Kannan Contraction Operator on the Domain of r . -Cesàro Matrix in ℓ t . and Related Prequasi Ideal with an Application of Nonlinear Difference Equations

Abstract

In this article, the sequence space ${\left(\Xi \left(r,t\right)\right)}_{\upsilon }$ has been built by the domain of $\left(r_l\right)$ -Cesàro matrix in Nakano sequence space ${\ell }_{\left(t_l\right)}$ , where $t=\left(t_l\right)$ and $r=\left(r_l\right)$ are sequences of positive reals with $1\le t_l<\infty$ , and $\upsilon \left(f\right)={\sum }_{l=0}^{\infty }{\left(\left|{\sum }_{z=0}^l{r}_z{f}_z\right|/{\sum }_{z=0}^l{r}_z\right)}^{t_l}$ , with $f=\left(f_z\right)\in \Xi \left(r,t\right)$ . Some topological and geometric behavior of ${\left(\Xi \left(r,t\right)\right)}_{\upsilon }$ , the multiplication maps acting on ${\left(\Xi \left(r,t\right)\right)}_{\upsilon }$ , and the eigenvalues distribution of operator ideal constructed by ${\left(\Xi \left(r,t\right)\right)}_{\upsilon }$ and $s$ -numbers have been examined. The existence of a fixed point of Kannan prequasi norm contraction mapping on this sequence space and on its prequasi operator ideal are investigated. Moreover, we indicate our results by some explanative examples and actions to the existence of solutions of nonlinear difference equations.