Smaleʹs α-theory for inexact Newton methods under the γ-condition

The present paper is concerned with the convergence problem of inexact Newton methods. Assuming that the nonlinear operator satisfies the γ-condition, a convergence criterion for inexact Newton methods is established which includes Smaleʹs type convergence criterion. The concept of an approximate zero for inexact Newton methods is proposed in this paper and the criterion for judging an initial point being an approximate zero is established. Consequently, Smaleʹs α-theory is generalized to inexact Newton methods. Furthermore, a numerical example is presented to illustrate the applicability of our main results.