Some multiplier difference sequence spaces defined by a sequence ...

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176 Hemen DuttaThe studies on paranormed sequence spaces were initiated by Nakano [8]and Simons [9] at the initial stage. Later on it was further studied by Maddox[7], Lascardies [5], Lascardies and Maddox [6], Ghosh and Srivastava[3] and many others.2 Definitions and PreliminariesA sequence space E is said to be solid (or normal) if (x k ) ∈ E implies(α k x k ) ∈ E for all sequences of scalars (α k ) with |α k | ≤ 1 for all k ∈ N.A sequence space E is said to be monotone if it contains the canonicalpreimages of all its step spaces.A sequence space E is said to be symmetric if (x π(k) ) ∈ E, where π is apermutation on N.A sequence space E is said to be convergence free if (y k ) ∈ E whenever(x k ) ∈ E and y k = 0 whenever x k = 0.A sequence space E is said to be sequence algebra if (x k .y k ) ∈ E whenever(x k ) ∈ E and (y k ) ∈ E.Let p = (p k ) be any bounded sequence of positive real numbers andΛ = (λ k ) be a sequence of non-zero scalars. Let m, n be non-negativeintegers, then for a sequence F = (f k ) of modulus functions we define thefollowing sequence spaces:c 0 (F , ∆ n (m), Λ, p) = {x = (x k ) : limk→∞(f k (|∆ n (m)λ k x k |)) p k= 0},c(F , ∆ n (m) , Λ, p) = {x = (x k) : limk→∞(f k (|∆ n (m) λ kx k − L|)) p k = 0,
Some multiplier difference sequence spaces 177for some L ∈ C},l ∞ (F , ∆ n (m), Λ, p) = {x = (x k ) : sup(f k (|∆ n (m)λ k x k |)) p k< ∞},k≥1where (∆ n (m) λ kx k ) = (∆ n−1(m) λ kx k − ∆ n−1(m) λ k−mx k−m ) and ∆ 0 (m) λ kx k = λ k x kfor all k ∈ N, which is equivalent to the following binomial representation:n∑( ) n∆ n (m)λ k x k = (−1) v λ k−mv x k−mv .vv=0In the above expansion it is important to note that we take x k−mv = 0and λ k−mv = 0, for non-positive values of k − mv. Also it is obvious thatc 0 (F , ∆ n (m) , Λ, p) ⊂ c(F , ∆n (m) , Λ, p) ⊂ l ∞(F , ∆ n (m), Λ, p).The inclusions are strict follows from the following examples.Example 1 Let m = 2, n = 2, f k (x) = x 6 , for all k odd and f k (x) = x 2for all k even, for all x ∈ [0, ∞) and p k = 1 for all k ≥ 1. Consider thesequences Λ = (k 8 ) and x = ( 1 ). Then x belongs to c(F, ∆ 2 k 6 (2), Λ, p), but xdoes not belong to c 0 (F, ∆ 2 (2), Λ, p).Example 2 Let m = 2, n = 2, f k (x) = x 2 , for all k ≥ 1 and x ∈ [0, ∞)and p k = 2 for all k odd and p k = 3 for all k even. Consider the sequencesΛ = {1, 1, 1, ...} and x = {1, 3, 2, 4, 5, 7, 6, 8, 9, 11, 10, 12, ...}. Then x belongsto l ∞ (F, ∆ 2 (2) , Λ, p), but x does not belong to c(F, ∆2 (2), Λ, p).Lemma 1 If a sequence space E is solid, then E is monotone.3 Main ResultsIn this section we prove the main results of this article. The proof of thefollowing result is easy, so omitted.
Page 1 and 2: General Mathematics Vol. 17, No. 3
Page 3: Some multiplier difference sequence
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