ABSTRACT
We present in this chapter the convergence analysis for successive methods to solve xed points problems.
In this section, we are concerned with the problem of approximating a locally unique xed point x? of nonlinear equation
x = G(x); (12.1)
where G is a Frechet{dierentiable operator dened on a convex subset D of a Banach space X with values in D. Recall that x? is a xed point of equation (12.1) if
x? = G(x?) (x? 2 intD): (12.2) The method of successive approximations
xk+1 = G(xk) (k 0; x0 2 D) (12.3) has been used by several authors to generate a sequence fxkg converging to x? under various assumptions (cf. [139], [685]). The usual hypothesis is as follows
k G0(x) k q < 1 (x 2 D); (12.4) which implies that operator is a contraction on D and as such the existence of x? is guaranteed on D. The contraction hypothesis (12.4) however is too strong in general.
