Solutions of nonlinear difference equations in the domain of $(\zeta _{n})$-Cesàro matrix in $\ell _{t(\cdot)}$ of nonabsolute type, and its pre-quasi ideal

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Abstract We have constructed the sequence space $(\Xi (\zeta ,t) )_{\upsilon }$ (Ξ(ζ,t))υ , where $\zeta =(\zeta _{l})$ ζ=(ζl) is a strictly increasing sequence of positive reals tending to infinity and $t=(t_{l})$ t=(tl) is a sequence of positive reals with $1\leq t_{l}<\infty $ 1≤tl<∞ , by the domain of $(\zeta _{l})$ (ζl) -Cesàro matrix in the Nakano sequence space $\ell _{(t_{l})}$ ℓ(tl) equipped with the function $\upsilon (f)=\sum^{\infty }_{l=0} ( \frac{ \vert \sum^{l}_{z=0}f_{z}\Delta \zeta _{z} \vert }{\zeta _{l}} )^{t_{l}}$ υ(f)=∑l=0∞(|∑z=0lfzΔζz|ζl)tl for all $f=(f_{z})\in \Xi (\zeta ,t)$ f=(fz)∈Ξ(ζ,t) . Some geometric and topological properties of this sequence space, the multiplication mappings defined on it, and the eigenvalues distribution of operator ideal withs-numbers belonging to this sequence space have been investigated. The existence of a fixed point of a Kannan pre-quasi norm contraction mapping on this sequence space and on its pre-quasi operator ideal formed by $(\Xi (\zeta ,t) )_{\upsilon }$ (Ξ(ζ,t))υ ands-numbers is presented. Finally, we explain our results by some illustrative examples and applications to the existence of solutions of nonlinear difference equations.

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