The paper is concerned with solving iteratively systems of nonlinear equations by an inexact Newton method and by an inexact Newton-like method. The local convergence properties of these methods are discussed under weaker Lipschitz conditions than the affine invariant Lipschitz condition [see B. Morini
, Math. Comput. 68, No. 228, 1605–1613 (1999; Zbl 0933.65050
)], called center Lipschitz condition, respectively radius Lipschitz condition. The authors use, like other authors, an inexact Newton method and an inexact Newton-like method where a scaled relative residual control is performed at each iteration. The results obtained allow us to see how large the radius of the convergence ball is. Two concrete examples are given.