  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 $\kappa$-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 $\kappa$-average, and that $\kappa$ 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
$\kappa(u)$. A numerical test is carried out to corroborate the hypothesis
established in this work.
