The semilocal convergence of a third-order Newton-like method for solving
nonlinear equations is considered. Under a weak condition (the so-called γ-condition)
on the derivative of the nonlinear operator, we establish a new semilocal convergence
theorem for the Newton-like method and also provide an error estimate. Some numerical
examples show the applicability and efficiency of our result, in comparison to other
semilocal convergence theorems.