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Finite-dimensional realization of a Two-Step Newton-Tikhonov method is considered for obtaining a stable approximate solution to nonlinear ill-posed Hammerstein-type operator equationsKF(x)=f. HereF:D(F)⊆X→Xis nonlinear monotone operator,K:X→Yis a bounded linear operator,Xis a real Hilbert space, andYis a Hilbert space. The error analysis for this method is done under two general source conditions, the first one involves the operatorKand the second one involves the Fréchet derivative ofFat an initial approximationx0of the the solutionx̂: balancing principle of Pereverzev and Schock (2005) is employed in choosing the regularization parameter and order optimal error bounds are established. Numerical illustration is given to confirm the reliability of our approach.

We present a local convergence analysis of some families of Newton-like methods with frozen derivatives in order to approximate a locally unique solution of an equation in a Banach space setting. In earlier studies such as Amat et al. (Appl Math Lett. 25:2209-2217, 2012), Petkovic (Multipoint methods for solving nonlinear equations, Elsevier, Amsterdam, 2013), Traub (Iterative methods for the solution of equations, AMS Chelsea Publishing, Providence, 1982) and Xiao and Yin (Appl Math Comput, 2015) the local convergence was proved based on hypotheses on the derivative of order higher than two although only the first derivative appears in these methods. In this paper we expand the applicability of these methods using only hypotheses on the first derivative and Lipschitz constants. Numerical examples are also presented in this study.

We provide a new semilocal convergence analysis for the inexact Newton method in order to approximate a solution of a nonlinear equation in a Banach space setting. Using a new idea of restricted convergence domains we present a convergence analysis with the following advantages over earlier studies: larger convergence domain, tighter error bounds on the distances involved and an at least as precise information on the location of the solution. This way we expand the applicability of the inexact Newton method. Special cases and numerical examples are also provided.

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