A Newton-like method for generalized operator equations in Banach spaces

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, 1955) 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, 1968). The aim of the paper is to discuss the convergence analysis under local Lipschitz condition ∥Fx′−Fx0′∥≤ω(∥x−x0∥)$\|F'_{x}-F'_{x_{0}}\|\le \omega (\|x-x_{0}\|)$ for a given point x0$x_{0}$. 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.

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