The aim of this paper is to study the local convergence of the four order
iteration of Euler's family for solving nonlinear operator equations. We get the optimal
radius of the local convergence ball of the method for operators satisfying the weak
third order generalized Lipschitz condition with L-average. We also show that the
local convergence of the method is determined by a period 2 orbit of the method itself
applied to a real function.