On Modular Difference Sequence Spaces

Parmeshwary Dayal Srivastava, Atanu Manna

Abstract

In this paper, using $m$-th order difference operator $\Delta^{(m)}$ and a sequence $\{\alpha_n\}_{n=0}^{\infty}$ of strictly positive real numbers, sequence spaces $\Delta^{(m)}l_\alpha\{f_n\}=\{x\in s:(\Delta^{(m)}x)_j\in l_\alpha\{f_n\}\}$ and $\Delta^{(m)}l^\alpha\{g_n\} = \{x\in s :(\Delta^{(m)}x)_j \in l^{\alpha}\{g_n\}\}$ are introduced, where $x=\{\xi_j\}_{j=0}^{\infty}\in s$ and $ \{f_n\}_{n=0}^{\infty}$, $ \{g_n\}_{n=0}^{\infty}$ are sequence of Orlicz functions. It is shown that these are separable Banach spaces and dense $F_\sigma$-set of the first Baire category in $s$, the space of all real sequences with the Fr\`{e}chet metric. Some earlier results related to Baire category are obtained when the sequence $\{\alpha_n\}_{n=0}^{\infty}$ is chosen specifically.