Generalized Difference Spaces of Non-Absolute Type of Convergent and Null Sequences

and Applied Analysis 3 2. The Difference Sequence Spaces c 0 B and c λ B of Non-Absolute Type The difference sequence spaces have been studied by several authors in different ways see e.g. 12, 16–21 . In the present section, we introduce the spaces c 0 Δ , and c λ Δ , and show that these spaces are BK-spaces of non-absolute type which are norm isomorphic to the spaces c0 and c, respectively. We assume throughout that λ λk ∞ k 0 is a strictly increasing sequence of positive reals tending to∞, that is, 0 < λ0 < λ1 < · · · , lim k→∞ λk ∞. 2.1 Recently, Mursaleen and Noman 22 studied the sequence spaces c 0 and c λ of nonabsolute type, and later they introduced the difference sequence spaces c 0 Δ and c λ Δ in 21 of non-absolute type as follows: c 0 Δ { x xk ∈ ω : lim n→∞ 1 λn n ∑ k 0 λk − λk−1 xk − xk−1 0 } , c Δ { x xk ∈ ω : lim n→∞ 1 λn n ∑ k 0 λk − λk−1 xk − xk−1 exists } . 2.2 Here and after, we use the convention that any termwith a negative subscript is equal to zero, for example, λ−1 0 and x−1 0. With the notation of 1.2 we can redefine the spaces c 0 Δ and c Δ by c 0 Δ ( c 0 ) Δ , c Δ ( c ) Δ , 2.3 where Δ denotes the band matrix representing the difference operator, that is, Δx xk − xk−1 ∈ ω for x xk ∈ ω. Let r and s be nonzero real numbers and define the generalized difference matrix B r, s {bnk r, s } by bnk r, s : ⎧ ⎪ ⎨ ⎪ ⎩ r, k n, s, k n − 1, 0, otherwise, 2.4 for all k, n ∈ N. The B r, s -transform of a sequence x xk is B r, s k x rxk sxk−1, ∀k ∈ N. 2.5 We note that the matrix B r, s can be reduced to the difference matrices Δ in case r 1 and s −1. So, the results related to the matrix domain of the matrix B r, s are more general and more comprehensive than the consequences of the matrices domain of Δ and include them. 4 Abstract and Applied Analysis Now, following Başar and Altay 18 and Aydın and Başar 17 , we proceed slightly differently to Kızmaz 19 and the other authors following him and employ a technique of obtaining a new sequence space by means of the matrix domain of a triangle limitation method. We thus introduce the difference sequence spaces c 0 B and c λ B , which are the generalization of the spaces c 0 Δ and c λ Δ introduced by Mursaleen and Noman 21 , as follows: c 0 B { x xk ∈ ω : lim n→∞ 1 λn n ∑ k 0 λk − λk−1 rxk sxk−1 0 } , c B { x xk ∈ ω : lim n→∞ 1 λn n ∑ k 0 λk − λk−1 rxk sxk−1 exists } . 2.6 With the notation of 1.2 , we can redefine the spaces c 0 B and c λ B as c 0 B ( c 0 )