Local convergence of a family of iterative methods for Hammerstein equations

In this paper we give a local convergence result for a uniparametric family of iterative methods for nonlinear equations in Banach spaces. We assume boundedness conditions involving only the first Fréchet derivative, instead of using boundedness conditions for high order derivatives as it is usual in studies of semilocal convergence, which is a drawback for solving some practical problems. The existence and uniqueness theorem that establishes the convergence balls of these methods is obtained. We apply this theory to different examples, including a nonlinear Hammerstein equation that have many applications in chemistry and appears in problems of electro-magnetic fluid dynamics or in the kinetic theory of gases. With these examples we illustrate the advantages of these results. The global convergence of the method is addressed by analysing the behaviour of the methods on complex polynomials of second degree.