Solutions of nonlinear difference equations in the domain of (ζn)-Cesàro matrix in ℓt(⋅) of nonabsolute type, and its pre-quasi ideal

We have constructed the sequence space ( Ξ ( ζ , t ) ) υ , where ζ = ( ζ l ) is a strictly increasing sequence of positive reals tending to infinity and t = ( t l ) is a sequence of positive reals with 1 ≤ t l < ∞ , by the domain of ( ζ l ) -Cesàro matrix in the Nakano sequence space ℓ ( t l ) equipped with the function υ ( f ) = ∑ l = 0 ∞ ( | ∑ z = 0 l f z Δ ζ z | ζ l ) t l for all f = ( f z ) ∈ 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 with s -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 ( Ξ ( ζ , t ) ) υ and s -numbers is presented. Finally, we explain our results by some illustrative examples and applications to the existence of solutions of nonlinear difference equations.