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

  • Malafosse B. de

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 $E\in \{ c,s_{1}\}.$
Published
25.08.2019
How to Cite
deM. B. “Application of the Infinite Matrix Theory to the Solvability of Sequence spaces inclusion Equations With Operators”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, no. 8, Aug. 2019, pp. 1040-52, http://umj.imath.kiev.ua/index.php/umj/article/view/1496.
Section
Research articles