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The motive of this paper is to discuss the local convergence of a two-stepNewton type method of convergence rate three for solving nonlinear equations inBanach spaces. It is assumed that the first order derivative of nonlinearoperator satisfies the generalized Lipschitz i.e. $L$-average condition. Also,some results on convergence of the same method in Banach spaces are establishedunder the assumption that the derivative of the operators satisfies the radiusor center Lipschitz condition with a weak $L$-average particularly it isassumed that $L$ is positive integrable function but not necessarilynon-decreasing.

On the Local convergence of two-step Newton type Method in Banach Spaces under generalized Lipschitz Conditions