Kantorovich-type convergence criterion for inexact Newton methods

Assuming that the first derivative of an operator satisfies the Lipschitz condition, a Kantorovich-type convergence criterion for inexact Newton methods is established, which includes the well-known Kantorovich's theorem as a special case. Comparisons and a numerical example are presented to illustrate that our results obtained in the present paper improve and extend some recent results in [X.P. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math. 25 (2007) 231–242; W.P. Shen, C. Li, Convergence criterion of inexact methods for operators with Hölder continuous derivatives, Taiwan. J. Math. 12 (2008) 1865–1882].