Results on Controlling the Residuals of Perturbed Newton-like Methods on Banach Spaces with a Convergence Structure

In this study, we provide suucient conditions for controllingthe residuals of perturbed Newton-like methods, in such a way that the convergence of such methods is ensured. In order to achieve this, we apply a theorem due to Kantorovich on a setting of a regular space with a convergence structure. In this study, we are concerned with approximating a solution x of the nonlinear operator equation F(x) + Q(x) = 0; (1) where F is a Frechet-diierentiable operator deened on a convex subset D of a Banach space X with values in X, and Q is a non-diierentiable nonlinear operator with the same domain and values in X. We generate a sequence fx n g (n 0) using the perturbed Newton-like method scheme given by x n+1 = x n + n (n 0) (2) where the correction n satisses A(x n) n = ?(F(x n) + Q(x n)) + r n (n 0) (3) for a suitable r n 2 X (n 0) called residual. The importance of studying perturbed Newton-like methods comes from the fact that variants of Newton's method can be considered as procedures of this type. In fact, the approximations (2) and (3) characterize any iterative process in which the corrections are taken as approximate solutions of the Newton equations. This happens when approximation (2) is solved by any iterative method or when therein the derivative is replaced by a suitable approximation. In 4] we provided suucient conditions for the convergence of iteration (2) to a solution x of equation (1), by assuming that X is a Banach space with a convergence structure (to be precised later). Here we derive suucient conditions for controlling the residuals r n in such a way that the convergence of the sequence fx n g (n 0) to a solution of equation (1) is ensured. We also refer the reader to 5], 6], 8] and the references there for relevant work, which however is valid on a Banach space X without a convergence structure. The advantages of working on a Banach space with a convergence structure have been explained in some detail in 4], 6], 7].