Application of the infinite matrix theory to the solvability of sequence spaces inclusion equations with operators

Abstract

UDC 517.9
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 = $y/a=(y_{n}/a_{n})_{n\geq 1}\in E.$ In particular, $c_{a}$ denotes the set of all sequences $y$ such that $y/a$ converges. We deal with sequence spaces inclusion equations (SSIE) of the form $F\subset E_{a}+F_{x}'$ with $e\in F$ and explicitly find the solutions of these SSIE when $a=(r^{n})_{n\geq 1},$ $F$ is either $c$ or $s_{1},$ and $E,$ $F'$ are any sets $c_{0},$ $c,$ $s_{1},$ $\ell_{p},$ $w_{0},$ and $w_{\infty }.$ Then we determine the sets of all positive sequences satisfying each of the SSIE $c\subset D_{r}\ast (c_{0})_{\Delta }+c_{x}$ and $c\subset D_{r}\ast (s_{1})_{\Delta}+c_{x},$ where $\Delta $ is the operator of the first difference defined by $\Delta _{n}y=y_{n}-y_{n-1}$ for all $n\geq 1$ with $y_{0}=0.$ Then we solve the SSIE $c\subset D_{r}\ast E_{C_{1}}+s_{x}^{(c)}$ with $E\in \{ c,s_{1}\} $ and $s_{1}\subset D_{r}\ast(s_{1}) _{C_{1}}+s_{x},$ where $C_{1}$, is the Cesaro operator defined by $(C_{1}) _{n}y=n^{-1} \displaystyle \sum \nolimits_{k=1}^{n}y_{k}$ for all $y$. We also deal with the solvability of the sequence spaces equations (SSE) associated with the previous SSIE and defined as $D_{r}\ast E_{C_{1}}+s_{x}^{(c)}=c$ with $E\in \{c_{0},c, s_{1}\} $ and $D_{r}\ast E_{C_{1}}+s_{x}=s_{1}$ with $E\in \{ c,s_{1}\}.$