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In this paper, we are concerned with the semilocal convergence analysis of a Newton-like method discussed by Bartle (Amer Math Soc 6: 827-831, ) to solve the generalized operator equations containing nondifferentiatble term in Banach spaces. This method has also been studied by Rheinboldt (SIAM J Numer Anal 5: 42-63, ). The aim of the paper is to discuss the convergence analysis under local Lipschitz condition $\|F'_{x}-F'_{x_{0}}\|\le \omega (\|x-x_{0}\|)$ for a given point $x_{0}$. Our results extend and improve the previous ones in the sense of local Lipschitz conditions. We apply our results to solve the Fredholm-type operator equations. [ABSTRACT FROM AUTHOR]

It is well known that the rate of convergence of 5-iteration process introduced by Agarwal et al. [J. Nonlinear Convex Anal., 8 (1) (2007), 61-79.] is faster than Picard iteration process for contraction operators. Following the ideas of S-iteration process, we introduce a parallel S-iteration process for finding altering points of nonlinear operators. We apply our algorithms to solve a system of operator equations in Banach space setting. This work also includes convergence analysis of hybrid steepest-descent-like method and hybrid Newton-like method in the context of altering points. [ABSTRACT FROM AUTHOR]

In this paper, we design a new third order Newton-like method and establish its convergence theory for finding the approximate solutions of nonlinear operator equations in the setting of Banach spaces. First, we discuss the convergence analysis of our third order Newton-like method under the ω -continuity condition. Then we apply our approach to solve nonlinear fixed point problems and Fredholm integral equations, where the first derivative of an involved operator does not necessarily satisfy the Hölder and Lipschitz continuity conditions. Several numerical examples are given, which compare the applicability of our convergence theory with the ones in the literature. [ABSTRACT FROM AUTHOR]