  In the present paper, we consider the semilocal convergence problems of the
two-step Newton method for solving nonlinear operator equation in Banach
spaces. Under the assumption that the first derivative of the operator
satisfies a generalized Lipschitz condition, a new semilocal convergence
analysis for the two-step Newton method is presented. The Q-cubic convergence
is obtained by an additional condition. This analysis also allows us to obtain
three important spacial cases about the convergence results based on the
premises of Kantorovich, Smale and Nesterov-Nemirovskii types. An application
of our convergence results is to the approximation of minimal positive solution
for a nonsymmetric algebraic Riccati equation arising from transport theory.
