1. ρ1 ρ2

Let ρ = ρ1 + ρ2 . Then we have n  q(∆x + ∆y )  1 o k k p sup M tk k ρ k≥1  1 o  n  q(∆x )  1 o  ρ2 q(∆yk ) ρ1 k p p tk k + tk k sup M sup M ≤ ρ1 + ρ2 k≥1 ρ ρ1 + ρ2 k≥1 ρ ≤ 1. Now we have n n  q(∆x + ∆y )  o 1 o pk k k p J g(x + y) = q(x1 + y1 ) + inf (ρ1 + ρ2 ) : sup M tk k ≤ 1 ρ k≥1 n  q(∆x )  o 1 n o pk k p tk k ≤ 1 ≤ q(x1 ) + inf (ρ1 ) J : sup M ρ1 k≥1 n  q(∆y )  o 1 o n pk k p + q(y1 ) + inf (ρ2 ) J : sup M tk k ≤ 1 ρ2 k≥1 ≤ g(x) + g(y). Let η ∈ C, then the continuity of the product follows from the following equality. n pk n  q(η∆x )  o 1 o k p J g(ηx) = q(ηx1 ) + inf ρ : sup M tk k ≤ 1, ρ > 0 ρ k≥1   n n o pk q(∆xk ) o p1k = |η|q(x1 ) + inf (|η|r) J : sup M tk ≤ 1, r > 0 , r k≥1 where

1 r

=

|η| ρ



Theorem 3.3 Let p ∈ `∞ , then the spaces c0 {M, ∆, p, q}, c{M, ∆, p, q} and `∞ {M, ∆, p, q} are complete paranormed spaces, paranormed by g. Proof. We prove it for the case `∞ {M, ∆, p, q} and the other cases can be established similarly. Let (xn ) be a Cauchy sequence in `∞ {M, ∆, p, q}, where i j xn = (xnk )∞ k=1 for all n ∈ N. Then g(x − x ) → 0, as i, j → ∞.

214

Thai J. Math. 3(2005)/ B. C. Tripathy and B. Sarma For a given ε > 0, let r and x0 be such that

> 0 and M ( rx20 ) ≥ sup (pk )tk .

ε rx0

k≥1

Now g(xi − xj ) → 0, as i, j → ∞ implies that there exists m0 ∈ N such that g(xi − xj ) < Then we obtain q(xi1 − xj1 ) < n pk n inf ρ J : sup M k≥1

ε rx0

ε , for all i, j ≥ m0 . rx0

and

q(∆xik − ∆xjk ) ρ

!

1 p

o

tk k

o ε ≤ 1, ρ > 0 < . rx0

(8)

This shows that (xi1 ) is a Cauchy sequence in X. Since X is complete then is convergent in X.

(xi1 )

Let lim xi1 = x1 , thus we have lim q(xi1 − xj1 ) < i→∞

j→∞

ε , which imply that rx0

ε . rx0

q(xi1 − x1 ) < Again from (8), we have M

q(∆xik − ∆xjk ) g(xi − xj )

!

1 p

tk k ≤ 1.

These implies that M

q(∆xik − ∆xjk ) g(xi − xj )

! ≤ (pk )tk ≤ M

Thus we obtain q(∆xik − ∆xjk )
0 such that   pk  q(∆xk ) M1 tk → 0, as k → ∞. ρ   k) for all k ∈ N. Let 0 < δ < 1 be chosen. For yk ≥ δ we Let yk = M1 q(∆x ρ have yk yk yk < 0 such that pk    q(∆xk ) M1 tk → 0, as k → ∞ ρ1 and

  pk  q(∆xk ) tk → 0, as k → ∞. M2 ρ2 Let ρ = max{ρ1 , ρ2 }. The rest follows from the following inequality.       pk  pk pk  q(∆xk ) q(∆xk ) q(∆xk ) tk ≤ D M 1 tk + M 2 tk . (M1 + M2 ) ρ ρ1 ρ2 Thus c0 {M1 , ∆, p, q}∩c0 {M2 , ∆, p, q} ⊂ c0 {M1 +M2 , ∆, p, q}. The cases c{M, ∆, p, q} and `∞ {M, ∆, p, q} can be proved in a similar way.  It can be easily shown that:

217

Some Classes of Difference Paranormed Sequence Spaces Proposition 3.9 Z{M, p, q} ⊆ Z{M, ∆, p, q} for Z = c, c0 and `∞ . The following results can be obtained from the lemmas listed in section 2.

Proposition 3.10 Let h = inf pk and H = sup pk then the following are equivalent (i) H < ∞ and h > 0, (ii) c0 {M, ∆, p, q} = c0 (M, ∆, p, q) (iii) `∞ {M, ∆, p, q} = `∞ (M, ∆, p, q). Proposition 3.11 Let p, s be two sequences of strictly positive numbers. Then c0 {M, ∆, p, q} ∼ = c0 {M, ∆, s, q} if and only if there exists a sequence u = (uk ) of strictly positive numbers such that eq.(1) and eq.(2) hold. s−1 −p−1 k

Proposition 3.12 Let the sequence a = (ak ) = (skk pk c0 {M, ∆, s, q} if and only if eq.(3) and eq.(4) hold.

). Then c0 {M, ∆, p, q} ∼ =

s−1 −p−1

Proposition 3.13 Let the sequence a = (ak ) = (skk pk k ). Then   1 1 lim − = 0 implies c0 {M, ∆, p, q} ∼ = c0 {M, ∆, s, q}. k→∞ pk sk Proposition 3.14 Let fk = pskk for every k ∈ N. Let (fk ) and (fk−1 ) both be in `∞ . Then `∞ {M, ∆, p, q} ∼ = `∞ {M, ∆, s, q}(f ). Proposition 3.15 Let s = (sk ) ∈ `∞ . Then `∞ {M, ∆, p, q} ⊆ `∞ {M, ∆, s, q} if and only if eq.(5) holds. Proposition 3.16 Let s = (sk ) ∈ `∞ and c0 {M, ∆, p, q} ∼ = c0 {M, ∆, s, q} then c0 (M, ∆, p) ∼ c (M, ∆, q). = 0

References [1] H. Kizmaz, On certain sequence spaces,Canadian Math. Bull., 24(1981), 169176. [2] M. A. Krasnoselkii and Y. B. Rutitsky,Convex Functions and Orlicz Spaces, Groningen, Netherlands, 1961. [3] J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces, Israel J. Math., 10(1971), 379-390. [4] C. G. Lascarides, On the equivalence of certain sets of sequences, Indian Jour. Math., 25(1)(1983), 41-52.

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[5] I. J. Maddox, Paranormed sequence spaces generated by infinite matrices, Proc. Camb. Phil. Soc., 64(1968), 335-345. [6] H. Nakano, Modular sequence spaces, Proc. Japan Acad., 27(1951), 508-512. [7] W. H. Ruckle, FK- spaces in which the sequences of coordinate vectors is bounded, Canad. J. Math., 25 (1973), 973-978. [8] S. Simons, The spaces `(pν ) and m(pν ), Proc. London Math Soc., 15(1965), 422-436. [9] B. C. Tripathy, On some classes of difference paranormed sequence spaces associated with multiplier sequences, Internet J. Math. Sci., 2(1) (2003), 159-166. (Received 25 August 2005) Binod Chandra Tripathy1 and Bipul Sarma2 Mathematical Sciences Division Institute of Advanced Study in Science and Technology Paschim Boragaon; Garchuk Guwahati - 781 035; India. e-mail : 1 [email protected], [email protected], 2 [email protected].