Applications of the Summability Theory to the Solvability of Certain Sequence Spaces Equations with Operators of the Form $B\left( r,s\right) $

Abstract

In this paper we deal with sequence spaces inclusion equations (SSIE), which are determined by an inclusion where each term is a sum or a sum of products of sets of the form $\chi_{a}\left( T\right) $ and $\chi_{f\left( x\right) }\left( T\right) $ where $f$ map $U^{+}$ to itself, and $\chi \in \left\{ \mathbf{s},\mathbf{s}^{0},\mathbf{s}^{\left( c\right) }\right\} $, the sequence $x$ is the unknown and $T$ is a given triangle. Here we give characterizations of the (SSIE) $\chi_{x}\left( B\left( r,s\right) \right) \subset \chi_{x}\left( B\left( r',s'\right) \right) $ and of the (SSE) $\chi_{x}\left( B\left( r,s\right) \right) =\chi_{x}\left( B\left( r's'\right) \right) $, where $\chi=s,s^{0}$, or $s^{\left( c\right) }$ and $B\left( r,s\right) $ is the generalized operator of first difference defined by $B\left( r,s\right) _{n}y=ry_{n}+sy_{n-1}$ for all $n\geq 2$ and $B\left( r,s\right) _{1}y_{1}=ry_{1}$. We give an application to the spectrum of $B\left( r,s\right) $ considered as an operator from $\chi_{x}$ to itself, where $\chi=\mathbf{s}$, or $\mathbf{s}^{0}$. Then we apply these results to the solvability of the sequence spaces equation $\chi_{a}+\mathbf{s}_{x}^{(c)}(B(r,s)) = \mathbf{s}_{x}^{(c)}$ where $\chi =\mathbf{s}, \mathbf{s}^{0}$, or $\mathbf{s}^{\left( c\right) }$ and $x$ is the unknown.