The difference sequence spaces ℓ∞(▵), c(▵), and c0(▵) were studied by Kızmaz. The main purpose of the present paper is to introduce the space bvp consisting of all sequences whose differences are in the space ℓp, and to fill up the gap in the existing literature. Moreover, it is proved that the space bvp is the BK-space including the space ℓp. We also show that the spaces bvp and ℓp are linearly isomorphic for 1 ≤ p ≤ ∞. Furthermore, the basis and the α-, β-, and γ-duals of the space bvp are determined and some inclusion relations are given. The last section of the paper is devoted to theorems on the characterization of the matrix classes (bvp : ℓ∞), (bv∞ : ℓp), and (bvp : ℓ1), and the characterizations of some other matrix classes are obtained by means of a suitable relation.