ABSTRACT
SOME FURTHER APPLICATIONS OF THE NEWTON METHOD FOR FUNCTIONAL EQUATIONS *
In our previous papers [l]-[3] we have developed a theory of the Newton method for functional equations in normed spaces; certain applications of this theory were given as well. Later this method was studied in numerous papers by other authors [4]—[7]. In the notes [8, 9] I outlined a far-reaching generalization of this theory to functional equations in spaces normed by elements of semiordered spaces, using the majorant principle. It turns out that this principle provides a simpler and more exact construction of this theory for equations in normed spaces as well, and helps establish some further theorems about such equations. These matters are discussed in the present paper, along with some new applications of this theory to certain specific functional equations and, in particular, to the problems of finding approximate solutions to non-linear integral equations.
