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

Abstract

Abstract We have constructed the sequence space ( Ξ ( ζ , t ) ) υ $(\Xi (\zeta ,t) )_{\upsilon }$ , where ζ = ( ζ l ) $\zeta =(\zeta _{l})$ is a strictly increasing sequence of positive reals tending to infinity and t = ( t l ) $t=(t_{l})$ is a sequence of positive reals with 1 ≤ t l < ∞ $1\leq t_{l}<\infty $ , by the domain of ( ζ l ) $(\zeta _{l})$ -Cesàro matrix in the Nakano sequence space ℓ ( t l ) $\ell _{(t_{l})}$ equipped with the function υ ( f ) = ∑ l = 0 ∞ ( | ∑ z = 0 l f z Δ ζ z | ζ l ) t l $\upsilon (f)=\sum^{\infty }_{l=0} ( \frac{ \vert \sum^{l}_{z=0}f_{z}\Delta \zeta _{z} \vert }{\zeta _{l}} )^{t_{l}}$ for all f = ( f z ) ∈ Ξ ( ζ , t ) $f=(f_{z})\in \Xi (\zeta ,t)$ . 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 $(\Xi (\zeta ,t) )_{\upsilon }$ 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.

Keywords

• Kannan contraction mapping
• Minimum space
• Multiplication mapping
• Cesàro sequence space of nonabsolute type
• s-numbers
• Pre-quasi ideal