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The goal of this study is to investigate the local convergence of a three-step Newton-Traub technique for solving nonlinear equations in Banach spaces with a convergence rate of five. The first order derivative of a nonlinear operator is assumed to satisfy the generalized Lipschitz condition, i.e. the $κ$-average condition. Furthermore, a few results on the convergence of the same method in Banach spaces are developed under the assumption that the derivative of the operators satisfies the radius or center Lipschitz condition with a weak $κ$-average, and that $κ$ is a positive integrable function but not necessarily non-decreasing. Our new notion provides a tighter convergence analysis without the need for new circumstances. As a result, we broaden the applicability of iterative approaches. Theoretical results are supported further by illuminating examples. The existence and uniqueness of the solution $x^*$ are examined in the convergence theorem. In the end, we achieve weaker sufficient convergence criteria and more specific information on the position of the solution than previous efforts requiring the same computational effort. We obtain the convergence theorems as well as some novel results by applying the results to some specific functions for $κ(u)$. A numerical test is carried out to corroborate the hypothesis established in this work.