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_n∥  for all 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.