Error bounds of Newton type process on Banach spaces

Abstract

For solving a nonlinear operator equationF(x)=0 in Banach spaces, the Newton's method or Newton type methods are important numerical techniques. We use the properties of real equationt=?(t) majorizing an operator equationx=Gx to find a fixed point ofG as a solution of equationF(x)=0. Various type of operatorsG are considered in this paper. For a nonlinear operatorG, we would find a real function ? majorizing the operatorG and it will be related to a rate of convergence $$\omega (r) = \frac{{r^2 }}{{2(r^2 + d)^{1/2} }}.$$ It follows thatG has a fixed point as a solution ofF(x)=0. Practical limitations of error bounds like as in Potra and Ptak [5] are discribed.