# Altay B.

### On Some Euler Sequence Spaces of Nonabsolute Type

Ukr. Mat. Zh. - 2005. - 57, № 1. - pp. 3–17

In the present paper, the Euler sequence spaces $e_0^r$ and $e^r_c$ of nonabsolute type which are the $BK$-spaces including the spaces $c_0$ and $c$ have been introduced and proved that the spaces $e_0^r$ and $e^r_c$ are linearly i somorphic to the spaces $c_0$ and $c$, respectively. Furthemore, some inclusion theorems have been given. Additionally, the $\alpha-, \beta-, \gamma-$ and continuous duals of the spaces $e_0^r$ and $e^r_c$ have been computed and their basis have been constructed. Finally, the necessary and sufficient conditions on an infinite matrix belonging to the classes $(e^r_c :\; {l}_p)$ and $(e^r_c :\; c)$ have been determined and the characterizations of some other classes of infinite matrices have also been derived by means of a given basic lemma, where $1 \leq p \leq \infty$.

### On the Space of Sequences of *p*-Bounded Variation and Related Matrix Mappings

Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 108-118

The difference sequence spaces ℓ_{∞}(▵), *c*(▵), and *c* _{0}(▵) were studied by Kızmaz. The main purpose of the present paper is to introduce the space *bv* _{p} consisting of all sequences whose differences are in the space ℓ_{ p }, and to fill up the gap in the existing literature. Moreover, it is proved that the space *bv* _{p} is the BK-space including the space ℓ_{ p }. We also show that the spaces *bv* _{p} and ℓ_{ p } are linearly isomorphic for 1 ≤ *p* ≤ ∞. Furthermore, the basis and the α-, β-, and γ-duals of the space *bv* _{p} 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 (*bv* _{p} : ℓ_{∞}), (bv_{∞} : ℓ_{ p }), and (*bv* _{p} : ℓ_{1}), and the characterizations of some other matrix classes are obtained by means of a suitable relation.