We present local convergence results for inexact iterative procedures of high convergence order in a normed space in order to approximate a locally unique solution. The hypotheses involve only Lipschitz conditions on the first Frechet-derivative of the operator involved. Earlier results involve Lipschitz-type hypotheses on higher than the first Frechet-derivative. The applicability of these methods is extended this way and under less computational cost. Special cases and applications are provided to show that these new results can apply to solve these equations.