The Solution by Iteration of Nonlinear Equations in Uniformly Smooth Banach Spaces

Abstract

LetEbe a uniformly smooth Banach space and letT:D(T)⊂E↦Ebe a strong pseudocontraction with an open domainD(T) inEand a fixed pointx*∈D(T). We establish the strong convergence of the Mann and Ishikawa iterative processes (with errors) to the fixed point ofT. Related results deal with the iterative solution of operator equations of the formsf∈Txandf∈x+λTx, λ>0, whenTis a set-valued strongly accretative operator. Our theorems include the cases in which the operatorTis defined only locally. Explicit error estimates are also given.