In this paper we provide a local convergence analysis for multistep Newton-like method (1.3) in order to approximate a solution of the nonlinear equation (1.1) in a Banach space setting. A refined and more flexible than before local [4]-[7] local convergence analysis of multistep simplified Newton-like methods for approximating solutions of nonlinear operator equations in Banach space is provided, by approximating not only the differentiable (see [4]-[7]) but also the non differentiable part (see also [1],[2]). A numerical example is used where our results compare favorably with earlier ones [4]-[7].