The convergence properties are studied for some inexact Newton-like methods for solving nonlinear operator equations in Banach spaces. In practice the Newton method has two disadvantages: it requires computing exactly Jacobian matrices and secondly, it requires solving exactly the corresponding linear equations. In this paper such inexact Newton-like methods avoiding both disadvantages are developed using a new type of residual control. Under the assumption that the derivative of the operator defining the equation satisfies the Hölder condition, the radius of the convergence ball of the inexact Newton-like methods with the new type residual control is estimated, and a linear and superlinear convergence rate is proved. A slight modification of the inexact Newton-like method of R. H. Chan, H. L. Chang
and S. F. Xu
[BIT 43, No. 1, 7–20 (2003; Zbl 1029.65036
)] for solving inverse eigenvalue problems is proposed. A numerical example for illustrating the performance of the latter algorithm is presented.