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In this paper we introduce a semi-local theorem for the feasibility andconvergence of the inexact Newton method, regarding the sequence $x_{k+1} = x_k- Df(x_k)^{-1}f(x_k) + r_k$, where $r_k$ represents the error in each step.Unlike the previous results of this type in the literature, we prove thefeasibility of the inexact Newton method under the minor hypothesis that theerror $r_k$ is bounded by a small constant to be computed, and moreover weprove results concerning the convergence of the sequence $x_k$ to the solutionunder this hypothesis. Moreover, we show how to apply this this method tocompute rigorously zeros for two-point boundary value problems of Neumann type.Finally, we apply it to a version of the Cahn-Hilliard equation.

On the feasibility and convergence of the inexact Newton method under minor conditions on the error terms