New results on the sequence spaces inclusions equations involving the spaces w1 and w0

Abstract

Given any sequence $a=(a_n)_{n\geq 1}$ of positive real numbers and any set $E$ of complex sequences, we write $E_a$ for the set of all sequences $y=(y_n)_{n\geq 1}$ such that $y/a=(y_n/a_n)_{n\geq 1}\in E$, in particular, $c_a$ denotes the set of all sequences $y$ such that $y/a$ converges. In this paper, we use the well known sets $w_\infty =\{y\in \omega :\sup_n(n^-1\sum_{k=1}^{n}|y_k|) <\infty\}$ and $w_0=\{y\in \omega :\lim_{n\rightarrow \infty }( n^{-1}\sum_{k=1}^{n}|y_{k}|) =0\} called the spaces of strongly bounded and strongly summable to zero sequences by the Ces`aro method. Then we deal with the solvability of the (SSIE) of the form $w_\infty\subset \mathcal{E}+F_{x}^{\prime }$ with $F^{\prime }=c_0, s_1, or w_\infty$ and $w_0 \subset \mathcal{E}+F_{x}'$ with $F'=c_{0}, c, s_1, or w_\infty$, where $\mathcal{E}$ is a linear space of sequences. We apply these results to the solvability of each of the (SSIE) $w_\infty \subset w_0+F_x', w_ infty \subset bv_p+F_x', w_\infty \subset (c_0)_{R_t}+F_x', w_\infty \subset (c_0)_{C(\lambda)}+F_x'$ with $F'\in \{c_0, s_1, w_\infty\}$. These results extend some of those stated in [18, 15].