Abstract

In this paper we deal with special {\it sequence spaces equations (SSE) with operators}, which are determined by an identity whose each term is a {\it sum or a sum of products of sets of the form $\chi_{a}(T)$ and $\chi_{f(x)}(T)$} where $f$ maps $U^{+}$ to itself, and $\chi$ is any of the symbols $s$, $s^{0}$, or $s^{(c)}$. We solve the equation $\chi_{x}(\Delta )=\chi_{b}$ where $\chi$ is any of the symbols $s$, $s^{0}$, or $s^{(c)}$ and determine the solutions of (SSE) with operators of the form $(\chi_{a}\ast\chi_{x}+\chi_{b})(\Delta)=\chi_{\eta}$ and $[\chi_{a}\ast(\chi_{x})^{2}+\chi_{b}\ast\chi_{x}](\Delta)=\chi_{\eta}$ and $\chi_{a}+\chi_{x}(\Delta)=\chi_{x}$ where $\chi$ is any of the symbols $s$, or $s^{0}$.

Keywords: Sequence space; operator of the first difference; BK space; infinite matrix; sequence spaces equations (SSE); (SSE) with operators.

MSC: 40C05, 46A15

Pages:  39$-$52     

Volume  64 ,  Issue  1 ,  2012