  The motive of this paper is to discuss the local convergence of a two-step
Newton type method of convergence rate three for solving nonlinear equations in
Banach spaces. It is assumed that the first order derivative of nonlinear
operator satisfies the generalized Lipschitz i.e. $L$-average condition. Also,
some results on convergence of the same method in Banach spaces are established
under the assumption that the derivative of the operators satisfies the radius
or center Lipschitz condition with a weak $L$-average particularly it is
assumed that $L$ is positive integrable function but not necessarily
non-decreasing.
