An application of Newton's method to differential and integral equations (Q2711361)

Let \(F\) denote a nonlinear operator between two Banach spaces. In an earlier paper [J. Comput. Appl. Math. 79, No. 1, 131-145 (1997; Zbl 0872.65045)] the first author proved a semilocal convergence theorem for Newton's method applied to \(F(x) = 0\) under stronger conditions than those of the classical Kantorovich theorem. In essence, a Lipschitz condition for the second derivative is required and a third-degree polynomial is used as majorizing function. NEWLINENEWLINENEWLINEIn the present paper some comparisons of the convergence and error estimates for the two theorems are given. Then, as an application, these results are used to obtain existence and uniqueness information for the solution of scalar ordinary differential equations and nonlinear integral equations. A final section concerns regions where the solution is located and where it is unique.