formulations of Newton's method abound in the literature (see, e.g., [2], [6], [8], [12], [13]). Our aim here is twofold: to present in a unified setting a number of old and some new results that may be of particular interest in applications, and to discuss as illustrations two numerical techniques of solving boundary value problems for ordinary differential equations. These are the Goodman-Lance method [9], [15] and the so-called method of quasi-linearization [4], [14], which have attracted considerable attention in recent years.
Throughout the sequel we will denote by X and Y real Banach spaces, of finite or infinite dimension, and suppose that the basic equation to be solved, (i.l)
is given by a mappingf of an open subset U of X into Y which is, at least, continuously differentiable. This means that f has a derivative f'(x) at every point x ~ U such that f' is a continuous mapping of U into the Banaeh space L(X, Y) of continuous linear mappings of X into F. As usual, if(x) is defined as the unique element in L(X, Y) for which lim ~
in particular, ilf'(x)ll = sup{[jf'(x) h [I :11 h II ~< 1}. It follows then from the mean value theorem that there exists , for every point x o ~ U and every ~ > O, an open ball B(x o , r)C U with center x o and radius r such that (1.2) if(x2) --f(xl) --f'(xo)(Xz -xx)i <, ~l x2 --x a]l for anY two points Xl, x~ in B(xo, r).