General study of iterative processes of R-Order at least three under weak convergence conditions

We consider a family of Newton-type iterative processes solving nonlinear equations in Banach spaces, that generalizes the usually iterative methods of R-order at least three. The convergence of this family in Banach spaces is usually studied when the second derivative of the operator involved is Lipschitz continuous and bounded. In this paper, we relax the first condition, assuming that vertical bar vertical bar F ''(x) - F '' (y)vertical bar vertical bar omega (vertical bar vertical bar x - y vertical bar vertical bar), where. is a nondecreasing continuous real function. We prove that the different R-orders of convergence that we can obtain depend on the quasihomogeneity of the function omega. We end the paper by applying the study to some nonlinear integral equations.