Some Topological and Geometrical Properties of a New Difference Sequence Space

and Applied Analysis 3 2. ap Δ Difference Sequence Space In the present section, we introduce the difference sequence space ap Δ and emphasize its some properties. Although the difference sequence space λ Δ corresponding to the space λ was defined by Kızmaz 10 as follows: λ Δ {x xk ∈ ω : xk − xk 1 ∈ λ}, 2.1 the difference sequence space corresponding to the space p was not examined, where λ denotes the anyone of the spaces c0, c or ∞. So, Başar and Altay have recently studied the sequence space bvp, the space of p-bounded variation, in 11 defined by bvp { x xk ∈ ω : xk − xk−1 ∈ p } , 1 ≤ p < ∞, 2.2 which fills up the gap in the existing literature. Recently, Aydı́n and Başar 12 studied the sequence spaces ar0 and a r c, defined by ar0 { x xk ∈ ω : lim n−→∞ 1 n 1 n ∑ k 0 ( 1 r ) xk 0 } , ac { x xk ∈ ω : lim n−→∞ 1 n 1 n ∑ k 0 ( 1 r ) xk exists } . 2.3 Aydı́n and Başar 13 introduced the difference sequence spaces ar0 Δ and a r c Δ , defined by ar0 Δ { x xk ∈ ω : lim n−→∞ 1 n 1 n ∑ k 0 ( 1 r ) xk − xk−1 0 } , ac Δ { x xk ∈ ω : lim n−→∞ 1 n 1 n ∑ k 0 ( 1 r ) xk − xk−1 exists } . 2.4 Aydı́n 14 introduced ap sequence space, defined by ap { x xk ∈ ω : ∑ n ∣∣∣∣ 1 n 1 n ∑ k 0 ( 1 r ) xk ∣∣∣∣ p < ∞ } ; 1 ≤ p < ∞. 2.5 Define the matrix Δ δnk by δnk ⎧ ⎨ ⎩ −1 n−k, n − 1 ≤ k ≤ n, 0, 0 ≤ k < n − 1 or k > n. 2.6 As was made by Başar and Altay in 11 , we treat slightly more different than Kızmaz and the other authors following him and employ the technique obtaining a new sequence space 4 Abstract and Applied Analysis by the matrix domain of a triangle limitation method. We will introduce the sequence space ap Δ which is a natural continuation of Aydı́n and Başar 13 , as follows: ap Δ { x xk ∈ ω : ∑ n ∣∣∣∣ 1 n 1 n ∑ k 0 ( 1 r ) xk − xk−1 ∣∣∣∣ p < ∞ } ; 1 ≤ p < ∞. 2.7 With the notation of 1.3 , we may redefine the space ap Δ by ap Δ ( ap ) Δ . 2.8 Define the sequence y {yn r } which will be frequently used as the Br-transform of a sequence x xk , that is,