Solutions of nonlinear difference equations in the domain of (zeta _{n})-Ces\xe0ro matrix in ell _{t(cdot)} of nonabsolute type, and its pre-quasi ideal

Abstract

We have constructed the sequence space (Xi (zeta ,t) )_{upsilon }, where zeta =(zeta _{l}) is a strictly increasing sequence of positive reals tending to infinity and t=(t_{l}) is a sequence of positive reals with 1leq t_{l}<infty , by the domain of (zeta _{l})-Cesàro matrix in the Nakano sequence space ell _{(t_{l})} 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}} for all 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 (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.