# Iterated nets of weakly contractive operators

### Abstract

Let *K* be a subset of a Banach space *X*. An operator *T*:*K*→*K* is said to be weakly contractive, if for every point *x*∈*K* there is a number *C*(*x*) and for any sequence {*x _{n}*} with lim

*x*=

_{n }*x*there exists a number

*N*such that ∥

*Tx−Tx*∥ ≤

_{n}*C*(

*x*)∥

*x−x*∥ for all

_{n}*n*>

*N*. Every contractive operator is weakly contractive.

The investigation of transfinite iteration processes leads to the study of the sequence space Σ(

*X*) with Tikhonov's topology, to the study of fundamental and uniformly fundamental nets in the space Σ(

*X*). In Theorem 2, by using regular operators summing divergent sequences, necessary and sufficient conditions for the existence of a fixed point of a weakly contractive operator are given. In particular, if

*T*is a contractive operator, then by a theorem of Kirk and Massa [1],

*T*

^{Ω}(

*x*)=

*z*for all

*x*∈

*K*, and by Theorem 2

*T*(

*z*)=

*z*.