A Harmonic-Type Method for Nonlinear Equations in Banach Space

In this work, we investigate the local and semi-local convergence of a harmonic mean Newton-type fourth-order technique for estimating the locally unique solutions of nonlinear systems in Banach spaces. The local analysis is established in previous works under assumptions reaching the fifth derivative of the involved operator. Therefore, the applicability of the method is restricted to solving nonlinear equations containing operators that are at least five times differentiable. However, this method may converge even if these assumptions are not satisfied. Other limitations include the lack of a priori error estimates and the isolation of the solution results. The local analysis in this work is shown using only the first derivative of the method. Moreover, a priori estimates on the error distances and uniqueness results are provided based on generated continuity assumptions on the Fréchet derivative of the operator. Furthermore, the more interesting semi-local case not studied previously is developed by means of majorizing sequences. The analysis in both cases is given not in the finite-dimensional Euclidean but in the more general setting of Banach spaces. Some numerical tests are performed to validate the theoretical results further.